梶浦欣二郎教授論文集 Selected Papers of Professor Kinjiro KAJIURA

オホーツク海の夏季海況に就いて 梶浦欣二郎

緒言

オホーツク海の海況に就いては,襯測資料が少い爲め,現在迄余り研究されていない．こ
の報告は・主に昭和17年夏水路部と中央氣象毫とが協同し,中央氣象喜搬測船凌風丸によつて行
われた観測に基き,この海域全般の海況を明らかにしやうとしたものである.

§1　海水の層重欺態

オホーツク海内部及び太平洋の千島近海に於ける代表貼の温戯曲線を第1圖に示す.之れによれば，オホーツク海内部表暦水は陸水の流入,冬期生成された海氷の融解等のために低鹽(塩)分を呈し,主として日射のため夏期は相當に昇温している.75m乃至100m層を中心として水温が著しく低く,所謂中冷層が存在するが,これは宇田博士により輪ぜられた如く，冬期表層水が冷却され,又結氷に際して生じた高密度海水が封流により沈降したものである.中冷層と表層の聞には安定度の非常に大きい不蓮続層がみられるが,中冷層と下層との境界は明瞭でなく,150m層前後に於ては安定度が中性に近く・鉛直封流が割合に起り易いことを示している.200m以深になると水温,鹽分共ほぼ一様に増加し,800m乃至1000m層で水温が極大となる.この原因は後述する如く,深層に於ける太平洋水の流入によるものと思われる．千島列島聞の海峡はすべて水深1500m以下であるため,それ以深は3000m以上の底迄ほとんど水温の変化がみられす,1.8度乃至1・9度を示し,オホーツク海固有の水塊が存在するものと考えられる.

§2水温,鹽分の分布

a)水温,鹽分の水干分布
表面に於ける鹽分双ぴ,年変化を考慮して補正を加えた水温の分布を書いたものが,第2圖である.表層では,すで指摘されている様に，中部千島近海に低温,高鹽分海水が存在する.
これはオホーツク海水が太平洋に流出するに際し,海峡附近で渦動か発達し,海水の鉛直混合が盛んに行われるためと思われる.中部樺太沖に見られる低温,高鹽分水は,鉛直断面に於ける水温，鹽分の分布を併せ考えると，地形性の湧昇流によるものらしい.南樺太東方海面に広がつている32%以下という低鹽分海水は，厚さ10m乃至25mに及び,その起源は明らかでないが,今迄の観測資料によれば.日本海側にはかかる低鹽分水は見られず，又陸水の影響と考えるには余り広範園に存在し過ぎる様である.あるいは融水により生じた低鹽分水が,風のためこの海域に集積したものかも知れぬ．北樺太の近海も著しく低鹽分(30%以下)であるが,これは予想される通り,黒龍江排水の大部分其の他シベリアの陸水が多量に流入する爲めであらう.カムチヤツカ半島南部近海には周囲より低温,高鹽分水がみられるが,之は観測当時東風が卓越していたため，海岸近くに湧昇流か起り,鉛直渦動が発達したためではなからうか.宗谷海峡より高温,高鹽分の■島海流末枝が流入し,南千島近海に迄及んでいるが,10m層になると,北海道北沖約50浬附近に冷水があらはれ,下層水の上昇が考えられる.25m以深になると,水温,鹽分の分布は表暦とやや趣きを異にし,中冷層のそれに近くなる.一般に表層に於ては顯著なる海流は認められず,中部千島附近よリオホーツク海側の冷水の太牛洋側に流出している事が目立つのみである.
中冷層に於ける水温分布を知るため,最低水温の分布図を書いたものが第3図である.これによると,北千島附近よ1りは太平洋側の温暖水が流入し,中部千島附近よりは冷水が太平洋に流出している・又新知島北方の高温なるは150°E附近を北上する暖流末伎の影響であり,南千島,北海道北沖の高温なのは宗谷暖流の存在を示すものと考えられる.最低温層の深度は,54°E以北50m,以南は千島沿海を除き大■75mで,一搬に北より南に向うにつれ深くなつているが,特にカムチヤツカ半島の南端附近では深さ125mとなり,冷水が太平洋より流入する暖水の下に潜入している有様がうかがえる・塩分は一般に周辺部に多く,52°N,149°Eを中心として低塩分海域(約0.3%過少)が存在する・深さ20Omに至る迄はほぼ同様な水温,塩分の分布傾向を示すが,カムチヤツカ南端附近で沈降した低温・低塩分水は深くなるにつれ次第に北千島北方一帯に広がる.300m深前後では150°Eを境として,樺太側がカムチヤツカ側より約0.5℃低水温となり,塩分も少く,特に樺太側の南部が低温,低塩分である．この傾向は深層に迄あらはれ,500m以深になると水温,塩分の変化は少いが,それでも南部は北部よりやや低温,低塩分を示し中部千島附近にそれが著しい・800m乃至1000m層に至ると,北千島附近の海峡より温暖なる太平洋水が流入し,等深線に平行に北西に流動する様に見える.そのため中及南千島の北方のみ周園より約0.1℃低温,0.1%低塩分となつている・1200m層に到ると水温変化はほとんどなくなつている.　ここに注目すぺきは，200mより1000mの深層に至る迄ボ一ソリ海峡の南方は周園の太平洋水より低温,低塩分(約0.3℃低温，0.2%低塩分)を示し,上層とは逆にオホーツク海の冷水がこの海峡を通して太平洋に流出していることである．
b)水温,塩分の鉛直断面に於ける分布
　一般に深さ25m前後に顯著なる水温躍層が存在し,50m乃至150mの間は0℃以下の低温を示している．然し第4図よりも知れる様に,南部カムチヤツカ西沖では,この躍層が明瞭を欠き,表暦は鉛直混合の盛んなことが判る．千島沿海を除いては,一般に南西部で等温,等塩分線が最も降下しているが，これは深層流か海底地形に從つたために現れたものか,或はこの附近は,冬期生成される氷原の縁■に当たり,最も■流が発達するため,深層迄その影響が及んでいるのか,どちらかであらう．千島沿海では,すでに水平分布の項で述べた如く,中部千島近海の200m乃至300m以浅に於ける海水の鉛直渦動による混合,北千島近海に於ける太平洋水の流入と,その下層への冷水の沈降がみとめられる．叉宗谷海峡の底層よりは,水温2℃乃至3℃の日本海系水が流入し,100m乃至150m層を北海道北岸にごく接近して南東に流れ,北海道東端145°E附近にて二枝に分れ，一枝は南干島北沖に達し,一枝は北に向を■じて中知床岬の東方海面に達する.又亜■湾内には水温0℃以下の冷水が10m層附近に存在し,暖水と冷水とは,南樺太と北海道の間の海域で復雑に交錯している．新知島北方の約400m以深で等温,等塩分線共に降下していることは第5図より知れる通りであるが，この原因としては、100m暦前後に於ける暖水の流入のため，北方より流来したオホーツク海の冷水がその下に沈降すること，及び深層に於けるオホーツク海水がボーソリ海峡より流出する際に，一部が新知島にさえぎられて沈降する事が考えられる．中部千島東沖では上昇流のみられることはすでに述べたが，これは等温，等塩分線がそのあたりで凸出していることより知り得る．叉水平分布図にも現はれているが，北部オホーツク海56°N，144°E附近に，周園と全く異り表面より150mの海底に至る迄，水温1・5℃前後，塩分:3．9%前後でほとんど変化のない海水が存在する．この様な特異な海水が常にこの附近に存在することは，他の時期に行われた観測資料よりも確認し得る．その成因としては，附近に海堆が存在するため，海水の水平流動が相当ある時には，ここに渦動が発達し海水の鉛直混合が充分行はれる事が考えられる．千島の諸海峡を通して，その両側の海水の混合は，200m以深ではあまり顯著ではない．千島列島に沿う太平洋側の鉛直断面図をみると，北海道南方約150浬附近に，高温，高塩分なる黒潮末枝が明瞭に認められ，又暖流の一分枝は得撫島南方150°E附近にもみられる．寒流は南千島，北海道沿海，上述の二暖流分枝の中間，及び中部干島の南方の三枝が存在する．然し太平洋側寒流域にみられる中冷層は，オホーツク海内のもの程低温ではなく，水温0℃乃至1℃で100m深前後に存在する．太平洋側に於ける上述の結果は宇田博士の研究と大旨一致している．

§3　海水の流動

a)　力學的地形図　400m層を基準とし，深さ100m即ち中冷層に於ける海水の流動を表わしたものが第6図である．800m層を基準とした図もこれと大旨同様な傾向を示すが，流速はやや過大となる．図には0．01ダイナミツクメーター毎に等量線を引いてある故，50°Nに於いて，その間隔が180浬あるとき1■/秒の流速を示す．一般にオホーツク海内部は非常に密度流が微弱なることが知れる．南西オホーツク海に於いて，海水は中知床岬の沖合より等深線にほぼ平行して東北東に流れ，北千島附近より西流した海水と152°E，49°N近傍で衝突し，一部は南下して中部千島の諸海峡より太平洋に流出し，残りは北に■向し，50°Nより53°N迄の間はカムヤツカ半島のごく沿海を除き，一般に北流となる．このため北方より海岸沿いに南下したる海水は流路を■じ北樺太の東方海面に環流を生ずるに至る．太平洋側の流れはすでに§2にて述べた通りになつているが，中部千島より流出したる海水は，カムチヤツカ東岸を流下した海水と共に直ちに南下を始め，南千島沿海を流れる寒流とは直接関係を持たない様に見える．
b) σtの水平分布　今100m以深の海は平行ソレノイド場であると仮定して海水流動を推察すると，200m深のあたり迄は力學的地形図によるものと大旨同様な流動を示すが，それ以深になると，ボーソリ海峡より流出するオホ一ツク海水以外に千島列島の両側の海水の蓮絡はほとんどなくなる．然し深さ1000mに至ると再び僅かながら北千島附近より太平洋水の流入を認め得る．
c)　力學的断面図　鉛直断面に於ける比容の分布よりみれば，10m乃至25m深に存する密度不連続層はほぼ水平に位置し，流動の緩慢なることを示している．
d)計算流速　密度流としては，表層より100m深附近迄はほとんど等速のところが多く，毎秒数■以下である．然るに，海面に於いては海岸近くではあるが0．2節程度の流れが観測されている．
これより考えるとオホーツク海表層の海流の原因は風が主となつているらしい．
e)等密度面上の塩分分布　イセントロピツク面(Isentropic　Surface)上の塩分の分布を調べれば、種々の海水の水平混合の有様が理解される。第7図はσt=26.00 の面を示しているが，この面は表面乃至50mに位置する.これによるとオホーツク海内の塩分は32.5%前後，太平洋側は32.9%以上で，両海水の混合状態はすでに§2にて述べたことを裏書きしている.
叉亜庭湾内の陸水系低塩分水南千島近海の日本海系暖水，中部樺太東沖の湧昇流による高塩分水のみられることも，既述の通りである.
ボーソリ海峡の南方に暖流系水が接近しているが，300mより深くなると，逆にこの海峡よリオホーツク海の低塩分水が太干洋に■散している.一般に200m以深では太平洋側がオホーツク海側より高鰹分で，余り水平混合を起していない.

§4　水色及び透明度の分布

第7図は，水色，透明度の分布を示し，今迄の研究と大旨一致している.この海域の大部分は水色4であり，水色番号の大きいのは，すべて鉛直渦動が発達していると見なされる海である。又透明度は大旨水色と相伴って変化している。47°N〜48°N，145°E〜151°Eの範園に存在する周囲より特に清澄な海水は，例年見られるらしいが成因は明らかでない.又54°N附近にも清澄な海水が存在する.
　結語　極く海岸近くの海水流動については，観測■の数が少い故確かなことは解らないが，オホーツク海の夏期一般海況はやや明瞭になつたと思われる．この研究では，今迄考えられていた東樺太海流の存在は認められず，又左旋の大環流なるものも考えられなかつた．然し，千島近海に於ける海水の流動については，ある程度既往の研究と一致し，上暦200m深附近迄は海峡の両側の海水は互に蓮絡しているが，それ以深になるとボーソリ海峡を通つて流出するオホーツク海水を除き，ほとんど相互の流動はみられない．千島沿海には又沈降流等も存在し今後の研究に待つ問題が多い．其の他，北千島より流入する太平洋水の消長は，オホーツク海東部の海況に大きな影響を及ぼすであらうから，158°E附近を北上する暖流分枝の消長とオホーツク海の海況の変動との關係等も研究か望まれる．
　この研究を終るに当り，終始御指導を賜わつた日高教授に御禮申上げます．

　　　　　　　　　　　　　　　　　　　[参考文献]
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　　　　　　　　中央気象台■文彙報　第10雀第2號，1936
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On the Hydrography of the OKhotsk Sea in Summer. (Abstract) By K.Kajiura.

In this paper,the general oceanoglaphic conditions of the Okhotsk Sea were investigated on the basis of obserations made in summer of 1942 on board the ship「Ryofu-maru」, which was conductcd by the Hydropraphic Department in cooperation with the Central Meteological Observatory.
It was found that a conspicuous discoontinuity of temperature exists at the depth between 10m and 25m, and the mininmm tenperature of about -1.5℃ is at the depth of about 100m.
Also, an examimtion was made with regard to the state of flowing of water in and out of this region, and it was found that the Pacific water flows into this area through the North Kurile and the Okhotsk water flows out through the Central Kurile.
　It was also found that, within this area,the primary current is generally weak, and the current drifted by wind prevails.

The Velocity Distribution of Wind Currents in the Eastern Part of the Equatorial Pacific. By K.KAJICTRA (Geophysical Institute, Tokyo University)

Abstract

Wind driven currents in a baroclinic ocean are discussed following the ideas of SVERDRUP and REID, and the current velocities in the upper layers of the eastern equatorial Pacific are calculated on the assumption that the inertial forces and the lateral friction are negligible. According to the present theory, currents may be devided into two parts, of which one is the pure drift currents and the other is the relative currents, and it is found that the results are in good agreement with the present knowledge of the currents near the equator. In these regions, the relative currents are much stronger than the pure drift currents and the north-south component of the velocity is very small compared with the east-west component.

Introduction

Recent theories on the wind driven ocean circulation have shown the circulation in the upper layers of the oceans being the result chiefly of the stresses exerted by winds. The stream lines of oceanic mass transport derived from solutions of a vertically integrated vorticity equation account for many of the gross features of the general circulation (MUNK,1950).
Mass transport methods, however, add only little to the knowledge of the vertical structure of currents. In this respect, HIDAKA (1950) treated this problem by using the velocity components themselves instead of the mass transport, but the results were not satisfactory. Different approach to this problem is attempted here which may be successful in a region where lateral stresses of the currents can be neglected.
According to the ideas of SVERDRUP (1947)and REID(1948),the pressure field in a baroclinic ocean can be completed by the knowledges of the wind stresses and the vertical structure of water-mass. Therefore the problem to be solved is reduced to find currents under the influence of wind stresses in the known pressure field. This problem is similar to that discussed by EKMAN(1906).

Theory

In a region where lateral stresses and field accelerations can be neglected, the equations of stationary motion of water can be written;
式(1)

Here u and v are the x and v components of velocity where x- and y- axes are directed toward the east and north, respectively, with the origin on the equator, and the z- axis is taken to be positive downwards. And \psi\ is the latitude, taken positive to the north of the equator,\omega\, the angular velocity of the Earth’s rotation,\rho\, the density,p, the pressure and \mu\, the eddy viscosity.
Writing

[Fig.107_021_02]

where i^{2}=-1, the equations(1) are reduced to,

式(2)

In this equation \alpha\ includes \rho\ and \mu\ both of which must be functions of z.
The variation of the density with depth, however, is too small to modify the results and the knowledges of the eddy viscosity are not enough at present to draw reliable conclusions as to the variation of a with depth. Therefore, on the assumption ofαto be constant in the vertical direction, the general solution of the equation (2) is easily obtained as follows

式(3)

where A and B are arbitrary constants.
Introducing the boundary conditions such that [Fig.107_021_05] at the surface and W=O at a sufficiently great depth, in which [Fig.107_021_06] and \tau_{x}\ and \tau_{y}\ are x and y components of the wind stresses, W can be determined definitely when F is known.
In a baroclinic ocean it is reasonable to assume the horizontal velocity and the horizontal pressure gradient vanish at a moderate depth below the sea surface. If d is the selected depth of no motlon, the vertically integrated equations of motion from the surface to the depth,d, on the assumption that [Fig.107_021_07] at the depth,d, are

式(4)

K.Kajiura：The Velocity Distribution of Wind Currents.

where [Fig.107_022_01]
The equation of continuity is,

式(5)

By cross-differentiation of the two equations of (4) with the aid of (5), it follows that

式(6)

where \tau_{x}\ and\tau_{y}\ are assumed to be independent of x,and \psi\is taken to be y/R, in which R is the radius of the earth.
Substitution of (6) into (5) leads to

式(7)

Here the continental boundary is assumed at x=x0 where no mass transport is present across the boundary.
By insertion of both (6) and (7) into (4)the following equations are obtained,

式(8)

Assuming the Reid’s model of the vertical structure of mass, such that
[Fig.107_022_06]
where [Fig.107_022_07] is the elevation of the sea surface, the pressure at any depth may be determined by a integration of the hydrostatic equation.
From the condition that the pressure gradient vanishes below a moderate depth, it follows that,

式(9)

when the depth of no motion is assumed to be sufficiently great compared with the depth of the upper homogeneous layer.
By use of the above equations (9),the following relations are obtained,

式(10)

and from the equations(8)and(10),it follows that

式(11)

where C is a constant of integration.

The pressure distributions in

式(12)

Similar expressions hold for [Fig.107_023_02]
If the equations (8), (9), (10), (11) and (12) are combined together, the pressure gradient at any depth can be computed from the wind stresses alone.

Formulae for the current velocities in the upper layers.
The arbitrary constants A and B in the equation (3) can be easily determined from the boundary conditions as the horizontal pressure gradient does not vary with depth in the upper homogeneous layer and it vanishes at the depth, d.
For the sake of simplicity, the Lower boundary is replaced from z=d to z=\infty\.
This simplification may not greatly modify the result.
The current velocities in the upper layers are given by,

式(13)

In the equations 13) the first term of the right hand side represents the pure drift current similar to that discussed by Ekman and the second term represents the relative current which depends on the distribution of density.
The calculation of currellt veloeities.
The density distribution being related to the stress gradient, it is desirable to determine the accurate distribution of wind stresses, but it is impossible at present because of the lack of available observations. Therefore, we assume the sinusoidal stress distribution in the equatorial region of the North Pacific similar to the one assuiizecl by Reid (198),such that

[Fig.107_023_04]

and

[Fig.107_023_05]

where
[Fig.107_024_01]
This formula for \tau_{y}\ is applied only to the north of 4°N and it is assumed that \tau_{y}\ decreases linearly from 4° N to the equator and it vanishes in the south latitude.
Numerical values used in calculation are as follows;
[Fig.107_024_02]
Fig.1 is the vector representation of the wind stress and Fig.2 shows the variation of currents with latitude. The drift current, depending on the absolute value of the wind stresses, varies its intensities according to the value of \tau_{y}\, while the relative current which is mainly determined by the gradient of \tau_{x}\ in the north-south direction is only little affected by the presence of \tau_{y}\ except
near the coastal region.
Fig.3shows the velocity distribution in the upPer layers. The drift current decreases exponentially with depth and the depth of frictional influence depends on the latitude and the value of eddy viscosity, so that the currents at the depth deeper than about 100m are al most negligible except in the very vicinity to the equator, when \mu\ is assumed to be 10^{2} C.G.S. In the surface layer, the drift current seems to make narrower the counter current and the maximum velocities of the counter current appear in the sub-surface layer.
The maximum calculated velocity of the counter current is about 20cm/sec which is only half of the expected value of the velocity of the counter current.
This discrepancy may be ascribed to the under-estimate of the wind stress in this region, or the neglect of the thermodynamic effect. On the whole, relative currents prevail compared with the drift currents. This conclusion is, however, not always reasonable as the drift current is developed by the temporary wind but the relative current is maintained by the stationary wind stress and is influenced only little by the temporary wind. Just under the equator, the above theory fails and other factors such as lateral stresses, field accerelations and the thermodynamic processes must be taken into consideration.
Divergence and convergence of currents on the surface can be computed from the following equation;
[Fig.107_026_01]
The calculation shows that no remarkable regions of divergence or convergence exist, except in the very vicinity of the equator.

Concluding remarks.
In the region discussed in this paper, the neglect of lateral stresses and inertia terms may not introduce serious result, because the horizontal stresses are rather small compared with the vertical stresses, even when the lateral eddy viscosity is taken to be 10^{3} c.g.s. except along the boundary line between the equatorial current and the counter current, where both stresses may become equal in magnitude.
When the drift current and the relative current are co-existing, the stationary state cannot be reached in a baroclinic ocean without some mechanism which re-establishes the distribution of density of water disturbed by drift currents.
Lateral mixing processes and thermodynamic processes may be the factors which take part in this mechanism.1n the present paper, however, the assumption of homogeneous water in the upper layer diminishes these difficulties and simplifies the treatment of phenomena.
Further investigations about the model of a water-mass are desirable but it seems very difficult to treat it exactly. The formation of water-mass is chiefly the result of thermodynamic processes such as evaporation and precipitation.If the thermodynamic processes act uniformly over the large area without any dynamic disturbances, a water-mass is formed in which the density of water may become a function of the depth only. When wind begins to blow under this circumstance and when the unequality of the wind velocity exist, the surface water is accumulated somewhere, result：in the cause of the horizontal pressure gradient.This pressure gradient, reaching to the very bottom of the ocean, may produce the movement of the water in deep layers which changes the distribution of density. Thus the horizontal pressure gradient caused by the surface elevation will be cancelled soon by the re-distribution of density, and when the stationary state is reached, the water in the deeper layers will be at rest. In these ways the inner pressure gradient which becomes negligible at a moderate depth, may be generated.
The present paper is only a first step to study this interesting problem of the wind current in a baroclinic ocean, and many objections may be raised against the looseness of the treatrnents. Nevertheless we believe that the density distribution in the ocean may play an important role for the distribution of velocity of the wind current.

Acknowledgement.
The author wishes to express his sincere thanks to Prof. K. Hidaka for his support and encouragement during this work.
References.
Ekman, V. W.(1906):Beitrage zur Theorie der Meeresstromungen. Ann.d. Hydro.u.Marit. Meteorol.,34, pp.566-583.
Hidaka. K.(1950):Drift Currents in an Enclosed Ocean, Part 1. Gpophysical Notes(Geophysical Institute, Tokyo Univ.).3, No.23.
Munk, W. H.(1950):On the Wind-Drived Ocean Circulation・J. Meteorol.,7, PP・79-93.
Reid, R.0.(1948a):The Equatorial Currents of the Eastern Pacific as Maintained by the Stress of the Wind.J.Marine Res.,7, PP.74-99.
Reid, R.O.(1948b):A Model of the Vertical Structure of Mass in Equatorial Wind-Driven Currents of a Baroclinic Ocean..」Marine Res.,7, PP.304-312.
Sverdrup, H. U.(1947):Wind-Driven Currents in a Baroclinic Ocean;with Application to the Equatorial Currents of the Eastern Pacific. Proc.Nat. Acad. Sci.,33,pp.318-326.

The Effect of Winds on the Stratified Lake Water By K.KAJIURA (Geophysical Institute, Tokyo University)

Abstract

The currents and the accumulation produced by winds in the water consisting of two layers of different densities are investigated on the assumption that the eddy viscosity is negligible along the boundary of two water masses. The computed profile of the boundary surface is in food agreement with observation.

lntroduction

The currents and the accumulation of water in an enclosed sea and in a lake produced by the wind stresses have been investigated by many authors.
Most of them assumed, however,that the water is homogeneous in spite of the fact that the actual ocean and lakes are generally in stable stratification and there exist remarkable discontinuities of density below the surface in summer. In the homogeneous water, the horizontal pressure gradient due to the surface elevation caused by the wind reaches to the very bottom, and the whole water makes an circulation, en the other hand, in the non-homogeneous water, the movement of water changes the inner distribution of density and the inclination of isopleths gradually compensates the pressure gradient exerted by the surface deformation. Therefore in the stationary state, the features of the water movement in the homogeneous water and in the non-homogeneous water may be quite different. Moreover, the eddy viscosity is generally expected to decrease greatly owing to the increase of stability at the boundary of two water masses of different density. A. Defant (1932) and H. Arakawa (1935) discussed this problem, but they made an approximation of small amplitude of disturbances along the boundary of two water masses which may not be accepted when the elevation or the depression of the boundary surface becomes large. E. Palmen (1941) investigated the effect of the density stratification upon the elevation of the sea surface and concluded that when the considerably large area is concerned, this effect may be neglected because it needs very long time for the water movement to reach a stationary state.
In a small lake, however, the density stratification modifies the pattern of the vertical circulation of the water as the stationary state is easily attained.

Theoretical Consideration

We consider the motion of water in a canal consisting of two layers of different density. The equations of motion in a stationary state is,

式(1)

where the x-axis is the horizontal co-ordinate lying on the undisturbed surface and perpendicular to the coast with the origin at the coast and the z-axis is directed upward. u is the velocity along the.the x-axis, \rho\ is the density, \mu\ is the eddy viscosity, p is the pressure, and g is the acceleration due to gravity.
As equations of continuity we take

式(2)

where \zeta\ is the surface elevation, h is the depth to the bourldary of different water masses, and d is the depth to the bottom. Suffixes 1 and 2 are used to represent the quantities in the upper and the lower layer, respectively.
The boundary conditions are

式(3a,b)

where T is the constant wind stress exerted on the surface of water. When \mu_{2}\ is assumed to be neglegible along the internal boundary, the upper water feels no friction at the boundary surface, and the condition (3b) is reduced to

式(3c)

In a stationary state the water in the lower layer is at rest because the frictional effect of the upper water is interrupted at the boundary and the pressure due to the surface elevation is adjusted by the depression of the boundary surface. This condition is satisfied by the following relation,

式(4)

where H is the mean thickness of the upper layer.
The first equation of (1) is integrated from the depth of the boundary to the surface, taking the second equation into consideration,

[Fig.107_030_01]

and with the equations (3a) and (4) it follows;

式(5)

This equation can be integrated easily and results in,

式(6)

where [Fig.107_030_04] and A is a constant of integration. The positive sign in the second term of the right hand side is taken, because the elevation must be positive at some points.
The constant A is to be determined from the conservation of mass such that

[Fig.107_030_05]

where l is the width of the canal.
The slope of the surface is given by,

式(7)

At the point x=-A/b where no elevation of water exists, the slope is the same as that in the homogeneous water of the depth H, and to the up-wind side of this point, it is steeper and to the lee-wind side it is smoother.
From the equation (1) it foflows that

式(8)

where B and C are constants of integration.
The equations (3a),(4) and (5) yield,
B=h,
and the equation (2) becomes,

[Fig.107_030_08]

Assuming\zeta\ ＜＜ h・we obtain [Fig.107_030_09]
Therefore we have,

式(9)

which shows that the vertical distribution of velocity is similar to thet in the
homogeneous water of depth h with smooth bottom.

The comparison with the observation

The above theory is applied to Lake Suwa, which is about 6〜7 meters deep and 4 km wide. According to J. Masuzawa (1950),there exists in this lake a thermocline in summer at the depth of about 5 meters and this stratification is not destroyed by the mixing caused by the prevailing winds of about 5〜10 m/sec which blows regularly in every afternoon. He found the very small eddy duffusion coefficient (about O.02 c.g.s.) near the boundary layer of different water masses, and this fact supports the assumption used in the present theory such as \mu_{2}\=0 at the boundary.
Fig.1 shows the observed distribution of temperature after Masuzawa in the morning and in the afternoon. In both cases the isotherms are almost coincident near the lee-wind coast, because the observations at St.46 and St.47 where carried soon after the wind set in and the stationary state was not reached yet. Fig.2 shows the calculated inclination of the boundary surface. The inclination of the isotherms of 22℃〜23℃ in the up-wind side in Fig.1 and that of Fig.2are in good agreement.
The numerical constants used in the computation are as follows：
[Fig.107_031_04]

Conclusion

In a small lake the circulation of water due to stationary wind stress is greatly modified by the stratification of water of different density. The surface elevation of water is related to the thickness of the layer of lighter, upper water and the depth of the lake itself iS of little importance to the elevation, when it is sufficiently deep.
The slope of the surface is not constant but relate to the square root of the distance from some fixed point which is determined from various factors such as the width of the lake,l, depth of the upper layer, H, density difference, \alpha\, and the stress of the wind, T.
In a transient state, the movement of the lower water is chiefly governed by the hydrostatic pressure due to the surface elevation and the circulation pattern may be similar to the one in the homogeneous water.
The auther wishes to express his hearty thanks to Mr.,T. Masuzawa for his kind advice.
References
A.Defant:Beitrag zur theoretischen Limnologie.Beitr. Phys.freien Atmos.,19 143-150,(1932).
荒川秀俊:吹送流による湖水の循環 海と空15 437-444, (1935). (H.Arakawa:Circulation of Lake Water due to Wind Stresses).
E・Palmen:Zur Frage des Windstau in einem geschichtetem Meer. Soc.＆Scient. Fenn.,Comm. Phys.一Math. XI Nr.7,1-11,(1941).
J.Masuzawa:On the stratification of lake water in summer, Lake Suwa, Nagano Pref.(in Japanese). Rep. Res. Inst. Nat. Res. No.15 (1950).

On the Influence of Bottom Topography on Ocean Currents. By Kinjiro KAJILTRA (Geophysical Institute, Tokyo University)

Abstract

Topographic effect on currents has been discussed by the method of small perturbation. An equation for the vertical motion is derived which includes a wide range of phenomena, and it is shown that the character of the deflection of currents by the bottom deformation is complicated since the different kinds of waves become dominant according to the scale of topography, and, moreover, since the horizontal shear of a current takes part in the phenomena. When the scale of the bottom deformation is very small, the stationary internal gravity waves prevail under a certain condition and when the scale is large, planetary waves develop in a horizontal plane.

1. Introduction

It is a well known fact that the distribution of density is considerably modified when an ocean current flows across a submarine ridge. Theoretical attacks to this problem, however, are very scarce owing to difficulties of mathematical analyses. In his very extensive studies on the character of ocean currents in homogeneous water, V. W. Ekman (1923) first examined the effect of bottom topography on slope currents, the velocity of which is constant with depth except in a layer influenced by the bottom friction. His conclusions are：The currents generally tend to follow the bottom contours in higher Iatitudes and in the northern hemisphere, they deviate to the right, passing over an elevation and to the left, over depressions of the bottom, and vice versa in the southern hemisphere. He extended his analysis later (1932) on the basis of hydrodynamic equations including inertial terms as well as frictional terms, but the major features were almost as before. Similar studies of the problem have been made by many meteorologists concerning to an air flow over a mountain.P・Queney (1947,1948) has developed an extensive study of perturbations of air flow and discussed the mountain effect, taking the vertical stability of the atmosphere into account. Recently, the influerlce of a large scale mountain range on westerlies has been discussed by many authors (J. Charney and A.Eliassen・1949・D・Colson,1949, B・Bolin,1950, etc.)on the basis of the vorticity equation. In these studies, frictional effects are neglected and the general flow is assumed to be zonal and uniform. According to the vorticity equation in a barotropic atmosphere, the horizontal divergence or convergence due to the effect of topography results in an increase or decrease of vorticity,respectively, and the deflection of stream lines can be expected on the downstream side which is followed by the planetary waves of Rossby type .
On the other hand, the effect of bottom topography on relative current in non-homogerleous water has been examined qualitatively by H. U. Sverdrup (1941). The conclusion of his analysis is that the observed deflections of currents, which seem to be maximum at the summit of the bottom ridge, may not be caused by bottom friction as was assumed by Ekman, but they are caused by modifications of the density distribution due to the effect of topography. In his theory, the assumption that the isopicnic surface must rise when a watermass approaches a ridge, includes essentially the existence of a vertical velocity as well as a horizontal velocity near the bottom, and this is inconsistent with the 「law of the parallel solenoids」 which is the fundamental concept in his discussion.
In these circumstances, V. B. Stockman (1947,1948) reviewed the former studies and discussed the topographic effect, using the concepts of mass transport and lateral frictions. He emphasized the prilmary importance of the distribution of wind system over the ocean which is the main catlse of ocean currents, and concluded that the deflection of currents by the bottorn topography may not be related to the Coriolis force, but is connected with the vorticity in the windfield, such that when a current crosses a sea-bottom elevation, the flow-lines deviate over its summit in the same direction as the vorticity in the wind field and just the opposite above the central part of a depression. Though his conclusions are very suggessive, his deduction of results seems to be very doubtful in respect to the procedure of simplifying the equations of motion, by means of
taking mean values of various quantities.
At any rate, it is a usual experience that the currents, flowing close to the coast or flowing slowly, are apt to follow the bottom contours. It is, however, not so exactly ascertained that in which direction they will turn when ocean currents pass over an elevation or depression of the bottom. Therefore, in the present paper, the effect of bottom topography on ocean currents has been reexamined on the basis of the hydrodynamic equations including inertial terms, taking the density distribution into consideration as far as possible.

2.Basic Equations of Motion

We assume that the field is stationary and may be divided into two parts, of which one is the basic field and the other is the field of small parturbations caused by irregularities of the bottom. In spite of the defects involved in the approximation of small perturbations, the qualitative features of a flow pattern may be estimated. We write the pressure, p=po+p1, the density \rho\= \rho\0十+\rho\1, the horizontal components of velocity u=uo十u1, v= v1 and the vertical component of velocity\omega\=\omega\1, where subscripts 0 and 1 refer to the quantities in the basic state and the disturbed state, respectively.
The basic current is assumed geostrophic and zonal defined by the following equations:

式(1)

and

式(2)

where the co-ordinate x, y and z are directed to the east, to the north and vertically upward, respectively, with the origin situated at the bottom, and λ=2(\omega\ sin (ψ) is the Coriolis parameter and、g is the acceleration due to gravity.
The equations of motion for stationary perturbations are:

式(3)

式(4)

and

式(5)

provided that the frictional and other small terms be neglected and that the basic current u0 is assumed to be independent of x.
When the fluid is incompressible and individual density is conservative, we have:

式(6)

and

式(7)

Cross differentiating (3) and (4) with respect to x and y, and taking (6)
and (7) into account, we obtain:

式(8)

and from (3) and (5),we have:

式(9)

Eqs. (8) and (9) may be simplified when adequate assumptions are introduced.
When the horizontal gradient of density is very small and the basic current maybe considered to be almost uniform , we have, from (8),neglecting the variation of velocities In the y-direction,

式(10)

and from (9),

式(11)

where U is the uniform basic velocity.
Here we shall transform v1 and \omega\1 into v’and \omega\’ by the following equations:

式(12)

From equations (1) and (2) ,we have,

[Fig.107_036_06]

where P_{0B} is the bottom pressure. The assumption of almost uniform basic current is satisfied when the condition [Fig.107_036_07] is fulfilled, in other words, when the baroclinic effect is neglected. Since the relative currents prevail in the ocean, this assumption can not be accepted. Our discussion, however, will be valid if the effect of the vertical shear of velocity does not play a dominant role. Thus, U may be given, regardless of (1) and (2), in our discussion.
Writting [Fig.107_037_01] both of which are assumed to be constant,we have from (10) and (11) , under the condition [Fig.107_037_02]

式(13)

and

式(14)

where　[Fig.107_037_05] , and the term containing [Fig.107_037_06] is neglected,sinces, the coefficient of [Fig.107_037_07], is sufficiently small.The orders of magnitude of various parameters are tabulated in Table1.Here, we assume k_{\beta\} and ks to be positive, since we are chiefly concerned with steady and stable perturbations.

Table 1. Order of magnitude of various parameters.
[Fig.107_037_08]

In (14),we can see that when k ＞＞ k_{\beta\}, k_{\beta\^{2}} may be neglected and the equation will hold for any direction of a basic current.

3. Solution of the equations

A formal solution of (14) is,

式(15)

where [Fig.107_037_10] and A and」B are constants of integration.
Since p^{2} is positive when [Fig.107_037_11] and negative when [Fig.107_037_12], p is real or pure imaginary according to the value of k. Substituting (15) into(12),we hare,

式(16)

Here, boundary conditions for the vertical velocity icl are given by

式(17)

and

式(18)

where h is the height of the bottom deformation which is assumed to be independent of y corresporlding to the assumption that the velocity variation ln v-direction is negligible.
In the case when the bottom deformation is expressed by h=h0coskx and the mean depth is H, we have from (16) ,(17) and (18),as a particular solution:

式(19)

from (10) and (19),

式(20)

from (6),(19) and (20),

式(21)

and from (7),(19),(20) and (21),

式(22)

where [Fig.107_038_06]. ln equations(19) to (22),p should be understood as:

式(23-26)

Thus we can easily verify that the wave Iength in the horizontal direction of the disturbances are the same as that of the bottom configuration, and when k＞ks, that is, when the bottom deformation is of very small scale, the amplitude of v1 is very small compared with those of ul and \omega\1, and the disturbances may
be considered to be confined to the x-z plane. When [Fig.107_038_08], the amplitude of v1 is still small compared with that of u1, and the variation of velocities in the vertical direction is periodic if the condition, [Fig.107_038_09],is satisfied.When [Fig.107_038_10], that is, when bottom deformation is large, the amplitude of v1 is large compared with that of \omega\1 and the disturbances may be corlsidered to pe almost * Strictly speaking, small deformation of the free surface is expected, as in the case of unusual internal waves. This effect, however, will be very small and may be neglected in this discussion.
horizontal, and when k ＜ k_{\beta\}, which is the case of a very large scale deformation, \omega\1 ＜＜ v1 and the disturbances are almost horizontal so far as the depth of water, H,is not so great.
These solutions coincide with those obtained by Queney (1947) if the depth H tends to infinity, and k_\beta\ tends to zero.
In general, when the condition pH ＜＜ 1 is fulfilled,(that is, [Fig.107_039_01] )the disturbances may be simply written as：

式(27-30)

These solutions indicate that u1, and v1 may be considered as uniform in the vertical direction if the deformation of large scale is concerned. The above solutions can also be derived from (10) assuming that the water is homogeneous, u1and v1 are independent of z, and w1 is a linear function of z, satisfying the boundary conditions (17) and (18). In (30),the condition [Fig.107_039_03] is assumed, and the density distribution is just the same as suggested by Sverdrup. The effect of horizontal variation of density, however, is ineffective to the general pattern of the current, because we have assumed that the basic current is almost independent of the baroclinity of water.ln this respect we cannot discuss the influence of the density modification on the current pattern, which was the main part of Sverdrup’s theory. From (20), it is found that the stream lines deviate to the left near the summit of the bottom deformation in the northern hemisphere if k>ks and just the opposite if k ＜ k_{\beta\}. This is illustrated in Fig.1.

4.Perturbations caused by the parabolic deformation of the bottom

Let us now investigate the case when bottom deformation is such that

[Fig.107_040_01]

and

[Fig.107_040_02]

We may write h by the Fourier integral as follows：

式(31)

Then, from (16),(17) and (18),we have：

式(32)

Again, from (10) and (12),we have：

式(33)

Both of these integrals can be evaluated by the method of contour integration in the complex plane.
Here we make the following transformations

式(34)

and

式(35)

where n is a positive integer. We may write approximately under the conditions [Fig.107_040_08]

式(36)

where

[Fig.107_040_10]

and

[Fig.107_040_11]

K.Kajiura:On the Influence of Bottom Topography on Ocean Currents.

In general, the conditions [Fig.107_041_01] are satisfied, and all these quartities are real.
Furthermore, we rewrite the terms of sine and cosine in exponential forms such as

式(37-38)

Thus, we must evaluate these integrals separately in each region, taking a suitable path of integration. The path may be taken from -R to R, excluding poles, on the real axis and upper or Iower semi-circle of radius R, so as to make the integral on the semi-circle to vanish whe:n R→\infty\. If we take the principal values in Cauchy’s sense, and substitute the conditions that v’=0 and w’=0 at x=-\infty\, definite integral values can be obtained. (See Lamb, H.：Hydrodynamics.6th Ed. P.408)
The results are:

式(39-41)

and

式(42-44)

where n0 is the largest positive integer less than ksH/\pi\, abbreviations are used：

[Fig.107_042_02]

[Fig.107_043_01]

From these solutions, we can conclude that \alpha_{n}\ and \delta_{n}\, which exist only when ksH > \pi\, contribute to establish the stationary internal waves in the down-stream side of the deformation of the bottom, and no conspicuous disturbances occur in the up-stream side. And these internal waves prevail in the current pattern when the scale of deformation, a, is sufficiently smal1(ak_{\lambda\}＜＜1), on the other hand, they may be negligible compared with the horizontal perturbations when the scale of deformation is large. These features can be expected already from the discussion of perturbations for the sinusoidal bottom profile.
For aks ＜ 1, the terms containing \gamma_{n}\ and \delta_{n}\ are negligibly small compared with those of \alpha_{n}\ and \beta_{n}\ an, and, at a sufficiently large distance downstream, the terms of \alpha_{n}\ for n is near no predominate, since the terms of \beta_{n}\ will die out.
The leading term whern0 =1 is

式(45)

and the stream lines are given by

式(46)

For ak_{\lambda\}〜1, the terms of \alpha_{n}\ and \beta_{n}\ are rather small compared with the terms of \gamma_{n}\ and \delta_{n}\ , so that in this case, the whole terms in the equations must be taken into account.
For ak_{\lambda\} ＜＜ 1, we may reasonably neglect the terms containing \alpha_{n}\ ,\beta_{n}\ \gamma_{n}\ and \delta_{n}\ and the solutions reduce to：

式(47-48)

and

式(49)

These solutions can be derived from the original equations if the several assumptions are introduced such that the water is homogeneous and the variation of u1 and v1 in the vertical direction is negligible. In these solutions, we can see at once that in the northern hemisphere, the stream lines deviate to the right by the planetary effect and it is followed by Rossby waves. The deviation of stream line is illustrated qualitatively in Fig.2.

5. The Effect of horizontal shear of a basic current.

When the scale to be considered is large, and the basic current is constant in a vertical, we can assume from the above discussion that the horizontal velocities of perturbations are uniform in the vertical direction and the vertical velocity is a linear function of z. Therefore, on the assumption that the water is homogeneous and [Fig.107_044_03] ,eqation (8) becomes：

式(50)

provided that h is independent of y.
In a region of strong currents, it may be justified that \beta\ is negligible compared with d^{2}U,/dy^{2}. On these assumptions, we will discuss a very simple case, such as U=U0 cosk1y and h=h0cosk2x. A particular solution of the equation is then：

式(51)

where H is the depth of water.
In case when k1 is several times greater than k2,(k1=mk2),k2^{2} may be neglected compared with k1^{2}, and (51) reduces to :

式(52)

From the above solution,we can see that it is very important to take the horizontal shear of a current into account, if it is comparable with Coriolis parameter.
The larger the ratio of the scale of the bottom deformation to that of currents is, the greater the effect of the shear is.
For example, in case of
[Fig.107_045_01]
becomes 3 and the order of magnitude of m^{2}U0k1 is almost comparable with that of\lambda\,(〜10^{-4} sec^{-1}).
The current pattern is shown qualitatively in Fig.3.

6. Conclusions.

We have discussed above that when we are investigating the effect of bottom topography on a uniform basic current, it is important to consider the relation of the scale of the bottom configuration to various parameters such as ks,k_{\lambda\} and k\beta\, each of which is the representative scale of the motion of the stratified water on the rotating earth. When the scale of bottom deformations are very small, such that k_{\lambda\}a ＜＜ 1, the parameter ks, which is related to the vertical gradient of the density, plays a dominant role and the perturbation may be almost two dimensional in the vertical plane, and, moreover, if ksH >\pi\, the internal gravity waves occur downstream, the wave length of which is 2H^{0}/n in the vertical direction and 2\pi\/\alpha_{n}\ in the horizontal direction.
When the scale of bottom deformations is very large such that k\lambda\a ＞＞ 1, the parameter ks, which is related to the latitudinal change of Coriolis parameter, plays a dominant role and the conclusions are the same as those obtained on the assumption of homogeneous water, such that the stream-lines deviate to the right in the northern hemisphere, accompanied by the Rossby waves.
In the case of an intermediate scale of phe:nomena, the parameter k\lambda\, which is related to Coriolis parameter itself, must be taken into account and the perturbations are considered as the ccmbined effects of various factors.
On the other hand, it was Shown that, if the basic current is not uniform, the horizontal shear of velocity may sometimes play an important role in the discussion of the topographic effect upon the currents. Thus, the deflection of currents may be related to the horizontal gradient of velocity and Coriolis parameter.
It must be remembered, however, that the whole discussions are made under the assumption that the steady perturbations always exist, regardless of values of various parameters, so that the discussions of stability and instability of these flow patterns are entirely ommitted.

7.Acknowledgements.

The author is indebted to Mr. H. Kusunoki, of the Low Temperature Science Institute, Hokkaido Univ., who generously communicated him the papers of V. W. Stockman. He also wishes to thank Dr. M. Nakano, who kindly read the manuscript before printing.

References
Bolin, B.,1950: 0n the influence of the earth’s orography on the general character of the westerlies. Tellus,2,(3), pp.184-195.
Charney, J. and Eliassen, A.,1949:A numerical method for predicting the perturbations of the middle latitude westerlies. Tellus,1,(2), pp.38-54.
Colson, D.,1949:Air flow over a mountain barrier. Trans. Amer. Geophys. Union,30,pp.818-830.
Ekman, V, W.,1923:Uber Horizontalzirkulation bei winderzeugten Meeresstromungen.
Ark. f. Mat., Astr. och Fys., Bd.17, No.26.
Ekman, V. W.,1932:Studien zur Dynamik der Meeresstromungen. Beitr. zur Geophys., Bd.36, S.385-438.
Queney, P.,1947: Theory of Perturbations in.Stratified Currents with Applications to Air Flow over Mountain Barriers. Misc. Report No.23, University of Chicago.
Queney, P.,1948:The problem of air flow over mountains;a summary of theoretical studies・Bull. Amer. Met. Soc.,29,(1), PP.16-26.
Stockman, V. B.,1047: Effect of Bottom Topography on the Direction of Currents in the Sea. Priroda.11, pp.10-23.(Defence Research Board, Canada., T 57 R.) Stockman, V. B.,1948:Effect of bottom topography on the direction of the transport of water set up by the wind or the muss-field in a non-homogeneous ocean. Dok. Acad.
Nauk. SSSR,59,(5), pp.889-892.(DRB, Canada. T 56 R.) Sverdrup, H. U.,1941:The influence of bottom topography on ocean currents. Applied Mechanics, Th. von Karman Anniv. Volume.

THE AGRICULTURAL AND MECHANICAL COLLEGE OF TEXAS Department of Oceanography and Meteorology College Station, Texas A FORCED WAVE CAUSED BY ATOMOSPHERIC DISTURBANCES IN DEEP WATER Kinjiro Kajiura Technical report 133-1 Project 133 - Reference 56-26T

Research conducted for the United States Department of commerce, 、Weather Bureau, Contract Ilo・ cwb-8717, through the Texas A＆M Research Foundation. Presentation of material in this report is not considered to constitute final publication.

October, 1956

LIST OF FIGURES
Figure 1 Schenatic representation of each tern of Q
Figure 2 Air-pressure and mind stress distribution
Figure 3 Wind stress curl
Figure 4 Surface elevation due to pressure, \zeta_{1}\
Figure 5a Surface elevation due to wind stress
Figure 5b Surface elevation due to mind stress in sections parallel
to the y-axis
Figure 6 Deformation of the sea surface due to a model storm
Figure 7a Distribution of the xcomponent of the total flow
Figure 7b Distribution of the y component of the total flow
Figure 8a Distribution of \zeta\ along atraight lines (ky=0. +-0.04)
parallel to the x-axis
Figure 8b Distribution of Qx and Qy along atraight lines
(ky=0. +-0.04) parallel to the x-axis
Figure 9 Surface elevation, \zeta_{1}\ ,due to air pressure,
[Fig.107_048_01] , along the x’ and y
axis (ak=1)

ABSTRACT

The two dimensional forced wave in deep homogeneous water generated by a moving storm is discussed. The inverse barometric defororation of the sea surface is a fairly good approximation to the actual condition near the center of the storm, but the flow of water is quite different from that expected from the analogy of the one dimensional progressive wave. The geostrophic flow related to the wind stress curl is predomirant, and a permanent deformation of the water surface behind the storm is created, which is balanced by the geostrophic flow.
The critical case when the velocity of the storm movement is the same as the velocity of free waves in water, is briefly discussed. In this case, the two dimensional model gives quite a different picture coupared with the one dimensional model. It is found that a resonance condition does not obtain as in the simple one-dimensional problem for the critical speed. It is noteworthy that the two-dimensionality and the presence of Coriolis force will allow the establishment of an equilibrium height of the moving forced surge, even in the absence of bottom friction.

Introduction*

In deep rvater, it is generally believed that the response of the sea surface to the atmospheric pressure is inversely barometric and that the effect of wind stress is small compared with that of the former a t least in a one dimensional case. For a large scale wind system, however, the surface deformation due to stress becomes significant even in deep water.
Very few studies have been made to date on storm surges of two dimensions, because of mathematical difficulties, though the results derived from a two dimensional model might be quite different from those derived from a one dimensional model.
In this paper, a particular case of a two dimensional forced gave in deep homogenous water of infinite horizontal extent generated by a traveing circular atmospheric disturbance is studied and it is found for usual dimensions and speed of movement of a storm, that the quasi-steady solution for the surface deformation can be approximated, as is expected, by the equilibrium elevation of the sea surface near the center of the storm, and that the total flow is predominantly governed by wind stress curl and maintains geostrophic balance with a part of the deformation of the sea surface.
The present study excludes phenonena such as free inertiagravity waves and the readjustment of the density stratification in water which wouId play an important role in initial and boundary value problems.

*This work was originated from the necessity to understand the nature of waves due to atmospheric disturbances in an open sea so that we can formulate suitable boundary conditions at the edge of a continental shelf in connection with the study of storm surges in shallow water.

Basic Equations

The hydrodynamic equations of motion utilized are,

式(1-2)

and the equation of continuity is,

式(3)

Where x and y are the horizontal co-ordinates, t is time ,Qx, and Qy are the components of volume transport,Q,* in the x and y directions respectively, \zeta\ is the elevation of the sea surface from the undisturbed level,\tau_{x}\ and \tau_{y}\ are the components of wind stress , \tau\, in the x and y directions respectively, \zeta\ is the equilibrium height of the sea surface due to the atmospheric pressure, f is the Coriolis parameter, d is the depth of water, g is the acceleration of gravity, and \rho\ is the density of water.
The above equations are valid under the following assumptions:
1. The water is homogeneous.
2. Field accelerations are negligible.
3. The bottom friction is negligible.
4. \zeta\/d and d/ \lambda\ are both very small compared with unity where \lambda\ is the water wave length. This is the so-called long wave approximation.
Furthermore, in the following discussion the variation of the Coriolis parameter with latitude is neglected, and the depth is assumed constant.
[Fig.107_051_03] where u is the velocity vector in water and Z is vertical axis positive upward.

Derived Equations

From the equation (1), (2) and (3), we obtain*

式(4)

where curl [Fig.107_052_02] ,and for the total flow, we have

式(5)

and

式(6)

where \nabla^{2}\ is the Laplacian operator with respect to x and y.
Now we can split \zeta\ into two parts, \zeta\ = \zeta\1+\zeta\2 ,such that equation (4) consists of two equation as follows：

式(7)

and

式(8)

Here it is obvious that \zeta\1 represents the surface elevation due to the atmospheric pressure gradient modified by the effect of wind divergence, and \zeta\2 represents the elevation due to curl of wind stress.
Corresponding to the terms \zeta\1 and \zeta\2, we can split Q into two parts, Q = Q1+Q2 and from the equations (5) and (6), we have,

式(9-10)

* \tau_{x}\ and \tau_{y}\ stand for \tau’_{x}\/\rho\ \tau’_{y}\/\rho\ respectively.
式(11)

and

式(12)

where we assumed that the ocean is initially in the undisturbed state, and in the two equations (11) ,and (12) ,we used equation (8) for modification of the original form.
Equations (7) and (8) have the form:

式(13)

where

[Fig.107_053_04]

respectively.
This type of equation is sometimes called the Klein-Gordon equation [Morse and Feshbach,1953, p.138] which appears in the Quantum Iiechanics, and the fornal solution is given in terms of a Green’s function (see Appendix A), but the actual integration is very complicated except for a case of special type of the forcing function.
Crease [1956] discussed the solution for the special case of a one diaensional forcing function.
The equations from (9) to (12) are the Poisson type, if \zeta\1 and \zeta\2 are known from equation(13). Inspecting the right hand sides of these equations we can easily find that Q consists of two parts;a divergent term associated with the timely variation of the sea surface, and a rotational term which is in geostrophic balance with the part of the elevation.

The Equation for a Forced Wave

If the external force is assumed stationary with respect to the co-ordinate system moving with a constant velocity, V , in the positive x direction, in other words, if the external forcing function can be expressed by

式(14)

a forced wave equation is derived from the equation (13) by means of the transformation,

[Fig.107_054_02]

and the result is,

式(15)

where x’ = x-Vt .
This equation excludes free waves which may play an important role in initial and/or boundary value problems. However, It is sufficient to get a forced wave solution if we are interested in a stationary wave moving with the storm in an ocean of infinite extent.

A Forced Wave Solution
We proceed to find a forced wave solution under the following assumptions.
The depth of the ocean, d , is large enough that the condition V^{2}/\epsilon_{d}\ ＜＜ 1 is satisfied (see Appendix B).
The atmospheric pressure distribution is radially symmetric with respect to the moving storm center.
The wind distribution is assumed to be determined by the gradient wind formula. The effect of the movement of the storm - isallobaric effect - is neglected. Therefore, the distribution of wind stress is radially symmetric and has no radial component,(div\tau\ =0), (\tau\=\tau_{\theta\}\).
Tnus equation (15) can be written,

式(16)

where r , \theta\ are the polar co-ordinates with the origin at the moving storm center and \tehta\=0 on the positive x’axis. Also k^{2} = f^{2}/■

and

[Fig.107_055_01]

corresponding to, [Fig.107_055_02] respectively.

The solutions of homogeneous equation for equation (16) are given by modified Bessel functions of the first and ■ kind, I0(kr) and K0(kr), and the solution for the inhomogeneous equation is obtained by a standard method,such that

式(17)

where limits of integration are selected to satisfy the conditions, \zeta\ = 0 at r = \infty\ and \zeta\ = finite at r = 0 .
Solution (17) can be transformed by partial integraton,into a more convenient form for the practical computation,such that

式(18)

and

式(19)

Therefore, the elevation ,\zeta\2, is given by,

式(19’)

where \zeta\2 is assumed zero at the infinite distance in the positive
x direction.
In the solution for \zeta\1,the last two terms involving Bessel functions are considered to be the correction terms to the equilibrium elevation, and the magnitude of these terms depend on the shape and the scale of the air-pressure pattern. If the dominant wave nunber of \zeta\ is much larger than the wave number k (reciprocal of the Rossby’s deformation radius), these terms become very small.
In the solution for \zeta\2,we can easily find that the elevation due to curl \tau\ is inversely proportional to the speed of the storm movement and the depth of water. Therefore, in shallow water the effect of curl \tau\ increases and may exceed the barometric effect, provided the condition V^{2}/gd ＜＜ 1 is satisfied. Because w is radially symmetric. \zeta\2 is not radially symmetric with respect to the moving storm center. In general, the permanent deformation of the surface is produced behind the storm, unless the following condition is fulfiled;

[Fig.107_056_01]

In the limiting case of k → 0 , which corresponds to the condition that the wave number of \zeta\ is much larger than the wave number k , solutions (18) and (19) are reduced to,

式(20)

and

式(21)

The solutions (20) and (21) are good approximations to the solutions (18) and (19) in deep water as will be shown later in a numerical example [see Appendix C for a simple physical interpretation of eqs(20),■
The solutions of the equations from (9) to (12) are (see Appendix D),

式(22-24)

and

式(25)

where

[Fig.107_057_03]

and

[Fig.107_057_04]

The last term in each solution represents the component of nondivergent flow which contribute to the geostrophic balance viith the elevation of the surface and the remaining terms represent nongeostrophic flow which contribute to build up the deformation.These terms are schematically illustrated in Fig 1, where the dominant term is that in geostrophic balance with \zeta\2・Near the storm area,the total flow is approximated within a few percent of error by,

[Fig.107_057_05]

and

[Fig.107_057_06]

where the P2 term is in geostrophic balance with the elevation (\zeta\1 - \zeta\), and (near the center of the storm) the last term is about ten times larger than the remaining terms.

A Model Storm

The pressure distribution of our model storm is given by,

式(26)

where P is the atmospheric pressure, the suffixes \infty\ and o refer to the pressure at infinity and at the center of the storm, and R is a constant distance from the center where maximum wind is found [Hydrometeorological Report No. 31, 1954]
The conditions that the pressure and its gradient at r = a is continuous, determines k’ such that,

[Fig.107_058_02]

The gradient wind v_{\theta\} is governed by the equation,

式(27)

where \rho\’ is the air density and isallobaric terms associated with the movement of a storm are neglected. The solution of the equation (27) gives,

式(28)

where [Fig.107_058_05]

The wind stress formula is,

式(29)

where K is the friction coefficient for wind stress.
We can calculate curlZ \tau\ from the equations (28) and (29).
Nurmerical values used for evaluating \zeta\ and Q are:

[Fig.107_059_01]

Numerical Results

The variation of atmospheric pressure and stress with radial distance is shown in Fig 2, and the variation of curl \tau_{\theta\}\ with radial distance is shown in Fig 3. The value of curl \tau_{\theta\}\ changes sign at about kr = 0.08 and is discontinuous at r = a , but the absolute value is very small except near the center of the storm.
The surface elevation due to pressure [\zeta\1,given by equation (18)] is shown in Fig 4 together with the equilibrium elevation (\zeta\).
Inverse barometric elevation is a very good approximation within a few per cent of error near the center (4 per cent at the center). A very slight depression appears at a large distance from the center and approaches zero at infinity, which is not shown in the figure.
The deformatio of the surface due to wind stress is quite different from the effect of pressufe gradient as shown in Fig 5, and the order of deformation due to stress is about one-tenth of that due to air pressure near the center.This elevation, \zeta\2,is inversely proportional to the speed of a storm, V,and in the present example,\zeta\2 is about 7 per cent of \zeta\1 near the center.
The total deformation of the surface is shown in Fig 6 which shows an almost inverse barometric deformation near the center of the storm and an unsymmetric character of elevation in the drection of storm travel. On. the lee side of the storm the permanent depression exists of the order of about 13-14 per cent of \zeta\maxs.
The distribution of the total flow Q is given in Fig 7 in temsof Qx and Qy. The predominant term is the geostrophic flow balanced by the elevation \zeta\2 and so the height contours given in Fig 4 may be interpreted as stream lines of the total flow to a good degree of approximation.
About 10 per cent of the total flow is in geostrophic balance with the pressure gradient caused by the difference between \zeta\ and \zeta\1,and few per cent is associated with the divergence field due to the movement of the disturbance.
The very small value of the flow associated with the divergence field compared with the geostrophic flow is explained by the fact that the depth of water is very large and a relatively small [Fig.107_060_01] is enough to build the surface deformation.
Sections of \zeta\ and Q along lines parallel to the x-axis and through the maximun wind region are drawn in Fig 8. Due to the prevailing effect of the geostrophic balance between current and elevattion, the reIattions between \zeta\ and Qx are quite different from
that expected for one-dimensionaI progressive waves.

Discussion of the Results

If we remove the assumption that V^{2}/gd ＜＜ 1,we can see from equation (15) that the forced wave will deviate considerably from the equilibrium elevation due to the effect of the storm movement. The critical case of gd=V^{2} will be briefly discussed in Appendix E.
In this case, the Coriolis force and the two dinensionality of the waves are important factors in determining the elevation of the sarface.
If we take into account the deviation of the wind direction from the gradient wind due to the friction at the sea surface, it is easily seen from equation (4) that the air-pressure effect will be modified by the convergcnce effect of air flow, but this effect is rather small in deep water.
On the otaer hand, the absolute value of curl \tau\ will decrease and the corresronding surface elevation,\zeta\2, and the total flow, Q2 will be considerably less than those given in the above example.
If the absolute value of wind velocity is the same, the values of \zeta\2 and Q2 will decrease by sin^{2}\gamma\ per cent due to the deviation, \gamma\; of the wind direction from the isobar.
As a result of these modifications due to the surface friction of air flow, the deformation of water surface will approach more and more to symmetric pattern and the equilibrium elevation of water will become a very good approzimation in deep water.
The effect of the isallocaric wind in the will stregthen the wind in the right falf and weaken the wind in the left falf of the storm and correspondingly the field of flow in water will be modified to strengthen the flow in the right half and weaken the flow in the left half.
The assumption made in the basic equations that the field acceleration (non-linear terms) is negligible,is verified by examining the order of these terms compared with the other terms. From the Fig 7a,b, we can see,

[Fig.107_061_01]

Therefore,

[Fig.107_061_02]

because, in general, [Fig.107_061_03] for deep water. Thus, we can neglect non-linear terms compared with the Coriolis term, at least in deeo water.
In the present paper, we discussed only the forced wave traveling with the storm, and the density of the water is assumed constant.
These two assumptions are not fulfilled in the actual ocean, where the phenomena are not quasi-steady, nor the density of water uniform.
The stratification of water in the ocean will affect the formation of the geostrophic flow through the readjustment of density distribution, and, moreover, the free waves of surface and internal modes, which are neglected in this discussion, play an important role to disperse the energy transferred from a storm to a wider region*.
The rigorous discussion should be based on the solution of an initial and boundary value problem, taking both inertia-gravity waves and density variation into consideration. The study in this direction is now continued.

*According to Veronis [1956], the ratio of baroclinic to barotropic energy is dependent on the scale of the flow and, when the scale is smaller than the barotropic radius of deformation as in the case of this paper, more energy goes into the baroclinic mode.
And the time of duration of the wind stress is an important factor in the generation of non-geostrophic motions.

REFERENCES

Crease,J.: Propagation of long waves due to atmospheric disrurbances on a rotating sea, Proc. Roy. Soc.,A.233,556-569, 1956.
Horse, P. M.,and H. Feshbach:Methods of Theoretical Phisics,McGraw-Hill,New York,p.138,1953.
Proudman, J.:Dynamical Oceanography, John Wiley ＆Sons, Inc.New York,p.295,1953
U.S.Weather Bureau: Analysis and synthesis of hurricane wind patterns over Lake Okeechobee,Florida,Hydrometeorological Rep.No.31,1954
Veronis, G.:Partition of energy between geostrophic and non-geo-strophic oceanic motiions,Deep Sea Res.,3,(3),157-177 1956.

Appendix A

Green’s function in the two dimensional infinite given by the solution of the foflowing equation:

[Fig.107_064_01]

The solution is,

[Fig.107_064_02]

where [Fig.107_064_03]
This Green’s function shows that the disturbance travel faster than the free wave velocity [Fig.107_064_04]
The solution for equation (13) is given by,

[Fig.107_064_05]

The limit of integration must be determined to cover all x0,y0, t0 provided c\tau\ > P.

Appendix B

For d=1500 fathoms, and V=40knots,the ratio, V^{2}/gd, is O.02.
If the velocity of the storm, V,is small, the assumption, V^{2}/gd ＜＜ 1,is valid even in shallow water. For example, say, for V= 10knots, V^{2}/gd=0.02, we have, d = 60 fathoms.
See also Appendix E.

Appendix C

The solutions (20) and (21) can be understood by the following simple interpretation.
In deep water the free wave velocity is much larger than the velocity of the storm Movement, the whole sequences may be considered as quasi-steady and quasi-geostrophic. Therefore, the atmospheric pressure gradient is balanced by the deformation of the sea surface \zeta\1 :

式(20)

Wind stress accelerates a water body such that,

[Fig.107_065_02]

andthe Coriolis force due to the flow Q2 wil be balanced by the pressure gradient due to the surface deformation,\zeta\2.
Thus, we have,

[Fig.107_065_03]

and from these equations together with the equltion of continuity, we have

[Fig.107_065_04]

and the solution is,

式(21)

The deviation from the geostrophic balance is included in the solutions (18) and (19) but the numerical example shows that, in deep water, the correction is small.

Appendix D

By virtue of the radially symmetric characterof \zeta\1,the right hand side of equation (9) can be written in the polar co-ordinate as follows:

[Fig.107_066_01]

Corresponding to tale above expression, we put

[Fig.107_066_02]

Then equation (9) can be split into three equations

式(1)’

式(2)’

and

式(3)’

The solution of equation (1)’ is given by,

[Fig.107_066_06]

The solution of equation (2)’ can be constructed from the elementary solutions of the homogeneous equation, r^{2} and 1/r^{2}. And the solution of equation (3)’ can be constructed from the elementary solutions of the homogeneous equation,r and 1/r.
If we take into consideration that P1 and P2 must be finite at r = 0 and r = \infty\ , we can obtain the required solutions,

[Fig.107_067_01]

and

[Fig.107_067_02]

The other equations can be solved similarly, if we take into consideration that [Fig.107_067_03] is radially symmetric in equations (11) and (12).

Appendix E

For a critical case of gd=V^{2},equation (5) becomes,

式(1)’’ [Fig.107_067_04]

The solution of the above equation under the conditions \zeta\ = 0 for y=+- \infty\ is given by,

式(2)’’ [Fig.107_067_05]

where

[Fig.107_067_06]

Now we study the barometric effect only. Then,

[Fig.107_067_07]

Therefore, if we assume,

[Fig.107_067_08]

we have the following solution

式(3)’’ [Fig.107_068_01]

The second term in the bracket is the correction term to the equilibrium elevation and its magnitude depends on the scale of the storm, a , relative to Roesby’s deformation radius, 1/K .
For aK = 1 , which corresponds to a=200km, [Fig.107_068_02] =V=40 knots, and f =10^{-4}/sec, we have at the center of the storm,

[Fig.107_068_03]


On the other hand if we consider the case when all the quantities are constant in the y-direction, which is not the one dimensional case in a strict sense because we retain the quantity, Qy, the equation becomes,

式(4)’’ [Fig.107_068_04]

and assuming [Fig.107_068_05],we have

式(5)’’ [Fig.107_068_06]

At the center of the storm with aK= 1 , we have

[Fig.107_068_07]

For the case of the one dimensional forced wave without the Coriolis force, [Fig.107_069_01] gives the resonance condition and the elevation becomes indefinitely great [Proudman, 1953, P.295].
From the above consideration, it is clear that the Coriolis force and the two dimensionality of the waves are important factors in determining the elevation of the surface, and the deviation of the surface deformation from the equilibrium height becomes more and more conspicuous as the ratio [Fig.107_069_02] increases.
The solution (3)" for aK=1 computed numerically along the x’ and y axes is shown in Fig 9. The deviation of the surface deformation from the equilibrium shape is shown clearly.

ACKNOWLEDGEMENTS

This study is a part of the research on storm surges which is being continued, under the supervision of Mr. R.O.Reid, to whom the author is indebted for many valuable suggestions. The author also wishes to express his appreciation to Mr.J.D. Cochrane and Dr. G. . Heumann for their advice concerning the form of the manuscript. He is grateful to Mr. T. Williams for drafting of figures and Mrs. H. Proctor for the preparation of the report.

LIST OF SYMBOLS(Main symbols only)

[Fig.107_070_01] [Fig.107_070_02] ,vertical component of stress curl

d depth of water

[Fig.107_070_03] [Fig.107_070_04] ,divergence of stress vector

f Coriolis parameter

[Fig.107_070_05] external forcing function, eq (13) or (16)

g acceleration of gravity

[Fig.107_070_06] modified Bessel function of the first kind (order 0 and 1 respectively)

k [Fig.107_070_07], reciprocal of Rossby’s deformation radius,
eq (16)

K stress coefficient, eq (29)

[Fig.107_070_08] modified Bessel function of the second kind (order 0 and 1 respectively)

P1,P2 coefficients in eqs (22) and (23)

Q volume transport vector

Q1,Q2 volume transport vector corresponding to \zeta\1
and \zeta\2 resrectively

Qx,Qy components of volume transport in the x and y directions, respectively

r,\theta\ polar co-ordinates with the origin at the center of the moving storm; \theta\ is measured from the
positive x’一axis counterclockwise

S1,S2 coefficients in eqs (24) and (25)

t time

V velocity of the storm movement

w [Fig.107_070_09], vertical velocityof the movement of the
water surface due to curlz \tau\

x,y Cartesian co-ordinates, in which the positive x is in the direction of the storms movement.

x x -Vt , x co-ordinate relative to the moving storm

z vertical co-ordinate positive upward

\zeta\ elevation of the sea surface from the undisturbed level

\zeta\ equilibrium elevation of the sea surface due to atmospheric pressure

\zeta\ 1 elevation of the sea surface due to barometric effect including the effect of the wind divergence

\zeta\ 2 elevation of the sea surface due to wind stress curl

\xi\ [Fig.107_071_01]

\rho\ density of water (\rho\’ density of air at sea level)

\tau\ wind stress vector

\tau\x,\tau\y components of wind stress in the x and y directions, respectiveiy

\tau_{\theta\}\ tangential component of wind stress in the polar co-ordinate
[Fig.107_071_02] [Fig.107_071_03] Laplacian operator in the horizontal plane

Equation (16) is written as

式(A)

and the solutions of the homogeneous equation are given by modified Bessel functions of order zero, I0 (kr) and K0 (kr). The function I0 (kr) is finite at kr = 0 and infinite at kr = \infty\, and the function K0 (kr) is infinite at kr = 0 and zero at kr = \infty\. Therefore, corresponding to the boundary conditions that \xi\ is zero at r = \infty\ and \zi\ is finite at r = 0, we can construct a Green’s function such that

[Fig.107_075_02]

and

[Fig.107_075_03]

where [Fig.107_075_04]

From the above Green’s function, we have,

[Fig.107_075_05]

Some formula are listed below:

[Fig.107_075_06]

for large value of z,

[Fig.107_075_07]

A FORCED WAVE CAUSED BY ATMOSPHERIC DISTURBANCES IN DEEP WATER Kinjiro Kajiura The Agricultural and Mechanical College of Texas Department of Oceanography and Meteorology Technical Report 133-1 - Reference 56-26T October,1956

ERRATA AND SUPPLEMENT

Page Line or Equation Number Change

iii 5 Figure 5a should read, 「Surface elevation due to wind stress parallel to the y-axis.」

iii 6 Figure 5b should read, 「Surface elevation due to wind stress"

iii 11 and 14 Insert comma to read, 「(ky = 0, +-0.04)」

3 4 Should read, 「[Fig.107_082_01] stand for [Fig.107_082_02] respectively.For the total flow, we have.…」

3 Footnote Omit footnote.

4 4 「where we assumed that the ocean is initialy in the undisturbed State, and the last two equations (11) ,…」

4 eq (13) [Fig.107_082_03]

6 2nd para. 「The solutions of the homogeneous equation…… of the first and third kind,…」

6 last line, 2nd para.; 「… at r = C. (See Appendix F) 」

8 last two equations; [Fig.107_082_04]

8 last line; 「…… remaining terms except on the line ky = O,where [Fig.107_082_05] 」

9 eq (26) [Fig.107_083_01]

9 7 「Hydrometeological Report No,31,1954.The formula (26b) is assumed for r>a instead of the formula (26a) in order to have the finitet total volume of water due to the equilibrium elevation.

[Fig.107_083_02]

The formula (26a) becomes 〜R/r for a large value of r and the above integral becomes infinite. However, this formula is a very good approximation to a pressure pattern within a hurricane. Therefore,the distance a may be chosen to bo equal to the approximate radius of a storm area.

10 Numerical values : [Fig.107_083_03]

11 4 「… of storm travel. Behind the storm, there remains a permanent depression of the order …」

11 4th and 5th paragraphs to be replaced as follows:
The very small value of fhe flow associated with the divergence field conpared with the geostrophic flow is due to the assumption of deep homogeneous water. For the same surface deformation, the geostrophic total flow is prportional to the depth but the total flow associated with divergence field is not directly related to the depth and we can expect the predominant effect of the geostrophic term. This situation is quite different from the case of shallow water, where the total flow associated with a divergence field may overcome the geostrophic flow and thus the divergence term becomes very important in determinin a flow pattern.
Sections of \zeta\ and Q along lines through the maximum \tau_{x}\ region and parallel to the x-axis are shown in Figure 8, together with the section along the x-axis. Due to the prevailng effect of th e geostrophic balance between total flow and elevation, the relation between \zeta\ and Qx along the line, ky =0.04, is .quite different from that expected for one-dimensional model in which the condition Qx = V\zeta\ must be satisfied due to the equation of continuity.

21 List of Symbols should be placed in front of report, following page iv

Figure 1: (5) ■ Term should be turned 90° to the left.

Figure 8b: [Fig.107_084_01] should read [Fig.107_084_02] ,and the value of [Fig.107_084_03] should be plotted instead of
[Fig.107_084_04] , as follows:
[Fig.107_084_05]

EFFECT OF CORIOLIS FORCE ON EDGE WAVES (II) SPECIFIC EXAMPLES OF FREE AND FORCED WAVES(1) BY KINJIRO KAJIURA Department of Oceanography and Meteorology, Agricultural and Mechanical College of Texas

ABSTRACT

Modification of edge waves due to Coriolis force is examined in terms of two specific problems:dispersion of an initial deformation of the water surface ; the forced wave due to a moving atmospheric pressure disturbance. In both cases the scale of the disturbance is an important factor, and the larger the scale the larger the effect of Coriolis force. In the northern hemisphere, the initial deformation of the water surface is split into two free edge waves in which the one moving to the left (facing the coast from the sea) has a larger amplitude than that moving to the right. The contribution of Coriolis force to the edge wave amplitude for an initial deformation having a horizontal scale of several hundred kilometers is about 20% at the center in the initial stage. In the case of forced waves, movement of the atmospheric pressure system to the left is more favorable for exciting resurgent edge waves thanmovement to the right. The contribution of Coriolis force is about 10 to 20% in the forced period and 20 to 40% in the amplitude for an atmospheric disturbance of several hundred kilometers in diameter which moves with a speed of about 20 to 30 knots.

INTRODUCTION

Resurgent water level disturbances caused by hurricanes traveling parallel to a coastline were analyzed from the standpoint of forced edge waves by Munk, et al.(1956),who obtained periods of resurgences in close accord with some actual observations. However, the range of conditions available was not sufficient to fully establish the veracity of the theory. Later, Greenspan (1956) extended the theory to a more realistic situation by treating the transient problem with a Gaussian distribution for the pressure deficit in the atmosphere. One of the interesting features of Greenspan’s analysis is that, for large scale atmospheric pressure disturbances of size comparable to a hurricane, only the fundamental edge wave mode is excited while all other modes are negligibly small if realistic values for bottom slope and storm propagational speed are taken. He concluded that hurricanes are usually too big to produce the maximum surge height which is possible for a given pressure deviation.
The theory of edge waves so far has been applied to specific problems withoutCoriolis force, and this approximation is quite proper if the scale in space and time is reasonably small. However, the effect of the earth’s rotation should be taken into account if the period or the horizontal scale of the external force respectively are comparable to or larger than either 1/f or the deformation radius C/f (where C is a representative wave velocity or storm speed and f is the Coriolis parameter).
Reid (1958) has discussed the modification of free edge wave modes due to Coriolis force and has presented some general considerations in regard to forced edge waves. In the present paper, two specific problems are investigated in detail on the basis of the fundamental edge wave mode. First, the dispersion of an initial deformation of the water surface is studied. The propagation of free edge waves to the right and to the left (facing the coast from the sea) becomes unsymmetrical due to the presence of Coriolis force, and a greater fraction of the total energy is propagated to the left than to the right in the northern hemisphere. Next, the influence of Coriolis force on forced waves due to a moving atmospheric pressure disturbance is discussed, making use of a model similar to the one utilized by Munk, et al.(1956). In this case, movement of the pressure system to the left is more favorable than movement to the right for the development of resurgent waves so long as the velocity of the pressure system is less than the maximum group velocity of the edge waves. The contribution of Coriolis force amounts to about 10 to 20% for the period and 20 to 40% for the amplitude in the resurgent edge waves. The effect of wind stress is not considered in the present paper.

(1)Contribution from Department of Oceanography and Meteorology, A＆M Colege of Texas, Oeeanography and Meteorology Series No.121.

Reptint from SEARS FOUND. JOURN. MAR. RES.

BASIC EQUATIONS

Assume that the coastline is straight and that the depth of water is linearly proportional to the distance offshore, with zero depth at the coast. Take the horizontal right-hand coordinates x and y so that the x-axis is directed offshore with the origin at the coast. Using the equation of continuity and the linearized equations of motion without friction [Reid,1958 :eqs.(1) to (3)], the equation for the surface elevation \eta\ (relative to the undisturbed level) due to atmospheric pressure disturbances can be shown to be

式(1)

where t is time, f the Coriolis parameter, g the acceleration due to gravity,\eta\ the equivalent height of water due to the atmospheric pressure deficit (relative to normal pressure), and s the bottom slope in the xdirection. The rotation \nabra^{2}\ is of conventional use with respect to xand y.
Auseful forced wave equation can be derived from (1), if the atmospheric disturbance moves with constant velocity V, parallel to the coastline, without change of pattern. Putting \eta\ = \eta\(x,\xi\) where \xi\ = y-Vt in (1) and assuming that all disturbances in the water are zero for t → \infty\ in the finite region, we find

式(2)

This relation is used in the second problem treated in this paper.

DISPERSION OF AN INITIAL DEFORMATION OF THE WATER SURFACE

Consider the situation in which the water body is initially at rest with a given surface deformation. For mathematical simplicity, we confine our attention to only the fundamental mode of edge waves by assuming a suitable deforlnation initially. specifically, the initial conditions are as follows:

式(3)

where Qx and Qy are the components of volume transport of water in the x and y directions respectively. The scale of the initial deformation is represented by a and corresponds to the distance, on the x or y axis, from the origin to the position where the height of the water surface is half the maximum height (see Fig.1). The unknowns \eta\ , Qx, and Qy for t > 0 can be expressed by the superposition of edge waves of the fundamental mode, in view of the form of the Fourier transform of \eta\ (with respect to y) at t = 0.
The normal mode edge waves may be found from (1), but they are not discussed here (Reid,1958). Edge waves of the fundamental mode for any longshore wave length (2\pi\/k) are given by

式(4a)

Figure 1. Graphs of the initial distribution of water level; upper graph is the relative water leveI along shore; lower graph is the relative water level offshore from the center of the initial disturbance.

式(4b)

式(4c)

where

式(5a)

式(5b)

式(5c)

Here Hi is the fundamental normal mode elevation (of unit amplitude at shore), Ui and Vi are the corresponding components of volume transport in the x and y directions respectively, and subscript j=1,2, or 3 correspond to the different frequencies \omega\1, \omega\2, and \omega\3. Later in this section, j=3 is omitted because there is no contribution from this mode to the surface elevation. Note that j=1and 2 correspond to waves moving in the positive and negative y directions respectively and that the phase velocity of the former is larger than that of the latter for the same wave length, though the group velocity is the same in both directions.
The functions \eta\, Qx, and Qy for t > 0 are formally given by the real parts of the following integrals

式(6a)

式(6b)

and

式(6c)

where、F(k) is the Fourier transform of \eta\ at t=0 and x=0, namely

式(7)

The weight functions p1 and p2 can be determined from the initial conditions such that

式(8a)

and

式(8b)

These weight functions p1 and p2 show that more energy goes to the negative y direction than to the positive y direction and also that the longer the wave length the larger is the difference in wave heights between two waves propagating in opposite directions, as shown in Fig.
2. The functions p1 and p2 are a special case of the pjn functions derived by Reid (1958).
Substituting (7),(8a) and (8b) into (6), we can split the elevation into four progressive waves:

式(9)

where

式(10)

式(11)

Figure 2. Graphs of the weight functions p1 and p2 versus relative wave number gsk/f^{2}, for the fundamental edge waves.

and

式(12)

In (9),\eta\1* and \eta\2* are the counterparts of the fundamenta1(Stokes) edge waves without Coriolis force traveling to the positive and negative y directions respectively, although the speeds in the positive and negative directions are different, as is evident in (12). On the other hand,\eta\1(f) and \eta\2(f) represent waves essentially due to Coriolis force.
At the center of the initial deformation x=y=0, it follows from (10) and (11) that initially(t=0)

式(13)

and

式(14)

provided that a ＜＜ 4gs/,f^{2}. The latter condition implies that the scale a must be much less than 4000 km at midlatitudes for s=10^{-3}. The ratio of the elevations \eta\1,2(f)/\eta\1,2(\delta\) is expressed approximately by fa/C0, where [Fig.107_090_05] is the ordinary edge wave velocity with the wave length \lambda\=8a. The ratio computed for several values of a are shown in Table 1, from which we can clearly visualize the effect of Coriolis force. In the case of the initial deformation with an effective radius of about 400 km (a=100 km), the contribution of Coriolis force is greater than 20% of the ordinary edge wave. If we compare the waves traveling in the positive y and in the negative y directions, the difference of initial heights of the two progressive waves may become 40% of the average height of two waves. However, the resultant deformation of the water surface does not appreciably deviate from the deformation for the case without Coriolis force so long as time is not great, because \eta\1(f) and \eta\2(f) cancel each other for initial stages.

TABlE 1.

Making use of the method of stationary phase, we can integrate (10) and (11) approximately for large values of t, such that

式(15a)

and

式(15b)

The above solutions are valid for [Fig.107_091_04].
For fIyI, there exists no stationary phase in the integral given by (11). Practically speaking, the second condition requires that the time must be very long (beyond a practical limit) if fIyI is not very small, because ft>10^{3}IyI/gst. Therefore, if the stationary phase is attained at all, the value of fIyI/gst is already very small, and the wave amplitude is almost negligible near the origin. However, formally looking into the argument of the cosine term in (15a) and (15b) we can find that modification of the dispersive character of edge waves due to Coriolis force is such that

local period : 式(16a)

wave length : 式(16b)

phase velocity : 式(16c)

group velocity : 式(16d)

where

式(16e)

式(16f)

式(16g)

and

式(16h)

where the subscript o indicates the quantity without Coriolis force and the upper sign and the lower sign in (16a, b, c) correspond to the subscripts 1 and 2 respectively. The above relationships are all consistent with the discussion based on the normal modes (Reid,1958) such that C1 > C2 and G1 = G2 for the same wave length. The ratio of the wave amplitudes at a common distance IyI to the right and to the left is given by

式(17)

as shown in Fig.3. The amplitudes for both waves decrease very rapidly with decreasing a due to the exponential factor involved in (15a) and (15b).
As a whole, the effect of Coriolis force increases with increasing values of \alpha\,(\alpha\ ＜ 1). This implies that, for a fixed time, the local wave length increases with distance from the origin, and consequently the effect of Coriolis force increases. However, for IyI > gst/f the condition of stationary phase does not exist and consequently the local wave length is undefined. It is probable that beyond this point, to the left, the water level decays monotonically to zero as in the outer part of the initial disturbance, but to the right the elevation is essentially zero beyond IyI = gst/f.
Note that the condition fIyI/gst＜1, which is required to establish the stationary phase, gives the maximum group velocity G1,2 = gs/f; this maximum value corresponds exactly to the maximum group velocity derived from the study of the normal modes [Reid,1958:eq.(42)].

Figure 3. Ratio of the wave amplitudes at a common distance IyI to the right and to the left at time t,due to an initial static mound of water.

FORCED WAVE DUE TO A TRAVElING PRESSURE DISTURBANCE

For atmospheric pressure disturbances,(2) can be solved generally by means of the Fourier transform technique. However, for mathematical simplicity, we assume an atmospheric disturbance of the form

式(18)

This is the same form as that assumed by Munk, et al.(1956), which is convenient since it implies that we can confine our attention to fundamental waves only. The wave characteristics derived from this model may be considered as fairly representative of the actual situation because, according to Greenspan (1956), the fundamental mode of the edge wave is the only mode effectively excited by a pressure disturbance of the scale comparable to a hurricane.
Since \eta\ and \partial\\eta\/\partial\t are assumed zero for t → -\infty\ (which corresponds to \xi\→\infty\ forV ＞ 0, and \xi\ → -\infty\ forV ＜ 0), the Fourier transform of (2) with the pressure disturbance (18) may be solved straightforward, and the solution for \eta\ is formally given by

式(19)

where7. ＜ 0 for V ＞ 0 and \tau\ ＞ 0 for V ＜ 0 and [Fig.107_094_02],
The other parameters are

式(20a)

式(20b)

式(20c)

The quantities ks and G represent respectively the wave number for the resurgence without Coriolis force and the maximum group velocity.
If we assume f=0 (G→\infty\), the solution for \eta\ reduces to the case discussed by Munk, et. al.(1956) and by Greenspan (1956).
Making use of the contour integral technique, we can express (19) for V > 0 in the form

式(21a)

式(21b)

where

式(22)

For V ＜ 0, we have a similar expression, provided IVI ＜ G:

式(23a)

式(23b)

If IVI ＞ G, the oscillatory part of the above solution is absent.
Asymptotically, for large values of [Fig.107_094_11], \phy\ is proportional to the atmospheric pressure disturbance \eta\. In fact, for IVI → 0, [Fig.107_094_12] for all values of x and I\xi\I.Thus, the forced response in the water consists of a train of resurgent edge waves of speed V behind the storm together with a disturbance restricted to the immediate neighborhood of the pressure system for which the water level is the direct effect of the atmospheric pressure. This result is qualitatively similar to that for the case without Coriolis force. However, if -V/G > 1,no resurgent wave exists in the present case. In other words, if the storm moves faster in the negative y direction than the maximum group velocity gs/f (this is very unrealistic), the resurgent waves do not develop. The case V = -G is the resonant condition and the water level increases indefinitely [Fig.107_095_01]. For the movement of the storm in positive y direction, no such condition exists.
The resurgent edge waves behind the storm are given by

式(24)

Therefore, the period of edge waves is given by

式(25a)

or

式(25b)

where T*(=2\pi\ lVI/gs) is the period of forced edge waves in the absence of Coriolis force. The contribution of Coriolis force in the forced period is approximately IVI per cent of T*if V is measured in meters per second, s=10^{-3} and f=10^{-4} sec^{-1}. Resurgences following a pressure pattern which moves to the right have a shorter period than that of forced edge waves without Coriolis force. The opposite is true of resurgences associated with a pressure pattern which moves to the left. If the scale of the pressure system is very large, the contribution of Coriolis force to the height of resurgent waves becomes large compared with that which occurs in the case without Coriolis force. Putting \eta\1 and \eta\2 as the amplitude of resurgent edge waves moving to the right and to the left (V ＞ 0 and V ＜ 0) respectively, and putting \eta\* as the amplitude of edge waves in the case without Coriolis force, we have

式(26)

Note that the ratio is independent of the slope of the bottom, and the effect of Coriolis force is measured by the number fa/IVI, which is what we anticipated in the beginning. For reasonable values of V and a (V〜 10 to 15 m/sec, a〜25 to 50 Km), the contribution of Coriolis force in the wave amplitude along the coast seems to be about 20 to 40%.

A somewhat similar integration is given by Lamb (1932:412).

The ratio of amplitudes\eta\1,\eta\2 and \eta\* to the equilibrium height \eta\0 along the coast is shown in Table II. There exists an optimum value of a for which the ratio becomes maximum if V and s are fixed.
The amplitude of edge waves decreases exponentially in the offshore direction, and the rate of decrease is larger for a storm moving in the positive y direction than that for a storm moving in the negative y direction.

TABlE II. RATIOS OF THE VARIOUS AMPLITUDES OF RESURGENT EDGE WAVES
RELATIVE To THE MAXIMUM EQUILBRIUM HEIGHT AlONG THE COAST FOR f=10^{-4}
SEC^{-1} AND s=10^{-3}. THE RATIO \eta\*/\eta\0 IS SHOWN IN ITALICS. UPPER FIGURES IN EACH BLOCK REPRESENT \eta\1/\eta\0 (STORM MOVING TO THE RIGHT) ;THE LOWER FIGURES REPRESENT \eta\2/\eta\0 (STORM MOVING TO THE LEFT).

CONClUDING REMARKS

In general, Coriolis force in the edge wave problem has the effect of producing unsymmetrical disturbances moving to the right and to the left along the coast, with waves moving to the left (negative y direction) from an initial static mound containing the larger fraction of total energy. For pressure-induced disturbances, a storm moving to the left is more favorable in the excitation of resurgent edge waves behind the storm than movement to the right. This is true so long as the speed of the storm does not surpass the maximum group velocity of the edge wave (order of 100 m/sec). The model utilized in this paper is not adequate for predicting changes in water level due to actual meteorological disturbance, because (besides the rather arbitrary pressure pattern in the present example) the effect of wind stresses is completely neglected. A brief investigation concerning the effect of wind stress (to appear later) shows that the stress may produce edge waves of the higher modes and that the cumulative effect of wind stresses parallel to the coast would remain behind the storm; this manifests itself as a geostrophic current system parallel to the coast (if the storm travels a long distance without change of pattern).

ACKNOWlEDGMENTS

This work is based in part on research conducted for the Texas A.＆ M.Research Foundation through sponsorship of the U. S. Weather Bureau (Contract Cwb-9281).

REFERENCES

GREENSPAN, H. P.
1956. The generation of edge waves by moving pressure distributions. J. fluid Mech.,1:574-592.
LAMB, HORACE
1932. Hydrodynamics,6th Ed., Cambridge Univ. Press.738 pp.
MUNK, WALTER, FRANK SNODGRASS AND GEORGE CARRIER
1956. Edge waves on the continental shelf. Science,123: 127-132.
REID, R.0.
1958. Effect of Coriolis force on edge waves (I) Investigation of the normal modes. J.mar. Res.,16:109-144.

高潮について 東京大学地震研究所　梶浦欣二郎

1.緒　　　　言

　最近十年間足らずの間に,世界各国において高潮に対する関心が急激に高まったようにみえる.その理由は,例えば日本では,1959年9月26日の伊勢湾台風の際に,高潮が原因となって名古屋を中心とする沿岸各地に大被害がひきおこされたので,防災的な見地から高潮の実態をよく知る必要にせまられたことがあげられるが,(1) ヨーロッパやアメリカにおいても同様なことがいえるようである.即ち,1953年1月31日から2月1日にかけて,冬期の低気圧が原因となってヨーロッパ北海に起った高潮の際には,オランダ附近で偏差約3mに達し,その時刻がほぼ満潮時と一致したために,オランダの大堤防も各所で決壊し,1421年の大洪水以来かってみなかったといはれる大洪水がおこって,南西部の島々は海水下に没するという大被害を生じ,その復旧には多大の苦労を要した.この被害の経験を二度と繰り返さないために,オランダ政府はその年の2月21日にデルタ委員会を設置してオランダ南西部の防災と開発に関する方策の研究を行い,委員会は壮大なデルタプランをつくり上げた.1957年頃からこの計画は着々と実行に移されつつあり,全計画の遂行には25年の歳月が見込まれている.(2) この高潮以来,北海をめぐる諸国で高潮に関する研究が多数発表きれている.(3) 一方アメリカでは1954年8月31日にハリケーンがニューイングランド地方を襲い,沿岸各地に災害をもたらしたが,特にナラガンセット湾の最奥部プロビデンス附近では最高水位約4.6mを記録した.この年には,数個のハリケーンが米国東海岸附近を通過し,ハリケーンの予報及びそれに伴う高潮の防御ということについて関心が高まったようである.(4)そこで,1955年の末にアメリカ気象局を中心とする国立ハリケーン研究プロジェクトが誕生し,その仕事の一部として高潮研究が強力に押し進められることとなった.(5)
　勿論以前から高潮に関する研究は方々で行なわれ,特に日本では世界的に有名な20世紀初頭の本多等による日本諸港湾の静振の研究(6) 以来,伝統的に海水振動の研究はよく行なわれ,1934年9月21日の室戸台風あるいは1950年9月3日のジェーン台風による大阪湾の高潮などにっいては多くの調査研究がある.(7) イギリスにおいても,海の長波理論と関連して,DOODSON,PROUDMAN其他によるすぐれた理論的研究があり,また北海をめぐる各国では種々な高潮予報方式が試みられた.特にオランダではSCHALKWIJKが予報技術についての研究をまとめて1947年に一応完成させた.(8) アメリカでは,昔から東海岸やメキシコ湾沿岸がしばしば高潮に見舞われているが,目立った研究はなされていなかった.
　ここで高潮研究の歴史を述べる余裕はないが,一般に世界の各国とも,予報に関してはその地域的な特性に応じた研究がなされ,得られた実験式は直接には他の国の高潮予報に応用出来ない.その理由は,第一に,外力として働く気象条件が異なること,第二に,海底及び海岸の地形が場所によって異なることである.例えば,アメリカのメキシコ湾沿岸や北東岸には一般に広い大陸棚があり,ハリケーンによって起る陸棚上の高潮が重要な問題であって,(9) 湾内の高潮は,外海に面したところの高潮にそれだけ余分に附加されたものと考えられる.これに対して日本では,台風によって起こる東京湾,大阪湾,有明海等のような湾内の高潮が問題となり,(10) 陸棚が余り広くないために湾口における高潮はそれほど高くならない.一方オランダやドイツ等のように,北海の奥部に位置する国々では,北海というかなり広くて限られた海の内部で冬季の低気圧がひきおこす高潮を対象とし,(11) イギリスでは,北海の外部から伝播して来る高潮にも注目している.(12) このように各国それぞれ特徴のある高潮研究が進められて来たが,最近に至って電子計算機の利用による流体力学方程式の数値解析法が発達し,世界中どこにでも利用出来るような高潮計算方式が開発されて来た.この場合に,個々の場合について変るのは具体的な気圧及び風の場の与え方と,地形の与え方とであって,階差方程式の解き方自体については一般性があり,どの国で発展した方式でも利用出来る.(13)

・本年4月22日に日本海洋学会と共催で開かれたシンポジウムで講演した要旨である.

2.高潮の定義

　ここで,高潮という言葉が,海洋における異常水位の現象のうちのどれを指しているかをはっきりさせておく必要がある.普通には,気象的擾乱によって起こされた,数時間から数日にわたる急激な水位の異常現象を高潮と呼ぶようであるが,研究上の方便としては,標準的な検潮儀で記録された実測潮位と,推算潮位(天文潮)との差に長周期水位変動についての補正をほどこしたものを偏差と呼び,このうちで気象的擾乱によるものを高潮とすることが多い.註(1)このように定義された高潮には,潮汐とか,地震による津波は入らないけれども,周期の長いうねりや静振は含まれる可能性がある.そこで見掛けの周期からみて,数十分以下の短周期振動は全部除外した方がよきそうである.註(2) 長周期の側についても同様であって数日以上の周期をもつ変動は高潮から除外してはどうであろうか.註(3) その他,例えば海流の短期変動に応ずる水位変動(16) などは,間接的には気象的擾乱が原因であっても,普通の意味での高潮とは言い難いし,数時間の周期をもつ湾や陸棚の海水自由振動は,その原因は気象的な擾乱であるかも知れないが,一度外力がなくなって自由振動の状態になってしまえば高潮とは呼びにくい.このようにして,高潮を厳密に規定することはむつかしいが,もう一つ,本質的に高潮の現象を他の水位変動ときり離して考えられない場合がある.それは,浅海に於ける潮汐と高潮との非線型結合の可能性であって,この点からいうと,厳密には水位の変動全体の中から,高潮の部分を取り出すこと自体が不可能となる.本文ではごく常識的な定義にしたがって,台風や低気圧によって起こった,数時間から数日にわたる割合に目立った異常水位の現象を高潮と呼んでおく.

註(1) これはstorm surgeと呼ばれ,storm tideというときには潮汐をも含めて異常水位全体を指すことがある.

註(2) 和達,広野(1954)は長周期のうねりを高潮に含ませているが,長いうねりの外不連緕線や気圧波に伴なう波をも除外する.これらの波は短周期だと沖合では長波の性質をもたない可能性がある.

註(3) 例えばGroves・G・などの取扱った数日から一週間程度の水面変動は高潮に入れない方がよかろう.

3.高潮の高さ

　台風やハリケーンのような激しい気象擾乱によってどの程度の高潮が起こるかを知るために,第1図には日本及びアメリカにおいて観測された高潮の高さを低気圧の最低気圧に対して示してある.註(4) この図でわかることは,同程度の台風とハリケーンとを比較すると,高潮の最高はメキシコ湾沿岸にあり,次いで北米東岸,最後に日本という順になるようで,これは地形によるものであろうと思われる.即ち,アメリカ南部メキシコ湾沿岸及び東岸のニューイングランド沿岸には広い大陸棚が発達しており,そこにナラガンセット湾のような細長い湾が附随しているが,日本の場合には陸棚が狭く,湾の入口近くでの高潮の高さがそれほど高くないことが主な原因である.一般に偏差が3mを越す高潮は珍らしく(北海の1953年の高潮で約3m),4mを越すものは余程特殊な場所に限られるようである.例えば日本の高潮で偏差3mを越すものは,室戸台風による大阪の高潮(3.1m),伊勢湾台風による名古屋の高潮(3.4m)であり,アメリカでは,最近の目立ったところで,1938年,1944年,1955年のニューイングランドの高潮及び1956年のメキシコ湾岸ルイジァナの高潮(19)があげられる.註(5)
　ある一地点における高潮の時間的変化は,低気圧の規模,進路,速さ等によって大きく左右され一概には言えないが,ほぼ数時間から半日位の間に目立った水位の昇降が起こり,その前後にはゆるやかな水位変化のあるのが普通で,時には静振のような振動が後に附随することもある.第2図には,海岸線に対して種々な相対進路をとるハリケーンがあった場合に期待される水位変動を,実例によって示してある.普通の低気圧によるものでは,数日にわたる水位変化がみられることも多いが,高潮の高さ自体はさほど大きくない.不連続線に関係した水位の異常変化は,高潮の中に入れるか入れないかには問題があるが,一般にかなり短周期である.
　広い大陸棚に面したほぼ直線状の海岸に対し,ハリケーンが海側から進行して来た場合,海岸に沿った各地の最高水位の分布が低気圧中心からの最短距離に対してどのようになるかを第3図に示す.勿論個々の場合についての違いは大きいけれども,平均的にはハリケーン進路の右側約数十粁のところで最大となるようで,中心から左右共に200km以上も離れると,高潮はかなり小さい.
　このような高潮の性質をみると、高潮の最高波高については観測点附近(数十粁以内)の風の状態が最も重要な原因と考えてよいようであり,今までに多く提出された実験式では,Hを高潮の波高,\sigma\ を観測点附近の最大風速,△pを気圧偏差としてH=a△p十6U^{2}の型のものを与え,α及びbを決定することが多い ．(20) この式は,高潮を気圧降下による海水吸い上げ作用と,風による海水の海岸への吸き寄せとによって大体説明出来ると考えていることに相当し,静的な見方と呼べよう.註(6)
　これに対して，動力学的な見方では,陸棚上あるいは湾内の海水は一つの振動系をなしているために，外力の見掛けの周期に応じて力学的な増幅作用が働き,共鳴する場合には湾奥の波高が静的な作用のみを考えた場合にくらべて遙かに大きくなり得ると考える.註(7)
　これらの静的あるいは動力学的な諸効果を明確に表現するのは流体力学の方程式であるから,次にはその最も簡単化した式によって話を進める.


註(4) 図中ハリケーンの最低気圧は公式から推定したもの,台風の場合には観測最低気圧が用いられているので,日本の高潮の諸点は全体として僅かながら右にずらして考える必要がある.メキシコ湾沿岸の高潮の高さには潮位の補正が行われていないが,潮汐の全振幅が30〜60cmの程度であるから大して問題にならないと考えられる.用いた資料は宮崎 (17)及びHarris (18) による.

註(5) インドにおいてもペンガル湾をサイクローンが北上するとき高潮が起る筈であるが詳細はわからない.

註(6) 実際にはa,bの中に平均的な動力学的効果が含まれていて,　aは必ずしも1ではないが,個々の低気圧についての動力学的効果は入っていない.

証(7) このような動力学的功果を含んだ高湖予報の実験式をつくる試みがWILSON (1959)(21)によってなされた.

4.高潮の方程式

　よく知られているように,長波に関する運動方程式と連続の式とは次のように書かれる.

式(1)

及び

式(2)

ここに，tは時間，水平面上に座標軸x,yをとり,\nabra\はx,yに関するベクトル微分オペレータ・kは鉛直上方に向う単位ベクトルとする．Vは深さに対して平均した水平速度ベクトル,\zeta\ は平水面上の海面上昇量，hは水深・gは重力加速度,fはコリオリ因数である. \zeta\* は大気圧の空間的平均値からの偏差を水面の上昇量として表わしたもの,\tau\は海面風が水面に及ぼす応力，\tau_{b}\ は海水運動が海底に及ぼす応力(密度で割った量を用いる)である.註(8) この式では海水運動によるエネルギーの消耗はすべて海底摩擦によることになるが,\tau_{b}\ の解釈を拡大して水平渦動粘性によるものを含めてもよい.
　さて，(1)は完全な運動方程式において,種々な仮定をおくことによって導かれたものであるから,その適用限界についての注意が必要である.仮定は,1)海水運動は小さく,その変化はゆるやかであるために,非線型の項はすべて省略出来る.註(9) 2)海水運動の鉛直加速度は重力速度に比して無視出来る．以上の仮定のうち 2)は波長の長い高潮のような海水運動ではごく海岸の近く(実際には水深と同程度の離岸距離)を除いては満足されていると思われるが,1)の仮定は水深がある程度ないと成立せず,遠浅な海岸で波高が水深に比して無視出来ない場合には非線型項を含む式を用いるべきである.また,防波堤などの設置によって,その開口部附近だけに速い流れの生ずるときには明らかに非線型の効果があらわれるものと予想される.このことは又浅海における潮汐と高潮との非線型結合についてもいえる.今は(1)、(2)についてのみ論じ,非線型項の影響については触れない.
　高潮の性質を調べるのに最も簡単なモデルは,海底摩擦も地球自転による転向力も省略し,現象が準静的に起こっていると考えた時で,(1)は

式(3) [Fig.107_103_01]

となり,高潮は,風の吹き寄せ効果と気圧低下による海水吸い上げ効果とで表わせる.問題を一次元に限り,x=0 から x=L までの陸棚を考えてその間では水深も風速も一定とし,x=Lから外は水深無限大とすると,(3)から

式(4) [Fig.107_103_02]

となる. (4)は水面の昇り高は深きに反比例し海域の広さ(風域より小さい場合)及び風の応力に比例することを示し,浅くて広い陸棚(又は湾)に海岸向きの風が吹けば高潮の高さが最も大きくなり易いことがわかる.風の応力は一般に風速の2乗に比例するから,高風速では僅かな風速の違いも高潮の高きの変化には大きく影響する.今△Uの風速変化によって変化する高潮の高きを△\zeta\ とすれば(\zeta\*を0として) △\zeta\’\zeta\ =2△U/U となり,10%の風速変化に対して20%の波高変化が起こることになる.
　地球自転による転向力が,高潮に対してどれくらいの影響をもつかは,無次元量 \alpha\=fL/Cの大小によってきまる.ここでLは海水運動の起こっている海域の長さ,Cは長波の平均伝播速度 [Fig.107_103_03],　fはコリオリ因数である.もし \alpha\ ＜ 1 ならば,地球自転の影響は考えなくてもよい.例えば, L〜60km, h〜26m, f〜0.8×10^{-4}sec^{-1} とすれば, \alpha\〜0.30となり波高に対して地球自転の影響がややきくという程度である.一方静振の周期Tに対しては,△Tを周期の変化として,ごく近似的に△T/T〜2/\pi^{2}\ (fL/C)^{2} となり,上述の例では△T/T〜0.02となって静振の周期に対して地球自転はほとんどきかないと見なせる.(22)
　次に動力学的な効果に関して,重要な無次元昼は,まず \beta\^{2}=V^{2}/c^{2} であり,ここにVは風域の移動速度(強制波の速度)をあらわす.よく知られているように,(23) 無限に長い水道中の強制波は(1-V^{2}/c^{2})^{-1} の割合で増幅され,　V/c=1 即ち強制波の波速が自由波の波速に等しくなると,増幅は無限大となる(勿論実際には摩擦の影響でこうはならない).これに対して,二次元的に拡がった強制波を考えると,増幅度は波の中心で [Fig.107_104_01] となる.今,地球自転の影響を考慮に入れると,一次元,二次元の場合ともにv/c=1においても無限大の増幅度を与えない.これは,V/c=1に近くなると転向力の効果が著しくなるためである.(24)
　以上は無限に広い海の場合であるが,陸棚上あるいは湾内の高潮について考えるときは,陸棚(湾)の固有振動周期Tと風域の移動による強制波の見かけの周期T*との関係が重要となる.今,L,L*をそれぞれ陸棚(湾)の長さ,及び風域の大きさとすると

[Fig.107_104_02]

という無次元量 \gamma\^{2} が1に近づくと,即ち,v/C〜L*/4Lで増福度が最大になることが期待される. \beta\^{2} と \gamma\^{2} という二つの無次元量のうち,どちらが高潮の場合に重要であるかはL*/Lの大小によってきまり,L*/Lの小さいときは \beta\^{2}が,L*/L ≧ 2では \gamma\^{2} の効果が卓越する.

第4図には,陸棚上をある大きさの風域が進行する一つのモデルに対して数値計算した結果(25)が示され,一般の増幅度の目安として,静的な上昇量に対して, \beta\^{2} 又は\gamma\^{2}が1になったときの増幅度が右端に附け加えてある.これによるとL*/Lが1以上では,最大増幅度は1.5〜2倍である.
　地形の影響に関しては,現象が複雑で簡単には述べられないが,定性的にいうと水深変化によって長波の部分反射,屈折,回折等が起こり,海岸では波が陸上にあふれるためにそのエネルギーの一部を失い,残りは反射される.複雑な過程によって生じた波は干渉し合うから全体としての波形は場所場所で相当に異なることが予想される.その上,風の応力の効果が水深に反比例することと,摩擦力も水深に関係することを考えると,個々の観測点について高潮の時間的変化を予測することは非常にむつかしい.
　現在の高潮研究は,いかにして数値計算による高潮予報を行なうかということに集中しており,まず基本的には(1)をどう取扱うかが考えられている.勿論実際計算に際して非線型の項を導入することはさして困難ではない.ここで問題は,1)基本の徴分方程式を如何なる階差方程式で代用し,時間,空間の格子間隔をどのようにとり,計算をどう進めるかという,数値計算技術の問題と,2)風及び気圧の場をどのように与え,註(10) 海面及び海底の応力に対してどのような式を用いるか,及び海岸と沖合との境界条件をどうするかという高潮に固有な問題とに大別される.これらの問題について今までにも研究はあるが,まだ完全なものとはいい難い．(26)
　ここでは,外海に於ける境界条件を如何に与えるかという問題に関連して,広い大洋上の海水運動について考えてみる.


註(8) 風の応力\tau_{y}\ はある一定の高さの風速Uの函数として [Fig.107_103_04] と与えられるが,k^{2} は厳密には常数ではなく,風速及び大気下層の安定度に関係する.　また海底の応力 \tau_{b}\ は,　\tau_{b}\=k^{2}V/V の形に書かれることが多いが,水面に風の応力が働く場合は流速の鉛直分布が変化するので表面応力に関係した項那必要となる.詳細は論じないが,海底摩擦が水深及び表面応力の水平分布によって変化することを念頭におく必要がある.

註(9) (波高)/(水深) ＜＜ 1及び (波高)・(波長)^{2} /(水深)^{3} ＜＜ 1が要求される.

註(10) 基本式が線型であれば,気圧及び風の場について「重ね合せ」の原理が適用出未るわけであるが,非線型となるとそうは行かない.

東京大学地震研究所　梶浦欣二郎

5.低気圧による大洋の海水擾乱

　今簡単のために,(1)及び(2)を変形し, \zeta\=\zeta\1+\zeta\2と置き,t=0 で\zeta\1-\zeta\2=0とする.
もし水深が一定であれば,　c^{2}-gh と置いて次の式が得られる.(\tau_{b}=0\)

式(5)

及び

式(6)

　もし海が極めて深いと,c^{2}が大きくなり,外力に対する海水の応答は迅速に行なわれて,(5)，(6)の左辺の第二項,第三項が第一項にくらべて省略出来,右辺では風の応力の効果が水深に逆比例するので，これも小さくなり,結局 \zeta\1〜\zeta\*,\zeta\2〜0となって水面変動は気圧の違いによる吸上げにのみ依存することになる.一般に風の応力の収束 (一▽・\tau_{s}\,) は気圧傾度の収束と同様な作用をなし,風の応力の渦度は,本質的に地球自転の転向力を媒介として水面変動にあずかる.このとき渦度は積分効果としてきくので,作用時間が長くなるほど,渦度の水面変動に対する寄与は大きい.今軸対称な風の場をとると,中心に吹き込む風は,中心の気圧降下による水の吸い上げ作用を幾分助けるように働き,風の回転成分は,海水を軸のまわりに循環させるように働く.この海水の循環運動に対して転向力がそれと直角方向に働くから,転向力に釣合うような水面傾斜がもたらされる.従って,転向力がなければ水面傾斜も起らないということになる.註(11)
　この考え方を実際の海洋に適用するには,海洋が成層をなしていることを考慮しなければならない.今簡単のために二層の海を考え,上層は下層よりも遥かに厚さが薄く、又上下層の密度の差は極めて小きい (10^{-3} 程度)とすれば,(5),(6)式で,水深hを上層の厚さh1 でおきかえることによって,\zeta\1及び \zeta\2 を近似的に躍層自体の変動量とみなすことが出来る.
　この場合には,h1 ＜＜ hであるから,躍層の変動に対する風の応力の収束と渦度の効果は大きくなるが,海面の変動に引き直してみると,まだ圧力傾度によるものにくらべて小さいのが普通である.しかしながら,海水運動について考えると,もし風系の移動速度がそれほど早くなければある一地点における風の作用時間が長くなるので,上層における循環流の流速は相当の大きさになり得る.即ち,大洋の上層では,圧値から傾度や風の応力の収束による海水運動にくらべて,風の応力の渦度による海水の循環流が卓越することが判る．
　こう考えて来ると,大陸棚の先端部で,海面の上昇量を \zeta\1 で近似するだけでは,境界条件として不十分なことは明らかで,何等かの方法で流れの状態をも考える必要がある.一般に,成層した海における順圧と傾圧との運動の分離可能なのは，水深と，層の厚さが一定の場合に限られ,陸棚傾斜のように水深の変化するところでは,両モードの運動が相互作用をする.従って、大陸棚の端の境界条件は,これを式であらわすこと自体がむつかしい問題で,未だ解決していないといえる.


註(11) 今の議論は線型理論の範囲において成立ち,もし遠心力が大きいと,勿論転向力がなくとも循環運動に対応する水面傾斜は出来る.ここで,風の向きと海水の循環運動の向きが同じであることは,一見有名なエクマン理論に反するようにみえるが,実はこの循環流は傾斜流に相当し,エクマン理論によつて計算される吹送流は,丁度転向力に釣合うような水面傾斜をつくるために行なわれる海水運動に相当する.

6.ハリケーン内の気圧と風との分布について

　ハリケーン及び台風の気圧分布についてよく使用される実験式は次のようなものである．

　　　高橋の式
　　　藤田の式
　　　Hydro.Met.U.S.

これらの三式は見掛け上異っているようであるが(p一p0)/(pn-p0)は r→\infty\ に対して等しく1-R/rに漸近する.今 p0,pn,R,という因数を同一とおいて比較すると第5図のようになり,実験式の違いは結局中心付近の気圧分布の違いということになる．アメリカでよく使われている式にはexponentialが入っているので,実測に合うようにp0,pn,Rを決めるにはややめのどうな点もあるが,Rは中心からほぼ最大風速地点までの距離を表わし,推定きれた中心気圧p0は割合に実際をあらわしているらしく,多くのハリケーンについて,これらの因数の値が計算されている.(27)
　風速に関しては信頼すべきものは少ないが,1944年9月のハリケーンに対して,アメリカ気象局の水理気象課で,各所の観測値を綜合してつくったモデルは第6図(a),(b)のようなもので,風速は中心からの距離の函数として4時間ごとに与え,進行方向に対して直角に右側と左側とでは異った値をとらせる.その上,風の吹き込み角も,場所の函数として与えられているが,この方の精度は良いとは言い難い.今,以上の図をもととして,風の応力の収束と渦度とを計算してみると第7図(a),(b)のようになる.収束,渦度共に強さは似たようなもので,風域全体の広さに対して,かなり狭い範囲(中心から約50海里以内)に集中していることが判る.

引用文献

1) 例えば,宮崎正衛,宇野木早苗,上野武夫：伊勢湾台風による高潮とその理論計算について.第7回海岸工学講演集,209-215,1960.
2) 例えば,「The Delta Plan」Information Dep’t of the Ministry of Transport and Waterstaat,15pp.,1958.
3) 例えば,H. Charnock and J. Crease：North Sea Surges. Science Progress,45,491-511,1957.
4) A.C. Redfield and A. R. Miller：Water levels accompanying Atlantic Coast Hurricanes. Met. Mono. Amer. Met. Soc.,2,1-23,1957.
5) H.F. Hawkins, Jr., L. G. Pardue and C. M. Reber：The National Hurricane Centor. Weatherwise,14,87-98,1961.
6) K.Honda, T. Terada, Y. Yoshida, D. Isitani：Secondary undulations of oceanic tides. Journ. College of Sci. Imp. Univ. Tokyo,24,113pp. and plates,1908.
7) 例えぱ,高潮一般について,K. Wadati and T. Hirono：Storm tides caused by Typhoons. Proc. UNESCO Symposium on Typhoons,31-48,1954.
8) W.K. Schalkwijk：A contribution to the study of storm surges on the Dutch Coast. Roy. Netherlands Met. Inst.,1258, pt.1,1947.
9) K.Kajiura：A theoretical and empirical study of storm induced water level anomalies. Tech. Rep. Texas A＆M Research Foundation, Ref.59-23F,97pp.,1959.
10) M.Miyazaki：On storm surges which recently struck the Japanese Coast. Ocn. Mag.,9,209-225,1957.
11) (7)を見よ.
12) J.R.Rossiter：The.North Sea surge of 31 January and 1 Februarv,1953. Phil, Trans. Roy. Soc.London, A.246,(915.),371-400,1954.
13) W.Hansen：Theorie zur Errechnung der Wasserstandes und der Stromungen in Randmeeren nebst Anwendungen. Tellus,8,287-300,1956. G.W.Platzman：A numerieal computation of the surge of 26 June 1954 on Lake Michigan. Geophysics,6,407-438,1958.
G.Fisher：Ein numerisches Verfahren zur Errechnung von Windstau und Gezeiten in Randmeeren. Tellus,11,60-76,1959.
気象庁：伊勢湾高潮の総合調査,気象庁技術報告第4号,昭和35年7月,287頁.
気象庁：東京湾高潮の総合調査報告,気象庁及び東京都,昭和35年10月,247頁.
気象庁：大阪湾高潮の総合調査報告,気象庁技術報告第11号,昭和36年3月,235頁.
14) W.L.Donn and W.T. McGuinness：Air-coupled long waves in the ocean. Journ. Met.,17,515-521,1960.
15) G.W. Groves：Day to day variation of sea level. Met. Mono. Amer. Met. Soc.,2,32-45 1957.
16) D.Shoji On the variation of the daily mean sea levels along the Japanese Islands. J.Ocn. Soc. Japan,17,(3),1961.
17) (10)を見よ.
18)D.L. Harris：Meteorological aspects of storm surge generation. Proc. A. S. C. E., Hydr. Div., Paper 1859,25pp.,1958.
19) D.L.Harris：Hurricane Audrey Storm Tide. Nat. Hurricane Res. Project, Report No. 23,19pp.,1958.
20) 例えば,気象庁：潮位表昭和36年.
21) B.W. Wilson：The prediction of hurricane storm-tides in New York bay. Tech. Rep.Ref.59-20F, Texas A＆M Research Foundation,120pp.,1959.
22) (9)を見よ.
23) J.Proudman：Dynamical Oceanography. John Wiley and Sons, Inc., New York,1953.
24) K.Kajiura Response of a boundless two-layer ocean to atmospheric disturbances：Dissertation, A＆M College of Texas,139pp.,1958.
25) R. O. Reid：Approximate response of water level on a sloping shelf to a wind fetch which moves towards shore. Tech. Memo. No.83, Beach Erosion Board,47pp.,1956.
26) (13)を貝よ
27) (18)を見よ.

UGGI TSUNAMI COMMITEE ON THE PARTIAL REFLECTION OF WATER WAVES PASSING OVER A BOTTOM OF VARIABLE DEPTH Kinjiro Kajiura Earthquake Research Institute, The University of Tokyo, Tokyo, Japan

Abstract

Theories of partial reflection of water waves are reviewed. The cases of strong and weak reflections are treated and it is shown that the local reflection modulus, which is similar to the reflection coefficient for the case of a discontinuous change of the depth, plays an important role in the process of partial reflection of long waves, and that the contribution of the one-time reflection alone is enough to estimate the reflection coefficient for weak reflection. Some particular examples are worked out, and the variation of the reflection coefficient is investigated with regard to the ratio of the width of a transition zone to the wave-length of incident waves. It is found that if the ratio is less than about 0.05, the reflection coefficient may be approximated by the well-known formula for an abrupt transition, and if the ratio is larger than about 1, the reflection coefficient is quite small so that Green’s formula gives a very good estimate of the transmitted wave amplitude. As for the waves of arbitrary wave-lengths, the analytical work of Roseau is introduced and amplified to give some numerical results. The two-dimensional propagation of long waves is also discussed on the basis of Tatarski’s method on the statistical treatment of the wave propagation through an inhomogeneous medium.

Introduction

In order to understand the physical process which governs the behavior of tsunamis, various factors should be taken into account, In particular, on continental and insular shelves, and in bays and estuaries, complicated free oscillations of water, including certain kinds of progressive boundary waves, may be excited by waves advancing from the open ocean, and one of the most important aims of tsunami hydrodynamics should be to clarify the response characteristics of a water body in shallow water in relation to the incident wave train. In the present paper, however, we are mainly concerned with the propagation of waves in an open ocean and the partial reflection of waves due to bottom irregularities, so that no fixed lateral boundary is taken into consideration explicitly;the eigen oscillations and boundary waves generated near the coast, and the complicated interference of waves due to total reflection at the continental boundary are not within the scope of the present discussion,
Even if we neglect the effect of energy dissipation, and confine our attention to a linear approximation, the propagation of water waves over a bottom of variable depth is known to be complicated by such effects as partial reflection, refraction, and diffraction, in addition to the complication due to the dispersive character of water waves in the range of intermediate and short wave lengths. If the wave length \lambda\ is much larger and the wave amplitude a is much smaller than the depth d of water, provided \lambda^{2}\/d^{3} is small, the equation governing the propagation of water waves conforms to the so-called wave equation which is a linear differential equation of hyperbolic type, and many of the general properties of wave propagation are considered to be common to various types of waves encountered in nature irrespective of the special nature of the waves. Thus, the knowledge obtained in studies of electromagnetic waves, waves along electric transmission lines and filters, electron waves in wave mechanics, elastic waves and sound, etc.,can be utilized with advantage to the understanding of wave propagation in water, at least for waves of small amplitude and long wave length.
Although the general properties of a linear differential equation of hyperbolic type is very well known mathematically, general solutions become very complicated if the coefficients of the equation are not constant and it is not easy to discuss the details of the solution for each particular case. This applies in the case where the medium of wave propagation is not homogeneous, and various methods have been developed to handle this situation.Among them, numerical analysis, which has made great progress by making use of high speed computers, seems to be very promising for the discussion of wave propagation in an inhomogeneous medium. However, analytical methods of approximation are very useful from the standpoint of the physical understanding of the process of partial reflection, and they also help in formulating the program of a numerical analysis, and in interpreting the results in physical terms.

Reprint from lUGG monograph n 24

The one-dimensional propagation of waves in a continuous medium has been studied very extensively since the 19th century(see Rayleigh, Theory of Sound,1926), and the reflection and transmission coefficients of waves are given for the case of an abrupt change of the medium, and also for the cases of certain particular types of the variation of the medium. The problem of wave propagation in an inhomogeneous medium may be discussed for the following three cases separately. Firstly, the case in which the variation of a medium is confined within a very small fraction of one wave length of the wave in question. The case may be called the strong ref-ection. The other limiting case may be adequately called the weak reflection when the variation of a medium is gradual so that the change of a quantity in one wave length is small enough compared with the quantity itself. Between these two limiting cases lies the case when the variation of a medium occurs within a distance of half a wave length or so. Techniques employed to solve the problem are generally different for these three cases.
In the discussion of the one-dimensional propagation of waves, an approximation of the continuous change of a medium by the abrupt, discontinuous change of a medium is permissible at least for the case of strong reflection, and the formulas for the partial reflection and transmission of waves can be easily derived. For the case of weak reflection, various approximate methods such as W-K-B method can be applied and the well-known Green’s formula for the amplification of wave height in relation to the change of depth in water usually corresponds to the first approximation in the system of successive approximation. In the third case where the transition of the medium occurs within a distance of half a wave length or so, the problem should be solved without approximation in many instances and the solution is given for each particular case of the variation of a medium separately.
Here, it should be pointed out that water waves are somewhat different from the other kinds of waves described by the wave equation, because the long wave is only an approximation to the real motion in water when the vertical acceleration of a water particle is negligible compared with the hydrostatic pressure, and in the neighborhood where the abrupt change of a medium occurs, wave motions may not obey the hydrostatic law. This situation may be easily understood by considering the case of onedimensional wave propagation, when an infinitely thin barrier extends from the bottom to a certain depth of water. The partial reflection of long waves cannot adequately be described by the conditions of the continuities of the total flow and the water surface at the location of the barrier without the vertical acceleration being taken into account.
Thus, there is a possibility that the formula derived on the assumption of long waves may not be applicable to the case when the depth actually changes discontinuously. In reality, however, the depth change in the ocean is very gradual, that is, tie slope of the bottom does not exceed a few degrees in most cases of interest and the formula derived on the long wave approximation seems to be valid as long as the wave length is large enough.
If the wave length is not very large compared with the depth of water, the properties of wave propagation change significantly, and the conclusions obtained for long waves should be modified accordingly. For immediate and short ranges of wave lengths, the problem of wave propagation over a bottom of variable depth has been investigated by many able mathematicians and within the frame-work of a linear potential theory the problem of waves on a linearly-inclined bottom is said to have been solved completely (Stoker, Water Waves, Chap.5,1957). However, the partial reflection and transmission of waves over a varying depth is solved for very special cases only and it may be worthwile to give numerical result obtained so far.
As for the two-dimensional problem of wave propagation in a variable medium, the mathematical difficulty is so great that no adequate general theory yet exists as far as the author knows. For the case of weak reflection, however, it seems to be possible to discuss the two-dimensional propagation of long water waves over a bottom of variable depth from a fairly general point of view, by applying the method developed in the study of the propagation of electromagnetic and acoustic waves in the atmosphere, which is in the turbulent state, and in which the refractive index of the air fluctuates in random fashion. Tatarski (1961) has given a general treatment of the theory of wave scattering and of the theory of parameter fluctuations of short waves propagating in a turbulent atmosphere, and the results may be interpreted in terms of the statistical variations of the phase and amplitude in space of long water waves although the assumptions of weak reflection and randomness of the depth variation are severe restrictions to the application of the results to actual situations such as the propagation of tsunamis in the ocean.

Basic aproximate equations

Assume the fluid is ideal (no viscosity) and incompressible and the motion is irrotational. The disposition of the co-ordinate axes is taken in the usual manner, with the x, y plane the undisturbed water surface and the z-axis positive upward. The free surface elevation is given by z=r(x,y,t) and the bottom surface, by z=-h(x,y), We adopt the convention that letter subscripts attached to dependent variables denote differentiation with respect to the denoted independent variables, unless otherwise stated.
The equations governing the propagation of water waves of small amplitude are derived. in terms of the velocity potentia1\Phi\ in the following form:

式(1)

and

式(2)

式(3)

where g is the acceleration due to gravity and the subscript n denotes the differentiation normal to the bottom surface. The free surface elevation \eta\ can be determined from

式(4)

(2),and(4)are obtained by rejecting all but the linear terms in \eta\ and \Phi\ and their derivatives in the kinematic and dynamic free surface boundary conditions. The problem in this formulation belongs, from the mathematical point of view, to the classical boundary problems of potential theory.
On the other hand, the shallow water theory is an approximate theory which results from the assumption that the vertical component of the acceleration of water particles has a negligible effect on the pressure \rho\ so that the pressure p is approximately by the hydrostatic pressure. This kind of approximation is valid when it is assumed that the depth of the water is sufficiently small compared with some other significant length, such as, for example, the radius of curvature of the water surface, Thus the theory is sometimes called a long wave theory.
lf we assume the wave of small amplitude, the governing equations for long waves are

式(5)

式(6)

and

式(7)

where u and v are the velocity components in the x and y directions respectively. These equations are derived as the first order approximation of Euler’s original equations of motion and the equation of continuity by developing all quantities in terms of a small parameter \sigma\ (\sigma\ =d^{2}/l^{2} where d and l are respectively the representative lengths related to the depth acid the horizontal dimension).
The development of the higher order approximations are out of the present scope, but it is of some interest to mention that the solitary wave is obtained by the approximation of the second order. It is remarked that (5) to (7) are valid provided a/h ＜＜ 1, h /\lambda\ ＜＜ 1and a\lambda^{2}\ / h^{3} ＜＜ 1 where a is the amplitude, \lambda\ the wave length, and h the depth of water. The importance of the last parameter a\lambda^{2}\ / h^{3} is pointed out by Ursell(1953) in the discussion of the long wave paradox in the theory of gravity waves.
The details of the approximate theory may be found in a text book (see Stoker, Water Waves, Pt.1,1957).
In terms of the integrated velocity Uh = U and vh = V,(5),(6) and (7) may be
written as follows:

式(8)

式(9)

and

式(10)

One-dimensional propagation of long waves

For the propagation of long waves in a canal of variable cross section, the equation of motion is

式(11) [Fig.107_114_01]

and the equation of continuity is reduced to

式(12) [Fig.107_114_02]

where b is the width of a canal, Q( =bU) is the volume flux, and the variations of \eta\ and U across the canal are assumed to be negligible. If we set b constant, all the subsequent treatments will be reduced to the ordinary discussion of the one-dimensional propagation of waves in which the effect of the varying width is not taken into account.
The wave equation derived from (11) and (12) is

式(13) [Fig.107_114_03]

or

式(14) [Fig.107_114_04]

If we assume \eta\ and Q to be periodic with respect to time (e^{iwt}),(13) and (14) may be written

式(15) [Fig.107_114_05]

or

式(16) [Fig.107_114_06]

In later discussions, the time factor is ommittee for simplicity.
In general, the one-dimensional wave equation can be written for the case of
periodic waves in the form:

式(17) [Fig.107_114_07]

This is the equation encountered in various problems related to waves, and it is very well investigated in the field of applied mathematics. The general properties of a solution change according to the sign of k^{2}(x), and when k^{2}(x)is a slowly changing function of x, various methods of approximation are applicable, among which the socalled W-K-B method is as follows:
Set [Fig.107_114_08] and substitute into (17). If we consider [Fig.107_114_09] as a correction term in the corresponding equation of Riccati type,

式(18) [Fig.107_114_10]

the first approximation of the original equation is found to be

式(19) [Fig.107_114_11]

The second approximation is givenby

式(20) [Fig.107_114_12]

This solution constitutes an actual W. K. B, solution, and corresponds to Green’s formula for the long wave propagation without reflection.
First of all, we treat the case of strong reflection when the change of the medium is abrupt. The conditions at the discontinuity of the medium are given by the continuities of the elevation \eta\, and the volume flux Q. In terms of Q or \eta\ alone, the conditions are

式(21) [Fig.107_114_13]

or

式(22) [Fig.107_114_14]

where the subscripts 1 and 2 denote,respectively, the quantities in two regions x＞O and x=＞ 0 of the medium separated by the discontinuity at x=0 (Fig,1).
Since b and h are constant for each region, the solution for Q of (16), which gives progressive waves moving in the negative x direction for x ＜ 0, is given by

式(23) [Fig.107_114_15]

and

式(24) [Fig.107_115_01]

where

[Fig.107_115_02]

The boundary conditions (21) determine the relation between the constants A,Band C.
The reflection coefficient may be defined by

式(25) [Fig.107_115_03]

This formula is very well known and forms the basis of the subsequent discussion
(Lamb, Hydrodynamics,§176,1932).

Fig.1
[Fig.107_115_04]
- Abrupt transition.

Now, to discuss the partial reflection of the wave in general, it is more convenient to formulate the problem in terms of the parameter related directly to the reflection of the wave. Studies in this direction have already been persued by Pierce (1943)who obtained a first-order differential equation for the input impedance of a non-uniform electric line, and by Walker and Wax (1946) who derived a similar equation for the voltage reflection coefficient of non-uniform lines. The application of the method to the long water wave problem was made by Ogawa and Yoshida (1959).
Take solutions of the form:

式(26) [Fig.107_115_05]

and

式(27) [Fig.107_115_06]

where k^{2} =\omega^{2}\/(ghb^{2}). The substitution of (26) and (27) into (11) and (12) yields the equation for the reflection coefficient R defined by

式(28) [Fig.107_115_07]

as follows:

式(29) [Fig.107_115_08]

The reflection coefficient R defined by (28) is for Q of the waves moving in the positive x direction.The reflection,coefficient R’ for \eta\ can be easily defined in similar manner,

式(30) [Fig.107_115_09]

and the equation for R’ becomes

式(31) [Fig.107_115_10]

As is pointed out by Schelkunoff (1951), no meaning can be attached to the reflection coefficient in general when the incident wave is in an inhomogeneous medium for the simple reason that we cannot separate the total wave into 「progressive」 components.
A formal expression such as

[Fig.107_116_01]

in the form of an apparent「progressive wave」function in a homogeneous medium does not insure that it represents a wave with「progressive」physical characteristics. Thus, we shall restrict the definition of the reflection coefficient given by (28) or (30) to conditions where the incident wave travels in a homogeneous medium and impinges on an inhomogeneous medium of width l at x=x1. From [Fig.107_116_02] to x=\infty\ we assume another homogeneous medium to recognize the transmitted wave. Under some conditions, we may be able to let l approach infinity.
In general,(29) and (31) are valid, even if the variation of the medium is not gradual, but the equation is non-linear and it is not so simple to solve the Riccati equation unless R^{2} is small in comparison with unity, or, in other words, the solution is easily obtained for weak reflection only.
If we neglect R^{2} compared with unity, the solution of (29) is given by

式(32) [Fig.107_116_03]

where x2 is the position when R is assumed to be zero. The physical meaning of this expression becomes clear if we consider that [Fig.107_116_04] is analogous to the reflection coefficient (25). Essentially, the above solution (32) shows that the one-time reflection of progressive waves contributes to the main part of the reflected waves as a whole for the case of weak reflection (Ogawa and Yoshida,1959).
Since the expressions (26) and (27) formally correspond to the first order approximation (19) of the wave equation (17), it is probable that the second order approximation (20) may yield a better result in the discussion of the wave reflection. Actually, it turns out to be the case. The details of the development are given by Bremmer(1949, 1951) for electromagnetic waves;he showed that the solution of the wave equation can be expressed by a series of terms, with a simple physical meaning, the first of which constitutes the W-K-B approximation and the successive terms are interpreted as a successively higher times reflection of the incident waves, The outline of the method is given below in terms of long water waves.
Instead of (26) and (27), we assume

式(33) [Fig.107_116_05]

and

式(34) [Fig.107_116_06]

The substitution of (33) and (34) into (11) and (12) yields,

式(35) [Fig.107_116_07]

and

式(36) [Fig.107_116_08]

Now set

式(37) [Fig.107_116_09]

and

式(38) [Fig.107_116_10]

and consider \beta\(x) to be a quantity of order unity.
Then, we may write

式(39) [Fig.107_116_11]

and

式(40) [Fig.107_116_12]

These equations clearly show that the wave is partially reflected. The waves travelling in the positive x direction and in the negative x direction change their amplitudes respectively according to the partial reflection of the waves moving in the opposite direction, Thus \epsilon\\beta\1 and \epsilon\\beta\2 may be considered as a kind of local reflection modulus per unit distance at a particular location. The absolute value of the local reflection modulus thus defined is consistent with the well known reflection coefficient (25). This view enable us to interprete (32) as a consequence of the one-time reflection of incident waves.
Now, if \epsilon\ is less than unity, the solutions of (39) and (40) may be expressed by

式(41) [Fig.107_117_01]

and

式(42) [Fig.107_117_02]

and these series converge rapidly if \epsilon\ is small, that is to say, for the slowly changing depth.
Here we may put zero order terms to be constant so that

[Fig.107_117_03]

This is the case of no reflection and corresponds to the solution (20) for W-K-B approximation. The first order correction terms A(1) and B(1) are to be expected as representing secondary waves originating from reflection losses of the zero-order wave B(0) and A(0), respectively. In a similar manner,corresponding consideration applies to the connection between the (n-1)th term and the nth term because we may write

式(43) [Fig.107_117_04]

and

式(44) [Fig.107_117_05]

If we follow the same argument to the zero order wave A(0), B(0), it is seen
that the nth term in the above development represents the field contribution of a special wave that originates from \eta\ successive reflections of the zero order wave due to the inhomogeneity of the bottom. It can be shown from the mathematical point of view that the series solutions of the form (41) and (42) are actually the solutions of (39) and (40).
Here it may be noted that for weak reflection the deviation of the transmitted wave amplitude from that expected from Green’s formula is one order of magnitude smaller than the reflected wave, in other words, Green’s formula gives a very good estimate of the transmitted wave amplitude even if the partial refiection exists on a sloping bottom (Reid,1957). An elementary explanation may be given as follows:
Assume the incident wave amplitude A and the reflection coefficient \epsilon\. Then the reflected wave amplitude is of the order \epsilon\ A and the conservation of energy requires the transmitted wave amplitude x to be x^{2} A^{2}(1-\epsilon^{2}\). Therefore, we have x A(1- \epsilon^{2}\/2) for small \epsilon\,and the percentage variation of the transmitted wave amplitude is

[Fig.107_117_06]

This is one order higher than the reflection.
In connection with the above stated series solution in powers of \epsilon\,(41) and (42), we may have a wave equation explicity in cluding \epsilon\. If we set

[Fig.107_117_07]

where k^{2}=\omega^{2}\/(ghb^{2}) as before, the equations (15) and (16) may be transformed into

式(45) [Fig.107_117_08]

and

式(46) [Fig.107_117_09]

Assuming the local reflection modulus to be

式(47) [Fig.107_117_10]

(45) and (46) can be written as

式(48) [Fig.107_117_11]

and

式(49) [Fig.107_117_12]

It is to be noticed that the variable \xi\ is proportional to the number of wave lengths comprised between x=0 and x=x. In other words, the distance is measured with the aid of the local wave length as the unit. The quantity \epsilon\\beta\ may be interpreted as the reflection modulus of a transition zone per 1/(bk) and actually corresponds to the well known reflection coefficient (25) for the discontinuous change of the medium. These equations (48) and (49) are useful in understanding the physical process of the partial reflection because the solution can be expressed in terms of the reflection modulus, and this point is illustrated by an example later.
If we have discontinuous changes of the medium at some points, the boundary can be interpreted as the continuities of [Fig.107_118_01] or the continuities of [Fig.107_118_02].
The method of characteristics which is originally developed for the solution of non-linear waves such as shock waves (Courant and Friedrichs, Supersonic Flow and Shock Waves,1948) is quite useful to solve the linear wave problem by numerical means as is pointed out by Reid (1957).
Starting from the equations (11) and (12), the following set of equations is derived.

式(50) [Fig.107_118_03]

and

式(51) [Fig.107_118_04]

where

式(52) [Fig.107_118_05]

and

式(53) [Fig.107_118_06]

and

式(54) [Fig.107_118_07]

式(55) [Fig.107_118_08]

This set of equations (50) and (51) is quite similar to the set of equations (39) and (40), and Mx/M corresponds exactly to kx/2k ,showing the local reflection modulus per unit distance. The details of the discussion related to (50) and (51) are not given here.As shown by Reid (1957), the numerical computation of the wave propagation can be performed by means of a finite difference scheme and in this formulation the local reflection modulus will become

[Fig.107_118_09]

Instead of Mx/M wave length \lambda\ and the distance of ghe grid points　Δx selected for numeerical computation becomes an important factor.This kind of modification factor,sin(2\pi\Δx/\lambda\)/(2\pi\Δx/\lambda\),is common to the finite difference scheme applied to the continuous system and the waves defined by the finite difference equations derived from the equations of a continuous system exhibit behavior somewhat analogous to the waves in a discrete system such as the electromagnetic waves transmitted through a lattice structure.

A particular example of long wave propagation

As a simple example to illustrate the foregoing discussion, we assume a canal of constant width, the bottom profile of which is shown in Figure 2. The regions are

式(56) [Fig.107_118_10]

where

[Fig.107_118_11]

This model is well known for its simple properties of the wave propagation (Rayleigh, Theory of Sound §1486,1926).

Fig.2
[Fig.107_119_01]
-A transition of the finite width.

Take the equation for the volume flux (16)and assume b=unity for simplicity.
The wave equation is now written,

式(57) [Fig.107_119_02]

where

式(58) [Fig.107_119_03]

and

[Fig.107_119_04]

The solutions of (57) are

I) 式(59) [Fig.107_119_05]

II) 式(60) [Fig.107_119_06]

III) 式(61) [Fig.107_119_07]

where m^{2} = n^{2}-1/4. We are concerned with the renection of negatively moving waves in region III and consequently in region I we take only one term which represents the waves moving to the negative x direction.
Defining the reflection coefficient as

式(62) [Fig.107_119_08]

we can derive the formula

式(63) [Fig.107_119_09]

where \mu\=x1/x2.
For large values of \eta\, we may approximate

式(64) [Fig.107_119_10]

If m be imaginary, we may write im=m’, and

式(65) [Fig.107_119_11]

From (63), we see that with increasing n corresponding to the gradual transition from (I) to (III),the reflection diminishes until it vanishes when m log\mu\ = \pi\,i. e.,when

式(66) [Fig.107_120_01]

With a Still more gradual transition, the reflection revives, reaches a maximum, vanishes again when m log\mu\ = 2\pi\, and so on.
To see the situation more clearly, we may set

式(67) [Fig.107_120_02]

where \lambda\2 is the wavelength in region lII and l(=x2-x1) is the width of the transition (II). The critical value of 1/ \lambda\2 corresponding to n defined by (66) is then given by

[Fig.107_120_03]

(64) shows that IRI is inversely proportional to l/\lambda\2 apart from the oscillatory factor.
In Figure 3, the reflection coefficient decreases very rapidly as the ratio of the width of the transition to the wave length in deeper water increases from zero to about 0.5. Another fact is that the amount of reflection does not decrease monotonically, but shows small periodic fluctuations with respect to l/\lambda\2, which suggests interference between the two discontinuities in slope of the bottom, somewhat analogous to the more familiar interactions between discrete interface. A similar phenomenon is found in many cases of the transition of the finite width as in the case of the linear variation of a medium connecting the uniform medium on both sides (Albini and Jahn,1961).
Returning to the theory of wave propagation discussed in the previous section, we may write the solution of (49) corresponding to (59) to (61) as follows:

式(68) [Fig.107_121_01]

式(69) [Fig.107_121_02]

式(70) [Fig.107_121_03]

where

[Fig.107_121_04]

and

[Fig.107_121_05]

Setting [Fig.107_121_06] ,the equation for R is determined from (68) to (70) with the ai of the boundary condirions at \xi\=0 and \xi\ = \xi\1 ,such that

式(71) [Fig.107_121_07]

and

式(72) [Fig.107_121_08]

This is of course the same expression as (63) since、\epsilon\^{2} = 1/4n^{2}. Now, to see the advantage of the expressions (68) to (70) over (59) to (61),(71) is expanded in powers of \epsilon\ assuming a small value for \epsilon\. The first few terms in the expansion are

式(73) [Fig.107_121_09]

where the first term represents the effect of the one-time reflection, and the second term which is of order \epsilon\^{3} gives the effect of the wave reflected three times, and so on.
The reflection coefficient due to one-time reflection only is reduced to

式(74) [Fig.107_121_10]

(74) agrees with (64) and clearly shows that, for weak reflection, the one-time reflection makes the most important contribution to the reflection as a whole. (64) can also be obtained from (32) and indirectly proves that the formula (32) for weak reflection can be derived from the consideration of the one-time reflection only.
If we remember that \xi\1=2\pi\l/\lambda\ (\lambda\ is ghe representative wave length in the transition zone), (74) shows that lRl is maximum at l=(n-1/2)\lambda\/2 and minimum at l=n\lambda\/2 (n=1,2,.…) as a result of the interference of the reflected wave.

An example of continuously varying depth

Now, we treat the case where the wave equation can be solved exactly for continuously varying depth. Epstein (1930) and Yoshida (1949) discussed the partial reflection of waves where the wave equation can be transformed into a hypergeometric differential equation, and gave simple expressions for the reflection and transmission coefficients.
The bottom profile is assumed to be

式(75) [Fig.107_121_11]

where y=e
If x varies from -\infty\ to \infty\, h(x) varies from h1 to h2. If 1/lh3I ＜ I1/h1 -1/h2I ,the variation of h(x) is monotonic; otherwise h(x) first rises or falls to a maximum or a minimum and goes back to the final value (see Fig.8a, b) according to h3=0 or h3 ＜ 0 reopectively. For h3 ＜ 0, we restrict the range of h3 to [Fig.107_121_12] lh3I ＜ h1h2/lh1-h2I,because h3 outside of the above condition is of no practical interest in the wave problem. If h3→\infty\, we have

式(76) [Fig.107_121_13]

and the transition is approximately antisymmetric about x=0 as shown in Figure 3c. If h1 = h2,we have

式(77) [Fig.107_122_01]

and we have roughly a symmetric ridge or a valley in the distribution of h according to h3 ＞ 0, or h3 ＜ -h2/4

The reflection coefficient of the wave travelling in the negative x direction from x = \infty\ is given by

式(78) [Fig.107_123_01]

where

[Fig.107_123_02]

and

[Fig.107_123_03]

\lambda\1, and \lambda\2 are the wave lengths at the depth h1 and h2 respectiveoy, and l (=2\pi\/n) is the representative width of the transition zone, although actually h1 and h2 are reached at x= -\infty\ and x=\infty\, respectively.
For h3 ＞ 0 or n^{2}g Ih3I/\omega^{2}\ ＞ 4, \delta\ is always imaginary and for n^{2}g Ih3I/\omega^{2}\ ＜ 4, \delta\ is real. h3 ＞ 0 corresponds to a bank and n^{2}g Ih3I/\omega^{2}\ > 4 corresponds to a trench which is either shallow or narrow, or to the waves of relatively long periods. On the other hand, n^{2}g Ih3I/\omega^{2}\ ＜ 4 corresponds to the case of a deep or broad trench or to the waves of relatively short periods.
It is easily found that if n^{2}g Ih3I/\omega^{2}\ → \infty\, \delta\→i/2 and the reflection coefficient is reduced to

式(79) [Fig.107_123_04]

and further,if p and q are small,we have

式(80) [Fig.107_123_05]

which is in agreement with the well known formula (25) for the strong reflection.
On the other hand, if the transition is very gradual, so that p and q are large, we may approximate (79) by

式(81) [Fig.107_123_06]

The variation of the reflection coefficient with respect to l/ \lambda\2 (=q) is shown in Figure 5.
As is alreday shown for the case of a transition with the finite width (Fig.3),the decrease of IRI is very steep for small values of l/ \lambda\2 and IRI in Figures 3 and 5 are substantially the same. In practice,wemay take l/ \lambda\2 ＜ 1/20 for the applicable range of the formula (25) for strong reflection. For large values of l/ \lambda\2,however,it should be noticed that the reflection coefficient for the case of an infinite transition (Fig.5) caries as exp (2\pi\l/ \lambda\2) and shows no periodic fluctuaion with respect to l/ \lambda\2,in contrast to the case of a transition with finite width (Fig.3) where IRI varies only as (2 \pi\l/ \lambda\2)^{-1} and fluctuates with the period △l = \lambda\/2 (\lambda\2 is the representative wave length of the long waves in the transition zone).The large difference of the behavior of R for large values of l/ \lambda\2 has already been noticed by Schelkunoff (1951) who explaine it as follows:In a continuous model, the infinite regions (-\infty\ , -1/2) and (1/2, \infty\) have an increasingly pronounced effect on the reflection coefficient at x=\infty\ as l/\lambda\2 becomes large. In other words, the various sections of the medium become increasingly better 「matched」to each other and reflections are greatly reduced. It may be said that the fluctuation of R with respect to l/\lambda\2 for the caseof a finite transition is a result of interference of reflected waves, and for the case of a continuous, infinite transition, this interference of reflected waves extends over the whole region and, as a consequence, the resultant reflection coefficient R at x=\infty\ shows no periodic fluctuation but the value itself is greatly reduced.

Reflection of waves in the range of intermediate wave lengths

The equation of water waves for the intermediate and short wave lengths does not conform to the wave equation discussed in the preceding sections. The problem of partial reflection and transmission in this range of wave lengths should be treated either by means of a solution of the boundary problem of a potential theory (1) to (3) or a solution derived from the original equation of motion retaining the vertical acceleration of the water particles. The process involved to solve the problem is complicated but, in recent years, the wave on a uniformly sloping beach has been solved completely as far as the theory of waves of small amplitude is concerned and it is shown that for periodic waves, the solution is uniquely determined by assuming a progressive wave moving towards shore with a certain amplitude at infinity (Stoker, Water Waves, Chap.5,1957).
Although the actual breaking of waves near shore is a somewhat different matter, the solution given by the above formulation of the problem shows that the wave energy transported toward shore is exclusively absorbed at the edge of the beach and allows no partial reflection of waves back to infinity. If we allow the partial reflection of wave energy, the problem cannot be determined uniquely because we can choose reflected waves arbitrarily and the singularity of the solution at the edge of the beach may take care of the rest. Thus the question of the partial reflection is not settled by this sort of a solution. As for the partial reflection and transmission of waves, complete solutions are given for surface waves in deep water passing over an infinitely thin barrier (Dean, 1945;Ursell,1947;John,1948) and for waves through a submerged barrier in water of constant depth (Heins,1950), The finite width of the barrier is treated by Takano (1960), but the results are not satisfactory in the sense that the series representing the solution converge very slowly for waves of interest. The problem of wave propagation for a model of continuously varying depth which is similar to (76) has been solved for an arbitrary wave length by Roseau (1952). The outline of his theory is given below.
By means of the transformation*

式(82) [Fig.107_124_02]

*In this section \xi\,\eta\,and \zeta\ are defined differently from the previous sections.

with

式(83) [Fig.107_125_01]

The bottom profile is defined by

式(84) [Fig.107_125_02]

or

式(85) [Fig.107_125_03]

式(86) [Fig.107_125_04]

where \alpha\ is a constant. The geometry of the profile is shown in Figure 6,and the relation between the angle H at the maximum slope and \alpha\ is

式(87) [Fig.107_125_05]

where h1 and h2 are the asymptotic depth of water for negative and positive infinities respectively.

We can also define the representative width of the transition by putting \xi\ =+-\pi\. For this definition the length of the transition is given by

式(88) [Fig.107_125_07]

and

式(89) [Fig.107_125_08]

and the totallength of the transition zone is

式(90) [Fig.107_125_09]

If we define the representative slope by

式(91) [Fig.107_125_10]

the representative slope is found to be proportional to \alpha\
For this model of the bottom profile, the set of equations (1) to (3) in terms of \phi\(\omega\) can be transformed into the new set of equations. These equations are, in terms of new velocity potential \varphi\(\zeta\), given by

式(92) [Fig.107_125_11]

式(93) [Fig.107_125_12]

and

式(94) [Fig.107_126_01]

where

[Fig.107_126_02]

The solution of the above set of equations is sought in the form:

式(95) [Fig.107_126_03]

where H(u) and K(u) are analytic function of u, C and P the contours in the u-space, and l,m, p, and q are constants, the relations of which are determined later. If the integral is convergent, it is seen that \varphi\(\xi\,\eta\) satisfies the set of equations. The functional characteristics of H(u) and K(u) are investigated in detail and it is shown that

式(96) [Fig.107_126_04]

where +- \mu\ and +- \lambda\ are simple poles of H(u) and K(u) respectively and are given as the real roots of the equation

式(97) [Fig.107_126_05]

when h is set equal to h1 and h2 respectively.
Finally,it is shown that the asymptotic form of the solution (95) is reduced to

式(98) [Fig.107_126_06]

for \xi\ →+\infty\ ,and

式(99) [Fig.107_126_07]

for \xi\ →-\infty\
Now assume a progressive wave for \xi\ →-\infty\ ,so that in (99) we set

式(100) [Fig.107_126_08]

Then from (98) the reflection coefficient is expressed by

式(101) [Fig.107_126_09]

where the relations between l,m, p, q, H, and K are used.
It is noticed that this formula is quite similar in form to the reflection formula (79) given for long waves. In fact, for long waves,(97) may be approximated by

式(102) [Fig.107_126_10]

so that

式(103) [Fig.107_126_11]

and

式(104) [Fig.107_126_12]

where [Fig.107_127_01] and [Fig.107_127_02] are the wave numbers for\xi\→ -\infty\ and \xi\→\infty\ respectively. Making use of (88) and (89), the equations may be rewritten

式(105) [Fig.107_127_03]

Thus, we have

式(106) [Fig.107_127_04]

(106) is analogous to (79).The only difference is that in the present model l1 \neq\ l2.
The bahevior of the reflection coefficient with respect to 1/ \lambda\2, where l=l1+l2,is almost the same as the case of the former example, namely, for long waves travelling over a bottom of relatively steep slope, the reflection coefficient is reduced to the familiar formula

式(107) [Fig.107_127_05]

and for a bottom of very gradual slope, the reflection ill be

式(108) [Fig.107_127_06]

The general behavior can, therefore, be estimated from Figure 5. It is noticed that in this case, too, no particular intensification of the reflection due to the interference is found.
The reflection coefficient is shown in Figure 7 as a function of x,for various values of \omega^{2}\h/g and h1/h2. Fr.m Figure 7 it is seen that the variation.f the reflection coefficient with respect to x is larger for an intermediate wave than for long waves, at least in the range of large x, as is expected.
As a particular example, assume a model of a continental slope as follows:

l=100km and 50 km
h1=200 m, h2=4000m ; h1/h2=0.05
From(91),x is given by

[Fig.107_127_07]

For this model, the reflection coefficients R corresponding to particular values of \omega^{2}\h/g or period are shown in Table 1.

if we consider that, for the abrupt change of the depth from h2 to h1,the reflection coefflcient is 0.635, the reduction of the coefficient is considerable,in particular for the waye of 10 minutes in period or smaller.
The Roseau’s theory is valid for \alpha\ ＜ \pi\ only. Therefore, the limiting procedure does not yield a reflection coefficient for the abrupt change of depth, although practically, the value for the long waves at \alpha\ \pi\ is almost identical to the value expected from the ordinary theory. As is seen in Figure 7, the reflection coefficient R for intermediate waves is increasing with \alpha\ even at \alpha\ \pi\ , and, therefore, the limiting value for the abrupt change of the medium would be a little higher than the values given for \alpha\ = \pi\.
The situation may be examined by a method which makes use of the series form of the solution (see Takano,1960). The solution of the potential theory can be expressed by a series, each term of which satisfies the surface and bottom boundary conditions. Then, the boundary condition at the junction of two different depths can be satisfied by a suitable choice of the constant for each terms. The inspection of these series reveals that the particular effect of the boundary extends only to a distance comparable to the depth, because, except for the first term, the remaining terms all die out exponentially as e^{-knx} where kn is of order n\pi\/h. The detailed discussion is not attempted here, but it can be shown that the ordinary reflection formula for long waves (25) is the first approximation to the complete formula for reflection neglecting all terms involving exponential decay term in the horizontal direction.

The two dimensional propagation of long waves

From the equations for long waves (8) , (9), and (10), we may derive the twodimensional wave equation in the following form:

式(109) [Fig.107_129_01]

If we consider the weak reflection only, and assume that the horizontal scale to of all the inhomogeneities in the spatial distribution of the depth are greater than the wave length \lambda\,or more precisely, \lambda\/l0 ＜＜ l then the propagation of waves in an inhomogeneous medium is described by the equation

式(110) [Fig.107_129_02]

where \eta\ is taken to be periodic in time (~e^{i\omega\t}), and [Fig.107_129_03]
Setting [Fig.107_129_04]

we can derive the equations

式(111) [Fig.107_129_05]

and

式(112) [Fig.107_129_06]

We note that ▽S is of order k and the wave amplitude A can change appreciably only in distances of the order of the dimensions of the inhomoneneities in the medium l0.
Therefore,▽^{2}A/A is of order no greater than l/l0^{2}. lf we assume \lambda\ ＜＜ l0, the first two terms in (111) are dominant, and the amplitude term can be neglected, so that we have

式(113) [Fig.107_129_07]

The equations (112) and (113) form the equations of geometric optics, and if the variation of depth within one wave length is small compared with the total depth and if the change of wave amplitude is negligible between neighboring rays, the simple laws of refraction and shoaling apply and the wave height follows Green’s law just as if the neighboring two rays constituted solid lateral boundaries.
However, even if the condition \lambda\ ＜＜ l0 is met, the solution obtained from the equations of geometric optics becomes unsuitable for distances l that exceed the critical distance Lcr=l0^{2}/\lambda\. At such large distances, we can no longer neglect the diffraction of the wave by depth inhomogeneities, regardless of the smallness of the diffraction angle \theta\〜 \lambda\/1. Because, at a distance L from the obstacle, the size of the diffracted bundle will be of order \theta\L〜 \lambda\L/l,and for the geometric shadow of the obstacle not to be appreciably changed, it is necessary for the relation \lambda\l/l ＜ 1 to hold. In order to take into account diffraction effects, it is necessary to start from the wave equation.
The following treatment is essentially due to Tatarski (1961). Write (110) in the form

式(114) [Fig.107_129_08]

where [Fig.107_130_01]

and we call n the refractive index.
Set

式(115) [Fig.107_130_02]

Then we have

式(116) [Fig.107_130_03]

Set \phi\ = \phi\0+\phi\1 ; \phi\0 satisfies the equation

式(117) [Fig.107_130_04]

Substituting \phi\ = \phi\0+\phi\1 in (116) ,we have

式(118) [Fig.107_130_05]

In the case where l▽\phi\1I ＜= n1I▽\phi\0I, we can neglect the term.(▽\phi\^{2})
Finally we obtain the equation

式(119) [Fig.107_130_06]

which is valid when the condition

[Fig.107_130_07]

are met. Since I▽\phi0\l k=2\pi\/\lambda\ ,the second condition can be written in the form

[Fig.107_130_08]

and expresses the smallness of the change of \phi\1 over distances of the order of a wavelength. By using the substitution [Fig.107_130_09] (119) can be reduced tothe form

式(120) [Fig.107_130_10]

Since [Fig.107_130_11] ,the above equation is equivalent to the perturbation equation derived from (110) by assuming \xi\ = \xi\0+\xi\1 so that w=\xi1\ and [Fig.107_130_12].
We now find expressions for the amplitude and phase fluctuations of the wave.Since

[Fig.107_130_13]

we have

[Fig.107_130_14]

Therefore, we may write

式(121) [Fig.107_130_15]

and

式(122) [Fig.107_130_16]

The validity of these expression is given by the conditions \lambda\I▽S1I ＜＜ 2\pi\ and \lambda\I▽xI ＜＜ 2\pi\ rather than requiring the smallness of the perturbations x and S1 themselves.Since the inequality Ixl ＜ IS1I is usually satisfied, the method is valid for the conditions

式(123) [Fig.107_130_17]

As is well known, the solution of (120) has the form

式(124) [Fig.107_130_18]

and for the quantity [Fig.107_130_19] ,we obtain

式(125) [Fig.107_130_20]

where G is the free-space Green’s function,\Sigma\ is the area, and dS’ is the two-dimensional element. ,
Since [Fig.107_131_01] satisfies the equation

式(126) [Fig.107_131_02]

which is equivalent to (117)
Now G is given by

式(127) [Fig.107_131_03]

where H is a Hankel function of the first kind.
Sett’ng

式(128) [Fig.107_131_04]

(125) may be written formany;

式(129) [Fig.107_131_05]

In general, a stationary random function can be expressed in the form of a stochastic Fouries-Stieltjes integral with random complex amplitudes. Thus, we may write

式(130) [Fig.107_131_06]

and

式(131) [Fig.107_131_07]

and (129) becomes

式(132) [Fig.107_131_08]

Multiply by it [Fig.107_131_09] and integrate with respect to y. Taking the relation of the Dirac delta function

式(133) [Fig.107_131_10]

into consideration, the left hand side of (132) is t ransformed into

[Fig.107_131_11]

Thus, we have

式(134) [Fig.107_131_12]

Changing the order of integration and considering that the main effect observed at the point of observation comes from the region between the original source of waves and the point of interest, we have

式(135) [Fig.107_131_13]

For Plane monochromatic waves, we have [Fig.107_131_14], so that (128) gives

式(136) [Fig.107_131_15]

Making use of the formula

式(137) [Fig.107_131_16]

we can deduce the relation

式(138) [Fig.107_132_01]

This formula shows that the field inhomogeneities characterized by the wave number k2 (i,e., by the dimension 1=2 \pi\/k2) owe their origin to inhomogeneities of the medium with the same characteristic wave number k2.
Since x=Re \phi\1 and S1=Im\phi\1, we have

式(139) [Fig.107_132_02]

and

式(140) [Fig.107_132_03]

where an asterisk indicates the complex conjugate quantity. Denoting the spectral amplitudes of the random fields x(r1) and S1(r1) by da(k2,L),and \sigma\(k2,L), we may write from (138)

式(141) [Fig.107_132_04]

and

式(142) [Fig.107_132_05]

Since we have assumed k2 ＜＜ k or \lambda\ ＜＜ l,we may approximate [Fig.107_132_06], and (141) and (142) will reduce to (7.37) and (7.38) of Tatarski’s book (1961),exactly as is expected.The refractive index inhomogeneities with dimensions l which are located at a distance (L-x’) from the observation point appear with weight sin (\pi\\Lambda\^{2}/l^{2}), where \Lambda^{2}\= \lambda\(L-x’) is the square of the radius of the first Fresnel zone.
To obtain the relations between the spectral densities of the correlation functions, we take the following averages:

式(143) [Fig.107_132_07]

But, according to the general foxmula, we have

式(144) [Fig.107_132_08]

and

式(145) [Fig.107_132_09]

where Fn(k2,x’-x’’) is the two-dimensional spectral function of the refractive index and FA(k2,0) is the one-dimensional spectral denssity of the correelation function of the fluctuation of x in the plane x=constant.
Substituting (144) and (145) into (143),we obtain

式(146) [Fig.107_132_10]

We introduce new variables \xi\=x’-x" and 2\eta\ =x’+x". Since Fn does not depend on \eta\, the integration of (146) with respect to \eta\ can be carried out explictly,and yields

式(147) [Fig.107_132_11]

Now since Fn(k2,\xi\) falls off very rapidly to zero for k2\xi\>~ 1,we can assume k2^{2}\xi\/(2k) ＜＜ 1 in the important region of integration of (147). Moreover,we are interested in values of k2 which satisfy the condition 2\pi\/k2 ＜＜ L, we may consider the region \xi\ ＜＜ L only. Taking these simplifications into account,we have

式(148) [Fig.107_133_01]

Here, we may set

式(149) [Fig.107_133_02]

and we may replace L by \infty\ in (148) because of the special nature of Fn(k2,\xi\). Therefore, we have

式(150) [Fig.107_133_03]

In the same way, we can derive for the phase fluctuation,

式(151) [Fig.107_133_04]

Equations (150) and (151) relate the one-dimensional special densities of the correlation functions of the amplitude and phase fluctuations of the wave on the line x=L to the twodimensional spectral density of the correlation function of the refractive index.
It follows from (150) and (151) that the phase fluctuations are always larger than the fluctuations of logarithmic amplitude. Taking the relation k=2\pi\/\lambda\ into consideration、 we can discuss the following three cases separately, where l0 and L0 are the inner and outer scales of the inhomgeneities of the medium.
In the case where, [Fig.107_133_05], the spectrum of the correlation function of the amplitude fluctuations is concentrated near the point 2\pi\/l0. It follows from general properties of the Fourier transform that the correlation function has a characteristic scale of order lo. The amplitude fluctuations of the wave do not depend on the frequency of the wave but grow with the distance as L^{3}, while the phase fluctuations are proportional to the square of the frequency and to the distance L.
In the case where [Fig.107_133_06], the spectrum of the correlation function of the amplitude fluctuations is concentrated near the point [Fig.107_133_07]. It follows from this that the correlation function of the amplitude fluctuations on the Line x=L has a characteristic scale of order [Fig.107_133_08]. The form of the correlation function of the amplitude fluctuations depends on the concrete form of the spectral density \Phi\n(k2) of the refractive index fuuctuations.
In the case where、[Fig.107_133_09], the field of spectral density\Phi\n(k2) is not homogeneous nor isotropic for k2 ＜ 2 \pi\/L0. Therefore, FA and FS have me aning only in the region k2 ＞＞ 2 \pi\/L0,where

[Fig.107_133_10]

Thus, FA and FS are equal to one another and proportional to the square of the frequency and to the distance L traversed by the wave. However, the chief contribution to the spectrum of the correlation function of the amplitude fluctuations is made by large scale inhomogeneities in the interval ([Fig.107_133_11]) and it is not possible to discuss the behavior of the spectrum within the present formulation of the problem.
In all the cases considered, the largest wave numbers which participate in the spectral expansions of the amplitude and phase fluctuations of the wave are of order 2 \pi\/lo. It follows from this that the correlation functions of the amplitude and phase fluctuations of the wave change quite slowly over a distance of order lo.
For cylindrical waves diverging from a point source.,the similar procedure can
be applicable. Now, we have in (128)

式(152) [Fig.107_133_12]

Making use of the asymptotic form for the Hankel functions, we may transform (152) into

式(153) [Fig.107_134_01]

where it is assumed that x,x’and x-x’ are generally large compared with y, y’, and y-y’ because the main contribution of the waves comes from the area with the angle \lambda\/lo at the point of interest and its axis on x.
The manipulations similar to the case of plane waves yields

式(154) [Fig.107_134_02]

Field inhomogeneities characterized by the wave number k2 owe their origin to inhomogeneities of the medium with characteristic wave number k2L/x, or with dimension l’=lx’/L. These inhomogeneities are at the distance x’ from the wave source. The factor x’/L takes into account the magnification of the dimensions of the image due to illumination by a divergent ray bundle.
It is possible to discuss the spectral densities of the correlation functions of the amplitude and phase fluctuations in relation to the two-dimensional spectral density of the correlation function of the refract ive index (inhomogeneities of the medium), by means of a method similar to that for the case of plane waves. However, because of the limited application of the present model to the actual problem of tsunamis in an enclosed ocean, no further discussion is attempted here (see Tatarski, Chap.9,1961).

Summary and conclusion

The partial reflection of long waves in an one-dimensional canal of variable cross section is discussed. The basic equations are (11) and (12), Various methods to solve the wave equations (11) and (12) are introduced.(29) is the equation in terms of the reflection coefficient,(39) and (40) are the set of equations showing the mechanism of wave reflection explicitly and defining the local reflection modulus (47). (48) and (49) are the wave equations which include the local reflection modulus explicitly. The method of characteristics applied to a linear wave problem is introduced in (50) and (51),which show the mechanism of reflection clearly just as (39) and (40) do.
In general, the contribution of the one-time reflection of the incident wave is the most important part of the reflected wave as a whole and, for the case of weak reflection, it is enough to consider the one-time reflection (32) only. Moreover, for weak reflection, the transmitted wave is approximated by Green’s formula quite well even though the part of the wave is actually reflected continuously.
Some particular examples are discussed in detail and it is shown that for long waves, the reflection coefficient is approximated by the well known formula (25) for the abrupt change of the depth if l/\lambda\2 is less than, say,one-twentieth. With increasing values of l/ \lambda\2, the reflection coefficient decreases very rapidly and for l/\lambda\2 ＞＞ 1, the reflection is very small. The differences of the behavior of the reflection coefficients for large values of l/\lambda\2,in the cases of finite width of a transition and of continuous, infinite width of a transition, are considered to be a consequence of the different interference of the reflected waves in both cases.
The partial reflection of intermediate and short waves is treated, and the reflection coefficients are shown in Figure 7 as a function of l/\lambda\2 for particular values of \omega^{2}\h2g9 and h1/h2. In a model resembling a continental slope and an ocean, the reflection coefficient of waves of 10 minutes in period is considerably smaller than that of waves of 20 minutes in period.
The two-dimensional propagation of long waves over a bottom of variable depth is discussed briefly for the case of weak reflection following the method presented by Tatarski for the electromagnetic wave propagation in a turbulent atmosphere. The relation of the one-dimensional spectral density of the correlation functions of the amplitude and phase fluctuations of the wave on a line x=L to the two-dimensional spectral density of the correlation function of the refractive index defined by n^{2} = 1/(gh) is formulated.
If we consider the problem from the standpoint of the tsunami propagation in an enclosed ocean, the status of research on the two-dimensional propagation of waves is still quite unsatisfactory. The interference of waves due to total reflection at the continental boundary, which seems to be actually taking place in the later stage of the wave train, might be treated, from the statistical point of view, differently from the view introduced in this paper. It may be worthwhile to try numerical method of approach to the problems of tsunami propagation but due care must be taken in the selection of a grid system because we must distinguish between the strung and weak reflection within one grid interval, and. in addition, the grid interval △x must be small compared with the wave length (actually,[Fig.107_135_01] seems to be enough for the case of weak reflection).

References

Albini F.A.,and R. G. Jahn.1961:Reflection and transmission of electromagnetic waves at electron density gradients.
Applied Physics, 32,75-82.
Bremmer, H.,1949:The propagation of electromagnetic waves through a stratified medium and its W-K-B approximation for oblique incidence. Physica,15,593-608.
Bremmer, H.,1951:The W-K-B approximation as the first term of a geometric-optical series. Comm. Pure and Appl. Math.,4,105-115.
Courants R.,and K. O. Friedrichs,1948:Supersonic flow and shock waves. Inter-science Publishers, Inc.,New York.
Dean, W. R.,1945:0n the reflexion of surface waves by a submerged plane barrier.
Proc. Cambridge Phil. Soc.,41,231-238.
Epstein, P. S.,1930:Reflection of waves in an inhomogeneous absorbing medium.
Proc. Nat. Acad. Sci.16,627-637.
Heins. A. E.,1950:Water waves over a channel of finite depth with a submerged plane barrier. Canadian Math.,2, 210-222.
John.F.,1948: Waves in the presence of an inclined barrier. Comm. Pure and Appl.
Math.,1,149-200.
Ogawa, K.,and K. Yoshida.1959: A practical method for a determination of reflection of long gravitational waves. Records Oceanogr. Works in Japan. 5,38-50.
Pierce. J. R.,1943:Note on the transmission line equations in terms of impedance.
Bell Sys. Tech.,22.263-265.
Rayleigh, Lord,1926:Theory of Sound Vo1.1,2nd Ed., MacMillan and Co. Ltd.,
London.
Reid, R.0.,1957:Forced and free surges in a narrow basin of variable depth and width;
A numerica1 approach. Tech. Report Ref.57-25T, Texas A＆M Research
Foundation 60 PP.
Roseau, M.,1952:Contribution a la theorie des ondes liquides de gravite en profondeur variable. Publ. Sci. et Tech. du Ministere de 1’Air. N’ 275,90 pp.
Schelkunoff, S. A.,1951:Remarks concerning wave propagation in stratified media,
Comm. Pure and App1. Math.,4.117-128.
Stoker. J. J.,1957:Water waves. Interscience Publishers, Inc., New York.
Takano, K.,1960:Effets d’un obstacle parallelepipedique sur la propagation de la
houle. La Houille Blanche, N’3-1960.247-267.
Tatarski, V.1.,1961:Wave propagation in a turbulent medium. Translated by R. A.
Silverman from the Russian. McGraw-Hill Book Company, Inc., New York.
Ursell, F.,1947:The effect of a fixed vertical barrier on surface waves in deep water. Proc. Cambridge Phil. Soc.,43,374-382.
Ursell,F.,1953:The long-wave paradox in the theory of gravity waves. Proc.
Cambridge Phil. Soc.,49,685-694.
Walker. L. R. and Wax, N.,1946:Non-uniform transmission lines and reflection coefficients. J. Applied Physics, 17,1043-1045.
Yoshida, K.,1948:0n the partial reflection of long waves. Geophys. Notes,Geophys.
Ins.,Tokyo Univ.,N’31,14 pp.

ANote on the Generation of Boundary Waves of Kelvin Type* Kinjiro KAJIURA**

Abstract

The generation of Kelvin waves along a straight boundary due to the reflection of cylindrical waves is studied in a rotating sea of constant depth. The result is applied to the discussion of an internal wave of Kelvin type generated by an atmospheric disturbance. In a two-layer model of the ocean a simple formula is derived which gives the Kelvin wave height in terms of the quantities associated with an atmospheric desturbance. The effect of the wind very close and parallel to the coast is most favorable to excite the internal Kelvin wave travelling along the coast to the right (facing offshore in the northern hemisphere), the amplitude of which at the water surface is of the order of 10 cm for a wind field of reasonable scale.

1. Introduction

It is well known that the Kelvin wave is a phenomenon essentially related to the existence of a lateral boundary on the rotating earth and tidal theorists who studied oscillations of a water body on the earth were well aware that the reflection of long waves at a boundary is seriously complicated by the fact of the earth’s rotation (Lamb, Hydrodynamics pp.307-320,1932). Tidal oscillations in gulfs and rectangular basins on the rotating earth were throughly discussed by G.I. TAYLOR (1922) who could show that a train of Kelvin waves is reflected near the head of a gulf in such a way that wave crests progress counterclockwise (in the northern hemisphere). Recently, J. CREASE (1956) studied the diffraction of simple harmonic plane waves on the rotating earth due to a semi-infinite barrier and fourld that the wave of Kelvin type travels without attenuation to the right (facing offshore) along the shadow side of the barrier.
Later study (CREASE,1958) has shown that the waves propagating into a semi-infinite channel are modified by the rotation of the system to generate the waves of Kelvin and Poincare types.
These studies have clarified the behavior of plane harmonic waves near the boundary and suggest an interesting possibility that the existence of a straight boundary might transform cylindrical harmonic waves diverging from a source region through complicated reflection at the boundary into waves of Kelvin type at a large distance from the source region (to the right facing offshore in the northern hemisphere) where ordinary harmonic plane waves propagating parallel to the straight boundary will be nullified by the reflected waves which are \pi\ out of phase to the incident waves.
Since the effect of the earth’s rotation on simple harmollic waves depends on the magnitude of the parameter f/\omega\ (\omega\ :frequency of the water wave, f:the Coriolis parameter), ordinary free long progressive waves of the barotropic mode such as tsunamis are considered too small in period to be appreciably influenced by the rotation of the earth but, in tidal theories in which f/\omega\ is of order unity, the geostrophic effect is always taken into account because of its importance. The above discussion can equally be applicable to internal long waves (baroclinic waves) the periods of which may be considerably larger than those of barotropic waves of the same wave lengths, so that the effect of the earths rotation on these internal waves of long wave lengths of the order of 100km is expected to be very important.
The detailed discussions of the behavior of barotropic and baroclinic waves generated by an atmospheric disturballce ln a two-layer boundless ocean are already given by VERONIS and STOMMEL (1956), VERONIS (1956), KAJIURA(1958). And ICHIYE(1958) and YOSHIDA (1960) have treated the similar problem in a bounded ocean, in which the boundary waves of the baroclinic mode are already noticed.
In the present paper, the reflection of long water waves, especially of cylindrical waves, at a straight boundary in a semi-infinite sea of constant depth is studied and the result is utilized to investigate the possibility of the Kelvin wave generation by means of an atmospheric disturbance of a finite dimension near the boundary. Unlike the case discussed by YOSHIDA who treated harmonic disturbances only, the present discussion is more general in a sense that the distribution of the atmos- pheric disurbance can be arbitrary in a framework of long wave approximation.

* Rec. Dec.25,1961.
** Earthquake Regearch Institute, University of
Tokyo.

2.Fundamental equations

The equations governing the generation and propagation of long waves of small amplitude in water of uniform density on the rotating earth may be written, subject to certain linearizing approximatlons,

式(1) [Fig.107_137_01]

式(2) [Fig.107_137_02]

and the equation of continuity is

式(3) [Fig.107_137_03]

were t is time, x and y are right-handed horizontal co-ordinates, u and v are volume fluxes per unit sidth in the x and y direction respectively,\zeta\ is the elevation of the water surface from the undisturbed level, d is the constant depth of water, g is the acceleration due to gravity and f is the Coriolis parameter.
Fx and Fy are the components of an external force F of the atmospheric origin in the x and y direction respectively and the frictional force at the bottom is neglected. (see Fig.1).

The initial conditions is assumed to be

式(4) [Fig.107_137_06]

so that no disturbance exists in water before the application of external forces:

式(5) [Fig.107_137_07]

The straight boundary is assumed to lie on the y-axis with x-axis positive in the offshore direction. The boundary condition is given by no net transport of water across the boundary so that

式(6) [Fig.107_137_08]

For a large distance from the source region, waves should be divergent to be consistent with the physical setting of the problem.
From (1) to (3), the equation in terms of \zeta\ is derived as follows

式(7) [Fig.107_137_09]

where

式(8) [Fig.107_137_10]

The corresponding expresslons for the initial and boundary conditions are

式(9) [Fig.107_137_11]

and

[Fig.107_137_12]

where

式(10) [Fig.107_137_13]

*The last condition, [Fig.107_137_04], is derived from the equations of motion with the aid of (4) and (5).

Furthermore, \zeta\ =0 for [Fig.107_138_01] with finite t.
In the derivation of (7), the effect of the variation of the Coriolis parameter f with latitude is neglected which is small if the condition \beta\/2\omega\ ＜＜ f/c is satisfied where \beta\ is the latitude change of the Coriolis parameter and \omega\ is the representative frequency of the wave considered.
By means of the Green’s function G(x,y,t,x0,y0,t0) defined by

式(11) [Fig.107_138_02]

with the boundary condition,

式(12) [Fig.107_138_03]

the solution of (7) with the conditions (5), (9) and (10) can be constructed formally as follows:

式(13) [Fig.107_138_04]

where \delta\(x) is Dirac’s delta function and dS0, is the surface element and S is taken for the region of the atmospheric disturbance. The term representing the initial conditions vanishes because of the conditions (9) and the last term is related to the boundary condition (10). (MORSE and FESHBACH,1953, p.854).
It may be remarked here that, for a twolayer model of the ocean, waves of the barotropic mode and of the baroclinic mode can be described by the similar equation (7) provided that c^{2} is understood as the minimum phase velocity (or the maximum group velocity) for each mode respectively, and the importance of the Coriolis parameter in wave motion shows up very distinctly for the internal wave because of the large magnitude of f/c baroclinic compared with f/c barotropic for the same wave length.

3. Elementary considerations

If we confine our discussion to a free wave which is periodic with respect to time;
[Fig.107_138_05] is reduced to

式(14) [Fig.107_138_06]

and the volume flux is given by

式(15) [Fig.107_138_07]

and

式(16) [Fig.107_138_08]

where

式(17) [Fig.107_138_09]

The boundary condition (10) is written as

式(18) [Fig.107_138_10]

From (15) and (16), it is clearly noticed that the components of the volume flux are related not only to the surface slope in the corresponding direction but also to the transverse slope because of the effect of the rotation, and (18) shows that the wave number in the x-direction is related to the wave number in the y-direction in contrast to the case without rotation in which the wave numbers in x and y directions are independent of each other.
An elemetary solution of (14) is given by

式(19) [Fig.107_138_11]

where A is an arbitrary constant and Sm^{2} = k^{2}-m^{2}. If we take m real (periodic waves in the y-direction) the character of the wave changes from the Kelvin to the Poincare type according to the condition m^{2} ＜= k^{2} or m^{2} ＞ k^{2}.
When the period of a wave is larger than half a pendulum day, plane harmonic waves of Poincare type with uniform conditions along the wave crests cannot exist because k^{2} becomes always negative.
Now consider a simple example of reflection when a train of plane harmonic waves (k^{2}>= m^{2}) is incident upon a straight boundary.
The incident wave may be given by

式(20) [Fig.107_138_12]

and the reflected wave which satisfies (18) is given by

式(21) [Fig.107_139_01]

where

式(22) [Fig.107_139_02]

The reflected wave train shifts phase with respect to the incident wave by \phi\ but the amplitude renlaiIls the same. If we introduce the incident angle \theta\ by measuring the incident angle clockwise from the seaward normal to the coast-line, the phase angle \phi\ is expressed by

式(23) [Fig.107_139_03]

Thus, according to the sign of \theta\, the phase of the reflected wave either leads or lags with respect to the case of no rotation.
At \theta\=0, which is the case of normal incidence,\phi\=0 and no phase change occurs. For [Fig.107_139_04],we have \phi\=+-\pi\ and the incident wave \zeta^{in}\ and the reflected wave \zeta^{ref}\ cancel each other completely as is already remarked by CREASE (1958).

4. Reflection of a train of circular waves which is periodic in time

To represent divergent circular waves originated at a point source (x0,y0), we take the free-space Green’s function of (14) as follows:

式(24) [Fig.107_139_05]

where H0, is the zero order Hankel function of the first kind (MORSE ＆ FESHBACH,1953,P.891).
An integral representation of (24) is given by

式(25) [Fig.107_139_06]

To satisfy the boundary condition (18) at x=0, we take a source at the image point with respect to the boundary. Here we should take the phase change of the reflected wave (22) into consideration to account for the effect of the rotation. Thus,we have

式(26) [Fig.107_139_07]

where

[Fig.107_139_08]

Exact integration of (26) seems to be rather complicated but an approximate form of the integral may be derived for large values of kr with [Fig.107_139_09] by means of the method of steepest descents (See Appendix I).
The result is

式(27) [Fig.107_139_10]

where

[Fig.107_139_11]

and

式(28) [Fig.107_139_12]

The first term in the right hand side of (27) may be formally written as

[Fig.107_139_13]

and,denoting \tehta\ the incident angle of circular waves at (0,y), along the boundary, we have

[Fig.107_139_14]

This relation of the phase difference between the incident and reflected waves is exactly the same as that given for plane waves (23).
Therefore,the incident and reflected waves tend to chancel each other along the boundary for a large value of kr since \theta\ approaches +- \pi\/2.
The term G1^{k}, which comes from the contribution of the pole in (26) and represents the Kelvin waves generated by the complicated reflection of circular waves at the straight boundary, remains without attnuation for y ＜ 0 but for y ＞ 0 no such waves exist. This Kelvin wave depends on x0, and the ratio f/\omega\ so that the amplitude is negligible for a large value of x0 and/or for a small value of f/\omega\.

The boundary wave of Klelvin type generated by an atmospheric disturbance

To investigate the unattenuated boundary wave which is supposed to be generated by atmospheric disturbances, the contribution of Kelvin wave G^{k} for an instaneous point source is evaluated.
Since we know the Kelvin wave caused by an periodic point source, G^{k} may be expressed by

式(29) [Fig.107_140_01]

From the condition (28) on G1^{k}, the limits of integration with respect to \omega\ should be -\alpha\ and \alpha\ where

[Fig.107_140_02]

However, we may put \alpha\ →\infty\ for a large distance in the y-direction from the source, namely, for(x+x0)/Iy-y0I ＜＜ 1.
Thus, the integral (29) can be evaluated* to be

[Fig.107_140_03]

for

[Fig.107_140_04]

and

式(30) [Fig.107_140_05]

for

[Fig.107_140_06]

With the aid of (30), the contribution of boundary waves of kelvin type in (13) is given by (See Appendix II)

式(31) [Fig.107_140_07]

Now, we can take, without loss of generality, the region of the atmospheric disturbance to be completely enclosed by a rectangle formed by points(0,0),(b,0)(b,a),and (0,a). Thus, Fx and Fy are always zero on and outside of the lines formed by these points. Only limitation imposed on the region of atmospheric disturbances is (x+b)/lyl ＜＜ 1 as a consequence of the condition assumed to drive G^{k}. After some manipulations (Appendix III),(31) can be transformed to give

式(32) [Fig.107_140_08]

Substituting (30) for G^{k}, we have finally,

式(33) [Fig.107_140_09]

for y＜0
where

式(34) [Fig.107_140_10]

Here T stands for a kind of the weighted average in the x-direction of the y-component of the external force and the weight is maximum at the coastal boundary and decreases exponentially with distance from the boundary.
It is remarkable that only the y-component (parallel to the boundary) contributes to the generation of the Kelvin wave and no effect of the on-and off-shore component of the external force appears in the final result.
For an arbitrary distribution of external forces,(33) can be integrated rather easily by a graphical method. However, to show the situation more clearly by a simple model, we assume the distribution of the wind stress as follows:

式(35) [Fig.107_141_01]

for 0 ＜ t0 ＜＝ tm, and for 0 ＜ x0 ＜ b, 0 ＜ y0 ＜ a and Fy=0 for the outside of the above region.
Then the Kelvin wave on the boundary for a large distance to the right from the source region is given by*

式(36) [Fig.107_141_02]

Therefore, along the boundary, we have the following solutions for a ＜ ctm:

式(37) [Fig.107_141_03]

This solution is graphically shown in Fig.2.
Fora ＞ ctm,the intervals of t will be;

[Fig.107_141_04]

and [Fig.107_141_05]

The amplitude for (3) will be

式(38) [Fig.107_141_06]

and the remaining part will be the same expressions to the case of a ＜ ctm.
It may be noticed that for a ＜ ctm the maximum elevation of the water surface is given by

[Fig.107_141_07]

and this formula resembles to a wind set up formula in the theory of storm surges where the winds perpendicular to the coastline contribute to the elvation in contrast to the present case where the winds parallel to the coast is important.


* The line of integration is taken to be from -\infty\ +i\epsilon\ with Small positive quantity \epsilon\,to be consistent with the integral representation of Dirac’s delta function.

6. Interpretation in terms of a model of two-layer ocean

It may be of interest to interprete the result in the previous section in terms of a two-layer model of the ocean.
If we put R1 and R2 as the solutions of (7) for the representative elevation of the water level in the barotropic and baroclinic modes respectively, the surface elevation h may be expressed by (KAJIURA,1958)

[Fig.107_141_09]

where (\epsilon\=\rho\1’\rho\2 is the density ratio of water in the upper and lower layers. D1 and D2 are the thickness of the upper and lower layers respectively with the total depth D:D=D1+D2.
The minimum phase velocities of barotropic and of baroclinic waves are respectively

[Fig.107_142_01]

For the example in the previous section, the contribution h1 of the barotropic mode for the surface elevation h is

[Fig.107_142_02]

On the other hand, the contribution h2 0f the baroclinic mode for the surface elevation is

[Fig.107_142_03]

The density ratio \epsilon\ does not enter into the expression explicitly. The ratio of the contributions from the barotropic and the baroclinic modes is given by D1/D2. This shows that if D1/D2 is very small, the amplitude of the Kelvin wave of the barotropic mode is negligible compared with that of the baroclinic (internal) mode, although, these two modes of Kelvin waves have different velocities of propagation and will be completely separated for large values of t.
Take realistic values for the wind field as follows

I\tau\yI=5 c.g.s. (wind speed of about
13m/sec)
a=200km
D1=100m
D2=4000m
The surface elevation will be about 10cm and the deformation of the internal boundary will be about 40m. If we assume (1-\epsilon\)=2x10^{-3} the velocity of the internal Kelvin wave for the present model is c2〜1.4m/sec or about 100km/day. Thus, for the atmospheric disturbance with the duration of about 2days, the duration of the water level variation from the beginning to the end will be about 4 days.

* Because of the sharp decrease of the weight function on \tau_{y}\,b may be replaced by \infty\ without serious error.

7. Summary and concluding remarks

It is shown in§4 that the circular waves radiating from the point source are reflected at the straight boundary through complicated processes and generate boundary waves of Kelvin type at a large distance to the right (facing offshore in the northern hemisphere), the amplitude of which depends on f/\omega\ and the distance of the source from the boundary.
In§5 the boundary wave of Kelvin type generated by an atmospheric disturbance is discussed, and it is found that the external force in the y-direction, say, the wind stress parallel to the boundary, is important and the force in the x-direction, namely, on-or offshore winds do not contribute to the generation of Kelvin waves. Morever, the contribution of the parallel winds in the region close to the boundary is most effective as is manifested by the weight function exp (-fx0/c).
Therefore, the wind field of the distant region from the boundary does not play a role in the Kelvin wave generation, although the approximation employed does not permit the rigorous discussion if (IyI+y0)/(x十x0o) is not so large. In §6, an application of the result to a two-layer model of the ocean is attempted.
In the model employed, the Kelvin wave of the barotropic mode is small compared with that of the baroclinic mode if the ratio of the thickness of the upper layer to that of the lower layer is small. The order of amplitude of the internal Kelvin wave generated by an ordinary storm is about 10cm at the surface and the wave is of the solitary type with the duration of a few days.
In the application of the theory to the actual ocean, many factors, which are not adequately taken into account in the present paper, should be considered. For example, the structure of the water body in an real ocean may be very far from the two-layer model and in many places the existence of a continental shelf and a steep slope connecting the shelf to the deep ocean proper complicates the generation and propagation of boundary waves of various kinds. Actually, on the continental shelf, the boundary waves of Kelvin type might be only a minor part of the spectrum of varieties of boundary waves such as edge waves.
However, it may be of some interest to mention that an analysis of day to day variations of sea level along the Japanese Islands by SHOJI (1961) seems to indicate that the maximum or minimum of the daily mean sea levels appears to propagate sometimes to the west along the Pacific coast and in these occasions the variations of the mean sea level in the original area are caused by the wind field parallel and close to the coastline. This finding by SHOJI is in good qualitative agreement with the present theory.

References

CREASE, J.(1956):Long waves on a rotating earth in the presence of a semi-infinite barrier. J.
FLUID MECH.,1,86-96.
CREASE,J.(1958):The propagation of long waves into a semi-infinite channel in a rotating system
J.FLUID MECH.,4,306-320.
ICHIYE, T.(1958): The response of a stratified, bounded ocean to variable wind stresses. Ocn.
Mag.,10,19-63.
KAJIURA, K.(1958): Response of a boundless two-layer ocean to atmospheric disturbances. Dis-sertation, A＆M College of Texas,139 pp.
LAMB, H.(1932):Hydrodynamics,6th Ed., Cam-bridge University Press, Cambridge.
MORSE, P. M. ane H. FESHBACH (1953): Methods of Theoretical Physics, Vol.1, McGraw-Hill Book
Co., New York.
SHOJI, D.(1961): On the variation of the daily mean sea levels along the Japanese Islands.
Journ. Ocn. Soc. of Japan,17 141-152.
TAYLOR, G.I.(1922): Tidal oscillations in gulfs and rectangular basins. Proc. London Math.
Soc.,2nd Ser.20,148-181.
VERONIS, G. and H. STOMMEL(1956): The action of variable wind stresses on a stratified ocean.
J.Mar.Research,15,45-75.
VERONIS, G.(1956): Partition of energy between geostrophic and non-geostrophic oceanic motions.
Deep Sea Research,3,157-177.
YOSHIDA, K.(1960):The oceanic waves of days to month’s periods. Rec. Oceanogr. Works in Japan,5,11-24.

Appendix I

The integral defined by

式(I-1) [Fig.107_143_01]

may be transformed into,

[Fig.107_143_02]

where

[Fig.107_143_03]

The shaded area in Fig. I-1 is the region of convergence with a pole located at tan\theta\ = if/\omega\ and the line of integral \Gamma\ is assumed to avoid the pole to the right as shown in the figure. In the deformation of the contour from \Gamma\ to W, which crosses the saddle point on the \Theta\-plane and follows the steepest path according to the method of steepest descent (Endelyi;Asymptotic Expansions, Dover Publ.,
Inc.,1956,39-41). W crosses \Gamma\ in the upper half plane when -\pi\ ＜ \delta\ ＜ -\pi\/2 and the pole is crossed whentan^{2}\delta\ ＜ f^{2}/(kc)^{2}. Therefore
the contribution of the pole should be taken into account if (y-y0) ＜ 0 and (y-y0)^{2}/(x+x0)^{2} ＞ (\omega\^{2}-f^{2})/f^{2}.
The integral along the steepest path near the saddle point gives nothing but the asymptotic form of the Hankel function of the first kind with the coefficient evaluted at the saddle point. It is remarked here that even if k is imaginary (f ＞ \omega\), we can arrive at the similar conclusion by means of a slightly different transformation.

Appendix II

From (13) the integral in question is originally written as

式(II-1) [Fig.107_144_01]

The substitution of the boundary condition (10) yields

式(II-2) [Fig.107_144_02]

where

式(II-3) [Fig.107_144_03]

Now, by meanshave ofthe partial integration,we have

式(II-4) [Fig.107_144_04]

where \alpha\ and \beta\ are to be determined by the condition for G so that on x0=0 the region of integral is enclosed by a curve defined by

式(II-5) [Fig.107_144_05]

and

t0=0

as shown Fig. II-1.
The first integral in the right hand side of (II-4) may be written, in terms of a line element on the boundary as follows:

式(II-6) [Fig.107_144_06]

where ds is taken positive in the clockwise direction and l is the total length on the boundary from (\alpha\0,0) to (\beta\0,0).The second integral is identically zero because of the boundary condition and the third integral becomes, after exchanging the order of integration, and taking the initial condition into account,

式(II-7) [Fig.107_144_07]

Therefore (II-4) is reduced to

式(II-8) [Fig.107_144_08]

It can be easily shown from (30) that

式(II-9) [Fig.107_144_09]

since x ＜＜ ly-y0I and c(t-t0) = -(y-y0). Therefore we can conclude that the contribution of G^{k} on I is

式(II-10) [Fig.107_145_01]

Strictly speaking, in (II-6) and (II-7) we have divergent integrals sillce G and \delta\G/\delta\x0 include divergent terms on \Gamma\. However, we may apply the technique developed by Hadamard on the finite part of an infinite simple integral (Hadamard;lectures on Cauchy’s Problem, Dover Publ., Inc.,1952, p.133).

Appendix III

The fi rst term on the right hand side of (31) is trahsformed into

式(III-1) [Fig.107_145_02]

where d’= c(t-t0,)+y, y ＜ 0.
Taking the condition that Fx=Fy=0 along , x0=b and y0=0,(31) may be reduced to

式(III-2) [Fig.107_145_03,04]

The boundary values at x0=0 are cancelled with each other.
The last integral in (III-2) is tranformed, by making use of the explict representation of G^{k},into

[Fig.107_145_05]

and the exchange of the order of integration yields

[Fig.107_145_06]

Since Fx=Fy=0 at t0=0 and

[Fig.107_145_07]

we have finally

式(III-3) [Fig.107_145_08]

Here, it can be shown that the term in the bracket in the right hand side of (III-3) vanishes. Furthermore, the first term can be transformed by exchanging the order of integration to dive finally,

式(III-4) [Fig.107_145_09]

27.湾水振動におよぼす防波堤の効果 地震研究所梶浦欣二郎 (昭和37年11月27日発表一昭和38年3月29日受理)

§1.はしがき

　最近国土開発あるいは津波,高潮の災害を防ぐ目的で方々の湾において大防波堤の建設が計画され,防波堤建設によつておこる湾内潮汐の変化あるいは津波,高潮の波高変化を知るための模型実験が行なわれている.1).2).3) 又一方では,東京湾共の他における高潮についての数値実験が電子計算機を用いて行なわれ,その際には大防波堤建設によつて防波堤内部の高潮の波高がどの程度軽減されるかも計算された.4).5) ところで,模型実験あるいは数値実験で得られたものは個々の湾に対する結果であるが,これらの結果のうちどの程度までが一般的な結論であるかは明らかでない.このことは,例えば模型実験の結果が,その問題に最も適当と思われるパラメータについて表示されていないところにもあらわれている.
　今ここでのべようとするのは,Love (1959)6)の行つたと同様に,極めて簡単な湾のモデルを考え,防波堤の開口部では水理学で普通に用いられる水位流量の関係式を使用して理論的に潮汐,津波,及び高潮に対する防波堤の効果を計算し,その結果がどの程度定量的に今迄に行なわれた模型実験の結果を説明するか,また逆にどのような点で,今までの模型実験結果を一般的なものとみることが出来るかということである.勿論,模型実験の目的はこの様な一般的結論よりは個々の湾についての結果そのものであるが,どの様な点にその湾の特殊性があるかを知るには,簡単な理論的推察と実験結果とを対応させ,その違つている点にもつと重点的な考察を進めるべきであると思われる.
　また一方では,この様な簡単なモデルによる考察は,複雑な地形をした湾内の海水振動の数値実験を行なう場合に,どの程度の項まで含めた基礎方程式を用いるのが適当であるかの目安をも与えることが出来る.これに関係して線型方程式を基礎とした理論から導かれる港湾副振動についての性質を現実に適用することには限界があることが判る.即ち,防波堆の開口部が小さくなるとそこでの非線型の効果が最も重要となるために,Miles and Munk (1961)7) ののべた様な,防波堤の開口部を狭くすることは副振動を小さくすることにはならないという港湾副振動についてのパラドックスは防波堤開口部の面積が可成り大きくないと適用しがたい.

　1)　長崎海洋気象台「模型実験による有明海の潮せきに関する研究」第1報,研究時報12 (1960),261.第2報,研究時報13 (1961),83.
　2)　岩崎敏夫・三浦晃「気仙沼湾の津波の模型実験について」第8回海岸工学講演会講演集 (1961),51.
　3)　樋口明生・吉田幸三「名古屋港の水位変動の特性について」第9回海岸工学講演会講演集(1962),34.
　4)　宮崎正衛「東京湾高潮に対する防波堤効果の計算について」産業計画会議資料第217号 (1961),171.
　5)　宇野木早苗・磯崎一郎「防潮堤開口部の流速計算についての一考察」第9回海岸工学講演会講演集 (1962),7.
　6)　R.W. LOVE,「Tidal response of a bay with a constricted opening to the sea.」Thesis for M. S., A and M College of Texas, Dept. of Oceanogr.,(1959),66 p.
7) J.MILES and W. MUNK,「Harbor Paradox,」Proc., ASCE,87, No. WW 3,(1961), 111.B. L. MEHAUTE and B. W. WILSON,「Discussion,」Proc., ASCE,88, No. WW 2, (1962),173.

§2.基礎方程式

　今,湾水の運動が一次元的なモデルで近似出来るとし,長波の近似が許され,防波堤の開口部を除いては線型近似が出来また海底摩擦や渦によるエネルギー消耗はないと仮定すると,運動方程式及び連続の式は次の様に書ける.

式(1) [Fig.107_147_01]

式(2) [Fig.107_147_02]

ここで,Q：流量,\eta\：平水面からの水位上昇量,x：水平座標,　t：時間,g：重力加速度, h：水深,b：水路巾,.F：湾内水に作用する外力,であり,考えている湾のモデルは第1図に示すように,一様な短形断面をもつ水路が防波堤位置において連結されている.座標原点 x=0 は防波堤位置にとり,湾と外海との境界を x=-x1,湾奥の位置を x=x2とし,必要のある場合には外側の湾に対する量には添字1を,内側の湾に対する量には添字2をつけて区別する.今海水の運動による摩擦力は考えていないので,潮汐あるいは津波の問題を取り扱うときにはF=0とするが,高潮を考える場合にはF=\tau_{s}\/\rho\ (\tau_{s}\：風の海面応力,\rho\:水の密度)を採用する.
　防波堤開口部 (x=0) の条件としては,水理学で一般に使用される水位差と流量との関係式を利用する.即ち,

式(3) [Fig.107_147_03]

ここで,A0：防波堤開口部面積,　k：流量係数であり,複号は (\eta\1-\eta\2) の符号に順ずる.
　流量係数kの値は,開口部の形によつてかなり変化することが知られているが,厚みのない板のような薄い防波堤の場合にはk〜0.7程度のようである.8) この式 (3) 自体は,もともと定常流について求められたものであつて今考えるような時間的に変化する流れ,あるいは外力の作用下の流れにも適用出来るかどうかには考慮の余地があるが,完全な運動方程式に立ち帰つて各項の大いさを吟味すれば判るように,周期の長い波動の場合について，開口部の面積が水路の断面積にくらべてかなり小さいとき,また外力については防波堤開口部の奥行きが短いときには成立するとみられる.
　x=0 における条件が(3)で与えられると,後は問題に応じて適当な初期条件や境界条件を与えることによつて,外側の湾及び内側の湾に対する水の運動はよく知られた方法で解くことが出来る.今解法の点からみて二つの異つた場合,即ち,外海から伝わる擾乱が完全に周期的な波の場合と弧立波的な波の場合とを分けて考えることにする.高潮の場合のように周期的でない外力が湾水全体に作用するときは後者の方法が便利である.

8) VEN TE CHOW, Open-channel hydraulics (McGraw-Hill Book Co. Inc.,1959), Chap.17

§3.　湾ロで周期的な水位変動が与えられる場合の湾水振動一潮汐振動

　上述のモデルにおいて,湾口(x=-x1)の水位変動が周期的なものとして与えられる場合を考え,F=0とする.このとき境界条件は,

式(4) [Fig.107_148_01]

また湾奥での条件は,

式(5) [Fig.107_148_02]

であり,H：湾口水位変化の振巾,\omega\=2\pi\/T,T：外力の周期である.(5)は湾奥での完全反射の条件であるが,もし湾奥で完全反射が起こらないときにはこの式を変更する必要がある.今のところ,実際の湾奥条件をどのよらにとればよいかの問題は未解決といえるが簡単には,流量が水位勾配に比例するような条件式を使うのが普通である.しかしここでは完全反射の式を採用しておく.
　ここで与えた問題の最も簡単な場合として,湾口条件(4)を防波堤直前で与えたときの内湾の海水振動についてはすでにLoveによつて議論がなされている.今のべる理論では,それをやや拡げて外湾の効果をも含めてある.
　今(1),(2)式を(3),(4),(5)の条件のもとに解くと(3)が非線型効果を含んでいるためにやや複雑な解が得られる.しかし非線型の作用によつてあらわれる高調波はその振巾が小さいので省略することにすると基本周期のもののみからなる解となる.具体的には(3)のうちに含まれる

式(6) [Fig.107_148_03]

のような展開において右辺第1項のみを残すこと相当する.
　結局(1),(2)の解は(3),(4),(5)の条件と(6)の近似のもとに次のように与えられる.

式(7) [Fig.107_148_04]

式(8) [Fig.107_148_05]

ここでH1,\epsilon\1は決定すべき常数, [Fig.107_149_01]である.今求めようとする量は湾奥の振巾及び位相のおくれ \epsilon\2 であつてそれらは次の関係式によつて与えられる.

式(9) [Fig.107_149_02]

式(10) [Fig.107_149_03]

ここで,

式(11) [Fig.107_149_04]

式(12) [Fig.107_149_05]

式(13) [Fig.107_149_06]

式(14) [Fig.107_149_07]

S1:外湾の表面積、 S2:内湾の表面積

　(10)から,位相 \epsilon\2 については 0 =＜ \epsilon\2 =＜ \pi\12であり湾内振動の位相は常に湾口に比して遅れることが判り,その量はパラメータBによつてきまる.(cosk1x1 ＜ 0 のときは湾口に対する位相のおくれは　( \pi\+\epsilon\2) である.) (11)において,Bの大勢はB0によつて決り，(12)を考慮すると,Bは湾口での波動振巾に比例し,内湾の表面漬の2乗に比倒し,周期及び防波堤開口部面債の2乗に逆比例することが判る.このような傾向は,模型実験の結論ともよく一致し,防波堤開口部の条件が口の狭い場合には本質的に非線型であることを示している.湾の長さと深さとに関係した量k1x1,,k2x2もパラメータBに含まれ，特に(13)をみれば \beta\ が防波堤位置に関係した量であることは明瞭であり,もし防波堤位置を湾口と考えると \beta\=1となる.
　また(12)に含まれるH0は(14)で示されるように,防波堤に開口部がないと考えたとき,即ち外湾のみがあるときに,防波堤位置(外湾の湾奥)で期待される水位変化の振巾であつて,B0中には外湾に関する量はこれ以外には入つていない.即ちLoveの計算はH0を与え, \beta\=1とおいた場合に縮当する.
　(10)を用いて,パラメータBに対する \epsilon\2(位相)及びcos\epsilon\2(振巾)の変化を計算したものが第2図である.図からも明らかなように,パラメータBの増加とともに内湾の位相は90°まで遅れ,振巾は0まで減少する.

§4.　周期的でない外力の作用による湾水の振動一津波或いは高潮

　一次元波動の間題において,過渡的な現象を議論するのに便利な方法の一つとして特性曲線法がある.これは非線型の波動を取り扱うのに広く用いられているが,線型の波動の場合にも他の方法にくらべてきわめて有効に用いることが出来る.この方法では(1)(2)を次のように書き換える．
　まず,

式(15) [Fig.107_150_02]

式(16) [Fig.107_150_03]

とおき,水路の巾及び水深が一定であることを考慮すると,

式(17) [Fig.107_150_04]

式(18) [Fig.107_150_05]

となる.ここで [Fig.107_150_06] は長波の速度である.
　今初期条件として t=0における \eta\ 及びQ (即ちZ+及びZ-) をxの函数として与える.境界条件としては x=0で(3)を,x=x2では(5)を与える.　x=-x1の条件としては例えば外海からの進行波を問題にするときは,§2のモデルにおいて外湾は無限に長いと考え, x1\=infty\ とし,x=-x1 では波の反射がないとするのも一方法である.また湾内の高潮のみを問題にするときには,湾口x=-x1での条件を例えば \eta\1=0とする.第3図は津波が内湾に向つて進んで来る場合の一例を示す.
　ここで,防波堤開口部即ちx=0における条件が,Z+及びZ-を用いてどのように書けるかが問題になるが,そのまえにまず防波堤のない場合を考える.このときx=0で水路の断面積が突然変るために波の部分反射が予想される.これを表現すると,

式(19) [Fig.107_150_07]

であるから,\alpha\=(b2c2)/(b1c1)として,

式(20) [Fig.107_151_02]

式(21) [Fig.107_151_03]

となる.ここでxの正方向(湾奥に向う)に進む波については 2/(1十\alpha\)が透過係数,(1-\alpha\)/(1+\alpha\)が反射係数を与えている．
これに反し,防波堤のある場合には非線型の条件が入つているためにことがらが面倒になる,即ち反射あるいは透過の割合が波の振巾によつて異なるのである.この場合には,Zの代りに,

[Fig.107_151_04]

h1:外側の水路の水深,
A1:外側の水路の断面積

で定義される無次元量Yを使つて,

式(22) [Fig.107_151_05]

式(23) [Fig.107_151_06]

と書ける,ここで複号は(Y1+ + Y2-)の符号に順ずる.したがつて,水位 \eta\ 及び流量Qは

式(24) [Fig.107_151_07]

となる.また(17),(18)式は

式(25) [Fig.107_151_08]

式(26) [Fig.107_151_09]

と変形される.
　以上で特性曲線法による津波,高潮計算を行なう際の防波堤開口部の条件は明らかになつた．ここで注意すべきことは，長さの次元をもつパラメータが [Fig.107_151_10]という型で入つていろことであつて,前述の周期的な振動の場合のパラメータBとは異なっている.今の場合,例えば初期条件として入射波の波高 \eta\*を与えたとすると,ある一定の \eta\*/Kの値に対してなされた計算はそのままでは一般に他の条件の場合に拡張出来ないが結果の解訳にあたつて \eta\/h1 と [Fig.107_152_01] とがあるきまつた関係を満すかぎりでは異なつた初期波高の場合にも適用される.例えば初期波高が2倍になることは,防波堤の力学的な効果としては初期波高に変化なくA0が1/√2に減少することに相当している.

§5.有明海の潮汐模型実験結果との比較

　有明海は潮汐振動の大きいところとして日本では有名であるが,ここに大防波堤をつくつた場合に内湾の潮汐がどのように変化するかを調べる目的で長崎海洋気象台が中心となつて模型実験が行なわれた.1) このとき行なわれた実験は,第4図に示すようなI,II,III,IV,Vの位置に防波堤を建設する場合,防波堤の開口部を順次に狭めて行つた際に有明海沿岸の観測点で潮汐振巾及び位相がどのように変化するかを調べたものである.(このとき竹崎の南側にある諫早湾は図に示した線で締切られている.)この結果に対して上述の理論的考察をあてはめるために,ここでは有明湾口に位置する口の津の潮汐振巾及び位相を基準にとつて湾奥、部の竹崎,住の江の潮汐振巾及び位相の変化を実験結果からとり上げる.また,防波堤開口部の面積が湾の断面積に比してかなり小さくないと今の理論は適用出来ない筈のものであるから,開口部面積が約5×10^{4}m^{2} (全断面積に対する開口部面積の比は約0.1〜0.2)以下のところを議論する.開口部面積がかなり大きいときには,条件(3)のような水位流量関係から考えられる非線型の効果だけでなく,水位,流量の時間的変化が重要な役割を果していることが実験にもあらわれている.
　さて,理論計算にあたつて,まず有明海附近の海図から表1にあげた諸量を計算しておく.(ここで,水深としては,「海図水深」+1.5mを平均水面からの水深と仮定し,その算術平均として,全体の平均水深を計算した.)これらの量及び潮汐周期(12.4時間),湾口振巾(1.5m),流量係数(0.7)を用いると,湾口の漸汐振巾に対する湾奥の潮汐振巾の比,及び位相の遅れは容易に計算出来る.
第5図にはかくして計算された半日潮の潮汐振巾及び位相について,理論と実験との比較が示されている.口の津の漸汐振巾 \eta\0 を基準とした湾奥の住の江,竹崎の漸汐振巾\eta\/\eta\0 を防波堤設置場所I,II,III,Vについてくらべてみると,防波堤断面が5×10^{4}m^{2}以下では理論値と実験値とがかなりいい一致を示していることが判る.これに反して,断面積が10^{5}m^{2}では,その違いが大きく,初めに予想した通り,条件式(3)は不適当であることが判る.断面IIについては,理論値と実験値との差がかなり現われているが,この原因については明らかでない.I及びIIIの断面について,理論値と実験値とが,かなりよく一致しており,またIIとIIIの実験値がほとんど一致することから考えると,　IIの断面の実験には何等か特別な条件が加わつているとみられる.IVの断面については正確な実験値が示されていないので議論が出来ない.Vでは住の江の実験値が示されていないので,島原での値を加えてあるが,一般に実験値よりも理論値の方が小さく,防波堤の効果としては大きく現われている.この違いもいろいろの原因によると思われるが,特に流量係数(k)に関係しているかも知れない.
　位相については,II及びIIIの位置に防波堤がある場合について,口の津に対する竹崎の位相のおくれのみが示されているが,もともと実験において,位相を正確にきめることが困難であることを考慮に入れると,防波堤開口部の断面積が2.5×10^{4}m^{2}以下ではかなりよく合つているといえさうである.ただ理論では摩擦の効果が無視されているから,竹崎に対する住の江の潮汐位相のおくれのような場所による位相のずれは期待されない.現実には,有明海では摩擦の効果が相当に顕著であつて,潮汐の調和分解を行うと,半日潮の位相が湾奥に向うにつれて順次におくれていることが明瞭にみられる.
　理論では一次元の仮定と摩擦の効果を無視したことから考えると,個々の波の形の比較では,今のモデルによる結果と,現実をうつした模型とではかなりの違いが出るはずで,実験値と理論値で1割程度の違いは当然期待されてよい.そのことを考慮に入れると,実験値と理論値とは,防波堤開口部面積が小さいときにはよく一致しているといえそうである.即ち,§3でのべたパラメータ・Bの大小によつて,振巾,位相の変化の大勢を説明出来ることになる.第6図で示したように,有明海の規模の湾では防波堤が湾口近く設置されるほど,同一の開口部面積に対して内湾における潮汐減衰の効果が著じるしく,例えば口の津の半日潮汐振巾1.5mのとき,Vの位置の防波堤を建設して,開口部面積2×10^{4}m^{2}のとき,湾奥の漸汐振巾は30cm程度ということになり,防波堤のない,現在の状態では湾奥の潮汐振巾が約2.5mあることからみると実に1/8程度の振巾に減衰していることになる.このとき口の津の防波堤開口部を通る海水の最大流速は約7ノットに達すると思われる.

§6.気仙沼湾の津波模型実験との比較

　1960年5月のチリ地震津波によつて東北地方太平洋岸のかなり大きな湾の最奥部で津波被害があり,それを防ぐために各所で湾口防波堤を設置するという案が出されている.
その基礎資料の一つとして,気仙沼湾の湾口に防波堤を設置した場合に,果して津波の波高減衰に対して効果があるかどうかということが模型実験によつて調べられた.2)
　模型実験では短形水漕内に気仙沼湾と外海の一部とをうつし,これに単一な押し波又は引き波を進行させ,湾の入口に設けられた防波堤によつて湾内の波高分布がどうなるかが測定されている.第7図には気仙沼湾の地形図及びこれに対応すると仮定された理論的な一次元モデルを示してあり,このモデルについて§4の方法によつて計算を行なう.
　まず,湾口の問題を考える.一次元モデルで実際には二次元的な現象を扱かうのであるから,幾分基礎のあやふやになる点のあるのは止むを得ないが,気仙沼湾入口の地形及び一般に波長の長い波の湾口附近での行動から推察して,湾の巾の2倍の巾をもつた水路が外海を代表すると仮定し,ここを伝わる波が湾口で部分反射をおこしながら湾内に入射すると考える.
水深は,地形図からみて湾内で約10m,湾口直前の外海で約25mとなつているから,水深の比を1/2.5とする．即ち断面積の比はA2/A1〜1/5であり波の反射に関する量 \alpha\〜0.3がきまる.湾の長さに対しては,今湾口から湾奥まで長波の進むに要する時間をaとすると,湾水の基本振動周期は4aであり,aをパラメータとして外部から伝わつて来る弧立波の周期(波の初まりから終りまでの時間)を表現することにすると,湾の長さそのものは計算には直接入つて来ない.
　まず初めに,防波堤のない場合について,波がどのくらい続いて来襲すると,完全に周期的な波の来る場合と同様な湾内波高が得られるかをみるために,正弦波形の波について計算を行なつたものを第8図に示す.このとき波の周期 na が湾の固有振動周期に近いと振巾が大きくなることは当然期待されるが,今の計算では,海底摩擦がなくても,湾口から外部へ逃げる波のエネルギーが存在するために,平衡の位置には意外に早く達している.
即ち \alpha\=0.3の場合,来襲波が3波程度ですでに周期的な来襲波のときの理論値にほとんど合致していることがみとめられる.この理論値は妹沢(1935)9)の与えたと同様な方法によつて容易に求められる.
　今入射波をH1cos(k1x-\omega\t) とすると,湾奥の振巾は

式(27) [Fig.107_156_01]

であり,湾口での振巾は

式(28) [Fig.107_156_02]

で与えられる.ここで [Fig.107_156_03] である.
　一山一谷の波の場合には周期が湾の固有振動周期(4a,4a/3,4a/5,…) から離れたところで周期的な波の理論値より一般に大きくなつている.別の言葉でいうと,干渉の作用を受けない方が山が大きい.逆に共鳴周期の近くでは,繰り返しの波の干渉による振巾増大があるわけである.また (28) から判るように,湾口での波の振巾は入射波の振巾H1とは一般に異なつたものであり,湾の大きさや深さ,それに外海とのつながり具合等によつて変化するものであるから,津波の湾口における高さから,外海での津波を推定するときには注意を要する.
　次に,湾口に防波堤を設けた場合には来襲弧立波を次のように仮定する.

式(29) [Fig.107_156_04]

ここで [Fig.107_156_05] ,来襲波の周期 T=na である.この初期波はxの正方向に進む進行波と仮定して計算を進める.
　§4にものべたように,非線型の効果をもつ防波堤の存在する場合には,計算にあたつて来襲波の波高を与えておく必要がある.今は模型実験の場合に対応させて \eta\0=1.5mをとる.
　結果は第9図の通りであるが,ここでは,防波堤のないときの湾奥最大波高 \eta*_{head}\ を基準として防波堤のあるときの湾奥最大波高 \eta_{head}\ をあらわすことにし \eta_{head}\/\eta*_{head}\ がA0/A2 (A0:防波堤開口部面積,A2:内湾の断面積)によつてどう変るかが示されている.入射波の周期が湾の基本振動周期よりも短いとき (n＜4) にはこの比 \eta_{head}\/\eta*_{head}\は周期にかかわらず一定であり,n＞4 では入射波の周期が長いほど同一の開口部面積に対して防波堤の効果は少ないことがわかる.これからみて明らかなことは,湾水の基本振動周期に比して入射波の周期がきわめて長いときには,防波堤の波高減衰効果は望めないわけで,内水面のそれほど広くない港の港口防波現のようなものは,開口部面積が極端に小さくなければ長周期津波に対してあまり効果のなことは明らかである.
　図中には岩崎,三浦による模型実験の結果をも入れてあるが,これらの実験値はすべてn ＜ 4の範囲のものある.計算及び実験の両方において,湾内の波高比 \eta\/\eta\*は場所によつてあまり変らないことが判つているのて,模型実験結果として図に示されているものは,湾内各所で得られた波高比を平均したものである.但し,平均をとる際に,実際の気仙沼湾では湾奥附近に巾のいちじるしく狭くなつたところがあり,摩擦や反射の影響が顕著にあらわれていると思われるので,湾の最奥部数ケ所の実験値は平均から除外してある.
　この図をみると,ごく大体の傾向からいつて理論と実験との結果は合つているとみられる.即ち湾口防波堤の開口断面積の減少による湾内波高減少の割合は,今のべている簡単な理論から,かなり定量的にも推察出来る.
　一般に津波に対する湾口防波堤の効果という点からみると,例えべ1960年5月のチリ地震津波のような周期1時間に近い波に対しては,湾の基本振動周期が1時間程度ないと,あまり防波堤の効果が発揮されないが,三陸地方でしばしば経験される近地津波を考えると,周期30分程度あるいはそれ以下のことが多いので,同じ開口部面積をもつ防波堤の効果は,がなり小さな湾でもあらわれるであろう.

9) K.SEZAWA,「Growth and decay of seiches in an epicontinental sea,」Bull. Earthq. Res. Inst.,13(1935),476.

§7.高潮に対する防波堤の効果

　§4の方法によつて,高潮の場合に防波堤がどのような効果をもつているかが計算出来るが,その簡単な場合として巾b 及び水深h の一様な短形湾(断面積Aは勿論一定)の内部に防波堤を設置した場合について考える,即ち,第1図のモデルで外湾及び内湾は深さも巾も等しい場合である.今湾の主軸方向に一様な風W (時間的には変化する)が吹く場合に,湾内の高潮が防波堤の設置場所,防波堤の開口部の広さA0,風の吹め始めてから止むまでの時間T等によつてどう変るかを調べる.気圧の効果は今の議論からは一応除外する.
　まず,t=0で静止した水面を考え,風の応力は,

式(30) [Fig.107_157_01]

とする.ここで\tau\ は最大風速に対応する応力である.時間は長波の湾口(x=-x1) から湾奥 (x=x2)まで進むに要する時間aを単位としてはかり,風の継続時間は,T=2\pi\/\omega\=naとする.湾口では常に水位変動がないと仮定すると,\tau\に対して静的な釣合状態を保つ湾奥水位 \eta\ は,海底摩擦がなく湾奥で完全反射の条件が満されているとき,

式(31) [Fig.107_157_02]

であらわされる.
　§4にのべたように,防波堤位置における非線型条件の存在のために,パラメータKが重要な量であり,以下の計算では特にことわらないかぎり,\eta\/K=14 の場合が主として計算されている.即ち具体的にいうと,例えばA0/A=1/15,k=0.8とおくと \eta\/h1=0.08であり,(x1+x2)〜40km,h1〜10m 風力係数2×10^{-6} とすると,W (風速)〜30m/sec,\eta\〜80cm の程度である.この模型湾 (長さ40km,深さ10m)ではセイシの基本周期約4時間30分ということになる.
　さて,第10図には湾奥に於ける最高波高 \eta\m について \eta\m/\eta\ が,風の吹き続ける時間Tによってどう変化するかが示されている.まず,防波堤のない場合にはT=4aの近傍で,即ち湾の基本振動周期の近くで動力学的な増巾効果が最も著くしく,\eta\m/\eta\ は約1.5となり,T=12aのあたりになるとすでに平衡状態の水面上昇量と等しくなる.このことはすでによく知られていることであつてここで特にのべるまでもない.次に,湾の中央に防波堤のある場合(x1=x2),T=4aのあたりで波高最大値が防波堤のない場合にくらて約半分になつている.これに対してT > 8a となると \eta\m/\eta\〜1 となり,風の周期Tが湾の基本振動周期に比して長いと防波堤の効果はみとめられないことを示している.ここで比較のために,防波堤の開口部がない場合を考えると,湾奥での最大波高は,防波堤のない場合の約1/4程であり,共鳴増巾のおこる風の持続時間Tはaのあたりである.
　第11図には防波堤が湾中央 (x1=x2) にある場合について,防波堤位置の直前及び直後の最高水位 \eta\max と風の持続時間Tとの関係が示されている.これによると,T=4aでは,外側最高水位がやや大きく,内側最高水位がやや小さくなることがみとめられるが,T>6a 程度になるとその差が小さくなり \eta\max/\eta\ は約0.5である.
　第12図には第11図と同様な条件の場合に,防波堤開口部を通る流量の最大値を示してあるが,風の持続時間Tの変化による最大流量Qの変化はゆるやかである．これを,A0/A=1/15,k=0.8,最大風速30m/sec,h=10m,(x1+x2) = 40kmの場合について換算すると,開口部を通過する海水の最大流速は約5ノットとなる.
　第13図には防波堤の開口部の広さ(A0/A)の変化によつて湾奥の最高水位がどう変るかが示されている.勿論A0/Aの小さい程防波堤の効果が大きいが,風の持続時間Tが長いときには,防波堤の有効性は小さなA0/Aの範囲にかぎられることが判る.例えば T=12a ではA0/A〜1/15，T=4a ではA0/A〜1/3の程度で,すでに防波堤の効果がほとんどなくなつている.この図では \eta\ の値を一定(80cm)に保ち \eta\/Kの値は変化させてある.
　第14図には防波堤の位置が高潮の高さにどういう影響を与えるかを示してある.ここでは動力学的増巾効果の最も卓越すると思われる T=4a をえらんであるが,この場合，防波堤位置が湾の中央よりやや外海寄りにあるとき湾奥波高が最も低く,波高は防波堤のないときの約1/2である.防波堤位置の内側及び外側の最高波高については,予想される通り,防波堤が湾奥に近い程波高が高くなつている.
　第15図には,第14図と同様な場合について防波堤開口部を通過する流量が示されている.ごく湾の奥に位置するときを除いて,防波堤の位置による流量の変化は少い.第12図の説明にあつたと同じ例では,開口部を通過する海水の最大流速は約5ノットに達する.
　以上を要約すると,

　1) 防波堤のない場合には,風の持続時間がほぼ湾の基本振動周期と一致するときに,湾奥の水位上昇が最大 (定常値の約1.5倍) となるが,湾の中央部に防波堤が存在し,その開口部面積A0が全断面積Aの約1/15程度であれば,湾奥の水位上昇は約1/2に減少する.しかしA0/Aが1/3程度まで大きくなると,防波堤の効果は極めて小さくなる.
　2) A0/Aが約1/15の場合,風の持続時間が湾の基本振動周期の約3倍程度になると,湾の中央部に位置する防波堤の効果はほとんどなくなる.
　3) 防波堤位置が湾奥に近くなると,湾奥水位を低める防波堤の効果は減少する.例えば防波堤と湾奥との距離が湾の全長の約1/10程度になると,T=4a 程度の持続時間をもつ風に対して,防波堤の波高減衰効果がみとめられない.
　4) 防波堤の開口部を通過する海水の最大流量は,風の持続時間や防波堤の位置によつてあまり変化しない.湾の全長40km,平均水深10mのところに最大風速30mの風が吹き,A0/A=1/15,k=0.8とすると,開口部での最大流速は約5ノットとなる.
　以上のような結論を実際的な問題に適用するときには,かなりの注意が必要である.理論における一次元の仮定や,摩擦の省略以外に高潮の問題では第一に,湾口水位が常に変化しないというのは事実に反するし,風の吹き方は今仮定されたものよりはるかに複雑である．また風の作用は,水深に関係するから,一様水深を仮定している今の理論値を現実の高潮の高さにむすびつけることには困難がある.これらを考慮すると,この節の結論は定性的なものとみるべきであろう.

§8.むすび

　すでに述べて来たように,湾内の潮汐,津波,高潮等に対する防波堤の効果を調べるにあたつて,簡単な一次元モデルで防波堤開口部における非線型効果を考えた理論が,模型実験の結果をかなり良く説明することがわかつた.このことは,現実の湾の地形を考慮した二次元モデルによる数値実験では,局部的な非線型効果を考慮に入れることによつて,かなりいい精度で現実の状態を再現する可能性が強いことを示している.ここでは摩擦の効果について何等の考察をしていないが,摩擦による長波のエネルギーの減衰が重要な問題であることは勿論であり,それについてはさらに潮汐,潮流の観測値を利用して研究を進めることが望まれる.
　数値実験と模型実験とを比較してその優劣を論ずることは本文の目的ではないが,研究の目的によつて,また必要とする精度によつて,どちらの方法がより有利であるかがきまるものと思われる.例えば,湾の特性のごく大体を知るには実験を行なう必要がないであろうし,複雑な地形の湾や,海岸附近の波の行動を知るには現在のところ大規模な模型実験の方が有利であると思われる.
　おわりに,この研究において計算及び製図を手伝つていただいた神里秀子嬢に感謝する.

湾水振動におよぼす防波堤の効果 Effects of a Breakwater on the Oscillations of Bay Water. By Kinjiro KAJIURA, Earthquake Research Institute.

The effects of a large breakwater with a constricted opening on the oscillations of bay water as a whole are examined on the basis of a very simple one-dimensional mathematical model. The bay is separated into two parts by a breakwater with a narrow opening and the dynamical condition at the opening of the breakwater is assumed to follow the wellknown relationship in hydraulics between the volume flux through the opening and the difference of water levels across the breakwater.
Several different cases such as tidal oscillations, invasion of tsunamis, and storm surges are considered and the theoretical results are compared with those of model experiments as far as possible. It appears that a simple theoretical model presented here can explain with satisfaction the observed change of wave height at the head of the bay with respect to the change of the period of forcing functions and/or the area of the opening, as long as the opening is small. Thus, the most important factor in this kind of phenomena should be the following non-dimensional parameter:
B ; for tidal oscillations,
[Fig.107_161_01] ; for tsunamis and storm surges.
For details, see§4 and §3respectively.
The amplitude and phase changes of the tidal oscillation inside the bay can be estimated as a function of B from Fig.2. The effect of a breakwater on invading tsunamis of the solitary type is shown in Fig.9. The effectiveness of a breakwater on storm surges is seen in Figs.10-15. From these figures it is evident that the period of the oscillating water body in relation to the incoming wave period or the given period of external forces seems to be the decisive factor in determining the effectiveness of the barrier, in addition to the geometry of the bay, the area of the opening and the wave amplitude. The present theory is applicable to the cases where the opening is sufficiently small so that the nonlinear effects predominate in the motion of water near the opening.

BULLETIN OF THE EARTHQUAKE RESEARCH INSTITUTE Vol.41 (1963) pp.535-571 33. The Leading Waves of a Tsunami. By Kinjiro KAJIURA, Earthquake Research Institute, (Read March 23,1963. -Rcceived June 27,1963.)

Based on the lillear theory, the decay with distance of the first wave of a tsunami in an infinite sea of constant depth is discussed genera!ly. In particular, for the case of a uniform source distributed in a rectangular area (horizontal dimension :major axis 2a, minor axis 2b) the decay is approximately proportional to the following power of the distance r, except for a distance close to the source area (for comparison, the case of one dimensional propagation is also shown).

[Fig.107_162_01]

[Fig.107_162_02], t:time,H:depth,g;acceleration of gravity.
For a wave ln the direction of the minor axes of the source,a should be replaced by b.
The ratio of the leading wave heights in the directions of the major and the minor axes varies from b/a to 1 depending on the values of pa and pb,so that the directivity of the leading wave height due to the elongated source distribution disappears at a very long distance from the source area. The time interval between the leading wave crest and the second one increases in proportion to t^{1/3}.
The present theory can also be extended to the dispersive wave train in the later phases of a tsunami for an arbitrary source distribution.

1. Introduction

Many theoretical discussions have been advanced, mainly in Japan, to explain the generation of tsunami in water of uniform depth when a portion of the bottolm is dislocated vertically.1),2) Most of these linear theories are, however, connned to a rather limited scope of wave properties because of mathematical difficulties confronted in the elucidation of formal solutions. Thus, apart from the numerical approach adopted by several authors,3),4),5),6) common procedures utilized are either the so-called long wave approximation or, at the other limit, the deep water wave approximation, and the analytical discussions including the intermediate range of wave lengths are quite limited because of the transcendental character of the dispersion curve. For a long distance from the source, however, the behavior of a leading wave can be handled analytically by means of an integral analogous to the Airy integral,7),8),9),10) where the approximation involved is essentially similar to the one used to derive the wave equation near the wave front including the effect of curvature of the water surface, and the general characteristics of waves can be estimated from the analysis of this equation.11),12)
However, there seems to be some misunderstanding of the behavior of the leading wave at a long distance from the source. For example, the discussions of the first wave of a tsunami given by Kranzer and Keller(1959),13) and Takahasi (1961) 14) are not valid because of their incorrect application of the ordinary method of the stationary phase up to the wave where the Airy Integral should be considered, and also Van Dorn’s arguments15) on the amplitude decay of the maximum wave with distance is not accepted as evidence of good agreement between theory and observation because he did not compare the corresponding observcd wave at difrerent locations for which the theory predicts the decay law, and besides no distinction was made of the theoretical results for the one-dimensional and two-dimensional dispersion except for the factor r^{1/2}. Similar deficiency of understanding is obvious in Wilson’s discussion 16) of the amplitude decay and period increase of the leading wave.
From a general point of view, this kind of problem is a part of the general theory related to the Cauchy-poisson wave theory concerning the generation and propagation of water waves, and the fundamental properties of linear waves are considered to be well known 17). However, it seems to be worthhwhile to look into the ploblem anew with expression making use of a time dependent Green’s function. By this approach, the dispersive characteristlcs of generated waves and their relation to the nature of the source can be clearly understood.
In particular, this paper deals with the dependence of the dimension of a elevation (or depression) of the water surface on the dimension of a source just after the instantaneous bottom deformation, the distinction between the one and two dimensional decay of the leading wave height at a long distance from the source, and the directional difference of wave height generated by a non-axially symmetric wave source. The limitation of the long wave approximation for the leading wave is also discussed. Finally, the method is extended to treat the dispersive wave train in the later phases of a tsunami for an arbitrary source distribution.


1) Japanese Organization for Tsunami Investigatlon, The annotated bibliography of tsunamis (1962),(mimeographed report).
2) B.W.WILSON, L M. WEBB, and J. A. HENDRICKSON, 「The nature of tsunamis, their gelleration and dlspersion in water of finite depth,」NESCO Tech.Report No.SN 57-2 (1962),146,Appendix 1,2.
3) K.SANO, and K. HASEGAWA,「On the wave produccd by the sudden depression of a small portion of the bottom of a sea of uniform depth,」Bull.Japan Centr.Met. Obsev., 2 (1915),30.
4) T.MATUZAWA,「On the tsunami accompanying the earthqtlake: Pt.1., Two dimensional problem of incompressible water, Pt.2., Numerical comutation of integral,」 Zisin [ii], 1 (1948), 18-23, [ii],2 (1949),33-36, (in Japancse).
5) T.ICHIYE,「A theory on the generation of tsunami is by an impulse at the sea bottom,」Oceanogr. Soc. Japan,14(1958),41-44.
6) loc.cit.,2).
7) K.SEZAWA and K. KANAI,「On the transmission of tsunamis in a sea of any depth,」Bull. Earthq. Res. Inst 20 (1942), 254-264, (in Japanese).
8) R.TAKAHASI,「On the seismic sea waves caused by deformation of the sea bottom,」 Bull. Earthq. Res. Inst. 20 (1942),357-400,(in JapaIlese).
9) C.ECKART,`「The approximate solution of one-dimensional wave equatlons,」Rev. Mod. Phys., 20 (1948), 399-417.
10) loc.cit.,2).
11) G.H. KEULEGAN and G.W.PATTERSON,Mathematical theory of irrotatinal translation waves,」J.Res., Natl.Bureau Stds.,24(1940),47-101.
12) H. JEFFRRYS and B. S. JEFFREYS, Methods of mathematical physics,(Cambridge Press, Cambridge, England,1956),714.
13) H.C.KRANZER and J.B. KELLER,「Water waves produced by explosions,」J.Appl. Phys.,30 (1959),398-407.
14) R.TAKAHASI,「On some model experiments on tsunami generation,」Proc. Tsunami Hydrofynamics Conf., Univ.Hawaii (1961),(publication pending).
15) W・G・VAN DORN,`「Some characteristics of surface gravity waves in the sea produced by nuclear explosions,」J.Geophys. Res.,66 (1961),3815-3862.
16) loc. cit.,2).
17) J.J.ST0KER, Water waves (lnterscience Publishers Inc., New York,1957),569.

2. Fundamental equations and a time-dependent Green’s fuction

Assume incompressible water of constant depth H and take origin of the Cartesian co-ordinate (x’,y’,z’) at the undisturbed free surface with the vertical axis z’ upwards. The irrotational motion in homogencous water may be expressed by means of a velocity potential \psi\ where the velocitoy vector V’ is given by V’=grad \psi\. For convenience in the later discussions, physical quantities are written in nondimensional form unless otherwise noted by putting the independent variables as foliows:

[Fig.107_165_01]
(quantities with ■ime are to be the original form)

where t is time and g is the acccleration due to gravity. The nondimensional form of the derived quantities are:

[Fig.107_165_02]

where \rho\ is density of water, p’ is the anomaly of the surface atmospheric pressure from the mean value, and the vertical component of V is Written as w.
Within the limit of linear approximation (deformations z’ at the surface and at the bottom are assumed small compared with the wave length \lambda\’ and the depth of water H together with the condition [Fig.107_165_03] so that the boundary conditions are satisfied at the undisturbed surfaces), the kinematic and dynamic conditions at the free surface are,

式(1,2) [Fig.107_165_04]

and the bottom condition is,

式(3) [Fig.107_165_05]

where wb is the assumed bottom velocity corresponding to the bottom deformation and the partial differentiations are abbreviated by letter subscripts of the respective variables.
To find a velocity potential \psi\ satisfying (1),(2), and (3) together with suitable initial conditions, it is advantageous to derive a timedependent Green’s function G which should be a harmanic function in the variable (x,y,z) with a singularity of appropriate character at a certain point (xo,y0,z0) introduced at the time t=\time\ Hence,

[Fig.107_166_01]

and G should be a solution of the Laplace equation,

式(4) [Fig.107_166_02]

satisfying the free surface condition

式(5) [Fig.107_166_03]

and a bottom condition

式(6) [Fig.107_166_04]

Atx,y→\infty\, we require G, Gt, and their first derivatives with respect to space coordinates to be uniformly bounded at any given time t. At the point (x0,y0,z0) we require (G-1/R) to be bounded wllere R^{2}=(x-x0)^{2}+(y-y0)^{2}+(z-z0)^{2}. As initial conditions at the time t=\tau\, we assume

式(7) [Fig.107_166_05]

These conditions determine G uniquely.
Following the similar line described by Stoker18), the Green’s function for the case of three dimensional motion in water of finite depth can be derived: for O > z, z0 > -1,

式(8) [Fig.107_166_06]

where r^{2}=(x-x0)^{2}+(y-y0)^{2} and r^{2}=mtanhm. It is evident from (8) that G is symmetric with respect to (x0,y0,z0) and (x,y,z) and also t and \tau\, i. e.:

[Fig.107_166_07]

Making use of the Green’s formula together with the above Green’s function, we may write

式(9) [Fig.107_167_01]

where dS0=dx0dy0 and integral on the lateral boundary in water vanishes
because of the condition imposed on G.
The integration of (9) with respect to \tau\ from 0 to t and the substitution of thc conditions (1),(2),(5),(6), and (7) for \psi\ and G yield,

式(10) [Fig.107_167_02]

The first integral shows the contribution from the surface condition and the second one from the bottom condition.
At the surface z=0,(8) is simplified to give

式(11) [Fig.107_167_03]

and since the surface elevation \eta\ is given by (2), the substitution of (10) into (2) yields

式(12) [Fig.107_167_04]

where

式(13) [Fig.107_167_05]

and

式(14) [Fig.107_167_06]

or alternately,

式(15) [Fig.107_167_07]

and

式(16) [Fig.107_167_08]

or

式(17) [Fig.107_168_01]

It is clear that (13) is the contribution of the initial velocity and elevation of water at the surface,(14) or(15) is the contribution of the the surface pressure, and (16) or (17) is the contribution from the bottom deformation.
The expression (12) together with (13) to (17) is quite general and in essence, represents the application of the principle of superposition in a linear system starting from a point source solution. Since G can be computed without regard to external conditions at the source, it may be quite suitable for numerical computation.
For certain special source conditions with respect to \tau\, (13) to (17) can be simplified as follows with the aid of the Dirac’s Delta function \delta\ (\tau\)19):

(a) water is initially at rest with initial elevation Hs;

式(18) [Fig.107_168_02]

(b) water is initially at rest and applied pressure at the surface is impulsive at \tau\=0+, namely p=Is\delta\(\tau\);

式(19) [Fig.107_168_03]

(c) water is initially at rest and the deformation of the bottom is completed instantaneously at \tau\=0+ with the total deformation HB, namely wB=HB\delta\(\tau\);

式(20) [Fig.107_168_04]

(d) water is initially at rest and uniform velocity of bottom deformation is given for a time interval 0 ＜ \tau\ ＜ \tau\*,with the total deformation HB,namely [Fig.107_168_05];

式(21) [Fig.107_168_06]

(e) water is initially at rest and impulsive bottom motion is given at \tau\=0+ with no net deformation of the bottom after \tau\>0, namely

[Fig.107_168_07]

where IB is the maximum deformation of the bottom;
After slight modification of (17), we have

式(22) [Fig.107_169_01]

It may be possible to represent other kinds of source conditions in simplified forms too; for exarnple, an atomospheric pressure disturbance such as a pressure jump line moving with the constant velocity can be simplified, but further discusslons will not be attempted here.
The comparison of (a) to (e) shows that the instantaneous deformation of the bottom (c) is analogous to a given initial elevation (a), and the bottom impulse (e) can be treated as a surface pressure impulse (b).
The difrerence lies only in the evaluation of the Green’s function G at the surface (z0=0) or at the bottom (z0=-1) which amounts to the decrease of high frequency components by a factor 1/cosh m for the bottom source. As will be shown later in the evaluation of the Green’s function, the leading wave form for a large distance from the source is determined mainly by component waves of very low frequencies \gamma\ (\gamma\ ＜ \gamma\* ＜ 1) so that 1/cosh m is almost one and (a) and (c); (d) and (e) become identical and furthermore,(d) is reduced to (c) provided \gamma\* \tau\* ＜＜ 1.
For simplicity, We write,

式(23-26) [Fig.107_169_02]

for the cases of initial elevation Hs, initial impulse Is, sudden elevation of the bottom HB, uniform velocity of the bottom deformation HB/ \tau\*, respectively. The functions P,Q,R,and S are considered to be water waves generated by point sources of the particular characters at \tau\=O and are given by evaluating the corresponding form of the Green’s function (11) as follows:

式(27) [Fig.107_169_03]

式(28) [Fig.107_170_01]

式(29) [Fig.107_170_02]

式(30) [Fig.107_170_03]

where r=(x-x0)^{2}+(y-y0)^{2},and for S(\tau\=\tau\*), t should be replaced by t(=t-\tau\*).
The central problem of the wave generation theory in the present formulation is to evaluate P,Q, R,and S as well as possible. Now, returning from (23) to (26), it is easy to show that these expressions comform to the usual expressions derived on the basis of the FourierBessel transform of a potential function and source conditions from the beginning (Appendix I). The advantage of the present formulation may be seen in the separation of the source distribution and the wave dispersion characteristics of the medium.
The Green’s function G*in the two-dimensional motion (x, z plane) can be derived essentially by the similar method in which the singularity imposed on a fluid should be of the type such as (G*-lnR*) to be bounded. In (8), J0(mr) is then replaced by (cosmx)/m where x^{2}=(x-x0)^{2}. In the application of the Green’s formula, in (9), the line integral should be taken instead of the surface integral and 1/(4\pi\) is replaced by 1/(2\pi\). The subsequent deduction is the same and the result can be expressed by

式(31-34) [Fig.107_170_04]

where

式(35) [Fig.107_170_05]

式(36) [Fig.107_170_06]

式(37) [Fig.107_170_07]

式(38) [Fig.107_170_08]

and for S*( \tau\= \tau\*), t should be replaced by t(=t-\tau\*). It is remarked that (35) to (38) can also be derived from (27) to (30) by integration with respect to y0 from -\infty\ to +\infty\ with the aid of a formula,

[Fig.107_171_01]

19) I.N.SNEDDON, Fourier Transform (McGraw Hill Book Co., Inc., New York 1951), 542.

3. Surface elevation at the initial time (t→0) for the case of a sudden deformation of the bottom

For t→0,(29) may be witten as

[Fig.107_171_02]

The expansion of 1/coshm in terms of exponential functions and the integration term by term yield,

式(39) [Fig.107_171_03]

In (39), n can be interpreted as the amount of reflection of the source disturbance at the surface and at the bottom, and the summation of refiection for infinitely many times contributes to the total deformation of the surface.
For one dimensional waves,(37) becomes,

式(40) [Fig.107_172_01]

The initial surface elevations (40) and (39) due to a point source in one and two dimensional waves are shown in Fig.1, which indicates that the elevation is extended over the distance comparable to the depth of water, and no clear-cut wave front is formed.
Talking the origin of the polar-coordinate at the center of an axially symmetric bottom deformation HB, the elevation of the water surface given by (25) at t=0+,r=0 becomes

式(41) [Fig.107_172_02]

For one dimensional waves, the combination of (33) and (40) gives

式(42) [Fig.107_172_03]

particular examples are given below:
α) The uniform deformation of the circular area of the bottom;

式(43) [Fig.107_172_04]

(41) becomes

式(44) [Fig.107_172_05]

and for very small values of a (a ＜＜ 1),

[Fig.107_172_06]

b) The parabolic deformation of the circular area of the bottom;

式(45) [Fig.107_172_07]

(41) becomes

式(46) [Fig.107_173_01]

where \eta\1/HB is identical to (44) and

式(47) [Fig.107_173_02]

For very small values of a (a ＜＜ 1),

[Fig.107_173_03]

so that

[Fig.107_173_04]

c)The uniform deformation of the bottom (one dimensional propagation);

式(48) [Fig.107_173_05]

(42) becomes

式(49) [Fig.107_173_06]

and for small values of a (a ＜＜ 1),

[Fig.107_173_07]

The initial elevation of water surface at the center of the bottom deformation,(44),(46)and (49), is shown in Fig.2, from which it is found that the elevation of the surface at the center rcaches the height of the bottom deformation if the radius of the deformation is about 3 to 4 times the depth of water. On the other hand, for small scale deformation the surface elevation is proportional to the volume of the bottom deformation.

4. Approximate evaluation of the Green’s function

a)Shallow water waves (m ＜＜ 1, \gamma\=m):
In shallow water, it is usual to adopt the long wave approximation which assumes that the pressure in water is hydrostatic. In the present formulation, the assumption corresponding to the long wave is m ＜＜ 1 and \gamma\ =m.
The evalution of the Green’s function for this case is rather easy
and the results are,

式(50) [Fig.107_174_01]

and

式(51) [Fig.107_174_02]

If we shart from the beginning on the assumption of a long wave by expanding \psi\ in power series of z and retaining the first order terms only, we arrive at a two dimensional long wave equation with respect
to \zeta\,

式(52) [Fig.107_174_03]

where w is the bottom velocity,\zeta\=\eta\+p, and [Fig.107_174_04]. It should be noticed that S given by (50), where t is replaced by t (=t-\tau\), is nothing but a Green’s function of the wave equation (52) and can be used with advantage for the studies of storm surges20) and tsunamis21).
Near the wave front for a long distance from the source, we may approximate

[Fig.107_175_01]

so that (50) and (51) are reduced to

式(53) [Fig.107_175_02]

式(54) [Fig.107_175_03]

and

式(55) [Fig.107_175_04]

These expressions show that, for a fixed value of (t-r),P,Q,R, and S are all proportional to r^{-1/2} irrespective of the source characteristics and at the front, t=r, the degree of singularity increases from S to P or R and P to Q. However, for a long distance from the source, the usual long wave approximation presented here is not valid clear the wave front because the curvature of the water surface plays a role in the dispersion. In other words, it is necessary to assume \gamma\=m-m^{3}/6 in the neighborhood of r/t \simeq\ 1 and t >> 1.
For one dimensional waves, the same long wave assumption gives

式(56) [Fig.107_175_05]

and

式(57) [Fig.107_175_06]

(56) indicates that one half of the initial surface elevation moves without change of form in the positive and the negative directions respectively.

b) Wave form near the wave front (m ＜ 1, \gamma\=m-m^{3}/6):
For a long distance from the source,(30) may be replaced by

[Fig.107_175_07]

and then transformed into

式(58) [Fig.107_175_08]

It can be shown that the use of the asymptotic expression for J0(mr) results in the error of the order r^{-1} for S. Now, since r, t and \gamma\, m, are positive, the contribution of sin(\gamma\t +m\gamma\-\pi\/4) term to the integral for a long distance is of the order r^{-1}. Thus,(58) may be approximated by

式(59) [Fig.107_176_01]

For the leading wave of a tsunami (r/t \simeq\ 1) for a long distance from the source, it can be shown that the main contribution to S comes from small m so that it is possible to assume \gamma\=m-m^{3}/6 and sinh2m \simeq\ 2m.
Therefore, by replacing m^{3}t/6=u^{6},(59) becomes,

式(60) [Fig.107_176_02]

and

式(61) [Fig.107_176_03]

where Re[z] means the real part of z and

[Fig.107_176_04]

If we neglect u^{6} term in the power of the exponential in (61), we have,

[Fig.107_176_05]

and S is reduced to (53).
Since t is large and the variation of (6/t)^{1/3} with respect to t is small, we may replace the derivative with respect to t or r by the derivative with respect to p, and we may write,

式(62) [Fig.107_176_06]

and

式(63) [Fig.107_176_07]

For one dimensional waves, the asymptotic expression for R* can be derived parallel to the case of two dimensional waves. Thus, for large positive values of t and x,(37) is reduced to

式(64) [Fig.107_176_08]

putting q=p/3=(1/3)(6/t)^{1/3} (x-t), and m=(6/t)^{1/3}v, we have

[Fig.107_177_01]

The integral cal be identified as an Airy Integral Ci(q) where

[Fig.107_177_02]

and

[Fig.107_177_03]

Here, K_{1/3}(x) is a modified Bessel function and J_{1/3}(x),J_{-1/3}(x) are Bessel functions22). (64) is essentially similar to the integral discussed by Eckart23), and Hendrickson24), who treated the asymptotic behavior of the leading wave of the one dimensional tsunami generated by an initial elevation and a bottom deformation of the small scale respectively.
For convenience in later discussions, we define T*(p),

式(65) [Fig.107_177_04]

Then, it follows,

式(66) [Fig.107_177_05]

式(67) [Fig.107_177_06]

and

式(69) [Fig.107_177_07]

Comparing S and S*,R and R*(P and P*), Q and Q*, it is evident that, even if we take the factor r^{-1/2} due to geometrical spreading effect out of consideration, the decay law of amplitude with time for the leading wage is different for the one dimensional and two dimensional waves. Furthermore, the decay laws of the leading waves for the initial elevation and initial impulse are different. A mathematical reason for these differences lies in the fact that, in the evaluation of the integral, the factor in front of the oscillatory term, say, the amplitude spectrum for small values of m plays a deciding role.
For example, the decay law of the leading wave amplitude for a long distance from the source of the type of initial elevation or the sudden deformation of the bottom; namely the variation of the first maximum values of R and R* are (rt)^{-1/2} and t^{-1/3} respectively. The decay law of (rt)^{-1/2} for two dimensional waves was first noticed by Takahasi 25) but was somehow abandoned in his later paper26), and the decay law of t^{-1/3} for one dimensional waves is taken for granted for two dimensional waves as well 27),28), except for the factor r^{-1/2}. In the Appendix II, the decay law for one dimensional waves is derived by means of the superposition of two dimensional waves to show the difference of the decay laws clearly.
For the leading wave generated by a surface impulse, the amplitude decays proportionally to r^{-1/2}t^{-5/6} and t^{-2/3} for the two dimensional and one dimensional cases, respectively.
In both one and two dimensional waves, the wave form of the leading part of the wave train is completely determined by a parameter p so that, with the increase of t, the crest of the wave is retarded with respect to the reference point moving with the velocity of ordinary long wave [Fig.107_178_01], by a factor t^{1/3} and the surging part in front of this crest spreads outward. Furthermore, the time interval between the first crest and the second one increases proportionally to t^{1/3}.
T,Tp, end Tpp, for two dimensional waves and T*,Tp*, and T*pp for one dimensional waves are shown as a function of p in Fig.3 and Fig.
4 respectively. Numerical values of T(p) are origillally given by Takahasi 29), but the re-computation is carried out by a different method.
The result shows a, slight modification of the Takahasi’s values. Apart from the decay factor already mentioned, the wave forms for the two dimensional waves and the one dimensional waves are qualitatively similar provided the position of the first maximum is a little later for the one dimensional wave than for the two dimensional wave. As will be shown in the later section, the wave form for the leading wave of a tsunami at a very Iong distance from the source is considered to be well represented by T, and Tp* for the cases of the initial surface elevation or the bottom deformation and by Tpp and Tpp* for the case of the initial surface impulse. From the figures, it is found that for the one dimensional wave started from the initial elevation or the bottom deformation (Tp*), the leading wave has the maximum height for a very long distance from the source, but for waves started from the initial impulse (Tpp*), this is not the case, and in many cases of practical interest, the leading wave may not be recognized because of its low amplitude and long wave length. For the two dimensional wave (Tp, Tpp), the leading wave may not be the wave with the maximum height for very long distances from the source area, and the second or the later crest will have the maximum height. The intervals between the first maximum and the second one are the same for all Tp,Tp*,Tpp,Tpp*, and the numericcal value in terms of p is 5.7.
Qualitatively similar conclusions concerning the change of the wave form can be obtained on the basis of the one dimensional long wave equation including the effect of curvature of the water surface. According to Keulegan and Patterson 30), the velocity \omega\ of propagation of an element of volume of an intumescence satisfies the relation,

[Fig.107_181_01]

and the wave equation including the second order terms is given by

[Fig.107_181_02]

provided (3/2)\eta^{2}\ ＜＜ (1/3)\eta\xx or [Fig.107_181_03] (\eta\0:wave amplitude,\lambda\’:wave length, H:depth of water). This wave equation is essentially similar to the assumption \gamma\=m-m^{3}/6. The combination of these two equations and the integration with respect to x give,

[Fig.107_181_04]

From this relation for the velocity of propagation of a volume element of an intumesence moving in still water(say \eta\ > 0), we can easily find that the wave front moves faster than the「long wave velocity」 and the first crest (maximum point) is retarded with respect to the point of inflection where \eta\xx=0 and \omega\=1 as shown graphically in Fig.5.
As for the time interval between the first crest and the second, the different approach made by Munk 31) about the period increase of the conservative waves in general may be applicable. A simple solution for the tsunami given by Munk can be further simplified for the case of a constant depth as follows:

[Fig.107_182_01]

and

[Fig.107_182_02]

where \tau\ is the wave period at the time t and the distance x (all quantities are in the non-dimensional form), and a is an arbitrary constant. For a long distance from the source, the time interval between the first and the second crests is approximately equal to t-x and also to the wave period \tau\, so that we may put \tau\ \simeq\ t -x. Thus we have

[Fig.107_182_03]

Furthermore, since x/t \simeq\ 1,we can conclude that the period increase is proportional to t^{1/3}.
The effect of finite duration of the deforming motion at the bottom can be examined by means of (26), which may be written as

[Fig.107_182_04]

where

[Fig.107_182_05]

In terms of p,is replaced by

[Fig.107_182_06]

where

[Fig.107_182_07]

Judging from Fig.4, the variation of T with respect to p in the neighborhood of (Tp)max is almost linear in the interval (p-p0) ＜ 1, so that

[Fig.107_182_08]

Thus the assumption of the instantaneous deformation is valid in the evaluation of the first crest for the deformation with the time interval \tau\* provided (6/t)^{1/3}\tau\* ＜ 1.For one dimensional waves too, the same condition approximately holds.

c) The wave train in the later phase:
The asymptotic solution of (30) or (56) within the limit of applicability of the stationary phase method is given by,

式(70) [Fig.107_183_01]

where

[Fig.107_183_02]

and

式(71) [Fig.107_183_03]

In (71), primes of \gamma\0 indicate differentiation with respect to m and the sufffix 0 shows the values to be evaluated at m=m0, which is a real root of the equation, \gamma\’= \gamma\/t.
Neglecting the time derivative of the slowly varying amplitude compared with that of the carrier wave, we have by differentiating S with respect to t,

式(72) [Fig.107_183_04]

式(73) [Fig.107_183_05]

and

式(74) [Fig.107_183_06]

For one dimensional waves,(38) is reduced to

式(75) [Fig.107_183_07]

where

式(76) [Fig.107_183_08]

and the other functions can be derived straight-forwardly:

式(77) [Fig.107_183_09]

式(78) [Fig.107_183_10]

and

式(79) [Fig.107_183_11]

For the case of deep water waves, i. e. when the wave number m is much larger than unity and \gamma\ is approximated by m^{1/2}, the asymptotic solutions are reduced to well known formulas where

式(80) [Fig.107_184_01]

式(81) [Fig.107_184_02]

式(82) [Fig.107_184_03]

式(83) [Fig.107_184_04]

The frequency and wave number of the individual wave is fixed by \gamma\0=t/(2r), and m0=t^{2}/(4r^{2}) respectively. The decay laws with distance of the amplitude for a fixed wave number are r^{-1} and x^{-1/2} for the two dimensional and one dimensional waves, respectively.
In both one and two dimensional waves, it is evident that for a fixed location the amplitudes of P and Q increase with time but Q increases more rapidly than P because Q=P, and for Q weight is placed on high frequency components which arrive at the fixed location later.
In contrast, R has a maximum at some intermediate time, because, for very high frequency components of waves, the factor 1/cosh m0 dominates and the amplitudes of high frequency components are surpressed by the factor exp(-m0).

20) K.KAJIURA,「A theoretical and empirical study of storm induced water level anomalies,」 Tech. Report,Ref.59-23F,Dep’t of Ocn. and Met., Texas A and M,(1959), 97.
21) l.N. SRETENSKY and A.S. STAVROVSKY,「Computation of the height of tsunami waves along the coast,」Trans. Marine Hydro. Inst.,Acad.Sci.USSR,24(1961),23-43. (Transl. scripta technica, inc., for the A.G.U.)
22) S.MORIGUCHI, et al,Mathemalical Formulas, III (lwanami Book Co., ltd.. Tokyo,1959),231-232,(in Japanese).
23) loc. cit.,9).
24) loc. cit.,2) Appendix 2.
25) loc. cit.,8).
26) loc. cit.,14).
27) loc. cit.,15).
28) loc. cit.,2).
29) loc. cit.,8) Fig.13.
30) loc. cit.,11).
31) W.H.MUNK,「Increase in period of waves travelling over large distance, with application to tsunamis, swell and seismic surface waves,」Trans. Amer. Geopys. Un.,28 (1947),198-217.

5.Leading wave of a tsunami

Consider the leading wave of a tsunami for a long distance from the source area enclosed by -a ＜ x0 ＜ a, and -b ＜ y0 ＜ b where waves may be generated either by the deformation of the bottom or by the initial elevation of the surface. Taking the origin of the co-ordinate (x, y) at the center of the source area, we may assume [Fig.107_184_05] where

[Fig.107_184_06]

Geometry of the source point
(x0, y0) and the observing point(x, y) is illustrated in Fig.6, where, for simplicity, the observing point is placed on the x-axis, and \xi\ is taken as the distance of the observing point relative to the moving reference point where x=t. Assuming r \simeq\ x, we may write

式(84) [Fig.107_184_07]

where

[Fig.107_185_01]

Thus, p is independent of y0 for an observing point on the x-axis.
From (25) together with (62), waves generated by the bottom deformation are given by

式(85) [Fig.107_185_02]

Since Tp is independent of y0 for a long distance in the x-direction from the source, we may rewrite (85) with the aid of (84) into the form:

式(86) [Fig.107_185_03]

where

[Fig.107_185_04]

In general, we may put HB=O at the outer edge of the source area, p0=pa and -pa, so that the partial integration of (86) gives,

式(87) [Fig.107_185_05]

where

式(88) [Fig.107_185_06]

If we introduce an explicit distribution of HB, U(p*) can be computed as a function of pa and p* and the surface elevation \eta\ is determined from (87).
For small values of pa, Tp does not change significantly near the maximum of -Tp so that we may approximate (86) by

式(89) [Fig.107_185_07]

where

[Fig.107_185_08]

(89) shows that the shape of the source does not affect the height of the crest or trough of the leading wave for a very long distance from the source (pa ＜＜ 1). The wave height is proportional to the total volume of the original deformation of the bottom and decays approximately proportional to (rt)^{-1/2} which is already expected from the analysis of R. If the total volume V of the deformation of the bottom is zero, the leading wave at a very long distance from the origin decays inversely proportional to a higher power of the time than 1/2.
For one dimensional waves, we may derive similar equations. From (33) together with (66), the wave generated by bottom deformation is given by

式(90) [Fig.107_186_01]

and (90) may be transformed into

式(91) [Fig.107_186_02]

since

[Fig.107_186_03]

For very small values of pa,(90) is reduced to

式(92) [Fig.107_186_04]

where

[Fig.107_186_05]

particular examples are given in the followings:

a) The uniform deformaction of a rectangulacrarea of the bottom;

式(93) [Fig.107_186_06]

For waves in the x-direction, we may put

[Fig.107_186_07]

so that

[Fig.107_186_08]

(89) gives,

式(94) [Fig.107_186_09]

For small values of pa,

式(95) [Fig.107_186_10]

and for very large values of pa,

式(96) [Fig.107_187_01]

since T(p*+pa)→0 for large values of p*+pa. Here,(-Tp)max is located at p*=-0.75 and Tmax is located at p*=pa-2.3.

b) The elliptic deformation of an elliptic area of the bottom;

式(97) [Fig.107_187_02]

within the region enclosed by

[Fig.107_187_03]

For waves in the x-direction, we may write,

[Fig.107_187_04]

so that

[Fig.107_187_05]

(89) gives

式(98) [Fig.107_187_06]

where \alpha\ = p0/pa and T=T(p*-pa\alpha\).
For small values of pa, we have

式(99) [Fig.107_187_07]

and for large values of pa, we cannot reduce (98) into a simple relation as in the case of a rectangular source.

c) The uniform deformation of the bottom (one dimensional propagation);

式(100) [Fig.107_187_08]

We may write,

[Fig.107_187_09]

and

式(101) [Fig.107_187_10]

For the one demensional leading wave according to (101), \eta_{max}\/HB and p* for which \eta_{max}\ is attained are shown as a function of pa in Fig.7, which indicates that \eta_{max}\/HB is considered to be a linear function of pa for small values of pa,say, for pa ＜ 1, so that the leading wave amplitude varies proportional to a and t^{-1/3}. On the other hand, for large values of pa, say for pa ＞ 3, the leading wave height reaches 0.635HB and does not change with the scale a and the time t. The time interval between the reference point (moving with the long wave velocity √gH from the origin) and the arrival time of the leading wave crest given by p* implies that the crest is retarded with respect to the reference point as the wave travels further, showing the velocity of the crest movement to be smaller than　√gH.
For the two dimensional Ieading waves according to (94) and (98), Umax/(2bHB) and p* for with U becomes maximum are shown in Fig.
8,which indicates a trend with respect to pa which is qualitatively similar to the case of the one dimensional wave. As for the directional difference of the leading wave height, the ratio of the leading wave heights in the x- and y-directions for the same time and the distance is given by

式(102) [Fig.107_189_01]

Thus, it is easy to compute the ratio from Fig.8 if pa and pb are known.
For large values of pa and pb, say pa and pb > 3, the leading wave amplitude decreases as r^{-1/2}t^{-1/6} and the amplitude ratio approaches approximately to b/a. On the other hand, for small values of pa and pb, say pa and pb ＜ 1, the amplitude decreases as (rt)^{-1/2} and the directional difference of the leading wave amplitude disappears.
The leading wave generated by the surface impulse can be discussed along a similar line of argument and the surface elevation derived from (24) and (63) is given by

式(103) [Fig.107_189_02]

where

[Fig.107_189_03]

and for small values of pa, we have

式(104) [Fig.107_189_04]

where

[Fig.107_189_05]

For the one dimensional case, the elevation may be derived from (32) and (67)as follows:

式(105) [Fig.107_189_06]

and for small values of pa, we have

式(106) [Fig.107_190_01]

where

[Fig.107_190_02]

Thus the decay of the leading wage height for the case of the surface impulse is greater by the factor t^{-1/3} than that for the case of the initial surface elevation or the sudden deformation of the bottom. However, the directional difference of the leading wave heights follows a rule similar to the case of the initial surface elevation.

6.Dispersion of a wave train in the later phase

Making use of the approximate representation of the Green’s function, it is straight-forward to derive the expression of a wave train in the later phase, originating from an extended source area (Ix0I ＜ a,Iy0l ＜ b).
For a large distance from the source,([Fig.107_190_03]), we may approximate

[Fig.107_190_04]

Therefore, the substitution of (72) into (23), which represents the waves started from an initial surface elevation HS, yields,

式(107) [Fig.107_190_05]

For the special case of an axially symmetric source, HS \equiv\ HS(r0), it is easy to show that

式(108) [Fig.107_190_06]

where

[Fig.107_190_07]

In the derivation of (108), the following formulas are used:

[Fig.107_190_08]

and

[Fig.107_191_01]

In a similar way, the waves originating from the axially symmetric surface impulse, Is(r0), are given by

式(109) [Fig.107_191_02]

where

[Fig.107_191_03]

(108) and (109) are identical to the solutions given by Kranzer and Keller 32) who gave the detailed discussions of the wave characteristics derived from these solutions. However, it may be remarked that their conclusions related to the leading wave should be understood with some reservations because the leading wave should be treated as an asymptotic solution of the Airy phase.
It is noticed by examining (107) that for a source function different from axial symmetry, the computation of the elevation becomes complicated in the polar co-ordinate. On the other hand, in the rectangular co-ordinate for \eta\ in the x-direction,(107) can be integrated with respect to y0 without regard to the sine or cosine term since r0cos\theta\0 may be replaced by x0.
Hence,(107) may be written as

式(110) [Fig.107_191_04]

where

[Fig.107_191_05]

The integrals in the right hand side of (110) are nothing but a Fourier cosine and sine transform of W, and represent the amplitude modulation for the carrier wave. For\eta\ in the y-direction, the integration with respect to x can be performed first irrespective of the sine or cosine term slnce we may put r0cos\theta\0=y0.
Some specific examples are given below:

a) The uniform initial elevation of a rectangular area of the surface;

式(111) [Fig.107_192_01]

We have

式(112) [Fig.107_192_02]

Replacing x by y and a by b, the elevation in the y-direction can be obtained,

式(113) [Fig.107_192_03]

Thus, for the amplitude of the carrier wave, the ratio in the x-and the y-directions is

式(114) [Fig.107_192_04]

If we follow the same wave length m0, the amplitude ratio becomes (bsinm0a)/(asinm0b). In particular, if a and b are very small so that m0a, m0b ＜＜ 1, the directional difference of wave amplitudes in the x-and y-directions disappears. For deep water waves (\gamma\ \simeq\ m^{1/2}) and for the same distance x=y, the ratio of the maximums of the modulation amplitudes in the x-and y-directions becomes b/a and the arrival times of the corresponding maximum amplitudes are given by

[Fig.107_192_05]

Therefore, in the direction of the shorter axis, say in the y-direction if b ＜ a, the arrival of the modulation maximum is later and the wave length of the carrier wave of the maximum amplitude is smaller than those in the x-direction. In other words, the modulation has a larger amplitude and wave length in the direction of the shorter axis than those in the longer axis.

b) The uniform deformation of an elliptic area of the bottom;

Hs=constant.

within the area enclosed by

式(115) [Fig.107_192_06]

We have,

式(116) [Fig.107_193_01]

and the substitution of (116) into (110) yields

式(117) [Fig.107_193_02]

where the following formula is used:

[Fig.107_193_03]

In the y-direction, the wave train is given by

式(118) [Fig.107_193_04]

It is shown that the above expression is reduced to (108) if a=b, because the Hankel Transform of the circular source with constant HS is given by

[Fig.107_193_05]

Now, the ratio of the amplitude of the carrier waves is given by

式(119) [Fig.107_193_06]

This expression is qualitativery similar to the case of a rectangular source provided the sine term is now replaced by the Bessel function. Therefore the qualitative conclusions regarding the difference of the modulation in the x-and the y-directions are similar.
Similar treatments may be applicable to the cases of bottom deformation or with the initial impulse at the surface. For the case of bottom deformation, the result is obtained simply by replacing HS by HB and adding the factor exp(-m0). For the case of the impulsive generation of waves, the result is obtained by replacing HS by Is, changing the carrier waves from cosine to sine, and multiplying the factor \gamma\0 (\simeq\ m0^{1/2}).

32) loc. cit., 13).

7.Comparison with experimental data

a) One dimensional propagation
Prins 33) investigated waves generated by an initial local elevation or depression of uniform height in a two dimensional model. Within the range of H’sa’^{2}/H^{3} ＜ 1 (H’s:the initial height, a’:the half length of the elevation, H:the depth of water), the leading wave height \eta_{max}\’ is found to be proportional to H’s and the proportionality factor may be read off from Fig.7 of his paper.

Depth of water H: 2.3’
Elevation H’s : 0.1’,0.2’,0.3’

Table 1 shows the comparison between the results of the theory and the observation and it may be said that good agreement is obtained for the distance x’=25’. For x’=5’, the agreement is not as good as for x’= 25’,because the approximation made in the theory is not accurate for a short distance from the source.

b)Two dimensional propagation
Takahasi and Hatori 34) carried out a model experiment for the generation of waves by the deformation of a bottom portion of the elliptic shape. The conditions of the experiment are:

Depth H: 5cm and 17.3cm
Dimension of the source:2a’=90 cm,2b’=30 cm

The final form of the displaced bottom surface is approximately parabolic with the maximum height HB0 at the center, and the duration of the bottom motion is of the order of 0.1 sec. (the bottom motion is simulated by the deformation of a rubber memberance). The results which are of interest in the present discussion are shown in Table 2.
The theoretical values for the maximum elevation at the center are estimated approximately by taking (a’+b’)/2 as the representative diameter of the deformation of the type 2b in Fig.2. The observed values for the ratio of the leading wave heights in the directions of the major and the minor axes are the average over the distance 1m to 4m but the theoretical values are computed for the distance of 4m on the basis of Fig.8 curve Ilb.
The observed elevation at the center is about half of the theoretical value and the ratio of the observed wave heights in the major and minor axes for the distance of 4rn is larger than the theoretical values. The reason for this disagreement between theory and observation may lie partly in the inadequate representation of the theoretical model for the actual experimental model conditions, particularly with respect to the time dependence of the bottom motion, and partly in the uncertainty of the model date, which show considerable scattering when several simulations are recorded.

33) J.E. PRINS, 「Characteristics of waves generated by a local disturbance,」Trans. Amer. Geophys. Union,39 (1956),865-874.
34) R.TAKAHASI, and T. HATORI, 「A model experiment of the tsunami generation from a bottom deformation area of elliptic shape,」Bull. Earthq. Res. Inst.,40 (1962),873-883, (in Japanese).

8.Concluding remark

The theory developed in the present paper may be severely restricted in application from the practical point of view, because of the various assumptions made in the course of study, such as 1) the linear approximation in the equations,2) constant depth and no lateral boundary,3) the leading wave at long distances from the source,4) time dependence of the source to be of the Delta function type. Since the area of the tsunami generation lies mainly on the continental slope along the pacific Ocean, the assumption (2) should be removed to have a little more realistic picture of the tsunami. The coastal boundary and the continental shelf produce reflected waves and waves travelling along the boundary, so that the wave-train of the tsunami observed along the open coast will be quite different from that expected from the theory with no boundary.
Moreover, the irregularities of the ocean bottom and the existence of islands would change the wave-form significantly during the course of propagation by refraction and diffraction. Unless these factors are adequately taken into account, it seems impossible to understand the wave train of a tsunami clearly. For example, the prevalent period of about one hour observed along the coast of Japan at the time of the tsunami of the Chilean Earthquake cannot be considered to be of local origin, say the oscillation of the bay and the shelf nearby, but of some distant origin. However, it is unlikely that the dispersion is responsible for the wave train of this long period. The reflected waves along the coast of South America and North America, the boundary waves propagated along the coasts and/or the interference of the refracted waves along the course of propagation of the direct waves might be the cause.
As for the assumptions (3) and (4), the numerical computation with the aid of an electronic computor would remove the restriction. Here, the application of the principle of superposition would be very useful to examine various cases of the source condition.
Lastly, the assumption (1) may be justified for the ordinary cases when the deformation of the bottom is not so large compared with the depth of water and also the lateral extent of the source. However, in shallow seas, the condition may not be satisfied and the non-linear effect comes into play. At the coast, the run-up of the tsunami should be discussed separately.

Acknowledgement

The author wishes to express his appreciation of Prof. Takahasi’s enthusiasm for tsunami research which has stimulated his interest in tsunamis and the undertaking of the present study. He also thanks Miss H. Kamisato for her help in various phases of the study.

Appendix I

For example,(23) may be integrated first with respect to dS0 as follows:

式(I-1) [Fig.107_196_01]

Now, since r^{2} = r^{2} +r0^{2}+rr0cosθ0, where (r,0) are the co-ordinates of the point in question and (r0,\theta0\) those of the source point, in the polar co-ordinate with the origin at the center of the source area, we may expand

式(I-2) [Fig.107_197_01]

Thus,(I-1) is reduced to

式(I-3) [Fig.107_197_02]

where

式(I-4) [Fig.107_197_03]

If Hs(r0,\theta\0 \equiv\ Hs(r0):namely for an axially symmetric source,it is easily shown that

式(I-5) [Fig.107_197_04]

and

式(I-6) [Fig.107_197_05]

Appendix II

From the definition of R* and R,it is evident that

式(II-1) [Fig.107_197_06]

where x0 is put zero for simplicity and y0^{2}=r^{2}-x^{2}.
Writing r and x in terms of p and p* for given time t;

式(II-2) [Fig.107_197_07]

we have

式(II-3) [Fig.107_197_08]

and

式(II-4) [Fig.107_197_09]

Since R is very small for large positive values of p and the contribution of R to R*is confined to small values of p only, we may put Approximately,

式(II-5) [Fig.107_197_10]

And the substitution of (II-4),(II-5), and (62) into (II-1) yields,

式(II-6) [Fig.107_198_01]

Thus, R* is proportional to t^{-1/3}.

33.津波の第一波について (津波の発生,伝播に関する古典的理論) 地震研究所梶浦欣二郎

　線型近似の範囲内において,海面あるいは海底に与えられたある種の外的じよう乱に原因してお
こる水の波の問題は古くから研究されているが,ここでは時間に関係する型のグリーソ函数を利用
して,一様な深さの海におこる波の問題を統一的に考える.特に外的じよう乱が時間的に簡単な形
で与えられる場合(瞬間的な変動)には波の表現が簡単になり,波の性質を波源の特性(初期水位，
瞬間的な衝撃,瞬間的な海底変動等)と波源のひろがりによるものとに分けて考えることができる.
　今 rを波源中心からの水平距離,tを経過時間とすると,波源から極めて遠方における第一波の
減衰は次の量に比例する.

[Fig.107_198_02]

　これでみて判る通り,第一波については,一次元・二次元という伝播の相異,および波源の特性の相異が減衰にも関係している.これに反して,第一波と第二波との峯の時間間隔はすぺてt^{1/3}に比例して伸びる.
　一方,波源のひろがりの影響については,2aを波源の大ぎさとすると pa =(6/t)^{1/3}a (tおよびαは水深Hおよび重力加速度gを用いて無次元化した量)というパラメータが重要となり,paの大小によつて第一波の減衰の有様が異なる.pa ＜ 1では,波は波源から極めて遠方と考えることができて,減衰は上述の通りであるが,pa ＞ 3では波源にかなり近いところとなり,減哀は次のようになる.

[Fig.107_198_03]

　今,長軸2a,短軸2bという矩形の波源から出る波についてその第一波の波高の方向性を調ぺると,近距離では長軸方向の波高は短軸方向の波高の b/a であるが,遠距離では差がなくなる.
　定常位相法の利用できる後続波については,円形以外の波源からのものも簡単に調ぺることができて,たとえば楕円波源の場合には,波高のモジュレーションの長さおよび最大振巾は長軸方向よりも短軸方向の方が大きくなることが示される.
　得られた結果を実験値と比較すると,一次元伝播の場合には一致が極めて良いが,二次元伝播の場合にはそれほど良くない.

Physical processes Associated with the Kuroshio By Kinjiro KAJIURA Earthquake Reseach lnstitute,University of Tokyo

The Kuroshio is a strong current flowing along the western boundary of the North Pacific Ocean all the year round as a part of the general circulation of the ocean, but exhibits fiuctuations of a wide range of frequencies and size scales.
Physical processes corresponding to these fluctuations are quite complicated and not yet well understood because of our lack of knowledge about the non-linear coupling of various dynamical and thermodynamical processes.
To start a discussion, we need to distinguish the scale of phenomena in time and in space. If we formally separate water motion into the mean and fluctuations, the Reynolds stresses enter into the equations of the mean motion due to the non-linearity of the dynamical equations, and the energy exchange of the mean motion and the fluctuating motion as a whole is governed by the tensor product of the Reynolds stresses and the mean velocity strains. Since the Reynolds stresses are essentially related to the fluctuating motion, it is not possible to evaluate the energy flow from the mean motion alone, unless some assumptions are made.
Indeed, in the equilibrium range of the locally isotropic turbulence, the energy flow due to Reynolds stresses is from large eddies to small eddies and the eddy viscosity is formally related to the 4/3 power of the scale is question. In the oceanic scale, we do not know much about the role of Reynolds stresses and the energy might flow from the fluctuating motion to the mean motion (WEBSTER, 1961),but if we apply the statistical theory, the horizontal eddy viscosity seems to be at most of the order of 10^{3} c.g.s. in a climatological sense.
It is well known that, except for the motion with the period of a few days or less, the quasi-geostrophic balance is maintained in most of the oceanic motions except for the surface Ekman layer where the frictional influence is important and this is the basis of the dynamical calculation of ocean currents. However, the computed pressure field from temperature and salinity is only relative and we need to assume the depth of reference pressure somewhere around 1500 m or so. There is a possibility that the depth of no-motion do not exist and the flow may persist to the bottom of the ocean in the same direction as the surface current. Therefore, at the present state of art, it is very uncertain even to say whether the total mass transport of the Kuroshio changes seasonally or not. In deep water, the velocity might be small but its contribution to the total mass transport could be large because of the large thickness of the layer. Furthermore, if the flow reaches to the bottom, the influence of the bottom topography on the flow pattern is expected to be very large, particularly when the flow leaves the continental rise and passes onto the abyssal plain (WARREN,1963).
In the discussion of the circulation, an important concept is the vertical component Z of the absolution vorticity which is the tendency of water particle to rotate and can be separated into two parts;the rotation relative to the earth and the rotation of the earth itself. The Coriolis parameter f gives the vertical component of the planetary vorticity and the ratio of the relative vorticity to the planetary vorticity is given by the Rossby number U/(fL) where U and L are the representative velocity and scale of the motion respectively. If there is no frictional effect, the so-called potential vorticity Z/H (H;thickness of the barotropic layer) and the energy [Fig.107_200_01] of the barotropic fluid are conserved for an individual water column. In the presence of frictional effects, the potential vorticity of a water column may increase or decrease according to the input vorticity, and the change of the vorticity is furnished by the change of the relative and/or the planetary vorticity or by the change of the thickness H of the column. However, the total vorticity integrated over a closed basin is constant for the steady state. This implies that the total input vorticity from the surface of the ocean by the wind stress curl would be cancelled by the vorticity generated by the frictional effect at the lateral boundary (the bottom stress in the interior of the ocean may be negligible).
In the interior of the ocean, it seems reasonable to assume that the Rossby number is very small so that the input vorticity by the wind stress curl is balanced by the change of the planetary vorticity of a water column by the movement in the north-south direction (SVERDRUP,1947). By the continuity requirement of water mass, water return somewhere to the original position. By doing so the movement must also be adequate to generate vorticity which cancels the input vorticity by the wind stress. The relative vorticity and the transport of relative vorticity integrated over the boundary current as a whole is zero and the generated vorticity along the boundary by the friction is just sufficient to balance the change of the planetary vorticity of the boundary current as a whole. To satisfy these conditions, the flow should be in the western boundary (STEWART,1964).
Although the accurate expression of the friction is unknown, if the eddy viscosity of the order of 10^{7} to 10^{8} c.g.s. is assumed, it is possible to give a reasonable picture of the circulation including the western boundary flow. (MUNK,1950).
However, in the western boundary region, the relative vorticity may become comparable to the planetary vorticity, so that the possibility of the inertially controlled boundary current is investigated by neglecting the frictional effect* (CHARNEY,1955, MORGAN,1956). In homogeneous water with constant depth, the inertially controlled strong steady current along the western boundary can be possible only when the boundary current, is accelerated in the direction of flow, and the interior flow is westwards. For the eastward geostrophic flow in the interior, the possible stationary flow pattern may be wave-like and rapidly oscillating in space(FOFONOFF,1962). Another possibility is that the instability of the flow pattern makes the phenomenon always transient and only in a statistical sense ,the frictionally controlled boundary flow would be achieved with the eddy viscosity of the order of 10^{8} c.g.s..
In a two-layer model of the ocean, too, the westward flow is required in the interior of the ocean to establish the western boundary flow (VERONlS,1963).
Here, the density stratification has effects on the flow characteristics. First of all, the total mass transport of the boundary current is limited by the depth of the upper layer in the interior, so that the volume of the upper warm water may control the boundary transport which is otherwise determined by the wind stress curl in the interior of the ocean. Here, we see the interacting mechanism between the dynamical and thermodynamical processes. However, if the barotropic component of the boundary flow is large, the discussion would take a different course. As to the upper warm water mass in the southern half of the interior of the ocean, the thermal circulation in the interior of the ocean and its relation to the formation of the permanent thermocline is discussed by ROBINSON and STOMMEL (1959), WELANDER (1959), and ROEINSON and WELANDER (1963). They solved the heat equation containing three-dimensional advection as well as vertical diffusion in conjunction with the geostrophic equations of motion, and proposed the importance of vertical upward velocity below the layer of the permanent thermocline.
These discussions together with the actual wind stress distribution indicate that in the southern half of the Pacific Ocean an intense northward boundary flow can be formed. It is mainly inertially controlled and includes a frictional boundary layer in the shoreward. On the other hand, in the northern latitude where the geostrophic interior flow is eastward, the strong inertial flow leaves the coastal boundary and the assumption of the steady state may not be justified.
Here, different physical processes should be considered. Indeed, the Kuroshio forms the boundary of the warm southern water and the cold northern water, and the theory developed for the wind stresses should be modified to include the thermohaline processes. particularly, it is desirable to show how much warm water is given off to the north and how much is recirculated in the gyre by taking the meandering and cut-off phenomena of the currents into consideration. Only in a statistical sense could the mean state of the flow be inferred from the frictional theory of the general circulation.
The abyssal circulation of the ocean, which is the internal mode of motion, can also be interpreted by means of the vorticity concept. If we accept the vertical velocity at the interface of the two-layer ocean model, the flow pattern in the lower layer can be estimated by considering the vorticity change due to the stretching of the water column in the lower layer. For the uniform upward velocity at the interface (STOMMEL,1960), the flow pattern in the interior of the ocean is determined, and the concentrated western boundary flow is required to satisfy the continuity equation. However, in the lower layer, the interior flow is eastwards and the boundary flow cannot be of inertial type. This means that the western boundary flow in deep water, if exists as in a model suggested above, might be non-steady and fluctuate with time, so that only in a climatological sense the frictional boundary flow would be realized. A different approach to the internal mode of motion (convective circulation) is developed by TAKANO (1963) who also obtained the intense flow along meridional boundaries. At any rate, it is important to take the internal mode of motion into consideration in the discussion of the general circulation, and the importance of the topographic control of the motion in deep water should be kept in mind.
To examine the time-dependent motion and the stability of the flow pattern, the first step would be to investigate the wave-like motion. A well-known quasigeostrophic wave is the Rossby wave which moves to the west without change of form if no lateral boundary exists. The extension of the Rossby wave to include the wave travelling obliquely to latitude circles is suggested by FOFONOFF (1962) who also clarified the mechanism of the reflection of these waves at the meridional boundary. If we compare barotropic and barociinic waves in a twolayer model of the ocean with respect to the period wave-length relation, it is found that for a scale of several thousand kilometers, the barotropic wave has a period of several weeks and the baroclinic wave has a period of tens of years, so that it is conceivable that the ocean responds mainly in barotropic mode to the external forces of shorter periods but for the seasonal change, the baroclinic response may become about half of the barotropic response (VERONIS and STOMMWL,1956). However, near the equator, the responding baroclinic wave period becomes about a year so that the possibility of the baroclinic resonance exists. It should be mentioned that the situation requires more careful study because the period of the baroclinic wave depends on the latitude and the assumed density structure of the ocean, and moreover the existence of the lateral boundary plays a very important role (Rossby type waves cannot satisfy the reflection condition along the latitudinal boundary) and also the interaction of the barotropic and baroclinic modes may be expected at the boundary.
If the response of the boundary currents to the seasonal variation of the wind stress over the whole ocean is assumed to be mainly barotropic, the amplitude of the variation of the boundary current is roughly proportional to the amplitude of the variation of the wind stress. However, the position of the axis of the boundary current may (lCHIYE,1951, TAKANO,1965) or may not (VERONlS and MORGAN,1955) change with the change of the wind stress. It may be worth-while to investigate whether the current axis of the boundary current changes its position or not when the transport changes suddenly in a two-layer model of the ocean, in connection with the formation of the cold water mass shorewards.
As for the instability of ocean currents, results of several studies have been published (HAURWITZ and PANOFSKY,1950, STOMMEL: The Gulf Stream,1958, IWATA,1961) but our knowledge of the subject is still very far from satisfactory.
Particularly, we need to clarify the instability of flow in relation to the meander and cut-off phenomena when the strong flow leaves the boundary. Here, it may be mentioned that ROSSBY (1951) stated that it is not unreasonable to assume that the factor controlling the shape and behavior of jets must be fairly independent of their driving mechanism and derivable from quite general dynamic principles. He proposed a model of the Gulf Stream on the basis of the lateral mixing of a jet in which the momentum transport is constant and mass transport increases downstream (ROSSBY,1936) and also discussed the vertical and horizontal concentration of momentum in ocean currents by assuming the constant mass transport and the minimum transport of momentum (ROSSBY,1951, ICHIYE, 1960). These arguments are not exact but illuminate some phases of the dynamical behaviors of ocean currents.
As has been discussed so far, the understanding of the oceanic circulation and, in particular, the Kuroshio is very poor and we are just in the beginning of the formulation of a plausible model of the circulation. As in most of the geophyslcal phenomena, theoretical discussions can not go alone without the guidance of observational facts, because the basic form of the hydrodynamic and thermodynamic Iaws cannot be handled without suitable approximations. Thus, it is important to design observational programme suitable to the particular problem in mind. Here, it is necessary to consider the spectral density of the information compared to those of the so-called noises which seems to be very large in most cases (STOMMEL,1963). Therefore, to sort out desired information from the background of noises, continuous time series of observations for a long period, and/or the rapid survey covering wide areas is very desirable, particularly in the upper layers of the ocean and in the region of strong currents where the fluctuations of oceanographic conditions are supposed to be very large.

* CARRIER and ROBINSON (1962) appeared to have pushed the inertial theory to the extreme and their picture of the general circulation is quite unrealistic.

References

CARRIER, G.F. and A.R. ROBINSON,1960: On the theory of the wind-driven ocean circulation. J. Fluid Mech.,12,49-80.
CHARNEY,I.G,1955: The Gulf Stream as an inertial boundary layer. Proc. Nat. Acad.
Sci. Wash.,41,731-740.
FOFONOFF, N.,1962: Dynamics of ocean currents. The Seas, Interscience Publishers, New York.
HAURWITZ, and H.A. P.ANOFSKY,1950: Stability and meandering of the Gulf Stream.
Trans. Amer. Geophys. Union,31,723-731.
ICHIYE, T.,1951: On the variation of oceanic circulation (1st paper). Oceanogr. Mag., 3,78-82.
ICHIYE, T.,1960: On cirtical regimes and horizontal concentration of momentum in ocean currents with two-layered system. Tellus,12,149-158.
IWATA, N.,1961: Uber die Instabilitat der Meeresstromung im geschichteten Meer.
Contr. Mar. Res. Lab., Hydro. Office of Japan,2,135-170.
MORGAN, G.W.,1956: 0n the wind-driven ocean circulation. Tellus, 8,301-320. MUNK, W.H.,1950: On the wind-driven ocean circulation. J. Met.,7,79-93.
ROBINSON, A.R. and H. STOMMEL,1959: The oceanic thermocline and the associated thermohaline circulation. Tellus,11,295-308.
BOBINSON, A.R. and P. WELANDER,1963: Thermal circulation on a rotating sphere; with application to the oceanic thermocline. J. Mar. Res.,21,25-38.
ROSSBY, C.G.,1936: Dynamic. of steady ocean currents in the light of experimental fluid mechanics. Pap. Phys. Ocn. ＆Met., 5(1).
ROSSBY, C.G.,1951: On the vertical and horizontal concentration of momentum in air and ocean currents I. Tellus 3,15-27.
STEWART, R.W,1964: The influence of friction on inertial models of oceanic circulation.
Studies on Oceanography,3-9,(Prof. HIDAKA Anniversary Volume).
STOMMEL, H.,1958: The Gulf Stream. Univ. Cal. Press.
STOMMEL, H.and A.B. ARONS,1960: On the abyssal circulation of the word-ocean-I, Stationary planetary flow patterns on a sphere. Deep-Sea Research,6,140-154.
STOMMEL, H.,1963: Varieties of oceanographic experience. Science,139,566-572.
SVERDRUP, H.U.,1947: Wind-driven currents in a baroclinic ocean: with application to the equatorial currents of the Eastern Pacific. Proc. Nat. Acad. Sci.Wash., 33,318-326.
TAKANO, K.,1962: Circulation generale permanente dans un ocean. Rec.Oceanogr.
Works in Japan,6,59-155.
TAKANO, K.,1965: Periodic variation of the barotropic components of a wind-driven circulation in an ocean. Journ. Oceanogr. Soc. Japan,21,1-5.
VERONIS, G. and G.W. MORGAN,1955: A study of the time dependent wind-driven circulation in homogeneous rectangular ocean. Tellus,7,232-242.
VERONIS, G. and A. STOMMEL,1956: The action of variable wind stresses on a stratified ocean. J. Mar. Res.,15,43-75.
VERONlS, G.,1963: On inertially-controlled flow patterns in a \beya\-plane ocean. Tellus, 15,59-66.
WARREN, B.A.,1963: Topographic influences on the path of the Gulf Stream. Tellus, 15,167-183.
WEBSTER, F.,1961: The effect of meanders on the kinetic energy balance of the Gulf Stream. Tellus,13,392-401.
WELANDER, P.,1959: An advective model of the ocean thermocline. Tellus,11,309-318.

BULLETIN OF THE EARTHQUAKE RESEACH INSTITUTE Vol.42 (1964), pp.147-174 7. On the Bottom Friction in an Oscillatory Current. By Kinjiro KAJIURA, Earthquake Research Institute. (Read Dec.17,1963. -Received Dec.28,1963.)

Abstract

The frictional coefficient C of an oscillating turbulent flow is estimated on the assumption that the eddy viscosity is proportional to the amplitude of the bottom friction velocity and the height above the bottom. The dependence of C on parameters such as the amplitude of the vertically averaged horizontal velocity, the period of the oscillation, the depth of water, and the roughness length z0 (for the case of a rough boundary) or the molecular viscosity \nu\ (for the case of a smooth boundary)is shown graphically by choosing suitable non-dimensional parameters. The frictional coefficient for the case of a laminar oscillatory flow is also discussed.

1. Introduction

In many practical applications of the dynamical equations of motion, such as the case of a numerical experiment on long waves in shallow water, the law of bottom stress is assumed as something like [Fig.107_205_01] where u is the depth-mean velocity in a water column and C is the frictional coefficient. There are several estimates of C in tidal currents by means of the dynamical method (Taylor,1918 ; Grace,1936,1937; Bowden and Fairbairn,1952). On the other hand, the bottom friction is intimately related to the mean velocity profile or the turbulent velocity fluctuations near the bottom and there are some observations which show the logarithmic law of velocity profile near the bottom (Lesser, 1951; Charnock,1959). Furthermore, the turbulent velocity fluctuations in a tidal current are measured by means of an electromagnetic flow meter (Bowden and Fairbairn,1956; Bowden,1961) and the Reynolds stresses near the bottom are computed. From these results, it appears that the frictional coefficient (for depth-mean velocity) of a tidal current is about 1.5〜2.5×10^{-3}. However, it is not yet certain whether the same frictional law holds for a long wave of shorter periods, because it is plausible that the structure of the bottom frictional layer in such an oscillatory current may be different from that of the tidal current.
On the other hand, the frictional law of the steady turbulent flow in pipes and open channels is extensively studied by hydraulic engineers but the frictional law of the oscillatory flow seems to be hardly investigated. Recently, the decay of waves in shallow water has become a subject of coastal engineers and some experimental results are discussed in conjunction with the theoretical estimate based on the condition of a laminar flow (Biesel,1949; Eagleson,1962; Grosch,1962), and the transition from the laminar to turbulent boundary layer in an oscillating flow over smooth and rough bottoms is also studied experimentally (Li,1954; Vincent,1957;Collins,1963).
The present paper is a theoretical attempt, though admittedly not so complete, to find the frictional coefficient C of the oscillatory flow in a fully turbulent state by assuming a suitable relation between velocity shear and stress in water and to examine the dependence of the frictional coefficient on the amplitude and period of oscillating currents as well as to the depth of water.

2. Basic consideration

We consider that the motion is predominantly horizontal in onedirection and water is homogeneous with constant depth. Then, the linearized equation of horizontal motion may be written as

式(2-1) [Fig.107_206_01]

where t is time, x and z are the horizontal and vertical co-ordinates with the origin at the bottom of the water and the z-axis positive upwards, u is the horizontal velocity,\zeta\ is the elevation of the water surface from the undisturbed free surface, zh is the depth of water, and g is the acceleration due to gravity. The tangential stress in the x-direction is given by \rho\\tau\ with \rho\ the density of water.
Now putting formally

式(2-2) [Fig.107_206_02]

we may rewrite (2-1) in the form

式(2-3) [Fig.107_206_03]

In terms of the vertically averaged velocity,(2-3) becomes

式(2-4) [Fig.107_207_01]

where \tau\s and \tau\B are the surface and the bottom stresses respectively, (\tau\s = 0 for the present discussion) and u is the mean velocity defined by

式(2-5) [Fig.107_207_02]

The equation of continuity may be given by

式(2-6) [Fig.107_207_03]

and from (2-4) and (2-5) together with (2-2), it follows

式(2-7) [Fig.107_207_04]

or in terms of U,

式(2-8) [Fig.107_207_05]

The equation (2-7) is the common representation of long waves of small amplitude and if the bottom stress is given in terms of u, the solution can be found under given initial and boundary conditions.
The bottom friction velocity u*B defined by

式(2-9) [Fig.107_207_06]

may be approximated for a periodic motion in time (u*=u*Bcos(\sigma\t+\epsilon\)) by

式(2-10) [Fig.107_207_07]

where

式(2-11) [Fig.107_207_08]

and u*B is the amplitude of u*B (See Proudman; Dynamical Oceanography, §151, 1953).
Now, for the relation between the velocity shear and tangential stress, we resort to the usual formulation of a turbulent flow. On the ground of dimensional analysis, the turbulent eddy viscosity Kz in neutral stability may be put formally proportional to the characteristic velocity of turbulence ＜v^{2}＞^{1/2} and the effective size of the turbulent eddy \Lambda\ such that

式(2-12) [Fig.107_208_01]

and

式(2-13) [Fig.107_208_02]

Here, we assume that the effective size of the turbulent eddy is proportional to the height above the bottom and the characteristic turbulent velocity＜v^{2}＞^{1/2} is proportional to u*B 1). More specifically, we assume

式(2-14) [Fig.107_208_03]

where k is von Karman’s constant (=0.4) and z0 is the roughness length of a rough surface. Similar assumption for the eddy viscosity is already used in the discussion of the atmospheric frictional layer near the ground (e.g., Ellison,1956). For the case of a smooth surface, z0 should be understood as the thickness of a laminar sub-layer. Approximating the frictional velocity in general 2) by

式(2-15) [Fig.107_208_04]

(2-12),(2-13),and (2-15) give

式(2-16) [Fig.107_208_05]

1) The size of turbulent eddies may decrease near the free surface because of the presence of the free surface z = zh, and the characteristic turbulent velocity may not be constant throughout the vertical column of water but a function of the velocity shear and the height above the bottom as assumed in the Prandtl’s mixing-length theory.
However, a refined distribution of Kz does not seem to be worth while to try in the present crude discussion unless more definite knowledge about the turbulent structure of an oscillatory flow is obtained.
2) Essentially, u* is not the friction velocity in an ordinary sense, but is a quantity defined by (2-15) and coincides with the bottom friction velocity uB* at the bottom. For convenience, we call u* the friction velocity in the present paper.

3. The frictional coefficient for the case of a turbulent flow over a rough bottom

If the motion is assumed to be periodic in time, we may put

式(3-1) [Fig.107_209_01]

where Re means the real part of the complex quantity. In the following discussion, we use the complex amplitude u* with the prime dropped for simplicity unless otherwise stated. The same rule applies for other dependent variables too.
From (2-3) and (2-16) together with (2-15), the equation for the friction velocity u* is derived

式(3-2) [Fig.107_209_02]

where

式(3-3) [Fig.107_209_03]

Under the stress free condition at the surface:

式(3-4) [Fig.107_209_04]

the solution of (3-2) becomes

式(3-5) [Fig.107_209_05]

where

[Fig.107_209_06]

and ch is determined by

式(3-6) [Fig.107_209_07]

Here,

式(3-7) [Fig.107_209_08]

and Jn(Y) and Nn(Y) are Bessel and Neumann functions of the order n and ch=a + ib. For small values of Y,Zn(Y,ch) can be expressed in series form as follows:

式(3-8) [Fig.107_210_01]

and

式(3-9) [Fig.107_210_02]

where r=0.5772.... (Euler’s constant).
The values of a and b computed numerically from (3-6) are shown in Fig.1 as a function of yh.



For small values of yh, say yh ＜ 1,

式(3-10) [Fig.107_210_04]

and

式(3-11) [Fig.107_210_05]

For large values of yh, say yh ＞ 3,

式(3-12) [Fig.107_210_06]

Therefore, it follows:

式(3-13) [Fig.107_211_01]

and

式(3-14) [Fig.107_211_02]

where H1^{(2)}(Y) is the Hankel function of the second kind. (3-13) shows that for small yh the stress \tau\ decreases linearly with respect to depth from the bottom to the surface, and (3-14) shows that the decrease of stress is exponential for large values of y.
Now, in terms of y,(2-16) may be transformed into

式(3-15) [Fig.107_211_03]

and the substitution of (3-5) into (3-15) yields

式(3-16) [Fig.107_211_04]

where the boundary condition at the bottom such that u=0 at y=y0 is taken into consideration. It can be easily shown that for small values of y (y ＜ 1), the velocity profile in terms of z is given by

式(3-17) [Fig.107_211_05]

In terms of y,(2-5) becomes

式(3-18) [Fig.107_211_06]

and the substitution of (3-16) yields

式(3-19) [Fig.107_211_07]

Since in general y0 is very small and (y0/yh)^{2} ＜＜ 1,(3-19) is reduced to

式(3-20) [Fig.107_211_08]

where

[Fig.107_212_01]

and

[Fig.107_212_02]

For small values of yh, it follows,

式(3-21) [Fig.107_212_03]

and for large values of yh,

式(3-22) [Fig.107_212_04]

The frictional coefficient C is now defined by

式(3-23) [Fig.107_212_05]

where u is the amplitude of u. 3) Therefore, the substitution of (3-20) yields

式(3-24) [Fig.107_212_06]

Taking (3-21) and (3-22) into consideration, it is found that for yh ＜ 1, the frictional coefficient is almost independent of wave period and amplitude and only a function of zh/z0, or in other words, the steady fiow condition is applicable to the oscillatory flow in this range. On the other hand, for yh ＞ 3 the frictional coefficient is a function of y0

3) Strictly speaking, the instantaneous friction coefficient C should be defined by
[Fig.107_212_07]
but for the periodic motion, the following approximation may be introduced;
[Fig.107_212_08]

only so that C is independent of the depth but dependent on wave period and amplitude.
Combining the two relations,(3-20) and (3-24), C (amplitude of C) can be given as a function of u/(\sigma\zh) with zh/z0 as a parameter as shown in Fig.2a. For small values of u/(\sigma\zh), C increases with decreasing values of u/(\sigma\zh) for a fixed value of zh/z0, and for u(\sigma\zh) larger than, say 10^{2},C stays almost constant. Fig.2b shows the corresponding phase angle I\theta\I as a function of u(\sigma\zh) with zh/z0 as a parameter, and it is seen that the phase angle is noticable only for small values of u(\sigma\zh) where the increase of C is observed.
For a fixed values of u(\sigma\z0) , the increase of zh/z0 is followed by the decrease of C and the increase of I\theta\I as shown Fig.3a and Fig.3b. However, for large values of zh/z0, C becomes constant for a fixed value of u/(\sigma\z0), showing the independence of C on zh. In this range, C can be given as a function of u/(\sigma\z0) as shown in Fig.3c. Thus, the frictional coefficient C increases with decreasing u, decreasing period of oscillation in the range of u/(\sigma\zh) ≦ 10^{2}, and with decreasing depth of water for large values of u/(\sigma\zh).

4.The dependence of \tau_{B}\ and u on U (the pressure gradient term)

The substitution of (2-10) and (3-20) into (2-4) yields

式(4-1) [Fig.107_216_01]

where

式(4-2) [Fig.107_216_02]

Since \alpha\1 > 0, and \alpha\2 ＜ 0, it follows that \delta\ ＞ 0.
Putting u*B/(\sigma\zh) = \eta\, we have u*B = (3 \pi\/8)\eta\\sigma\zh\theta\e^{i\delta\}, if U is assumed real, and it follows from (4-1),

式(4-3) [Fig.107_216_03]

Remembering that \alpha\1 and \alpha\2 are functions of yh and y0, and \eta\=10/y^{2}h by definition, we can solve U/(\sigma\zh) in terms of \eta\ with y0 or zh/z0 as a parameter.
In particular, for small values of \eta\, we have

式(4-4) [Fig.107_216_04]

so that

式(4-5) [Fig.107_216_05]

On the other hand, for large values of \eta\, we have

式(4-6) [Fig.107_216_06]

so that

式(4-7) [Fig.107_216_07]

Furthermore, from (3-20) and (4-1), it follows,

[Fig.107_216_08]

so that u/U \simeq\ 1 for small values of \eta\ and u/U \simeq\ \eta\^{-1} for large values of \eta\. The phase difference between U and u is given by \delta\+\theta\.
As shown in Fig.4a, the ratio u/ U of the mean velocity to the velocity with no friction stays almost unity for small values of U/(\sigma\zh) but decreases for increasing values of U/(\sigma\zh) as is expected from the simple consideration of the frictional effect. In other words, in the dynamical equation of motion the acceleration of water dominates over friction for, say, U/(\sigma\zh) ＜ 10^{2} as in the case of ordinary gravity waves without friction, but, with the increase of U/(\sigma\zh), the frictional effect becomes increasingly important and in the limiting case of U/(\sigma\zh) ＞ 10^{5} the wave is in a state of balance between the pressure gradient force and the frictional force. Fig.4b shows the phase difference \delta\, between U and u*B, which varies form l\theta\l to \pi\/2 with increasing U/(\sigma\zh). Taking Fig.2b into consideration, the phase difference between U and u increases from zero to \pi\/2 with the increase of U/(\sigma\zh).
It is understood that the dynamics of the unsteady river flow such as the flood wave can be treated as a quasi-steady state if U/(\sigma\zh) is very large and tidal waves in rivers may be considered as half inertial and half frictional to balance the pressure gradient.

5. The frictional coefficien.t for the case of a turbulent flow over a smooth bottom

For a smooth boundary, some modification of the results in the previous section is necessary, because the bottom surface is located at z=-z0 and the roughness parameter z0 should be replaced by the thickness of the laminar sub-layer with the kinematic viscosity \nu\. Thus, (3-16) is replaced by

式(5-1) [Fig.107_218_01]

where uB is the velocity at the top of the laminar sub-layer: y=y0.
Within the laminar sub-layer, we may put approximately,

式(5-2) [Fig.107_218_02]

so that

式(5-3) [Fig.107_218_03]

The Reynolds number N defined for the laminar sub-layer may be assumed constant and estimated to be 11.6 on the basis of an experimental formula of the velocity profile for a smooth circular tube 4).
Taking (5-3) into considerationr and neglecting the contribution of the velocity in the laminar sub-layer to the mean velocity, we may write in place of (3-20),

式(5-4) [Fig.107_219_01]

where

式(5-5) [Fig.107_219_02]

Therefore we have, in place of (3-24), for the frictional coefficient Cs,

式(5-6) [Fig.107_219_03]

Two limiting cases of yh, are easily derived as follows:

式(5-7) [Fig.107_219_04]

and

式(5-8) [Fig.107_220_01]

Making use of the Reynolds numbers such as

式(5-9) [Fig.107_220_02]

with

式(5-10) [Fig.107_220_03]

we have

[Fig.107_220_04]

and

[Fig.107_220_05]

so that (5-7) becomes

式(5-11) [Fig.107_220_06]

and (5-8) becomes

式(5-12) [Fig.107_220_07]

(5-11) together with (5-6) shows that for small values of yh, say \beta\zh ＜ 0.2L^{1/2}, the frictional coefficient C, is a function of a Reynolds number L only, and is equivalent to the case of the steady turbulent flow over a smooth bottom. On the other hand (5-12) shows that for large values of yh, say \beta\zh ＞ 0.6L^{1/2}, the frictional coefficient C, is a function of a Reynolds number M only 5) and the depth of water zh is not an important factor in the determination of Cs.
From a practical standpoint, it seems to be easier to understand the relation by taking a Reynolds number based on the mean velocity instead of the friction velocity. From (5-9) and (5-4) we may put

式(5-13) [Fig.107_221_01]

In particular, for two limiting cases of yh, this conversion can be made without regard to \beta\zh.
In Fig.5a and Fig.5b, C, and \psi\ are shown as a function of L* with \beta\zh as a parameter. For \beta\zh ＜ L*^{1/2}, the curves of different \beta\zh converges to a single curve and l\psi\I approaches zero, indicating the dependence of Cs only on L*. For \beta\zh > L*^{1/2} the effect of the period of oscillation (\beta\zh) shows up distinctly in this representation. Another representation of the relation is made with the aid of the Reynolds number M* in Figs.6a,6b which do not depend on \beta\zh explicitly for large values of \beta\zh/L*^{1/2} and are simpler than Figs.5a and 5b.


4) For small values of y, we may derive in place of (3-17),

[Fig.107_219_05]

and substituting the relation (5-3), we have

[Fig.107_219_06]

Comparing the result with the Nikuradse’s experimental formula for a smooth circular tube, namely,

[Fig.107_219_07]

we have
N=11.6 (uB*is put equal to uB*).
Strictly speaking, N may also be a function of \sigma\ for an oscillatory flow but we assume tentatively N constant.
Returning to the definition of the eddy viscosity (2-14), we have at the top of the laminar sub-layer,

[Fig.107_219_08]

Thus, we have implicitly assumed that the viscosity jumps to the eddy viscosity of about 5 times the molecular values at z=z0, so that the velocity gradient is not continuous, although the velocity itself and stress are continuous at the top of the laminar sub-layer.

5) A different definition of a Reynolds number is possible, for example,

[Fig.107_220_08]

where l is a kind of excursion distance defined by the friction velocity. Then, it is easily seen that

[Fig.107_220_09]

6.The frictional coefficient for the case of a laminar flow

Although the frictional damping of water waves for the case of a laminar flow has been well investigated theoretically (Proudman and Doodson,1924;Biesel,1949), it may be worth-while to discuss the problem from a viewpoint of the frictional coefficient.
For the case of a laminar flow, with the kinematic viscosity \nu\ we
may put in place of (2-16),

式(6-1) [Fig.107_223_02]

Then, the equation corresponding to (3-2) is

式(6-2) [Fig.107_223_03]

The solution of (6-2) under the stress free boundary condition at the surface (z=zh) is given by

式(6-3) [Fig.107_223_04]

where

[Fig.107_224_01]

Under the condition that

u=0 at z=0,

(6-1) may be integrated to yield

式(6-4) [Fig.107_224_02]

Therefore the mean flow becomes,

式(6-5) [Fig.107_224_03]

Now from (2-4), we may put

式(6-6) [Fig.107_224_04]

so that by substitution of (6-5) into (6-6) we have

式(6-7) [Fig.107_224_05]

Therefore, from (6-5) the mean velocity is given by

式(6-8) [Fig.107_224_06]

and from (6-4) the surface velocity us becomes

式(6-9) [Fig.107_224_07]

Formally,we may write (6-7),(6-8) and (6-9) in the form:

式(6-10) [Fig.107_224_08]

式(6-11) [Fig.107_224_09]

式(6-12) [Fig.107_224_10]

and, it follows

式(6-13) [Fig.107_224_11]

with

式(6-14) [Fig.107_225_01]

For large values of \beta\zh, it is found that

式(6-15) [Fig.107_225_02]

so that (6-10) becomes

式(6-16) [Fig.107_225_03]

and (6-14) becomes

式(6-17) [Fig.107_225_04]

For small values of \beta\zh, we have

式(6-18) [Fig.107_225_05]

so that (6-10) becomes

式(6-19) [Fig.107_225_06]

and (6-13) becomes

式(6-20) [Fig.107_225_07]

and

[Fig.107_225_08]

Fig.7 gives a and \theta\1 of (6-10) as a function of \beta\zn which shows that a increases from 0 to 1 and the phase angle decreases from \pi\/2 to \pi\/4 with increasing \beta\zh. Fig.8 shows b,c and \theta\2,\theta\3 as functions of \beta\zh in which Iu/usI is also shown. From the figure, it is found that the ratio of the mean current to the surface current varies from 1 to 0.66 and the minimum value lies somewhere around \beta\zh 〜 1.0. The phase angle relative to U decreases from \pi\/2 to 0, but the difference of \theta\2 and\theta\3 is always small. Fig.9 is a graph of D and \theta\4 of (6-13) as a function of \beta\zh which shows that D decreases and \theta\4 increases with increasing values of \beta\zh.
The frictional coefficient Cl defined by

式(6-21) [Fig.107_227_01]

can be written with the aid of (6-13) in the form,

式(6-22) [Fig.107_227_02]

Making use of the Reynolds number L*or M*, defined in (5-13), we
have

式(6-23) [Fig.107_227_03]

or

式(6-24) [Fig.107_227_04]

Therefore, the frictional coefficient is inversely proportional to the Reynolds number and also a function of \beta\zh. In particular, for large values of \beta\zh, [Fig.107_227_05] ,(c. f., Fig.6) and for small values of \beta\zh,Cl=3/L*.

7. Application of the laminar frictional formula

The advantage of writing the bottom stress in the form (6-10) or (6-13) may be seen in the following discussions.
If we are to solve (2-8) for the periodic wave in time (progressive waves), [Fig.107_227_06] , by introducing the relations (2-2) and (6-10) we have the characteristic equation as follows:

式(7-1) [Fig.107_227_07]

where

[Fig.107_227_08]

Since \sigma\ is assumed real, we have

式(7-2) [Fig.107_227_09]

For small friction, the solution may be reduced to

式(7-3) [Fig.107_228_01]

and

式(7-4) [Fig.107_228_02]

Therefore, the phase velocity c is given by

式(7-5) [Fig.107_228_03]

In particular, for large values of \beta\zh, a=1 and \theta\1=\pi\/4 so that

式(7-6) [Fig.107_228_04]

and

式(7-7) [Fig.107_228_05]

For free seiches in an enclosed basin, the wave decays with time so that the discussion should be modined by putting \sigma\=\sigma\1+i\sigma\2=\sigma\*e^{i\epsilon\}.
The essential change lies in the estimation of tanh(1+i)\beta\zh in which\beta\ is no longer real but [Fig.107_228_06].
However, for the case of small friction, we may assume

[Fig.107_228_07]

and may use \beta\* in place of \beta\. Then, for waves periodic in space (standing waves),\zeta\=\zeta\0 cosmxe^{i\sigma\t}, the characteristic equation becomes

式(7-8) [Fig.107_228_08]

where \sigma\0=c0m and E*=a/(√2 \beta\*zh). For small values of E*,we have

式(7-9) [Fig.107_228_09]

and

式(7-10) [Fig.107_228_10]

For large values of \beta\*zh,α=1 and \theta\1=\pi\/4, so that

式(7-11) [Fig.107_229_01]

and

式(7-12) [Fig.107_229_02]

These results are the same as the results given by Biesel(1949), at least for shallow water waves with \beta\zh >> 1 6). Furthermore, for \beta\zh \simeq\ 2, (7-11) and (7-12) coincide with the result given by Proudman and Doodson (1924) by a complicated numerical method (See also Defant; Dynamical Oceanography, Vol.2, p.159,1961) as shown by Platzmann and Rao (1964).
The essential parameter for the damping and period increase of seiches is \beta\zh as in the case of progressive waves and an important characteristic of the laminar friction is that the bottom frictional stress \tau\B is not in phase with the mean velocity u except for very small values of \beta\zh, so that, if we put the stress in the form -fzhu, the coefficient f is related to the period of oscillation and includes the term of phase difference.
In terms of the horizontal displacement of the water mass \xi\, (2-7)
can be written as

式(7-13) [Fig.107_229_03]

and if f is real and independent of the period T of a seiche, which seems to be realized in the case of a turbulent flow with u/\sigma\zh) > 10^{2}, the rate of period increase \Delta\T/T is proportional to T^{2} and u^{2}. However, for the case of a laminar flow with large values of \beta\zh, [Fig.107_229_04] , and the period increase \Delta\T/T is proportional to T^{1/2}. The difference of the period increase due to friction for the laminar and turbulent flow is already mentioned by Platzman and Rao (1964).

6) For small \beta\zh, his method of approximation is not accurate, because of the development in terms of \nu\^{1/2}.
7) According to experiments, the actual value of f seems to be several times greater than that derived on the basis of the linearized laminar theory. The discrepancy of the f value between theory and experiment is not yet resolved, although an attempt is made by Grosch (1962) without success to explain the discrepancy by introducing non-linear terms in the equation of the boundary layer.

8.Discussion

The approximation of long waves which is assumed throughout this paper requires that the wave length should be large compared with the depth of water, namely, mzh ＜ 1 and \sigma\/m =√gzh,where m=2\pi\/\lambda\. Taking these conditions into consideration,

[Fig.107_230_01]

Or, in terms of the wave amplitude a, we may write for the case of u \simeq\ U,

[Fig.107_230_02]

and

[Fig.107_230_03]

This shows that the approximation of long wave is roughly satisfied for u/(\sigma\zh) > 1 if (a/zh) = 0.1 (c. f., Fig.2a, and Fig.2b).
The critical Reynolds number for the transition from laminar to turbulent boundary layer in an oscillating wave motion is investigated experimentally by several authors. Li (1954) studied the case of an oscillating plate and proposed a Reynolds number for a smooth surface,

[Fig.107_230_04]

in which d’ is the total excursion of water particles relative to the plate just outside the bottom boundary layer. For long waves, this condition is approximately equivalent to (3\pi\/8)M*=566. Vincent (1957) argued on the basis of his experimental results for progressive waves that Li overestimated the range of laminar conditions at least for the case of a rough bottom. Both of these results are estimated from the observation of dye streaks, watching the transition from laminar to turbulent appearance. Recently, based on a different principle to determine the critical Reynolds number, Collins (1963) gave (3\pi\/8)M*=160.
For \nu\=10^{-2}cm^{2}sec^{-1}, g=980 cm sec^{-2}, the criterion of Collins can be written as a ≧ 0.75(zh/T)^{1/2} (c.g.s. unit). Therefore, a wave of, say,10 minutes in period may be considered to have a turbulent boundary layer if a > 1cm for the depth of 10m. This shows that tsunamis and seiches would have a turbulent boundary layer in shallow water even if the bottom is smooth. It may be of some interest to note that Cl (laminar) and Cs (turbulent) curves (c.f., Fig.6a) intersects at about M*=300 for large values of [Fig.107_231_01].

References

BIESEL, F.,1949: Calcul de l’amortissement d’une houle dans un liquide visqueux de profondeur finie. La Houille Blanche, Sept-Oct.,1949,630-634.
BOWDEN, K. F., and FAIRBAIRN, L.A.,1952: A determination of the frictional forces in a tidal current. Proc. Roy. Soc. London, A.214,317-392.
BOWDEN, K. F., and FAIRBAIRN, L. A.,1956: Measurements of turbulent fluctuations and Reynolds stresses in a tidal current. Proc. Roy. Soc. London, A,237,422-438.
BOWDEN, K. F.,1962: Measurements of turbulence near the sea bed in a tidal current. J.Geophys. Res.,67,3181-3186.
CHARNOCK, H.,1959: Tidal friction from currents near the sea bed. Geophys. J., Roy. Astr. Soc.,2,215-221.
COLLINS, J.I.,1963: Inception of turbulence at the bed under periodic gravity waves. J.Geophys. Res.,68,6007-6014.
DEFANT, A.,1961: Physical Oceanography. vol. II, Pergamon Press.
Eagleson, P. S.,1962: Laminar damping of oscillatory waves. J. Hydr. Div., Am. Soc. Civil Engrs,88,155-181.
ELLISON, T. H.,1956: Atmospheric turbulence. Survey in Mechanics, Cambrige Univ. Press,400-430.
GRACE, S. F.,1936: Friction in the tidal currents of the Bristol Channel. Mon. Not. Roy. Astr. Soc. Geophys. Suppl.,3,388-395.
GRACE, S. F.,1937: Friction in the tidal currents of the English Channel. Mon. Not. Roy. Astr. Soc. Geophys. Suppl.,4,133-142.
GROSCH, C. E.,1962: Laminar boundary layer under a wave. Phys. Fluids,5,1163-1167.
LESSER, R. M.,1951: Some observations of the velocity profile near the sea floor. Trans. Am. Geophys. Union,32,207-211.
LI, H.,1954: Stability of oscillatory laminar flow along a wall. Tech. Memo. No.47, Beach Erosion Board, Corps of Engrs., U. S. Army, Washingtom, D. C.
PLATZMAN, G. W., and D. D. LAO,1964: The free oscillations of Lake Erie. Oceanographic Papers dedicated to Prof. Hidaka,(in press).
PROUDMAN, J.,1953: Dynamical Oceanography. Methuen＆Co. Ltd., London.
PROUDMAN, J., and A. T. DOODSON,1924: Time relations in meteorological effects on the sea. Proc. Lond. Math. Soc.,24,140-149.
TAYLOR, G.I.,1918: Tidal friction in the Irish Sea. Phil. Trans. Roy. Soc., A,220, 1-33.
VINCENT, G. E.,1957: Contribution to the study of sediment transport on a horizontal bed due to wave action. Proc. Conf. Coastal Eng.,6th, Univ. of Florida,326-335.

7.時間的に変化する流れに伴なう水底摩擦 地震研究所　梶浦欣二郎

　渦動粘性係数を,水底摩擦速度と水底からの高さに比例すると仮定して摩擦係数を計算し,底面が粗および滑らかな場合のそれぞれについて,摩擦係数が流れの振幅,周期,水深に関してどう変るかを適当な無次元量をつかつて図示した.浅海の潮流については,摩擦係数はz0/z (z:水深,z0:粗度長さ)によつてきまり,　z0/zのもつともらしい値に対して摩擦係数は2×10^{-3}程度となるが,周期の短かいセイシュなどでは摩擦係数が大きくなる.層流の場合の摩擦係数も簡単に議論してあり,底の滑らかな場合に,水深がかなりあると,摩擦係数に関係するレイノルズ数は水深zhをつかつた [Fig.107_232_01] ではなくて箋界層の厚さ \delta\ をつかつた [Fig.107_232_02] が適当であることが示され,層流摩擦係数と乱流摩擦係数とがほぼ等しくなるのは M*=300 のあたりになる.

BULLETIN OF THE EARTHQUAKE RESEARCH INSTITUTE Vol.46 (1968), pp.75-123 5. A Model of the Bottom Boundary Layer in Water Waves. By Kinjiro KAJIURA, Earthquake Research Institute. (Read Sept.12,1967. -Received Nov.18,1967.)

Abstract

On the basis of an eddy viscosity assumption compatible with the two-layer model (wall and defect layers) of the turbulent boundary layer, a method of calculating various characteristics of the wave boundary layer, such as the friction coefficient, universal profiles of the stress and velocity in the defect layer have been developed.
Numerical results show that amplitude and phase of the friction coefficient increase with the decrease of the amplitude of the reference velocity and/or the period of wave. The criteria for the smooth-rough and laminar-turbulent transitions are also suggested from the analogy to the case of steady flow.
Most of the experimental data on the friction coefficint and the velocity profile in a wave boundary layer seem to be favorably compared with the prediction derived from the present model.

1. Introduction

The knowledge about bottom friction under waves is required to understand various phenomena related to wave modification and sediment movement in shallow water. Although the case of laminar flow has been clarified to a considerable degree by Iwagaki et al.(1967), the structure of the turbulent frictional boundary layer under waves is not well understood, so that the friction law applicable to the oscillatory turbulent flow is quite ambiguous.
Bagnold (1946) was the first to give an empirical formula of the friction coefficient of the quadratic friction law for an oscillatory flow in the presence of artificial ripples. He found that the friction coefficient decreases with the increasing excursion distance; namely with increasing period and/or the amplitude of velocity oscillation. Putnam and Johnson (1949) took Bagnold’s formula into account and adopted 0.01 as a reasonable value for the friction coefficient for wind waves and swells in their discussion of the dissipation of wave energy in shallow water. However, the field observations of wave decay in shallow water by Bretchneider (1954) and by Iwagaki and Kakinuma (1965) indicated much larger values than 0.01 for the friction coefficient, and, in particular, the results by the latter showed the tendency of the friction coefficient to decrease with the increase of wave period. On the other hand, laboratory experiments on wave decay over a sandy bottom were presented by Savage (1953), Inman (1962), and Zhukovets (1963). According to Savage (1953), the wave decay depends on the stage of the development of sand ripples compatible to the existing water wave, and the energy dissipation is smaller in the final, stable state compared with the initial, unstable state of sand ripples. This indicates the important difference between the artificial ripples and the natural sand ripples.
In the ordinary method of bottom stress estimation by means of the energy dissipation of waves, the phase difference between the reference velocity and the bottom stress is not explicity taken into account, although the energy dissipation is related to the in-phase components of the bottom stress and the reference velocity. It is well known that, for a laminar case, the bottom stress and the water particle velocity just outside the wave boundary layer have the phase difference of \pi\/4, and, for the turbulent case, too, the phase difference is expected if the boundary layer is very thin. Indeed, Lukasik and Grosch (1963) found a thin wave boundary layer in their field observation of ocean swell. Since the bottom friction and the phase difference between the bottom stress and the reference velocity are intimately related to the velocity profile near the bottom, it is very important to study the structure of the wave boundary layer. However, this kind of study is very rare, except for the observation in tidal currents, because of the difficulty of observation. In this regard, Jonsson’s contribution (1963) is important. According to his experiment on a turbulent wave boundary layer, the velocity profile near the bottom is logarithmic and the phase difference exists between the bottom shear stress and the reference velocity outside the frictional layer. Based on the experimental result, he proposed formulas for the friction coefficient and the boundary layer thickness. Later he (Jonsson,1967) summarized the empirical knowledge and attempted to delineate the viscous-turbulent and smooth-rough flow regimes in wave boundary layer for short period wave motions. Recently, Yalin and Russel (1966) also discussed the friction over a rough surface under waves and pointed out the importance of the phase difference between the reference velocity and the bottom stress, by indicating the dependence of the bottom stress on both the reference velocity and the surface slope.
To understand the frictional processes in the turbulent oscillatory flow, a theoretical model was presented by the author (Kajiura,1964, hereafter referred to as [1]). The basic idea was to consider the average state of turbulence over one wave period and to adopt the assumption of the eddy viscosity analogous to that for the steady turbulent flow.
This theory enables us to estimate the thickness of the frictional boundary layer and the velocity profile in an oscillatory flow.
Consequently, the frictional coefficient is given as a function of certain non-dimensional parameters constructed from known quantities of wave and bottom conditions.
In the present paper, some modification of the model presented in Paper[1] is attempted by introducing the concept of wall and defect layers, which seems to be adequate for the case of short period waves, in which the thickness of the bottom frictional layer is very thin compared with the total depth of water, and the results are compared with the experimental findings published so far.

2.Elementary considerations on the bottom friction

It may be instructive at first to consider some diagnostic relations derived from the equation of the oscillatory motion in the frictional layer. If we assume that the total thickness \delta\ of the bottom frictional layer is very small compared with the wave length of the potential wave considered, and the non-linear effect is negligible (except turbulence), the equation of oscillatory mean motion in the layer may be considered predominantly horizontal and is given by

式(2-1) [Fig.107_235_01]

where t is time, z is the vertical co-ordinate directed upwards with the origin at the bottom, u is the horizontal velocity, and \tau\ is the horizontal shear stress with \rho\ the density of water. U is the horizontal velocity near the bottom derived from the potential theory of waves and is given by

式(2-2a) [Fig.107_236_01]

where \partial\p/\partial\x is the horizontal pressure gradient near the bottom. Since the thickness \delta\ is assumed small compared with the wave length, it is reasonable to put

式(2-2b) [Fig.107_236_02]

The boundary conditions are

式(2-3a) [Fig.107_236_03]

式(2-3b) [Fig.107_236_04]

It should be noticed that in this formulation of the problem, the total depth of water H does not play a role in the processes within the frictional layer if \delta\ ＜ H, provided that \partial\p/\partial\x or U is given at the top of the frictional layer. In the extreme case of a shallow water wave, however, \delta\ may be limited by the depth of water H.
If we formulate the energy equation from (2-1), the energy dissipation E within the bottom frictional layer per unit area per unit time is given by

式(2-4) [Fig.107_236_05]

(2-4) may be transformed to

式(2-5) [Fig.107_236_06]

Partially integrating the first term and substituting (2-1) into the second term of the right hand side of (2-5), we have

式(2-6) [Fig.107_236_07]

where \tau\B is the bottom stress and the conditions (2-2b),(2-3b) are taken into account. If averaged over one wave period, the second term of the right hand side vanishes because of the periodicity of the motion, and we arrive at

式(2-7) [Fig.107_237_01]

where ＜＞ denotes the time average over one wave period.
Now, we define the vertical average velocity u by

式(2-8) [Fig.107_237_02]

and write the oscillatory quantities with the angular frequency\sigma\ in complex form:

式(2-9a) [Fig.107_237_03]

式(2-9b) [Fig.107_237_04]

式(2-9c) [Fig.107_237_05]

where ^ denotes the real amplitude, \varphi\ and \theta\ are the phase lead of u and \tau\B relative to U. The phase lead \theta\’ of \tau\B relative to u is given by

式(2-10) [Fig.107_237_06]

The integration of (2-1) with respect to z for the whole frictional layer yields

式(2-11) [Fig.107_237_07]

and the substitution of (2-9a, b, c) into (2-11) gives

式(2-12) [Fig.107_237_08]

式(2-13) [Fig.107_237_09]

where r=u/U. It is interesting to notice in (2-13) that the components of the potential velocity U and the mean velocity u in phase with the bottom stress \tau\B are equal, so that the mean energy dissipation ＜E＞ given by (2-7) is written in the form

式(2-14a,b) [Fig.107_237_10]

The two limiting cases, when i) the pressure gradient force is almost balanced by the frictional force and ii) the pressure gradient force is almost balanced by the inertial force, can be examined easily by means of (2-12) and (2-13). In the first case, which is analogous to a steady channel flow, we may consider \delta\/H→1, r→0, \theta\’→0, so that \tau\B/\rho\→U\sigma\H, \theta\→ \pi\/2. In the second case, which is encountered for relatively short period waves, we may take \delta\/H→0, r→1, so that \verphi\→0, \tau\B/\rho\→U\sigma\\delta\ ×{(1-r)^{2}+\verphi\^{2}},and tan\theta\=(1-r)/\verphi\.
Although \delta\ is a convenient length scale to discuss the structure of the frictional layer, it is advantageous to introduce more definite length scales in analogy with the displacement thickness and the scale of the defect layer commonly used in the steady turbulent flow. Furthermore, it is convenient to define the friction velocity u* and the friction coefficient C in parallel to the case of steady turbulent fiow. Thus, we define the following quantities:
the amplitude of the bottom friction velocity u*B:

式(2-15) [Fig.107_238_01]

the modified friction velocity u*:

式(2-16) [Fig.107_238_02]

the wave displacement thickness \delta\*:

式(2-17) [Fig.107_238_03]

the scale of the defect layer\Delta\:

式(2-18) [Fig.107_238_04]

the wave friction coefficient 1) C:

式(2-19) [Fig.107_238_05]

Taking (2-9a,c) into account,(2-19) may be written as

式(2-20) [Fig.107_238_06]

and

式(2-21) [Fig.107_239_01]

Or, by the combination of (2-16) and (2-19),

式(2-22) [Fig.107_239_02]

In terms of the friction coefficient C, the mean energy dissipation ＜E＞ given by (2-14a) becomes

式(2-23) [Fig.107_239_03]

In ordinary definition of the drag coefficient Cf/(=2 \tau\B/(IUl U)) without taking the phase difference between the bottom stress and the reference velocity into account, the wave energy dissipation due to bottom stress is given by

式(2-24) [Fig.107_239_04]

(i.g., Putnam and Johnson,1949) and the comparison of (2-23) and (2-24) shows that

式(2-25) [Fig.107_239_05]

By inspecting (2-1) and (2-9a, c) together with (2-17) and (2-18), it is easily found that

式(2-26a,b) [Fig.107_239_06]

and making use of the definitions of u*B and C, we may write (2-26a, b) in the form:

式(2-27a) [Fig.107_239_07]

式(2-27b) [Fig.107_239_08]

If we remember that U/\sigma\ is the amplitude of the horizontal movement of a water particle near the bottom when the frictional effect is absent, and C is of the order less than unity, the thickness \delta\* is found to be very small compared with the wave length of the potential wave under consideration so that the assumption of (2-2b) is justified.
Now,(2-26a) can be transformed to

式(2-28) [Fig.107_240_01]

because of the condition (2-2a). The left hand side of (2-28) is analogous to the pressure gradient parameter for the case of the steady turbulent boundary layer. For relatively shallow water when \delta\* is limited by the total water depth H,(2-28) becomes

式(2-29) [Fig.107_240_02]

which is analogous to the expression for the bottom stress in a steady channel fiow. In this case,(2-27a) gives C=(\sigma\H)/U and the phase \theta\ increases up to \pi\/2 with the increase of U/(\sigma\H). On the other hand, for the deep water case when \delta\* ＜＜ H, the main frictional effect is limited to a small portion of the water column and the frictional processes are more likely similar to the case of the turbulent boundary layer. This type of flow is realized for U/(\sigma\H) ＜＜ 1/C. In the present paper, we forcus our attention to the frictional layer of the boundary layer type. For U/(\sigma\H) ＞ 1/(10C), it is better to define the quadratic friction formula in terms of the mean velocity u instead of U, to conform to the ordinary friction formula of a steady channel flow for large values of U/\sigma\H). This case was already discussed in Paper[1].
To solve (2-1) in an explicit form, we need to have another equation connecting the velocity u and the stress \tau\. If we introduce the vertical eddy viscosity Kz by

式(2-30) [Fig.107_240_03]

the combination of (2-1) and (2-30) gives the basic equation

式(2-31) [Fig.107_240_04]

where \tau\/\rho\ is replaced by u*with the aid of (2-16).
For the case of a laminar oscillatory motion in deep water, Kz is the molecular kinematic viscosity \nu\ and the solution of (2-31) with the condition (2-3b) is given by

式(2-32) [Fig.107_240_05]

where

式(2-33) [Fig.107_241_01]

In terms of velocity u and the outer velocity U, we have, from (2-30), (2-32) and (2-3a),

式(2-34) [Fig.107_241_02]

and

式(2-35) [Fig.107_241_03]

The wave displacement thickness \delta\* now becomes

式(2-36) [Fig.107_241_04]

With a Reynolds number R defined by

式(2-37) [Fig.107_241_05]

the friction coefficient C derived from (2-19) and (2-35) is

式(2-38) [Fig.107_241_06]

If we consider that the important quantities, related to the bottom frictional processes in deep water for the case of a smooth bottom, are U,\sigma\,and \nu\, dimensional analysis indicates that the friction coefficient C in the turbulent case, too, can be expressed as a function of R alone.
For the case of a rough bottom, we usually introduce some length scale D and neglect \nu\, so that the friction coefficient is expected to be given by a function of U/(\sigma\D). To find the explicit functional form for C, we have to assume a suitable Kz. This will be done in the next section where the assumptions on Kz applicable to the case of a steady turbulent flow are translated into the mean state of the oscillatory turbulent flow.

1) In Paper[1],C(1) is defined in terms of u in the form

式(A) [Fig.107_238_07]

However, for the deep water case, u approaches U, so that the friction coefficient becomes identical in both forms,(A) and (2-19),if we replace u by u.

3. Assumptions on the structure of an oscillatory turbulent flow

(a) THE CASE OF A SMOOTH BOTTOM
Following the concept of the wall and the defect layers established for the case of a steady turbulent boundary layer (see, for example, Clauser,1956, Mellor and Gibson,1966), we assume the frictional layer of the oscillatory flow to consist of 3 parts: the inner layer, the overlap layer, and the outer layer. The inner and overlap layers are collectively called the wall layer in which the direct effect of the bottom boundary is felt, and the overlap and outer layers are the defect layer where the defect velocity profile is governed by the overall characteristics of the frictional layer. The characteristic of the overlap layer is that the velocity profile is expressed by both the wall and the defect forms. In the overlap layer for the case of the steady equilibrium turbulent boundary layer, Prandtl’s mixing length theory seems to be applicable (Mellor and Gibson,1966) so that the eddy viscosity can be written in the form Kz=Klu*Iz, with the definition of u*as \tau\/\rho\=Iu*Iu*. In the outer layer, eddy viscosity is assumed to be proportional to the outer velocity and the displacement thickness. In the inner layer, the effective viscosity varies from the molecular viscosity to the eddy viscosity corresponding to the base of the overlap layer in the range 5 ＜ u*z/\nu\ ＜ 30 (Mellor,1966).
In the reformulation from the eddy viscosity assumption valid for the case of steady turbulent flow to the case of oscillatory flow over a smooth bottom, we make some simplifications. Thus, we introduce the thickness DL of the viscous sub-layer, in a conventional way, by

式(3-1) [Fig.107_242_01]

We neglect the transition region between the viscous sub-layer and the overlap layer, so that, in the present case, the viscous sub-layer and the inner layer are identical. The eddy viscosity Kz then assumes the form:

式(3-2a-c) [Fig.107_242_02]

and

式(3-2d) [Fig.107_242_03]

where k and K are universal constants and d is the upper limit of the overlap layer. Although we have no reliable information on N, k, and K for oscillatory turbulent flow, we tentatively assume

式(3-3) [Fig.107_242_04]

which may be compared with those for the case of steady turbulent flow: N=11.6, k=0.41, K=0.016. Transforming (3-2d) by making use of (2-27a, b), we have

式(3-4) [Fig.107_243_01]

The critical condition, when the overlap layer disappears (DL=d) is given by R^{2}=(kN)/(KC), and the formal substitution of (2-38) yields R=240.
At this point, it is remarked that if the variation of stress with height is taken into account, it might be better to replace u*B in (3-1) and (3-2b, d) by the modified friction velocity u* with the definition \tau\/\rho\=u*u* instead of (2-16). However, this modification has not been attempted because of the uncertainty of the analogy of the eddy viscosity assumptions between steady fiow and oscillatory fiow. It is expected that when the thickness of the frictional layer becomes comparable to the thickness of the viscous sub-layer, the results obtained on the assumptions of (3-1), and (3-2a, b, c, d) are somewhat different from those derived from the modified version of the eddy viscosity assumptions. Also, the assumption (3-2d) in the outer layer is only the first approximation, and in reality the eddy viscosity may vary with height following some unknown law.

(b) THE CASE OF A ROUGH BOTTOM
For the case of a rough bottom, the eddy viscosity in the defect layer is the same as for the case of a smooth bottom, but, in the wall layer, we introduce the roughness length z0 (or Nikuradse’s equivalent sand roughness D, D=30z0), although, in reality, the equivalent roughness D is not well understood physically. There is also a question about the origin of the vertical coordinate, namely, the level of the equivalent bottom surface.
If we follow the ordinary formulation of the eddy viscosity in the wall layer, the overlap layer extends to z=z0, and Kz is given by

式(3-5) [Fig.107_243_02]

This eddy viscosity assumption is expected to be adequate for the case (\delta\*/D)^{2} ＞＞ 1. However, to include the case when (\delta\*/D)^{2} ＜ 1,we assume the alternative form of the eddy viscosity as follows:

式(3-6a,b) [Fig.107_243_03]

where DR is the height of the inner layer in which the eddy viscosity is constant. From the requirement that, for( \delta\*/D)^{2} >> 1, the velocity profile above DR derived on the assumption (3-6a, b) coincides with that derived on the assumption (3-5), we take (see§4,(4-40))

式(3-7) [Fig.107_244_01]

Furthermore, since the inner layer exists in a statistical sense only because of the presence of roughness elements, we assume

式(3-8) [Fig.107_244_02]

and, consequently, it follows

式(3-9) [Fig.107_244_03]

For small values of \delta\*/DR, the friction coefficient is essentially governed by Kz, in the inner layer, so that, by replacing \nu\ in (2-38) by αku*BDR, we have

式(3-10) [Fig.107_244_04]

Roughly speaking, the range of applicability of (3-10) is given by \delta\*/DR ＜ 1 which can be transformed, by means of (3-10), into U/(\sigma\DR) ＜ (\alpha\k)^{-2},and the substitution of numerical values yields U/(\sigma\z0) ＜ 685.
On the other hand, the condition of the vanishing overlap layer is given by U/(\sigma\DR) ＜ k/KC^{1/2}), and taking (3-10) into account, we may estimate the critical value of U/(\sigma\z0) somewhere around 10^{3}. The delineation of various flow regimes, such as the smooth or rough bottoms, and the viscous or turbulent flows will be discussed later (see§5,(c)).

4. Derivation of the velocity profile and the friction coefficient

(a) THE CASE OF A SMOOTH BOTTOM
The substitution of the eddy viscosity assumption (3-2a, b, c) into (2-31) yields

式(4-1a) [Fig.107_244_05]

式(4-1b) [Fig.107_244_06]

and

式(4-1c) [Fig.107_245_01]

For the deep water case when the bottom frictional layer is very thin
compared with the total depth of water, the boundary conditions at the
bottom and the outer edge of the frictional layer are

式(4-2a) [Fig.107_245_02]

式(4-2b) [Fig.107_245_03]

The conditions between the inner and overlap layers at z=DL and
between the overlap and outer layers at z=d are determined by the
continuity of u and u*. However, the continuity of u may be replaced
by the continuity of \partial\u*/ \partial\z in view of equation (2-1), so that we put

式(4-3) [Fig.107_245_04]

In the inner layer, a solution of (4-1a) is given by

式(4-4) [Fig.107_245_05]

where.A is an integration constant,[Fig.107_245_06] and the suffix L denotes
the quantity at z=DL. Then (2-30) yields

式(4-5) [Fig.107_245_07]

where the boundary condition (4-2a) is taken into account.
In the outer layer, a solution of (4-1c) with the condition (4-2b) is
given by

式(4-6) [Fig.107_245_08]

where [Fig.107_245_09] and the suffix d denotes the quantity at z=d.
From (2-30), then, the velocity u is

式(4-7) [Fig.107_245_10]

Where U is the reference velocity outside the frictional layer.
In the overlap layer, it is convenient to change the independent
variable z to y defined by

式(4-8) [Fig.107_245_11]

where d is defined by (2-18). Then,(4-1b) is reduced to Bessel’s differential equation and the solution can be written as

式(4-9) [Fig.107_246_01]

where y=ye^{-i\pi\/4}, and the suffixa denotes the quantity at z=a within the overlap layer. In later use of (4-9),we put a=d or DL according to the situation and, in particular, DL is abbreviated by L when used as a suffix. It is mentioned here that the substitution of (3-4) into (4-8) yields

式(4-10) [Fig.107_246_02]

Zn(y, c) is a cylindrical function defined by

式(4-11) [Fig.107_246_03]

where Jn(y) and Hn(2)(y) are the Bessel function and the Hankel function of the second kind of order n, respectively and c(=c1+ic2) is a complex constant. Zn (y, c),for n=0,1, satisfies the following relations:

式(4-12a) [Fig.107_246_04]

式(4-12b) [Fig.107_246_05]

For small values of y(y^{2}/4 ＜＜ 1), Zn(y, c) may be approximated by

式(4-12c) [Fig.107_246_06]

式(4-12d) [Fig.107_246_07]

where \gamma\=0.5722… (Euler’s constant).
The velocity u is obtained by the substitution of (4-9) into (2-30) with a suitable change of variables. With ua as the velocity at the level z=a, we have

式(4-13) [Fig.107_246_08]

Applying the boundary conditions (4-3) at z=d to (4-6) and (4-9) with a=d, the constant of integration c can be determined by the following equation

式(4-14) [Fig.107_247_01]

Substituting ud/u*d derived from (4-7) into (4-13) with a=d, replacing u*B/(Kd\beta\d) by its equivalent expression -2i/(kyd), and taking (4-14) into account, we may derive, in the overlap layer,

式(4-15) [Fig.107_247_02]

Now, applying the boundary conditions (4-3) at z=DL to (4-4) and (4-9) with a=DL, we have

式(4-16a) [Fig.107_247_03]

with

式(4-16b) [Fig.107_247_04]

and the substitution of uL/u*L obtained from (4-5) into (4-15) with a=DL, yields

式(4-17) [Fig.107_247_05]

It should be noticed that if the eddy viscosity Kd in the outer layer is assumed in the form of (3-2d) even after the overlap layer disappears, we can derive the same equation as (4-17) from (4-4) and (4-7), provided that

式(4-18) [Fig.107_247_06]

From (4-4) with z=0 and (4-17), it follows

式(4-19) [Fig.107_247_07]

Now, since we may put

式(4-20) [Fig.107_247_08]

we can express the friction coefficient given by (2-22) in the form

式(4-21) [Fig.107_248_01]

On the other hand, the Reynolds number R defined by (2-37) can be expressed as

式(4-22) [Fig.107_248_02]

Therefore, we may compute the friction coefficient C as a function of R alone.
The formula (4-19) may be simplified for two limiting cases;namely, i) tanh \beta\LDL→1 and ii) tanh \beta\LDL→ \beta\LDL. The range of transition in terms of \delta\L/DL from the condition i) to ii) is approximately given by

式(4-23) [Fig.107_248_03]

This condition can be interpreted that if the main part of the frictional layer is in the viscous sub-layer, the frictional law reverts to the laminar law. Indeed, for large values of \beta\LDL (small values of \delta\L/DL), (4-19) is reduced to

式(4-24) [Fig.107_248_04]

which is equivalent to (2-38) derived for the case of a laminar flow.
For small values of \beta\LDL (large values of \delta\L/DL),(4-4) and (4-19) are approximated by

式(4-25) [Fig.107_248_05]

式(4-26) [Fig.107_248_06]

where N and B are given by (3-1) and (4-16b) respectively. Furthermore, in this case, we may approximate B by

式(4-27) [Fig.107_248_07]

with the aid of (4-12c, d). Thus, the friction coefHcient is given by

式(4-28a) [Fig.107_248_08]

式(4-28b) [Fig.107_248_09]

where

式(4-29a) [Fig.107_249_01]

式(4-29b) [Fig.107_249_02]

These formulas (4-28a, b) are analogous to those given in Paper[1], (5-8),where the approximations of (4-25) and (4-27) were made from the beginning and c1 and c2 were neglected.
If yL is very small, further simplification of (4-28a, b) is possible.
By neglecting Q^{2} compared with (P-In yL/2)^{2}, we have

式(4-30a) [Fig.107_249_03]

式(4-30b) [Fig.107_249_04]

Taking (4-22) into account, we may transform (4-30a) into

式(4-31) [Fig.107_249_05]

(b) THE CASE OF A ROUGH BOTTOM
The difference of a rough bottom from a smooth bottom lies in the eddy viscosity assumption in the inner layer. Thus, the discussions for the case of a smooth bottom can be translated into the case of a rough bottom by the following transformation in (4-4),(4-5),(4-16a, b),(4-17) and (4-19):

式(4-32a) [Fig.107_249_06]

式(4-32b) [Fig.107_249_07]

式(4-32c) [Fig.107_249_08]

where the suffix R denotes the quantity at z=DR. Thus, in place of (4-21) and (4-22), we have

式(4-33) [Fig.107_249_09]

式(4-34) [Fig.107_249_10]

and. L(yR) is obtained from L(yL) by the transformation

式(4-35) [Fig.107_250_01]

(4-33) and (4-34) show that the friction coefficient C can be computed as a function of U/(\sigma\DR) alone.
In particular, for large and small values of y*R/(2,√\alpha\), approximate formulas become

式(4-36) [Fig.107_250_02]

式(4-37) [Fig.107_250_03]

in place of (4-24) and (4-26) respectively, and (4-36) is now equivalent to (3-10) as is expected.
If the ordinary eddy viscosity assumption (3-9) is adopted, we have, in general, from (4-15),

式(4-38) [Fig.107_250_04]

where B should be evaluated at yB, namely, at z=z0, instead of at yR(or z=DR). Since we assume that, for small values of [Fig.107_250_05] ,(4-37) is equivalent to (4-38), we have the condition

式(4-39) [Fig.107_250_06]

The substitution of the approximate formula for B into (4-39) yields

式(4-40) [Fig.107_250_07]

which is the condition for a assumed in (3-7). Taking this condition into consideration, the friction coefficient C can be written in the form (4-28a, b), provided that yL is replaced by yB and P by P’ where

式(4-41) [Fig.107_250_08]

This, in turn, means that the friction coefficient for the case of a smooth bottom can be expressed by the formula for a rough bottom, if we put

式(4-42) [Fig.107_250_09]

Taking (4-29a) and (4-41) into account, (4-42) becomes

式(4-43) [Fig.107_250_10]

Thus, the equivalent roughness D for the smooth bottom with the laminar thickness DL is given by

式(4-44) [Fig.107_251_01]

If yB is very small, we can deduce the equations similar to (4-30a, b) for a rough bottom. Taking (4-34) into account, it follows,

式(4-45) [Fig.107_251_02]

where

式(4-46) [Fig.107_251_03]

The formula for the length scale \Delta\ given by (2-27b) can be derived easily from (4-45) as

式(4-47) [Fig.107_251_04]

It is mentioned here that (4-45) and (4-47) are analogous to the semiempirical formulas proposed by Jonsson (1963) for the friction coefficient and the boundary layer thickness 2) (see§6,(a) ii). Futhermore, for the derivation of the friction formula for a smooth bottom, Jonsson (1967) assumed D〜0.435DL in place of (4-44).

5. Numerical results

(a) VERTICAl、 DISTRIBUTIONS OF SHEAR STRESS AND VELOClTY
At first, let us summarize the principal results of the preceding section for the case of a smooth bottom and express them in terms of y and yL for the convenience of numerical computation. The universal constants, k, K, N, and consequently yd are assumed known.
With reference to the bottom friction velocity u*B, the vertical distribution of stress given by (4-4),(4-6), and (4-9) are written, with the aid of (4-16a, b) and (4-20), as

式(5-1a,b) [Fig.107_252_01]

where

式(5-2a) [Fig.107_252_02]

式(5-2b) [Fig.107_252_03]

式(5-2c,d) [Fig.107_252_04]

式(5-2e) [Fig.107_252_05]

and

[Fig.107_252_06]

式(5-2f) [Fig.107_252_07]

The corresponding velocity distribution given by (4-5), (4-7) and (4-15) can be transformed into

式(5-3a,b) [Fig.107_252_08]

where

式(5-4a) [Fig.107_252_09]

式(5-4b,c) [Fig.107_252_10]

If yd is smaller than yL, the overlap layer vanishes and the stress and velocity distribution in the outer layer, yL≦y, given by (4-6), and (4-7), together with (4-18), become

式(5-5a) [Fig.107_252_11]

式(5-5b) [Fig.107_253_01]

For the case of a rough bottom, similar expressions can be given by replacing yL by yR. In this case, the definition of m and n in (5-2f) should be replaced by

式(5-6) [Fig.107_253_02]

To compute Zn(」,), n=1,2, defined by (4-11),we need numerical values of Jn(y), H^{(2)}n(y), and c.
Jn(y) and H^{(2)}n(y) may be found, for example, in the 「Table of Functions」by Jahnke and Emde (1945),pp 244-268, where Jn(y*) and H^{(2)}n(y*) are given. For small values of y, Zn(y) can be approximated by (4-11c, d). The constant c(=c1+ic2) determined by the condition (4-14) is shown in Fig.1 as a function of y, where it is seen that the amplitude [Fig.107_253_03] decreases rapidly with the increase of y, so that for large values of yd (yd≧2) , Zn(y) may be closely approximated by H^{(2)}n(y) and yd does not play an important role in the friction coefficient.
In the following computation, we have assumed

式(5-7a) [Fig.107_253_04]

so that it follows from (4-10), and Fig.1,

式(5-7b) [Fig.107_253_05]

If 2w look into the profiles of the stress and defect velocity in the defect layer (y > yL or yR),the important quantities are G(yL), G(yR), F1(y), and F2(y), among which G may be called the profile factor and FI and F2 are the universal profiles of the shear stress and the defect velocity, respectively. The amplitudes G and phases \psi\ of the profile factors G(yL) and y(R) are shown in Fig.2 as functions of yL or yR respectively, where it is seen that the variation of the amplitudes of G(yL) and G(yR) are relatively small and the deviation from \pi\/2 is only a few percent even for the condition of y1 or yR=yd. The universal profiles F1(y) and F2(y) are shown in Fig.3a, b where the amplitudes and phases are shown as a function of z/\Delta\(=y^{2}/10). It is seen that, in the lower part of the overlap layer, the variation of F1(y) is small and the corresponding variation of F2(y) is logarithmic. In the outer layer, F1(y) and F2(y) are given explicitly by

式(5-8a) [Fig.107_255_02]

and

式(5-8b) [Fig.107_255_03]

If the overlap layer vanishes, the stress and defect velocity are given by (5-5a) and (5-5b) respectively, and it is found that the profiles with respect to z/\Delta\ are the same as (5-8a),(5-8b),apart from the constant factors depending on yL or yR.
The amplitudes and phases of the factors S(yL,0),S(yR,0), T(yL, 0),and T(yR,0) are shown in Fig. 4a, b as functions of yL or yR, where it is seen that for small values of yL and yR, pairs of S(yL, 0) and S(yR,0),T(yL,0) and t(yR, 0) coincide and, for large values of yL and yR, pairs of S(yL,0),and T(yL, 0), S(yR, 0) and T(yR, 0) coincide with each other. A(yL) and A(yR) are shown in Fig.5 for the convenience of computing S(yL, y), S(yR, y),T(yL,y),and T(yR,y), where phases of A(yL) and A(yR) are equal for the same values of yL and yR.

(b) FRICTlON COEFFIENT
From (5-3a) and (4-21) the friction coefficient C for the case of a smooth bottom is given by

式(5-9a) [Fig.107_257_01]

and the Reynolds number R is expressed by

式(5-9b) [Fig.107_257_02]

For the case of a rough bottom, we have

式(5-10a) [Fig.107_257_03]

and the parameter U/(\sigma\z0) is given by (4-34), namely

式(5-10b) [Fig.107_257_04]

In Fig.6a, b and Fig.7a, b, the friction coefficient C for the cases of smooth and rough bottoms are shown as functions of R and U/(\sigma\z0) respectively. In these figures, it is clear that the amplitude C and phase \theta\ of the friction coefficient increase with the decrease of the parameters R or U/(\sigma\z0),and for small values of these parameters, C=1/R for the smooth case and C=1.70×{U/(\sigma\z0)}^{-2/3} for the rough case with the phase approaching \pi\/4 in both cases, as is already expected from (2-38) and (3-12). For large values of the parameters R or U/(\sigma\z0), the friction coefficient can be calculated easily from (4-31) or (4-42). Here it is remarked that if yd is assumed to be, say,1, instead of 0.7, the amplitude of the friction coefficient increases about 10 percent and the phase decreases about 20 percent for large values of the parameters R or U/(\sigma\z0).

(c) DELINEATION OF THE FLOW REGIMES
In the case of a steady turbulent pipe flow, a measure of whether the bottom is considered hydrodynamically smooth or rough seems to be given by the ratio D/DL centered around 1(Rouse,1937),and the range of transition may roughly be given by

式(5-11) [Fig.107_257_05]

According to Colebrook and White (1937), the beginning of transition is controlled by large grains, so that the lover limit of (5-11) may better be represented by the maximum grain size Dm as Dm/DL〜O.4. This is interpreted that if the roughness element is completely submerged in the viscous sub-layer, the bottom is hydrodynamically smooth and if the roughness element is much larger than the thickness of the viscous sublayer, the bottom is considered rough. In the present formulation, the eddy viscosity in the inner layer (3-10a) becomes equal to the molecular kinematic viscosity\nu\ at DR/DL=(\alpha\kN)^{-1}〜0.56. In terms of D, this condition gives D/DL〜1 and is consistent with the empirical result for steady turbulent flow (5-10).
For the transition between the viscous and turbulent flows we use similar reasoning, and consider the ratio of the wave displacement thickness \delta\* to the thickness of the viscous sub-layerDL. If (\delta\*/DL)^{2} ＜＜ 1, the main part of the frictional layer is submerged in the viscous sublayer, so that the bottom friction would be governed by a viscous law such as (2-38) for the case of a smooth bottom. Appreciable deviation of the friction coefficient from that governed by a viscous law wouid be expected if (\delta\*/DL)^{2} >> 1. Therefore, in both smooth and rough bottoms, the transition of the friction coefficient from the laminar type to the turbulent type may be found in some range of \delta\*DL, say

式(5-12) [Fig.107_260_01]

To show these criteria (5-11) and (5-12) in a figure, it is convenient to write

式(5-13a) [Fig.107_260_02]

式(5-13b) [Fig.107_260_03]

and express the amplitude of the friction coefficient C as a function of R with M as a parameter, as shown in Fig.8. Then, it is easy to delineate the zones of the smooth-rough, and viscous-turbulent transitions according to (5-10) and (5-12), since we may write

式(5-14a) [Fig.107_260_04]

式(5-14b) [Fig.107_260_05]

In Fig.8, it is found that the Reynolds number for the viscous-turbulent transition over a smooth bottom is given by

式(5-15) [Fig.107_261_02]

and the critical Reynolds number, R=113, determined by Collins (1963) seems to be in the middle of the above range. If we make use of (3-12),we may transform (5-14b), for the condition \delta\*/DR ＜ 1, as

式(5-16) [Fig.107_261_03]

and (5-12) becomes

式(5-17) [Fig.107_261_04]

On the other hand, Fig.8 also shows that the relation defined by (5-11) may be roughly approximated by 10^{2}≦ M ≦10^{3} in the range 10^{2} ＜ R ＜ 10^{4}.
Therefore, from a practical point of view, it appears possible to use

式(5-18) [Fig.107_261_05]

as a rough measure for the development of a rough turblent flow in the whole range of R.
For the smooth-rough transition, the condition

式(5-19) [Fig.107_262_01]

shown in Fig.8, it is found that (5-19) is not very different from (5-11), at least in the range 2×10^{2} ＜ R ＜ 10^{3}, so that (5-19) can be used as a criterion of smooth-rough transition in a limited range of R. Jonsson (1957) proposed essentially the same criterion as (5-11).
For the viscous-turbulent transition over a rough bottom, various criteria have been proposed on the basis of experimental data. However, the transition was defined differently and we are not certain which of the criteria proposed so far is most reliable. In terms of the present notations, Li (1954) and Manohar (1955)gave, from their oscillating plate experiments,

式(5-20a) [Fig.107_262_02]

式(5-20b) [Fig.107_262_03]

Later Kalkanis (1964) criticized the above equations and gave some evidence that

式(5-20c) [Fig.107_262_04]

for the whole range of the rough bottom. It is interesting to notice that the criterion (5-20c) is near the lower limit of the transition range defined by (5-18).
On the other hand, Vincent (1958) gave the relation

式(5-21) [Fig.107_262_05]

from the experiment on progressive waves for the period range of 1 to 5 sec. According to him, the beginning of turbulence over a rough bottom is found for very low values of M, say 15 to 30. If (3-10) is assumed to hold in this limiting case, it may be possible to approximate (5-21) roughly by

式(5-22a) [Fig.107_262_06]

since we can transform (5-22a) into

式(5-22b) [Fig.107_263_01]

Comparing (5-22b) with (5-21), the value of R_{\delta\*} seems to be somewhere around 15 to 20. In terms of R and C,(5-22a) is written as

式(5-22c) [Fig.107_263_02]

Jonsson’s criterion (1967) for the fully developed turbulence over a rough bottom is

式(5-23) [Fig.107_263_03]

where「\delta\」is the height scale of the boundary layer defined by Jonsson and is closely approximated by 0.23\Delta\ of the present paper. Transforming (5-23),we have

式(5-24a) [Fig.107_263_04]

or, in terms of R and C,

式(5-24b) [Fig.107_263_05]

If we substitute (3-10) into (5-24b),we may derive

式(5-24c) [Fig.107_263_06]

which is somewhat similar to the relation (5-20b),if the range of period is around 2 second.
The question of transition between the viscous and turbulent fiows over a rough bottom is not settled up to present, but it seems interesting to notice that if (3-10) is assumed to hold, most of the criteria proposed can be approximated by the form

式(5-25) [Fig.107_263_07]

with [Fig.107_263_08]


2) In particular, the wave boundary layer thickness 「\delta\」 defined by Jonsson is

式(B-1) [Fig.107_251_05]

Transforming (4-47) in the form of (B-1), we have, with constant factor n,

式(B-2) [Fig.107_251_06]

Neglecting the second term in the bracket in the left hand side as small, we have, by comparison of (B-1) and (B-2),

式(B-3) [Fig.107_251_07]

式(B-4) [Fig.107_251_08]

Since k=0.4, (B-4) yields ns=0.1 and (B-3) becomes

式(B-5) [Fig.107_251_09]

6.Comparison with experimental data

(a) FRICTION COEFFICIENT
i) Bagnold (1946) gave the drag coefficient「k」from his ingenious experiments using an oscillating plate. He used artificial ripples with two different pitches (p=10cm,20cm) but kept the pitch-height ratio p/h as 6.7/1. The ripple trough sections consist of circular arcs meeting to form sharp crests at an angle of 120°. Similar ripples were examined for a steady turbulent flow by Motzfeld (1937),who gave the equivalent roughness D as D=4h. Therefore, we assume the roughness Iength as z0=(4p)/(6.7×30)〜0.2,0.4cm. As noticed by Jonsson (1967), the drag coefficient「k」defined by Bagnold corresponds to 1/3 of the ordinary drag coefficient Cf and, from (2-25),(4/ \pi\)「k」corresponds to Ccos\theta\.
Since the range of Bagnold’s experiment was 152.5×U/(\sigma\zo) ≧ 12.5, we may compute the theoretical friction coefficient C and \theta\ easily from (3-10), where DR now becomes 2h. According to Bagnold,「k」becomes almost constant for U/(\sigma\p) ＜ 1. This result may be understood by the following model. If we assume the thickness of the inner layer DR or the equivalent roughness D decreases for U/(\sigma\p) ＜ 1 with the decrease of the relative amplitude of the horizontal motion of water U/\sigma\ compared with the ripple pitch p, we may put, in place of (3-6a),

式(6-1) [Fig.107_264_01]

Substituting this eddy viscosity in place of \nu\ of (2-38), we have, in place of (3-10),

式(6-2) [Fig.107_264_02]

and the phase \theta\ is \pi\/4. Thus, C becomes constant irrespective of U/(\sigma\z0) for U/(\sigma\p ＜ 1, if p/h is kept constant. If relevant numerical values are substituted in (6-2), we have C=0.125 and Ccos\theta\=0.089.

In Table 1,(4/\pi\)「k」estimated from Fig.3 of Bagnold’s paper is compared with the theoretical estimates where it is seen that the agreement between(4/\pi\)「k」and Ccos\theta\ is very good for U/(\sigma\p) ＞ 1 and fairly good for U/(\sigma\p) ＜ 1.
ii) Jonsson (1963) measured the velocity distribution in a turbulent oscillatory flow over an artificially roughened surface, by making use of the oscillating water tunnel (Test No.1). The roughness on the bottom surface was provided by a two-dimensional elevation of the triangular shape,0.6cm high and 1.7cm apart. The velocity was measured by a small propeller type current meter of 0.5cm diameter.
His experimental conditions were U/\sigma\=285 cm, U=211cm/sec, T=2\pi\/\sigma\ =8,39sec, and from the measured velocity profile, the roughness length was estimated to be z0 = 0.077cm. Later, Carlsen (1967) added new data (Test No.2) on the velocity distribution for the case U/\sigma\ =179cm, U=153 cm/sec, T=7.20 sec, z0=0.21 cm.
The analysis of Jonsson’s velocity profile will be given in the next sub-section, but here, in Table 2, the estimates of the friction factor (Jonsson’s friction factor is equal to 2C), phase \theta\, the energy dissipation ＜E＞ and the thickness of the wave boundary layer「\delta\」by Jonsson and Carlsen are compared with the prediction based on the present theory, according to which, C and \theta\ can be found as a functiion of [Fig.107_265_01].
In Table 2, it is found that the overall agreement is satisfactory but the theoretical values of the phase \theta\ are somewhat larger than the experimental estimates. It may be remarked that Jonsson’s semiempirical formula for the friction factor reads, in terms of the notations in the present paper,

式(6-3) [Fig.107_266_01]

On the other hand,(4-45) becomes, by substitution of relevant numerical values,

式(6-4) [Fig.107_266_02]


The difference between (6-3) and (6-4) is roughly about 10 percent in the friction coefficient. Fig.9 shows the comparison of the friction coefficients by Bagnold and Jonsson with the prediction based on the present model.
iii) Yalin and Russel (1966) measured bottom shear stresses due to long waves and proposed a formula

式(6-5) [Fig.107_266_03]

where \alpha\ and \beta\ are experimental constant,u is the velocity measured at a point 4.5cm (〜H/4) below the mean water level in a channel of depth H(=8.5cm), s is the surface slope and g is the acceleration due to gravity. The experimental ranges of periods T and wave heights Hmax are: 2min.＜ T ＜ 20min., 0.5cm ＜ Hmax ＜ 2.5cm and two kinds of equivalent roughness D (5mm and 3mm) are used. If we make use of the potential theory of waves, the experimental range was approximately given by 6 ＜ U/(\sigma\H) ＜ 147. Referring to the discussion ln §2,in this experimental range of U/(\sigma\ H),the depth H plays an increasingly important role with increasing U/(\sigma\H), so that it is better to compare the experimental result with theoretical prediction given in Paper[1]. If we assume u in the experiment is the mean velocity u in Paper[1], and approximate gs by (i\sigma\u), which is a good approximation, at least when the thickness of the frictional layer is thin compared with the total depth 3),(6-5) may be transformed to

式(6-6) [Fig.107_267_01]

where IuI is approximated by uu with u=8/3\pi\u. Therefore, the friction coefricient C 1) defined in Paper[1],(3-23) becomes

式(6-7a) [Fig.107_267_02]

式(6-7b) [Fig.107_267_03]

Yalin and Russel put

式(6-8a) [Fig.107_267_04]

式(6-8b) [Fig.107_267_05]

and mentioned that the diffrerences of \alpha\, \beta\ for the two relative depths were difficult to identify because of the scatter of data points. The values of C(1) and \theta\, computed from (6-7a, b) with the above numerical values may be compared with the theoretical prediction given in Paper [1]-Fig.2a, b, for the experimental range 5 ＜ u/(\sigma\H) ＜ 130 (in Paper[1], the water depth is written as zh instead of H). For small values of u/(\sigma\H),we can use Fig.7a, b of the present paper, which give the revised version of the friction coefficient of Paper [1] for the deep water condition.Taking the relations [Fig.107_267_06] into account, we may find C(1) and \theta\ from Fig.7a, b.
Sample estimates of the friction coefficients are given in Table 3, in which we see that the phases of the friction coefficient are reasonably close, but that the theoretical predictions of C(1) are consistently higher than the empirical estimates by (6-7a) This is the opposite trend to the case given in Table 2. However, taking the uncertainty of the replacement of (6-5) by (6-6) and the determination of \alpha\ and \beta\ into consideration, we may say that the overall agreement is fairly good. For u/(\sigma\H);≧10^{2}, the friction coefficient becomes constant, as shown in Fig.2a of Paper [1],and in agreement with one of the conclusion given by Yalin and Russel. However, for small values of u/(\sigma\H),(6-5) becomes inadequate for the expression of the bottom stresses. Their statement that for small values of T the second term of the expression (6-5) is important, is misleading, because \alpha\ and \beta\ defined by (6-5) can no longer be expected to be constant for small values of u/(\sigma\H).

(b) VELOClTY PROFILE
i) As already mentioned in the preceding sub-section, Jonsson (1963) measured the velocity distribution in a turbulent oscillatory flow. However, the oscillatory motions in the experiment were not perfectly sinusoidal due to various experimental limitations, so that his velocity data are subjected to harmonic analysis and velocity amplitudes and phases for the fundamental mode are extracted. The contribution of the higher mode oscillations is found to be about 10 percent of the fundamental mode at each level and seems to show some regularity in the vertical direction, but in the present analysis, only the fundamental mode oscillation is examined.
The experimental conditions for the fundamental mode oscillations are somewhat different from Jonsson’s original estimates and it is found that

式(6-9) [Fig.107_268_02]

where U is the mean value of velocity amplitude at three higher levels (height above the crest of a roughness element: 23cm,20cm,17cm) and z0 is estimated by the profile analysis of the velocity amplitude.
For the condition (6-9), the friction coefficient is

式(6-10) [Fig.107_269_01]

Now, we compare the observed velocity profile of the fundamental mode with the theoretical one, which is based on (5-3a, b) and can be expressed, with the aid of (5-6), in the form

式(6-11a,b) [Fig.107_269_02]

If we abbreviate (6-11a, b) by

式(6-12) [Fig.107_269_03]

it follows

式(6-13a) [Fig.107_269_04]

式(6-13b) [Fig.107_269_05]

Therefore, the theoretical values of amplitudes and phases of u/U as a function of z can be computed readily. Namely, for the condition (6-9) and (6-10), we have

式(6-14) [Fig.107_269_06]

The computed values of amplitudes and phases are shown in Fig.
10a, b together with the experimental value. The zero level of the experiment is adjusted, following Jonsson, to be 0.25 cm below the crest of the roughness element. From the figure, it may be concluded that the overall agreement between theory and experiment is good in both amplitude and phase, suggesting that the estimated roughness length z0 and the height of the overlap layer d are also reasonable. It is mentioned here that if we increase z0 from 0.05 cm to 0.077 cm as Jonsson estimated, the velocity profile moves upwards in the figure, and if we assume a larger value of K (or d) the level of the maximum amplitude moves upwards. The discrepancy of the detailed shape of the profiles may indicate the limitation of the present theory and to some extent the deficiency of the experiment in the lowermost levels.
ii) Kalkanis (1964) measured the velocity distribution in a turbulent oscillatory flow induced by the periodic oscillation of a flat plate at the bottom and presented data on the amplitude and phase relative to the movement of the plate, namely,f1 and f2 in (6-12). The empirical formulas proposed by Kalkanis seems to indicate that the experiment may not be in the range of a fully developed turbulence, because the length scale [Fig.107_270_03] plays a very important role, in contrast to our expectation that the important length scale does not depend on \nu\ for a fully developed turbulence over a rough bottom.
Now let us re-examine the data in the light of the present theory.
Since most of his data are found to be in the outer layer, we concentrate our analysis for the outer layer only, where we have from (5-8b) and
(6-11b),

式(6-15a) [Fig.107_270_04]

式(6-15b) [Fig.107_270_05]

where

式(6-15c) [Fig.107_270_06]

If yR exceeds yd, the expression for W becomes, according to (5-5b) and (5-9a),

式(6-15d) [Fig.107_270_07]

C,\theta\,G,\psi\,\Delta\, and yR can be determined from given experimental conditions, if the roughness length z0, is estimated from the actual roughness.
For the three dimensional roughness h(3) of grains, it is reasonable to assume z0=h(3)/30, but, for the two-dimensional roughness h (2), there is uncertainty about the estimation of z0. Furthermore, there is a problem of the zero-plane adjustment. Therefore, we omitted the data for the two-dimensional roughness and examined data for the smooth bottom and for the three dimensional roughness with the zero plane adjustment of 0.7h(3) below the crest level (Kalkanis assumed an adjustment of 0.2h(3).
It is mentioned here, that Jonsson (1966) made a somewhat analogous analysis of Kalkanis’data for the two-dimensional roughness.
For the case of a smooth bottom, the normalized amplitude factor f1/Amp [W] and the phase,f2 are shown in Fig.11a, b respectively, as a function of z/\Delta\, where the origin of z is displaced 0.06 cm (2×10^{-3} ft.) below the actual bottom to obtain better agreement between the data and the prediction. The theoretical line for f1/Amp [W] can be easily drawn by using (6-15a). The theoretical line for f2 given by (6-15b) is a linear function ofz/\Delta\ with the constant factor Phase [W] at z/\Delta\=0.
In the experimental range of Kalkanis, the values of phase[W] turn out to be about 0.5 but in Fig.11b, the constant factor is assumed to be 1.35 instead of 0.5.
Now,in Fig.11a, it is seen that the experlimental values for z/\Delta\ ≦ 0.2 are in agreement with the theoretical line. However, for larger values of z/\Delta\, the decrease of the experimental values with height is slower than the theoretical expectation, suggesting the increase of the eddy viscosity with height. Similar tendency can be observed in Fig. 11b where the increase of f2 with height becomes smaller than the theoretical expectation for z/ \Delta\ > 0.2. Furthermore, in view of the necessity to make the adjustment in both the zero plane level and the initial phase angle, some factors not included in the present model seem to exist in the inner layer for the case of a smooth bottom.
For the three dimensional roughness, the data are divided into two groups with M > 1500 and M ＜ 1500, since, for smaller values of M, the fully developed turbulence may not be realized as discussed in§4(c).
In Fig.12a, b and Fig. 13a, b the amplitude factor f1/Amp [W] and the phase f2 for M ≧ 1500 are shown,respectively,together with the theoretical lines. The theoretical line for f2, is drawn with the initial phase angle of 0.25 which is the average value of Phase [W] in the experimental range.
For M > 1500, the experimental values of f1/Amp [W] are in reasonable agreement with the theoretical expectation as seen in Fig.12a, but the phase f2 (Fig.12b) indicates more rapid increase with height than the theoretical values and still shows some dependence on M. For M ＜ 1500, the experimental values of the amplitude factor scatter in a wide range as shown in Fig.13a, and show very rapid decrease with height for M ＜ 1000, suggesting that the fully developed turbulence may not be established.
The corresponding phase f2 for M ＜ 1000 indicates a very slow increase with height as shown in Fig.13b.
Judging from Fig.12a, b and 13a, b, the fully developed turbulence in the outer layer for a moving plate experiment seems to be established for M > 2000, but the increases of the phase is somewhat faster than the theoretical expectation. For M ＜ 1000, it seems that the turbulence in the outer layer is distinctly different from the fully developed case and cannot be explained by the present model. It is remarked that the empirical formulas for f2 given by Kalkanis seems to give better agreement with data than the present prediction.

(c) ONSET OF SAND MOVEMENT UNDER WAVE ACTION
The bottom frictional stress and the movement of sand grains on the bottom of water are expected to be closely related. Following the concept of the critical tractive force producing sand movement in a steady channel flow, we may consider the following relation to hold for the onset of grain movement under wave action:

式(6-16) [Fig.107_274_01]

where (g*=g(\rho\’一 \rho\)/\rho\, g is the acceleration due to gravity, \rho\ and \rho\’ are the density of water and grain, respectively, D is the grain diameter, and \psi\ is the angle of repose. The complicated relations between forces acting on an individual grain and the mean bottom shear stress are absorbed in the factor I. Although it is essential to clarify the role of I for the understanding of the mechanics of grain movements, we discuss the problem only from the phenomenological point of view. For the case of a steady channel now, I seems to be a function of u*D/\nu\ and, for the fully developed turbulence (large values of u*D/\nu\), I is practically constant at about 0.05. Near the transition between the hydrodynamically smooth and rough bottom, namely near u*D/\nu\〜12, I reaches a minimum of about 0.03 and below this Reynolds number, I gradually increases up to about 0.15. (See, for example, Ishihara (Ed.), Applied Hydraulics, II(1),P.18, Fig.2.1.12,1957). Therefore, it is probable that I becomes a constant of the order 0.05 for a turbulent oseillatory fiow over a rough bottom.
Now(6-16) is transformed into

式(6-17) [Fig.107_275_01]

and, for a rough turbulent flow, C is given as a function of U/(\sigma\z0) as shown in Fig.7a. If we approximate C in the range 3×10^{2} ＜ U/(\sigma\z0) ＜ 10^{4} by

式(6-18) [Fig.107_275_02]

and substitute into (6-17), assuming D=30z0, we have

式(6-19) [Fig.107_275_03]

An empirical formula given by Bagnold (1946, Eq.4a) for the initial movement of sand grains is of the form

式(6-20a) [Fig.107_275_04]

with

[Fig.107_275_05]

This formula can be transformed to

式(6-20b) [Fig.107_275_06]

where the value of A^{2} in his experimental range (0.33cm ＜= D ＜ 0.09cm) is 0.417＜ A^{2} ＜ 0.232. Therefore, the expected value of I in (6-19) becomes 0.026 ＜ I ＜ 0.046 for the initial movement of grains. A similar empirical formula was presented by Sato and Tanaka (1962) for the general movement of grains. Their formula can be reduced to the form (6-19) where I/0.11=0.6, so that I becomes 0.066.
For the general movement of grains,more direct comparison between the experimental data and the theory is made by using the data by Manohar (1955), Goddet (1960), Sato and Tanaka (1962), Horikawa and Watanabe (1966).
As discussed in§5-(c), the lower limit of the rough turbulent flow may be delineated by M > 60 for R ＜ 10^{2}, and a Iittle larger value of M for R > 10^{2}. However, for simplicity of the discussion, we assumed M=60 as the criterion of smooth-rough transition in the whole range of the experiments by Goddet, Sato and Tanaka, Horikawa and Watanabe.
For the data of Manohar, we followed the criterion given by Manohar that [Fig.107_277_01] is the critical value of the smooth-rough transition. In terms of M, therefore, the data by Manohar for M ＜ 120 are included in the case of a smooth bottom.
In Fig.14a, the data for a rough turbulent flow are shown as a function of U/(\sigma\D) with the theoretical curve transcribed from Fig.7a, assuming D=30z0 and I=0.06. In Fig.14b, the data for a smooth bottom are shown as a function of R with the theoretical curve transcribed from Fig.6a, assuming I=0.11.
As seen in Fig.14a, b, the experimental results seem to be in fairly good agreement with the theoretical prediction with I=0.06 for a rough turbulent fiow and I=0.11 for a smooth fiow. This, in turn, means that the theoretical prediction of the friction coefficient may be realistic.
It is remarked here that the data of Goddet for M ＜ 60 and R ＜ 10^{2} do not show any systematic deviation from the data for M ＞ 60 if plotted in Fig.14a for a rough bottom. This suggests the difficulty of separating the smooth and rough bottom for small values of M and R. Horikawa et al.(1966) made a similar analysis for the initial and general movements of grains on the basis of the result given in paper[1]. However, since the friction formula given in paper[1] was not adequate for small values of U/(\sigma\z0), their conclusions should be revised accordingly.


3) For small values of U/(\sigma\H),u〜U. For large values of U/(\sigma\H), say, for u/(\sigma\H) =10^{2},u/U〜0.9 expi(0.45) from Fig.4a, b of Paper [1], and it may be better to approximate gs by i\sigma\ U. At any rate in this situation, \beta\gsH is negligible compared with \alpha\ lul u.

7.Summary and conclusion

The eddy viscosity assumptions [(3-2a, b, c),(3-6a, b)] analogous to those for the steady flow in a turbulent boundary layer are introduced by defining various quantities in the oscillatory flow, such as the modified friction velocity u*[(2-16)], the wave displacement thickness \delta\*[(2-17)], and the vertical scale of the defect layer\Delta\[(2-18)]. The boundary layer is divided into three parts: inner, overlap, and outer layer, and two universal constants k(=0.4) and K(=0.02) are assumed. In particular, the effective viscosity in the inner layer is assumed somewhat differently from that commonly used for the steady flow and we introduce a constant Reynolds number N(=12) related to the viscous sub-layer and the roughness height DR(=15z0) (§3).
Making use of the eddy viscosity assumptions, the linearized equation of oscillatory motions in the boundary layer is solved for the case of a boundary layer thin compared with the total depth of water. The vertical distributions of shear stress and velocity, and also the friction coefficient can be derived straightforwardly(§5). In particular, in the defect layer, the shear stress and defect velocity can be expressed in universal forms, F1, and F2,(Fig.3a, b), if they are suitably nondimensionalized by means of Gu*B and \Delta\. The parameter G and the friction coefficient C, both of which are defined in complex forms, are functions of [Fig.107_278_01] only and are shown in Fig.2, Fig.6a, b, Fig.7a, b.
The comparison of the present theory with empirical results shows reasonable agreement in both the velocity profile and the friction coefficient (or energy dissipation). The critical tractive force for the movement of sand grains under wave action can also be formulated parallel to the case of steady turbulent flow, by making use of the present formula of the friction coefficient (§6).
The gross features of the boundary layer under waves are thus simulated satisfactorily. However,if we look into the details of the structure of boundary layer, it is obvious that the discussion based on a model assuming the average state of turbulence over one wave period cannot be sufficient. The instantaneous state of turbulence is completely left out of the consideration, and we are not certain how the instantaneous state of turbulence and the average state of turbulence as assumed in the present paper can be related. For the development of the study of a wave boundary layer, it seems essential, at first, to understand the generation and maintenance of turbulence on a more rigorous basis.
As for the rough surface, our knowledge is far from satisfactory.
Among many problems related to the rough surface, we need more definite knowledge about the roughness length z0 or its equivalent, and the location of the zero plane, in view of the fact that for small values of U/(\sigma\z0), the scale of the boundary layer \delta\* becomes comparable to the height of roughness elements.

Reference

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BRETSCHNEIDER.C. L.,1954, Field investigation of wave energy loss of shallow water ocean waves, Beach. Erosion Board, Tech. Memo. No.46,1-21.
CARLSEN, N. A.,1967, Measurement in the turbulent wave boundary layer, Basic Res., Progress,Rep. No.14, Coastal Engineering Lab., Tech. Univ. of Denmark.
CLAUSER, F. H.,1956, The turbulent boundary layer,Advances in Applied Mechanics, 4,1-51,Academic Press, New York.
COLEBROOK, C. F., and C. M. WHITE,1937, Experiments with fluid friction in roughened pipes, Proc. Roy. Soc. London A,161,367-381.
COLLINS, J.1,,1963, Inception of turbulence at the bed under periodic gravity waves,J. Geophys. Res.,68,6007-6014.
GODDET, J.,1960, Etude du debut d’entrainment des materiaux mobiles sous l’action de la houle, La Houille Blanche, No.2,122-135.
H0RIKAWA, K. and F. WATANABE,1966, A consideration on sand movement under wave action, Proc. 13th Conf. Coastal Engineering in Japan,126-134 (in Japanese).
INMAN, D. L.,1962, Flume experiments on sand transport by waves and currents, Proc. 8th Conf. Coastal Engineering, Mexico City,137-150.
ISHIHARA (Editor),1957, Applied Hydraulics, II(1), Maruzen Co., Tokyo,(in Japanese).
IWAGAKI, Y., and T. KAKINUMA,1964, Some observations on the modification of wind waves and swells in shallow water, Proc.11th Conf. Coastal Engineering in Japan, 49-55, (in Japanese).
IWAGAKI, Y., Y. TSUCHIYA, and H. CHEN,1967,0n the mechanism of laminar damping of oscillatory waves due to bottom friction, Bull. Disas. Prev. Res.Inst., Kyoto Univ., 16 (3),No.116,49-75.
JONSSON,I. G.,1963, Measurements in the turbulent wave boundary layer,10th congress, I.A. H. R., London、1,85-92.
JONSSON,I. G.,1966,0n the existence of universal velocity distributions in an oscillatory, turbulent boundary layer, Basic Res. Progress Rep. No.12, Coastal Engineering Lab., Tech. Univ. of Denmark.
JONSSON,I.G.,1967,Wave bundary layers and friction factors, Proc.10th Conf. Coastal Engineering, Tokyo, 127-148.
KAJIURA, K.,1964、 On the bottom friction in an oscillatory current, Bull. Earthq. Res. Inst.,42, 147-174.
KALKANIS, G.,1964, Transportation of bed material due to wave action,U.S. Army Coastal Engineering Research Center, Tech. Memo. No.2.
LI, H.,1954, Stability of oscillatory Iaminar flow along a wall, Beach Erosion Board, Tech.Memo. No.47. .
LUKASlK, S. J., and C. E. GROSCH,1963, Pressure-velocity correlations in ocean swells,J. Geophys. Res., 68, 5689-5699.
MANOHAR. M.,1955, Mechanics of bottom sediment due to wave action, Beach Erosion Board, Tech. Memo. No.75.
MELLOR, G. L., and D. M. GIBSON,1966, Equilibrium tubulent boundary layers,J. Fluid Mech.,24, 225-253.
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5. 波に伴なう水底摩擦境界層について 地震研究所梶浦欣二郎

　水粒子の周期運動によって水底に生ずる乱流境界層に関して,前論文(梶浦,1964)では,摩擦層が水面に及ぶ様な周期の長い波の場合を主として取扱ったが,ここでは境界層厚が水深にくらべて小さい場合を考える.
　まず,流速u,境界層外流速 \sigma\ ,摩擦応力 \tau\,水の密度 \rho\ から,摩擦速度u*(\tau\/\rho\=u*u*),排除厚 \delta\*( [Fig.107_280_01]) 等を定義する．ここで，角速度 \sigma\ で変化する量は複素数であらわし,振幅には へ をつける.(§2)
　次に,定常流の場合にならって,境界層を3層に分け,それぞれに対応する定数N(=12),k(=0.4),K(=0.02)を導入して有効粘性係数Kzを次の様に仮定する.

式(1-a-c) [Fig.107_280_02]

ここで,\nu\ は分子動粘性係数,u*Bは底面摩擦速度の振幅, zは水面からの高さ, D Lは層流底層の厚さ (u*BDL/\nu\=N) であり,中間層の上限の高さdは, [Fig.107_280_03] からきめる.
底面が粗な場合には,内層(1-a)の代りに,

式(1-d) [Fig.107_280_04]

とおき,z0 (Nikuradse の相当粗度 D=30z0)を粗度長さとすると,DR=15z0, 1/\alpha\=ln(DR/z0)と仮定する.(§3)
境界層内の運動方程式は

式(2) [Fig.107_280_05]

と近似出来るから,Kzを利用すると (2)を解いて流れの様子は完全にきめられる.(§4)高さの尺度 \Delta\ ([Fig.107_280_06]) を利用すると,中間層,外層での規格化された速度欠損(U 一u),承び応力 (u*) の分布は (z/\Delta\) のみの函数として表示出来る.また,複素摩擦係数Cを \tau_{B}\/\rho\=CUU で定義すると,その振幅C及び位相 \theta\ は水底が滑面か粗面かによって,それぞれ[Fig.107_280_07] 又は U/(\sigma\z0) の函数として計算出来るし,単位時間,単位面積あたりの底面摩擦による平均エネルギー消耗率 ＜E＞ は,

式(3) [Fig.107_280_08]

であらわせる.特に前論文との相違は R 又は U/(\sigma\z0) の小さいところであらわれ,滑面の場合には R ＜= 100 で C〜e^{i\pi\/4}/R，粗面の場合には U/(\sigma\z0) ＜= 700 で [Fig.107_280_09] となる.
粗面一滑面の遷移領域を定常流の場合にならって [Fig.107_281_01] と考えるとすれば,層流一乱流の遷移領域は, [Fig.107_281_02] と考えるのが適当と思われ,滑面についてのCollins (1963)の実験値は \delta\*/DL=0.85である.(§5)
理論流速分布はJonsson (1963)の実験値と振幅,位相ともにかなりよい一致を示し,摩擦係数はBagnold (1946)の実験値を説明し,また波による底面砂の移動限界についても満足すべき説明を与える.(§6)

BULLETIN OF THE EARTHQUAKE RESEARCH INSTITUTE Vol.48 (1970), pp.835-869 45. Tsunacmi Source, Energy and the Directivity of Wave Radiation. By Kinjiro KAJIURA, Earthquake Research Institute. (Read June 23,1970. -Received July 31,1970.)

Abstract

Based on the long wave approximation, a formula to compute the tsunami wave form on a flat bottom generated by a broad crustal deformation is introduced. The energy transfer from the sea bottom to the water is examined in relation to the duration of the bottom movement. If the duration is less than several minutes, the deformation may be considered to be abrupt as far as the tsunami is concerned. However, if the movement is completed in a few second, the energy transferred to the compressional water waves might be larger than the tsunami energy.
The two-dimensional propagation of tsunami generated by a simple source of rectangular shape is explained by the superposition of various elementary waves. If the deformation of the bottom is a simple uplift or depression, the wave decay with distance near the source deviates from that due to geometrical spreading from a point source: in the direction of the major axis, the decay is faster, indicating a strong effect of diffraction, and in the direction of the minor axis, the decay is slower, indicating the approach to the one dimensional propagation. The variation of the directivity coefficient is conspicuous only for azimuths less than \pi\/4 from the direction of the minor axis.
For long distances from the source, the directivity of energy radiation is proportional to the square of the lateral dimension of the source with respect to the direction of the observing point. The nondispersive and dispersive wave forms are compared for long distances from the source, and it is concluded that the long wave approximation may be applicable for [Fig.107_282_01] being a parameter related to the dimension of the source, distance and the depth of water.

1. Introduction

The generating area of a large tsunami is elongated more and more as the magnitude of the earthquake increases, and the existence of the directivity of wave radiation from a tsunami source has been recognized for a long time. For the causes of the directivity, the following two factors are considered to be most important: 1) the effect of the shape of a source, 2) the effect of the bottom topography in the vicinity of the source. Field evidences for the directivity of tsunamis were presented by Miyoshi (1955) and Hatori (1963).
As to the shape factor, several studies were undertaken both experimentally (Takahasi and Hatori,1962) and theoretically (Momoi,1962, 1963; Kajiura,1963). As to the topographic factor, Miyoshi (1968) proposed a simple theory based on wave refraction. Recent numerical experiments by Aida (1969) for the Niigata tsunami revealed clearly the importance of these two factors in the consideration of tsunami propagation.
In spite of the importance of the directivity of energy radiation of a tsunami, adequate consideration has not been paid to this effect in the estimation of the total tsunami energy by a conventional method.
If various errors in the estimation of tsunami energy are taken into account, the uncertainty of the estimated energy may amount to a factor of 10 for extreme cases. If the magnitude scale of the tsunami is to be defined in relation to the tsunami energy, more accurate determinations of tsunami energy seem to be essential (Soloviev,1969).
Here, a problem arises as to the attenuation of the wave height with distance. The region of tsunami generation is mostly on the periphery of the ocean, and the nearest coast which is most severely affected by a large tsunami is located in the distance comparable to the dimension of the tsunami source. In this region, therefore, the propagation characteristics of the tsunami would be like the one-dimensional one. For a moderately long distance, R, from the source the attenuation of the wave height may be approximated by [Fig.107_283_01], in accordance with the geometrical spreading if the gravity wave dispersion is negligible. However, the proportionality factor H0 would be a function of the direction because of the directional difference of wave radiation. In the estimation of the attenuation characteristic of the wave height with distance from field data, this is one of the factors responsible for the large scatter of data points by a factor of 3 or so in the wave height vs distance graphs (Soloviev,1965;Iida,1966). In actual cases, the situation is further complicated by the wave guide effect of a continental shelf.
In the present paper, a simple ease of tsunami generation on a flat bottom is examined theoretically based on the long wave approximation to shed some light on the wave propagation characteristics and the energy radiation pattern from an elongated tsunami source of a large scale.
Although various studies of this kind were made in the past and considerable knowledge has been accumulated, it seems worthwhile to discuss the problem from a unified standpoint.
At first, the effect of duration of the bottom movement on the total energy transferred to the Water body is examined. Next, the radiation pattern of the wave energy together with the directional difference of wave height is discussed. Lastly, for long distances from the source, asymptotic wave forms are computed and the validity of the long wave approximation is discussed by comparing the non-dispersive wave form with the dispersive one.

2. Wave generation by the bottom deformation

The generation and propagation of a tsunami in the sea of constant depth under various source conditions were already discussed by the author (Kajiura,1963, hereafter referred to as Paper I). We consider now the wave directivity for a tsunami source of large dimensions under the linear long wave approximation. The limitation of the long wave approximation will be discussed in Section 7.
The linear wave equation for the tsunami generation is,

式(2-1) [Fig.107_284_01]

where \zeta\ is the free surface elevation from the undisturbed level, c^{2}=gh with g the acceleration due to gravity and h the constant depth of water. Here t is time, \nabla^{2}\ is the Laplacian operator in the horizontal co-ordinates, and wB is the vertical velocity of the bottom movement, taken positively upwards.
A Green’s function G in the infinite domain for the wave equation (2.1) is given by (see, for example, Morse and Feshbach,1953),

式(2-2) [Fig.107_284_02]

where r is the position vector, the suffix 0 denotes quantities related to the source, and H(x) is a unit step function with H(x)=0, for x ＜ 0, and H(x) =1,for x ＞ 0.
Making use of (2-2), the elevation \zeta\ in the infinite domain can be written as,

式(2-3) [Fig.107_284_03]

where S covers the source area with ds0 the surface element.
Now we assume that wB is constant for a certain duration, 0 ＜ t ＜ T, and zero otherwise and the total bottom deformation is D(r0).
Then, it follows:

式(2-4) [Fig.107_285_01]

where \delta\(x) is Dirac’s delta function.
Substituting (2-4) into (2-3) and taking the origin of the co-ordinates at the observing point, we have

式(2-5) [Fig.107_285_02]

where

式(2-6) [Fig.107_285_03]

and

[Fig.107_285_04]

In the later discussion, we omit subscripts 0 for the source and i for \eta\ and \tau_{i}\ for simplicity, unless otherwise noted.
In the polar co-ordinates (r, \theta\), we may write (2-6) in the form

式(2-7) [Fig.107_285_05]

where

式(2-8) [Fig.107_285_06]

In (2-7) and (2-8), the integral is limited within the source region so that the integral limits,(r1・r2) and (\theta\1・\theta\2) are determined by the geometry of the source. Here we may call r\Theta\ as a directive source function relative to the observing point.
For a limiting case when T→0, we may transform (2-5) together with (2-7) into

式(2-9) [Fig.107_285_07]

If the observing point is in the source region (r1=0),\Theta\(0) in the right hand side is preserved.
If the source region (source dimension, L) is very far from the observing point (distance to the center of the source, R), namely L/R ＜＜ 1, we may approximate, by taking the rectangular co-ordinates with the x-axis in the direction of the source center.

[Fig.107_286_01]

Therefore,(2-6) may be approximated by

式(2-10) [Fig.107_286_02]

where

式(2-11) [Fig.107_286_03]

The limits of integration (x1,x2),(y1, y2) in (2-10) and (2-11), respectively, are determined by the geometry of the source and (x2-x1),(y2-y1) are of the order L with L/R ＜＜ 1. In this case the directive source function r\Theta\ in (2-7) is replaced by W(x). For a limiting case when T→0,we can derive in place of (2-9),

式(2-12) [Fig.107_286_04]

It is remarked that the approximations (2-10) and (2-12) deteriorate for the proximity of x=x1 or x2 unless W(x) approaches zero. Although the directive source function r\Theta\ always approaches zero as r→r1 or r2, the corresponding directive source function W(x) does not necessarily exhibit this feature unless (y1-y2) approaches zero as x→x1 or x2.
For a simple geometry of the source, the directive source function can be expressed by an analytical formula and the generated wave form can be computed easily either analytically or numerically. Source models used in the present paper are
Model A Rectangular shape with uniform (A1) or linear variation
(A2) of the bottom deformation,
Model B EIIiptic shape with uniform (B1), elliptic (B2),or parabolic
(B3) deformation of the bottom,
Model C Circular shape with the uniform bottom deformation.
Model C is a special case of Model B and widely used for the study of the tsunami generation problem in the past.

3.Computation of wave forms for rectangular models

To compute the wave form due to an uniform deformation (Model A1) or due to a linear variation of the deformation (Model A2) in a rectangular area of the bottom, it is convenient to make use of the method of superposition. An elementary solution can be taken in various way, but, here, we take an elementary source geometry as shown in Fig.1. The origin of the co-ordinate is at the observing point, and the nearest corner of the source, P, is at (d,\varphi\) or (xp,yp) with the source region in xp ＜ x ＜ \infty\ and yp ＜ y ＜ \infty\.

a) Uniform deformation D0
In Fig.1,we divide the source into two regions (1) and (2), and denote the solutions given by (2-7) as \eta\(1) and \eta\(2) for the regions (1) and (2), respectively. Then,\eta\(2) can be obtained from \eta\(1) by replacing \varphi\ by \pi\/2-\varphi\, or exchanging xp, and yp. Let us now write the combination of \eta\(1) and \eta\(2) as \eta\(\tau\/xp,yp), namely,

式(3-1) [Fig.107_287_01]

For the region (1), the directive source function can be written as

式(3-2) [Fig.107_287_02]

where r is the radial distance, and H(r-d) is a unit step function.
Substituting (3-2) into (2-7), and performing the partial integration, we have

式(3-3) [Fig.107_287_03]

(3.3) can be transformed into

式(3-4) [Fig.107_287_04]

where

式(3-5) [Fig.107_287_05]

and

式(3-6) [Fig.107_287_06]

If the apex P is in the second quadrant, we have

式(3-7) [Fig.107_288_01]

where

式(3-8) [Fig.107_288_02]

If the apex P is in the fourth quadrant, the solution is given by exchanging xp and yp in (3-7), and if the apex P is in the third quadrant, we have

式(3-9) [Fig.107_288_03]

Thus, the solution \eta\ for a rectangular source in a general position (the co-ordinates of 4 corners are (x1, y1),(x2, y1),(x2, y2),(x1, y2) with (x2 > x1, y2 > y1) can be expressed by,

式(3-10) [Fig.107_288_04]

The wave form \zeta\(\tau\) can be given by (5).
For the case of the instantaneous deformation, the wave form for the region (1) is derived directly from (2-9) as

式(3-11) [Fig.107_288_05]

and the resultant solution \zeta\(\tau\Ixp, yp) can be constructed as in the case of \eta\(\tau\Ixp, yp).

b) Linear variation of deformation
The linear variation of deformation is given by D=\alpha\(x- \beta\) in the region shown in Fig.1. In this case, the contribution of the constant part (-\alpha\\beta\) is the same as in the case of an uniform deformation provided that D is now put (-\alpha\\beta\). For the linearly varying part, the directive source function is given by

式(3-12) [Fig.107_288_06]

where x=rcos\theta\, \theta\1 and \theta\2 are defined by sin\theta\1 =yp/r and cos\theta\2=xp/r.
The integration gives

式(3-13) [Fig.107_288_07]

and

式(3-14) [Fig.107_288_08]

Substituting (3-13) into (2-7) and performing a partial integration, it follows:

式(3-15) [Fig.107_289_01]

where d^{2}=xp^{2}+yp^{2}, and J(\tau\I(d, xp) is given by (3-6).
If the apex P is in the second quadrant, the solution is

式(3-16) [Fig.107_289_02]

because of the antisymmetric character of the deformation with respect to the y-axis. The solution for the apex P in the fourth quadrant is given by

式(3-17) [Fig.107_289_03]

If P is in the third quadrant, we may write

式(3-18) [Fig.107_289_04]

For the case of the sudden deformation (T→0), the substitution of
(3-14) into (2-9) gives

式(3-19) [Fig.107_289_05]

and the solution when P is not in the first quadrant is constructed similarly to the case of \eta\(\tau\Ixp, yp).
To compute the wave form due to the combination of the linear deformation of the bottom, we should superpose the solutions for each elementary part of the bottom deformation, which, in turn, can be expressed by the combination of elementary solutions given in this section.

4. Energy exchange associated with the bottom deformation between the solid bottom and the overlying water

a) Static energy and tsunami energy
The energy flux dE/dt to the overlying water by the vertical movement of the bottom can be computed from

式(4-1) [Fig.107_290_01]

where pB is the bottom pressure and S is the total area of the deformation.
The total flux of energy Et to the water in the time interval 0 ＜ t ＜ T is

式(4-2) [Fig.107_290_02]

Now, under the long wave approximation, the pressure may be considered as hydrostatic:

式(4-3) [Fig.107_290_03]

where \rho\ is the density of the water, \zeta\ and \zeta\B are the vertical displacement of the water surface and the bottom, respectively. Substituting this pressure in (4-2), we may write,

式(4-4) [Fig.107_290_04]

where

式(4-5) [Fig.107_290_05]

and

式(4-6) [Fig.107_290_06]

Taking the relations \partial\\zeta\B/\partial\t=wB,and \zeta\B=D for t > T, into account,(4-5) is transformed into

式(4-7) [Fig.107_290_07]

where V is the total volume of the bottom deformation; [Fig.107_290_08].
The energy flux Es is independent of the time characteristics of the tectonic movement and may be called the static energy exchange. If the total volume of the deformation is zero or negative, it is evident that Es is negative, indicating the energy transmitted from the water column to the solid bottom. On the other hand, ED depends on the time characteristics of the deformation and is always positive, so that we may call ED as the dynamic energy of a tsunami. This kind of separation of the energy exchange is originally envisaged by Miyoshi (1954) who proposed the concept of the efficiency of tsunami by defining the efficiency as ED/Es.
It is immediately clear in (4-6) that: 1) if the deformation is very slow (quasi-static),\zeta\→0 and ED→0, and 2) if the deformation is instantaneous \zeta\〜\zeta\B and [Fig.107_291_01]. In case(2), the total energy transferred to the water is

[Fig.107_291_02]

showing that Et is equivalent to the potential energy to transfer the water mass \rho\V from the bottom to the surface. We denote the dynamic energy in the case (2) as EDO and investigate the dependence of the ratio ED/EDO on the duration T of the bottom movement.
Taking (2-5) into account, we may write (4-6) in the form:

式(4-8) [Fig.107_291_03]

Therefore,ED can be computed numerically,if \eta\ is found from (2-7) for 0 ＜ t ＜ T. For a uniform cicular model (Model C; diameter A, deformation D0), a formula to compute \eta\ given by Takahasi (1942) is used(Eq. (36) in Takahasi’s paper where \xi\ is used in place of \eta\). For a uniform rectangular model (Model A1; side lengths A and B, deformation D0),\eta\ is computed by the method given in§3. For Model A1, ED0=(1/2)\rho\gD0^{2}AB and for Model C, ED0=(1/8)\rho\gD0^{2}\pi\A. A limiting case of B→\infty\ for a rectangular model corresponds to the one-dimensional case and the energy ED for unit width can be derived easily (see Appendix A).
Fig.2a shows the variation of ED/EDO with cT/A. It is seen that a rectangular model with A=B is roughly similar to a circular model and with the increase of B, the variation of ED/ED0 approaches that for an one dimensional model. In general, the decrease of ED/E0 is gradual for small values of cT/A. For large values of cT/A, the decrease of ED/EDO is proportional to (cT/A)^{-n} with n=1 and 1.64 for the one dimensional and circular models, respectively, as shown in Fig.2b.

b) Compressional wave energy and tsunami energy
In principle, if the bottom is uplifted (wB > 0), the energy is, at first, transferred from the solid earth to the overlying water by generating compressional waves in the water and then partly converted to gravity waves. To examine the relative importance of sound waves and gravity waves in terms of the generated wave energy, a simple model is considered, taking into account the fact that the gravity wave energy ED can be approximated by ED0 for small values of cT/A(＜ 0.2).
We confine the attention to the vertical direction only by assuming a very large horizontal dimension of the bottom uplift. If we assume c/v ＜＜ 1and wB/v ＜＜ 1, the sound pressure p* at the bottom due to the bottom movement wB is given by

式(4-9) [Fig.107_292_01]

where the characteristic impedance of the sound wave is \rho\v with v the sound wave velocity. Therefore, the sound intensity I is,

式(4-10) [Fig.107_292_02]

and the total energy Ec transferred as sound in the time interval t is,

式(4-11) [Fig.107_292_03]

The pressure change due to a reflected sound wave from the water surface interferes with the bottom after t=2tc where tc is the travel time of the sound from the bottom to the free surface and

式(4-12) [Fig.107_292_04]

Since this reflected wave is a rarefaction wave if the initial wave is a compressional one, the resultant bottom pressure is p*=-\rho\vwB and the transfer of energy takes place from water to the solid bottom as far as the sound wave is concerned, in the time interval 2tc ＜ t ＜ 4tc. After the termination of the bottom movement, the energy exchange between the solid earth and the water stops if the solid earth is perfectly rigid.
The time sequence of the sound pressure p*, the variation of the free surface with both components of the sound wave and the hydrostatic displacement, the total sound energy Ec and the gravity wave energy ED are shown in Fig.3, where for the clarity of Fig.3c, the hydrostatic energy is shown for a fictituous value of c/v (=1/10). The variation of the free surface given in Fig.
3b is similar to the result obtained by Miyoshi (1959) in the general discussion of the generation of tsunamis in compressible water.
The energy Ec of the sound wave at the time of the termination of the bottom movement depends on the ratio T/tc because Ec oscillates between 0 and Ecmax (=2\rho\wB^{2}h). On the other hand, the total energy ED of a gravity wave generated by the bottom movement is

式(4-13) [Fig.107_293_01]

The ratio Ec/ED can be given by

式(4-14) [Fig.107_293_02]

and

式(4-15) [Fig.107_293_03]

The duration Tcrit, for which the ratio Ecmax/ED becomes unity is

式(4-16) [Fig.107_293_04]

For a depth h=2000m, Tcrit is 28 sec. This value of Tcrit indicates that, for the duration of several minutes of the bottom movement, the energy of the sound waves is negligible compared with gravity wave energy, and the opposite is true for a very short duration of the bottom movement.
It is very difficult to estimate the duration T of the bottom movement for actual earthquakes. From the study of the earthquake mechanism, the rupture velocity of an earthquake fault is considered to be of the order of S-wave velocity If the fault length is, say,400km, and the rupture velocity 4 km/sec, the duration is 100 sec. In this model, the duration of the displacement at a particular location is assumed instantaneous and the displacement propagates along the fault. On the other hand, in the model adopted in the present paper, the displacement occurs in a broad area at the same time, but the vertical displacement is completed in the time interval T. Because of the difference of the models used, it is not possible to compare directly the duration T in the present paper with the duration of the fault movement considered in the study of the earthquake mechanism.
According to the study by Mikumo (1964) on atomospheric pressure waves generated by the tectonic deformation at the time of the Alaskan Earthquake of March 28,1964, the deformation does not seem to have lasted much longer than 3 min. However, the lower limit of the duration could not be determined.
In view of the fact that the duration T of the seismic local source movement might be very small, the energy transfer to sound waves in the water due to the crustal deformation should be examined more carefully by taking the impedance of the crust into account.

5. Directivity of wave radiation for rectangular models(Model A)

The wave radiation due to the crustal deformation in a rectangular area of the bottom is discussed by using rectangular co-ordinates (x*, y*) and polar co-ordinates (R,\theta\) as shown in Fig.4 with the origin at the center of a rectangle. The side lengths of the rectangle are A and B in the x* and y* directions, respectively. The time is measured from the first arrival of the wave disturbances at the observing point and is non-dimensional if we write \tau\*=ct/A.
The method to compute \zeta\ is already shown in Section 3. The asymptotic solutions for long distances from the source are derived in Appendix B.

a) Wave form due to an uniform bottom uplift (Model A1)
In this model the vertical displacement of the bottom is D0 with the duration T. Now consider the wave form travelling along the x*-axis (see Fig.4). It is easily understood that if the length B becomes infinite, the wave form would be similar to that of the one-dimensional case; that is to say, the wave form travels without change of shape.
However, the effect of the finite length of B is felt as soon as the wave fronts originating from the side corners of the uplifted area arrive at the observing point. This sequence is shown in Fig.5, for B=3A and cT/A=0.2, where the distance x is measured from the edge of the source area; x=x*-A/2.
It is seen that, for x/A=0, the elevated portion of the wave form is the same as that for the one-dimensional model and the following depressed portion corresponds to the contribution from imaginary sources on both
sides of the real source;namely, D=-D0 for Iy*I > B/2,Ix*I ＜ A/2.
With the increase of x/A, the depression wave due to imaginary sources begins to overlap that of the one-dimensional wave, the interval \Delta\t of the arrival times of the two types of waves being given by [Fig.107_295_01].Therefore, the resultant wave form deviates more and more from the
one-dimensional case. However the maximum wave height does not change as long as c\Delta\t > T.
The directional variations of wave forms for the case of B/A=2, cT/A=0.2, and R/A=4, are shown in Fig.6. The variation of wave forms for \theta\=0 to \theta\= \pi\/2 is very conspicuous, reflecting the variation of the directional source function. It may be noticed that for \theta\=0 and \theta\=\pi\/2,the wave forms are rather similar, provided that the total duration of the elevated portion of the wave is longer and the maximum height smaller for \theta\= \pi\/2 than for \theta\=0. Judging from the shape of the directive source function, the main parts of the elevated portion. and also the depressed portion in the wave forms for\theta\=0 and \theta\= \pi\/2 become almost identical if we scale the distance R by the lateral dimension of the source area; namely, R*=R/B for B=0 and R*=R/A for \theta\= \pi\/2. The duration of the elevated portion of the wave form is roughly equal to the length of the source projected in the direction of the wave propagation.
However, the wave form is very sensitive to the assumed source characteristics and the gravity wave dispersion, neglected in the present discussion, modifies the wave form drastically for long distances from the source (see§7). It is then more advantageous to discuss the directivity of wave radiation in terms of energy than in terms of wave forms.

b) Directivity of the energy radiation (Model A1)
The total flux of gravity wave energy Ef for the unit width, transmitted in a particular direction may be computed approximately by the formula

式(5-1) [Fig.107_296_02]

since the wave can be considered as progressive after leaving the source area. The directively coefficient Q of the total wave energy flux at a distance R from the center of the wave source is defined by

式(5-2) [Fig.107_297_01]

where ED is the total energy of gravity waves. For a uniform bottom uplift in a rectangular area, ED is already shown in Fig.2a. The total energy fiux Ef is computed numerically by (5-1) after evaluating \zeta\.
The dependence of Q on the distance R*(R/B for \theta\=0, R/A for \theta\= \pi\/2) is shown in Fig.7 for B/A=1,2,3 and \theta\=0, \pi\/2. For large values of R*, Q becomes constant, but, for small values of R*,Q decreases for \theta\=0, and increases for\theta\= \pi\/2 with the decrease of R* except for the case of B/A=1. This nature of Q reflects the following features of energy propagation. Ef decreases inversely proportional to R* for R*> 4 but, for small values of R*, the deviation from this law becomes significant as B/A increases. In particular, for R* ＜ 1 and B/A ＞ 3,the energy flux Ef(0) (in the direction of the minor axis) does not vary much, indicating the approach of the wave form to the one-dimensional case. On the other hand, Ef( \pi\/2) (in the direction of the major axis) decreases much faster than the inverse of distance near the edge of the source area, indicating that the lateral dispersion of wave energy is stronger than the ordinary geometrical spreading.
Fig.8 shows the directivity coefficient Q at R/A=4, as a function of \theta\ for the length ratios, B/A=1,2 and 3. As is expected, with the increase of B/A the variation of Q with B increases. The minimum of Q lies somewhere around \theta\=3 \pi\/8 but in the range, \pi\/4 ＜ \theta\ ＜ \pi\/2, Q does not differ much from this minimum value. For the case of B/A=1, two local minimums of Q appear because of the symmetry of the wave form relative to the direction of \theta\= \pi\/4 but the variation of Q is, in general, very small. For B/A=3, the decrease of Q with \theta\ is very rapid for small values of \theta\, showing a very strong directivity. The ratios Q(\pi\/2)/Q(0) at R/A=4 are 0.326 and 0.187 for B/A=2, and 3 respectively. The variation of Q with \theta\ shown in Fig.8 suggests strongly that, in practice, it is rather difficult to estimate the total energy of a tsunami from the energy flux measured at a single location unless information on the source characteristics is available.

c) Rectangular source with the combination of formation (Model A2)
A model considered here has sides with the lengths l1,l2, and l3 whereby l1+l2+l3=2a and the heights are D0 and D0’. The bottom deformation D is limited within Iy*I ＜ b and the x*-dependence is given by

式(5-3) [Fig.107_298_01]

where [Fig.107_299_01]. The other parameters are [Fig.107_299_02].
As an example, the case when \gamma\=1 and l1=l3=2l2 is shown in Fig.9, where A=2a and B=2b.
The computed dependence of the wave form on the direction B is shown in Fig.10 for B/A=2, and R/A=4. The directional difference of the wave form is quite large, reflecting the variation of the directional source function. For example, at \theta\=3\pi\/8, the wave train is divided into two groups of crests and troughs, the second one of which is quite similar to the first one but the shape is inverted. This is due to the characteristics of the given source model, for which the directional source function vanishes in the central part of the source area as \theta\ approaches \pi\/2.
The dependence of the wave form on the ratio B/A is shown in Fig.11 for theta\=0 and R/A=4. It is noticed that, in general, the first crest height is smaller than the following trough depth and a small secondary crest appears after the major trough, in spite of the fact that the bottom deformation in the direction of the x*-axis is antisymmetric with respect to the center.
For long distances from the source, the wave pattern can be computed easily by means of (2-12) (see Appendix B), and in the present example, the height ratios become 2-28 and 0.45 for trough/1st crest, and 2nd crest/1st crest, respectively.
In general, a steep scarp of the fault produces a deep trough in the generated wave at long distances from the source.

6. Asymptotic wave form for long distances from an elliptic source (Model B)

Elliptic source models are defined by:
(B1) uniform deformation; 式(6-1) [Fig.107_299_04]
(B2) elliptic deformation; 式(6-2) [Fig.107_299_05]
(B3) parabolic deformation; 式(6-3) [Fig.107_299_06]

within the area bounded by

[Fig.107_300_01]

Now take the X-axis in the direction of the observing point with the origin at the center of the source. Then the directive source function W defined by (2-11) in the direction X can be written as

式(6-4) [Fig.107_300_02]

for Models B1, B2 and B3, respectively. a* and b* are defined by,

式(6-5) [Fig.107_300_03]

and

式(6-6) [Fig.107_300_04]

These expressions show that the directive source function is similar for all directions, provided that the parameters a* and b* corresponding to lengths of axes of the ellipse in the directions of X and Y are given by (6-5) and (6-6).
The wave form for long distances from the source, then, can be derived in the form (see Appendix C),

式(6-7) [Fig.107_300_05]

with

式(6-8) [Fig.107_300_06]

and for T→0

式(6-9) [Fig.107_300_07]

Here, t* is defined by{\tau\-(R-a*)}/a* and indicates the non-dimensional time from the first arrival of disturbances at the observing point. F(t*) and G(t*) are given by simple algebraic formulas (Model B2) or by the combination of the complete elliptic integrals of the 1st and the 2nd kinds (Model B1 and B3). It may be mentioned here that the wave forms due to models similar to B1, and B3 are computed by Momoi (1962,1963) by a completely different method. in which the high wave number components are truncated in the Fourier-Bessel expansion of the wave form.
(6-8) and (6-9) show that for large values of cT/a*,the maximum wave height is given by the maximum of F(t1*) and the height varies as b*√a*/R in contrast to the case when T→0, in which the maximum height varies as b*√a*R. The dependence of the maximum wave height on the duration T is shown in Fig.12, where the variation of \zeta_{max}\ on cT/a* for Model B3 is given for b*/a*=2,3, and 4. As is expected, the decrease of \zeta_{max}\ with the increase of cT/a* is larger for larger values of b*/a*.
If we compare the maximum wave heights in the directions of the major and the minor axes of the ellipse, the height ratio is found to decrease from (b/a)^{3/2} to (b/a)^{1/2} with the increase of the duration cT/a.
For the case of the sudden deformation T→0, the total flux of wave energy in a particular direction defined by (5-1) becomes

式(6-10) [Fig.107_301_01]

and, since the integral part is invariant with respect to the direction \theta\, the directivity function R(\theta\) defined by

式(6-11) [Fig.107_301_02]

becomes

式(6-12) [Fig.107_301_03]

For b/a=2,3, and 4, the dependence of R(4) on \Theta\ is shown in Fig.13.
The minimum value is found at \theta\= \pi\/2 where R(\pi\/2)=(a/b)^{2}. The general pattern of R(\theta\) given in Fig.13 is comparable to the directivity coefficient Q(\theta\) given in Fig.8. Both figures show that, roughly speaking, the variations are very rapid for small \theta\ and almost constant for \theta\ larger than \pi\/4, in spite of the difference of the source models used.
If we imagine a straight boundary perpendicular to the x*-axis (the direction of the minor axis of the source) at a distance xB* from the source center, the variation of the maximum wave height along the boundary can be expressed, by means of (6-7) and (6-9) as

式(6-13) [Fig.107_302_03]

where L*(=y*/xB*) is the non-dimensional distance along the boundary measured from the intersection of the x*-axis with the boundary.
Fig.14 shows the variation of wave heights for B/A=1,2,3, and 4. Compared with the case of b/a=1, which is a symmetric source model, the cases of b a=2 or larger show strong directivity, and the distance L* for which the local wave height decreases to 1/2 of the maximum height are 0.80,0.47,0.33 for b/a=2,3,4,respectively,in contrast to 3.87 for the symmetric case.

7. Comparison of non-dispersive and dispersive wave forms

For long distances from a source, the asymptotic wave form, taking the effect of the gravity wave dispersion into account, can be computed by the method developed in Paper I. Here, we compare the non-dispersive and dispersive wave forms for the case of an instantaneous bottom deformation of Model A1 and B3.
In both models the wave form in the x*-direction can be expressed by

式(7-1) [Fig.107_302_04]

where in Model A1, a=A/2 and b=B/2. In the case of a non-dispersive wave,\zeta\*is a function of t* only defined by

式(7-2) [Fig.107_303_01]

which is a non-dimensional time measured from the first arrival of nondispersive waves at the observing point.
In the case of dispersive waves, the scaled elevation \zeta\* is a function of p* and pa or t* and pa, and does not depend explicitly on R and b. The parameters p*and pa given in Paper I are defined in the present notations by

式(7-3) [Fig.107_303_02]

式(7-4) [Fig.107_303_03]

The last expression of (7-4) holds if we confine our attention to the main part of the wave train only (a(1-t*)/(3R) ＜＜ 1). The formula to compute \zeta\* for Model A1 and Model B3 can be derived from (94) and (98) in Paper I, respectively, by a slight modification.
Thus, we may write

式(7-5) [Fig.107_303_04]

where T(p) is defined by

式(7-6) [Fig.107_303_05]

and is shown in Fig.3a of Paper I.
In Fig.15a, b, the dispersive and non-dispersive wave forms \zeta\* are compared for Model A1 and Model B3, respectively. It is easily seen that Model A1 contains more energy in high wave number components than Model B3, so that the difference of the dispersive and non-dispersive wave forms is greater in Model A1 than in Model B3 for the same parameter pa.
It is remarkable that, for Model B3, two wave forms for the dispersive and non-dispersive cases are relatively similar up to about pa〜4.
This indicates that the non-dispersive assumption is satisfactory for large values of pa, but for pa=2, for example, the wave forms are different, showing the significance of dispersion. For Model A1, dispersive waves of high frequencies appear from the beginning, but the non-dispersive wave forms represents a kind of smoothed version of dispersive waves for large values of pa.
To see the applicability of the non-dispersive assumption which may be limited by the condition pa > 4, a diagram is constructed in Fig.16, where the relations between R/h, a/h. and pa are shown. The assumption of long distances requires the condition a/R ＜ 0.1, and the scale of the ocean is delineated by assuming R=20,000 Km and h=4Km. From this diagram, it is found that if a/h=20 or a=80Km, the non-dispersive assumption for Model B is satisfactory up to R/h〜800, or R〜 3200Km and the minimum value of pa is about 2. On the other hand, if a/h=5 or a=20Km, the same limiting distance becomes R〜50 Km and the assumption of a long distance is invalidated. The actual scale of the tsunami source L is somewhere around several tens Km to several hundred Km at the depth of 2〜3Km. To convert the scale L at depth h0 to the scale a at depth h, we should apply the relation [Fig.107_305_01].
Fig.16 indicates that, if the tsunami propagation is to be discussed, the long-wave equation can be used near the wave source of large horizontal extent,i.e., for tsunamis generated by near earthquakes in the vicinity of the Japanese coast, but it is not a good approximation for the tsunami propagation in the whole Pacific Ocean.

8. Concluding remarks

In the present paper, the following topics are discussed for a sea of constant depth: 1) the energy exchange between the solid bottom and the overlying water associated with the broad crustal deformation -in particular, the dependence of energy exchange on the duration of the bottom movement, 2) the variations of wave forms in two-dimensional propagation and the directivity of energy radiation, 3) the effect of the gravity wave dispersion.
Most of the conclusions obtained are rather sensitive to the assumed source characteristics and we should be careful when generalizing. This, in turn, suggests that the estimation of the source characteristics is essential to the prediction of a tsunami wave form.
To develop a method to solve the inverse problem, various factors not included in the present paper should be examined. One of the important first step seems to be the evaluation of the effects of the bottom topography near the tsunami source-the directivity of energy radiation would be modified significantly. The deep water tsunami signatures is expected to include the effects of multiple reflections on the continental shelf near the source and this problem will be discussed in a separate paper.
To obtain wave forms under conditions of a realistic topography and source, it is necessary to resort to a hydrodynamical-numerical method, where the starting equations of motion should be selected in relation to the scales of the source and the ocean area concerned. The inverse problem would then be solved by comparing wave forms computed on various probable source models with observations.

Acknowledgments

The author is grateful to Miss Ikuko Kato for various technical assistance.

References

AIDA,I.,1969, Numerical experiments for the tsunami propagation-the 1964 Niigata tsunami and the 1968 Tokachi-oki tsunami, Bull. Earthq. Res. Inst.,47,673-700.
HATORI, T.,1963, Directivity of tsunamis, Bull.Earthq.Res.Inst.,41,61-81.
IIDA, K.,1966, The Niigata tsunami of June 16,1964, General Report on the Niigata Earthquake of 1964, Pt.1-6,97-127, Tokyo Electrical Eng’ng College Press.
KAJIURA, K.,1963, The leading wave of a tsunami, Bull. Earthq. Res. Inst.,41,535-571.
MIKUMO, T.,1968, Atmospheric pressure waves and tectonic deformation associated with the Alaskan Earthquake of March 28,1964, J. Geophys. Res.,73(6),2009-2025.
MIYOSHI, H.,1954, Generation of the tsunami in compressible water, Part 1,J.Oceanogr. Soc. Japan,10,1-9.
MIYOSHI, H.,1954, Efficiency of the tsunami,J.Oceanogr. Soc. Japan,10,11-14.
MIYOSHI, H.,1955, Directivity of the recent tsunamis, J. Oceanogr. Soc. Japan,11(4), 151-156.
MIYOSHI, H.,1968, Re-consideration on directivity of the tsunami (1) (in Japanese), Zisin, ii,21,121-138.
MOMOI, T.,1962, The directivity of tsunami (1):the case of instantaneously and uniformly elevated elliptic wave origin, Bull. Earthq.Res.Inst.,40,297-307.
MOMOI, T.,1963, Some remarks on generation of waves from elliptical wave origin, Bull. Earthq. Res. lnst.,41,1-8.
SOLOVIEV, S. L.,1965, The Urup earthquake and associated tsunami of 1963, Bull. Earthq. Res. Inst., 43, 103-109.
SOLOVIEV, S.L.,1969,0n recurrence of tsunamis in the Pacific,IUGG Tsunami Symposium Hawaii,1969, Hawaii Univ. Press.
TAKAHASI, R.,1942,0n seismic sea waves caused by deformations of the sea bottom (in Japanese), Bull. Eurthq. Res.Inst.,20,375-400.
TAKAHASl, R. and HATORI, T.,1962, A model experiment on the tsunami generation from a bottom deformation area of elliptic shape (in Japanese), Bull. Earthq. Res. Inst.,40, 873-883.

Appendix A Wave in a one-dimensional model

The Green’s function for one-dimensional wave propagation is given by

式(A-1) [Fig.107_307_01]

where x and x0 are horizontal co-ordinates of the observing point and the source, respectively. Taking the time origin at the beginning of the bottom movement, we have the movement in 0 ＜ t0 ＜ T.
For the uniform deformation D0 of the bottom in the generating area delineated by a > x0 > -b, the elevation \zeta\(x,t) is given by

式(A-2) [Fig.107_307_02]

The integration of (A-2) is straight forwards and if x is inside the source region,

式(A-3) [Fig.107_307_03]

and if x is outside the source region (x > a),

式(A-4) [Fig.107_307_04]

where ct’=ct一(x-a), and A=a+b. Here the following abbreviation is adopted:

[Fig.107_307_05]

and

[Fig.107_307_06]

Now if the observing point is at the origin of the coordinate which is inside the source region and t ＜ T, we have

式(A-5) [Fig.107_307_07]

Therefore, the energy e transferred by the bottom deformation to the gravity wave at the origin is given by

式(A-6) [Fig.107_308_01]

Integrating this energy for the whole source region with the length A,we have

式(A-7) [Fig.107_308_02]

where

式(A-8) [Fig.107_308_03]

Appendix B Asymptotic solutions for long distances from a rectangular sources

For long distances from a source, wave forms can be computed from (2-10) or(2-11), which are written in terms of rectangular co-ordinate system (x, y) with the origin at the observing point and the x-axis directed to the center of the source area. However, to compute the directive source function, it is convenient to take right-handed rectangular co-ordinates (x*, y*) and (X, Y) with the origin at the center of a source. The x*-axis is taken parallel to an edge of the rectangle, and the X-axis is directed toward the observing point. The azimuth of the X-axis relative to the x*-axis is \theta\ and the distance to the observing point from the origin is R.

(a) Rectangular uniform source (Model A1)
If the uniform uplift of the bottom occurs in a rectangular area;

式(B-1) [Fig.107_308_04]

the directive source function W defined by (2-11) can be given, by projecting the corners of the source area on the X-axis, as follows:

式(B-2) [Fig.107_308_05]

Here, from the geometry of a rectangle relative to the (X, Y)co-ordinates, we have

式(B-3) [Fig.107_309_01]

式(B-4) [Fig.107_309_02]

式(B-5) [Fig.107_309_03]

and

式(B-6) [Fig.107_309_04]

with tan\theta\*=b/a.
The solution for \eta\ given by (2-10) is now transformed to

式(B-7) [Fig.107_309_05]

where \Delta\/a*=\Delta\’, and

式(B-8) [Fig.107_309_06]

For a limiting case of \Delta\’→0, we have

式(B-9) [Fig.107_309_07]

where \Delta\’→0 corresponds to the situation when \theta\=0 or \pi\/2, and b*=b or a, respectively.
For an instantaneous deformation (T→0), (2-11) gives, in terms of \xi\

式(B-10) [Fig.107_309_08]

and for a Iimiting case of \Delta\’→0,

式(B-11) [Fig.107_309_09]

(B-11) shows that \zeta\/D0 becomes infinite for \xi_{s}\→O or \xi_{e}\→0. This is due to the deficiency of the directive source function W(x) near the edge of the source as already mentioned in Section 2. In the interval,0 ＜ \xi\ ＜ \epsilon\ or -\epsilon\ ＜ \xi\ ＜ 0 with \epsilon\〜b^{2}/(2R), we should return to the exact formula.
For example, in the interval 0 ＜ \xi\ ＜ \epsilon\, we have

式(B-12) [Fig.107_309_10]

Since the elevation \zeta\ for a duration cT is given by substituting (B-9) into (2-5) with an appropriate change of \xi_{s}\, and \xi_{e}\, it is found that the elevation for \Delta\=0 with the duration cT is similar to the elevation from the instantaneous deformation if \Delta\ is put equal to cT and b* is equal to b ora in (B-10).

(b) Rectangular source with linear variation of deformation (Model A2)
A model defined by (5-3) can be transformed to

式(B-13) [Fig.107_310_01]

where

[Fig.107_310_02]

In this formulation, the contribution from, say, the first term in (B-13) to the directive source function W(X) in the direction X can be written as

式(B-14) [Fig.107_310_03]

where

式(B-15) [Fig.107_310_04]

and

式(B-16) [Fig.107_310_05]

Therefore, it follows:

式(B-17) [Fig.107_310_06]

and

式(B-18) [Fig.107_310_07]

Thus, the surface elevation for the case of an instantaneous deformation (T→0) is given by

式(B-19) [Fig.107_310_08]

where

式(B-20) [Fig.107_311_01]

and

式(B-21) [Fig.107_311_02]

The expression Ii holds provided that ti and ti* are positive. In ex plicit form, we have

式(B-22) [Fig.107_311_03]

In particular, for the wave propagating in the x*-direction (\theta\=0), in which Xi=Xi*and ti=ti*,(B-19) becomes

式(B-23) [Fig.107_311_04]

and the first crest \zeta_{c}\ and the following trough \zeta_{t}\, occur at t2=0 and t3 =0,respectively. Furthermore, the second crest appears at t4=0.
Therefore, we have,

式(B-24) [Fig.107_311_05]

It is easy to derive the ratio of the heights of the second crest \zeta_{c}\’, to the first crest \zeta_{c}\, in the similar way. Thus,

式(B-25) [Fig.107_311_06]

If we go back to the original parameters defined in (5-3), it follows

式(B-26) [Fig.107_311_07]

The relation between l2/l1 and \gamma\ with l\zeta_{t}\/\zeta_{c}\l as a parameter is shown in Fig. B-1, from which we can estimate the height ratio for given values of l2/l1 and \gamma\. For example, I\zeta_{t}\/\zeta_{c}\I = 2.28 for \gamma\=1 and l2/l1=0.5 which is an example given in Section 5-c. For this example,\zeta_{c}\’/\zeta_{c}\=0.45 from (B-25).

Appendix C EIIiptic source models

Let us transform the variables by putting

式(C-1) [Fig.107_312_01]

and

[Fig.107_312_02]

where x is the co-ordinate with the origin at the observing point.
Then, it follows:

[Fig.107_312_03]

and

[Fig.107_312_04]

The directive source functions (6-4) can be written as

式(C-2) [Fig.107_312_05]

with

式(C-3) [Fig.107_312_06]

and

式(C-4) [Fig.107_312_07]

with

式(C-5) [Fig.107_312_08]

Substituting these relations into (2-10) or (2-11), we have the solution of the form:

式(C-6) [Fig.107_312_09]

and for T→0,

式(C-7) [Fig.107_312_10]

It is remarked that for \tau\ ＜ R+a*, \beta\^{2} is negative and the lower limit of the integration in (C-6) and (C-7) should be put zero.
The integrals in (C-6) and (C-7) are composed essentially of the following kind of integrals:

式(C-8a) [Fig.107_313_01]

and

式(C-8b) [Fig.107_313_02]

If both l and m are 0 or even numbers, the integration can be performed without difficulty. The solution for Model B2 corresponds to this case.
On the other hand, if l and m are odd numbers, the solution is given in terms of the complete elliptic integrals.
Now,(C-8a) is transformed, by putting \lambda\=\alpha\sin\theta\ into

式(C-9) [Fig.107_313_03]

where

[Fig.107_313_04]

and (C-8b) is transformed by putting \lambda\^{2}=\alpha\^{2}sin^{2}\theta\+\beta\^{2}cos^{2}\theta\, into

[Fig.107_313_05]

where

式(C-10) [Fig.107_313_06]

(C-9) and (C-10) can be integrated by making use of the following recurrence formulas:

式(C-11) [Fig.107_313_07]

and

式(C-12) [Fig.107_313_08]

where K0(k^{2}) and E0(k^{2}) are complete elliptic integrals of the first and the second kind, respectively.
Performing the integration, the final form for (C-6) becomes,

式(C-13) [Fig.107_314_01]

The explicit form of F(t*) is as follows:
Model B1;

式(C-14) [Fig.107_314_02]

Model B2;

式(C-15) [Fig.107_314_03]

Model B3;

式(C-16) [Fig.107_314_04]

with

[Fig.107_314_05]

and

[Fig.107_314_06]

E2(k^{2}), E4(k^{2}), E6(k^{2}), K2(k’^{2}), K4(k’^{2}) are given by (C-11) and (C-12).
In the same way,(C-7) becomes

式(C-17) [Fig.107_314_07]

The explicit form of G(t*) is as follows:
Model B1;

式(C-18) [Fig.107_314_08]

Model B2;

式(C-19) [Fig.107_314_09]

Model B3;

式(C-20) [Fig.107_314_10]

The time dependence of F(t*) and G(t*) for Model B1, B2, B3 are shown in Fig. C-1 and Fig. C-2, respectively. Fig. C-2 shows that the maximum height appears earlier than the time corresponding to the wave travelling from the center of the source, but the lowest level (trough) appears very close to the wave corresponding to the rear end of the source(t*〜2). It is noticed that for Model B3, the crest height is larger than the trough depth, and for Model B2, they are equal. In particular, for Model B1, the maximum height appears at the front of the wave, and a depression of infinite amount occurs at the time corresponding to the rear end of the source. This behavior, however, is somewhat fictituous in view of an unsatisfactory approximation of the directive source function near these points.

45. 波源特性と津波エネルギー及び指向性 地震研究所　梶浦欣二郎

　長波近似をもちい,海底での大規模な地殻変動によって一様水深の海で発生する津波の波形を求める式を提出し,それを用いて,(1) 地殻変動によって海水に伝えられるエネルギーが,変動継続時間によってどう変るか,(2) 津波の2次元伝播による波形変化,及びエネルギー伝播の方向性,(3) 津波伝播における重力波分散の影響,を調べた.
　(1) については,継続時間が数分程度までなら,津波発生に関するかぎり,変動が瞬間的におこったとみなしてよいが,継続時間が実際に数秒程度であるとすれば,海中の圧縮性の波になるエネルギーの割台が津波のエネルギーより大きくなる可能性がある.
(2) については,簡単な矩形波源から出る津波の距離による波形変化,波形の伝播方向による違い,及びエネルギーの指向性を計算した.指向性係数の大きな変化は,波源の短軸から約45度以内の方向にかぎられる.また,波源の近傍(波源の大きさの程度)での波動エネルギーの距離による減衰は,長軸および短軸方向で著しく異なり,前者では単なる点源からの2次元的伝播による減衰より減衰が大きく,回折の影響が大きいことを示し,一方後者では減衰が小さく1次元的伝播に似てくる.楕円状の波源から遠方の点でのエネルギー伝播の方向性は,観測点と波源中心とを結ぶ方向に対して直角方向の波源のひろがり(横のスケール)の2乗に比例する.
　(3) については,同一の波源から出る分散性の波と,長波近似による非分散性の波の波形を比較し,[Fig.107_316_01] 程度では長波近似が有効であることが示される.ここでpaは波源のスケール a,水深h,波源からの距離Rによってきまるパラメータである.

Journal of the Oceanographical Society of Japan Vol.28, pp.260 to 277, December 1972 The Directivity of Energy Radiation of the Tsunami Generated in the vicinity of a Continental Shelf* Kinjiro KAJIURA**

Abstract

The radiation pattern of the tsunami generated by a broad crustal deformation on or near the continental shelf is examined analytically in the framework of a linear longwave approximation. Detailed discussion, however, is made only for a model of step-type bottom topography.
The proportion of the energy trapped on the shelf as edge wave modes relative to that radiated into deep water increases with the decrease of the long-shore dimension of the source.
The nearer is the source to the coastline, the greater is the rate of total edge wave generation.
However, the proportion of higher modes increases for the source near the shelf edge.
The proportion of the wave energy radiated in deep water, normal to the coastline, increases with the increase of the long-shore dimension of the source and/or the decrease of the depth difference between the shelf and deep water. Furthermore, this proportion increases with the increased distance of the source location from the coastline and approaches the value for the case without a shelf. For a source of the square shape, the directive difference of energy radiation in deep water is mainly caused by the refractive effect of the shelf edge. For larger long-shore dimension of the source, however, the geometric shape effect of the source is more important to cause the directive difference.

1. Introduction

In recent years, the measurement of a tsunami in deep ocean is one of the central interest not only for its usefulness in warning service but also from the standpoint of the better understanding of the tsunami.
However, from a viewpoint of finding quantitative information concerning the crustal deformation of the sea bottom, which is responsible for tsunami generation, there are mane difficulties to be resolved, even if deep ocean signatures of a tsunami are obtained at several stations. Since most of large tsunamis in the Pacific Ocean are generated in the vicinity of the continental shelf and slope along the circumPacific seismic belt, it is necessary to understand the important role played by the shelf near the source in the radiation of tsunami energy.
The shelf constitutes a wave guide and traps energy as edge waves, which have analogy in many other types of guided waves. In particular, the analogy between edge waves of long wave length on the shelf and Love waves in a stratified elastic half-space was first pointed out by SEZAWA and KANAI (1939) and the application of the Love wave theory to the problem of tsunami generation and propagation was attempted by OGAWA (1960)and NAKAMURA (1962);both treated the case of a point source in deep water.
On the other hand, the radiation of tsunami into deep water is affected by the multiple reflection and refraction on the shelf and slope.
As a consequence, the directive characteristics of radiation due to the source geometry (KAJIURA,1970) are modified in accordance with the shelf oscillations.
Although there are some successful attempts (AIDA,1969; HWANG and DlVOKY,1970) to compute the tsunami propagation numerically for the realistic bottom topography and source conditions, wave patterns obtained are the resultant of all the effects mentioned above, and it seems desirable to understand the individual effect more clearly for a simple topographic situation.
In the present paper, the role of the shelf in energy radiation is examined by clarifying the energy partition between the trapped and leaky waves and also the directivity of energy radiation in deep water. In addition, the radiated wave-form in deep water is discussed.
Very recently, SLATKIN (1971) tried to discuss the similar problem analytically for an exponential bottom profile, but the emphasis of his final analysis was placed on the leading wave along the coast, so that the problem of energy partition between trapped and leaky waves was left untouched.

* Received July 28,1972
** Earthquake Research Institute, University of Tokyo, Bunkyo, Tokyo 113, Japan

2.Formulation of the problem and the Green function

Let us consider a semi-infinite sea bounded by a straight coastal boundary and take the rectangular left-handed coordinates, with the x-axis along the coast and the y-axis directed offshore. Considering a continental shelf near the coast and deep water offshore, the depth h is assumed to increase monotonically in the y-direction with the constant deep water value h\infty\ for long distances (y > yl; yl=constant) from the coast (See, Figure 1).
The wave equation in the linear shallow water wave theory is given by

式(2-1) [Fig.107_318_01]

where t is time,\Delta\ is the gradient operator in the horizontal plane, and c^{2}=gh with g the acceleration due to gravity. \zeta\ is the elevation of the free water surface from the initial undisturbed level and wB is the vertical velocity (positive upward) of the bottom deformation with the condition wB=0 for t ＜ 0. The effect of wave dispersion neglected in the long wave theory is already discussed by KAJIURA (1970).
The boundary conditions for water waves in question are:(1) no flux of water through the coastal boundary,

式(2-2) [Fig.107_318_02]

and (2) the radiation condition (including \zeta\→0) at long distances from the source.
We take a Green function G(r, r0 I t, t0), which is the solution of (2-1), provided that the term -\partial\wB/\partial\t is replaced by -\delta\(r-r0, t-t0), and satisfies the boundary conditions (1) and (2), as well as the initial conditions:

[Fig.107_318_03]

Then, the solution of (2-1) can be written formally as

式(2-3) [Fig.107_318_04]

where r denotes the position vector and the suffix 0 indicates quantities related to the source and dSo is a surface element of the source area S.
The partial integration of (2-3) with respect to t0, taking the initial conditions for wB and G into account, yields

式(2-4) [Fig.107_318_05]

Since the Green function G is real, we may write

式(2-5) [Fig.107_318_06]

where t=t-t0 and G_{\omega\} has the property

[Fig.107_318_07]

with a star * indicating the complex conjugate quantity.
By means of the Fourier Transform of G_{\omega\} with respect to x(x=x-x0), it follows:

式(2-6) [Fig.107_319_01]

In (2-6), the path of integration \Gamma\(k= -\infty\ to k=\infty\) should be consistent with the outgoing wave for large values of ■, so that Im(k^{2}) ＜ 0 on \Gamma\ for ■ > O and \omega\ > 0.
The transformed Green function G\omega\k(y,y0 l \omega\, k) satisfies the equation;

式(2-7) [Fig.107_319_02]

where \omega\ is the angular frequency and k is the wave number in the x-direction. Since c^{2} is assumed to be a monotonically increasing function with respect to y, the homogeneous equation of (2-7) with the boundary conditions gives the eigen-value spectrum of a mixed type with the discrete spectrum for k^{2}c^{2}_{min} ＜ \omega^{2}\ ＜ k^{2}c^{2}_{\infty\} (c_{\infty\} : the maximum value of c, and c_{min}: the minimum value of c) and the continuous spectrum for \omega^{2}\ > k^{2}c^{2}_{\infty\} (MORSE and FESHBACH, 1953,Vol.1, p.766-768). The discrete spectrum corresponds to edge wave modes trapped in the vicinity of the shelf and the continuous spectrum corresponds to wave radiated into deep water after multiple reflections on the shelf as swell as the non-trapped waves along the shelf (lateral waves).
If Zs and Zr are solutions of the appropriate homogeneous equation of (2-7), each satisfying the boundary condition at y=0, and y=\infty\, respectively, the Green function G\omega\k may be written in the form (MORSE and FESHBACH, 1953,Vol. 1, p.832)

式(2-8a) [Fig.107_319_03]

where

式(2-8b) [Fig.107_319_04]

and

式(2-8c) [Fig.107_319_05]

Hereafter, we assume, without loss of generality, that Zs=1 at y=0. It should be noticed that A is a constant independent of y and Zr(y) is proportional to Zs(y) if A=0.
Since h=h_{\infty\} for y > yt, Zr may be written as

式(2-9) [Fig.107_319_06]

where \alpha\ isaconstant. In orderto make Zr(y) one-valued and bounded at y→\infty\, it is required that [Fig.107_319_07] on the real axis of k. Hence, we take branch cuts in the k-plane, running from k=\omega\/c\infty\ to i\infty\ and from k=-\omega\/c\infty\ to i\infty\ with \omega\ > 0, and assign the positive value to {(\omega\/c\infty\)^{2}-k^{2}}^{1/2} on the real axis in the interval -(\omega\/c\infty\) ＜ k ＜ \omega\/c\infty\. If we evaluate (2-6 )for x > 0, by the contour integral in the lower half of the complex k-plane, the waves of normal modes propagating along the shelf are given by the contribution of the real poles k^{2}=kn^{2} located in the range -(\omega\/c_{min}) ＜ k ＜ -(\omega\/c_{\infty\}).
Furthermore, the progressive waves radiated into deep water for a long distance from the source can be obtained by evaluating the contribution from the branch line integral in the 3rd-quadrant. In the discussion of the decay characteristics of the wave front on the shelf and/or the detailed properties of wave form in the neighborhood of the source, it is necessary to analyze the integral near the branch point more rigorously, including the complex poles, as has been done by ROSENBAUM (1960) for the case of sound pulse propagation in a twolayer liquid medium. However, in the present paper, we do not go into details, because we are mainly concerned with the total energy of trapped and leaky waves in long distances from the source.

Normal Mode Waves

If (\omega\/cmin)^{2} ＞ k^{2}> (\omega\/c\infty\)^{2}, Zr and Zs merge to the eigenfunction Zn at k^{2}=kn^{2} on the real axis:

式(2-10) [Fig.107_319_08]

Thus, in the limit of k^{2}→ kn^{2},(2-8c) becomes, by virtue of the Green’s formula,

式(2-11) [Fig.107_319_09]

where

式(2-12) [Fig.107_320_01]

Substituting (2-8a) into (2-6) by making use of (2-11), it is straight-forwards to evaluate the contribut{on G\omega\e of the normal modes to G\omega\ as follows:

式(2-13) [Fig.107_320_02]

where the contribution from the poles at k=-kn (kn > 0) is valid. If x is negative, the normal mode poles at k=kn are valid and G is given by (2-13), provided that x is replaced by Ixl.
Here it is mentioned that (2-13) is a formal representation of G\omega\e, and, in reality, the number of modes n is limited for a given value of \omega\;in other words, the lower cut-off frequency exists for an individual mode n.
If we make use of the energy integrals related to the eigen-function Zn, the phase and group velocities of mode waves can be given explicity (See, Appendix A).

Wave Radiated in Deep Water
For very long distances from the shelf (y/yl >> 1),we may express Zr(y) as (2-9), so that the contribution G\omega\d of the branch line integral in (2-6) becomes

式(2-14) [Fig.107_320_03]

Now putting

式(2-15) [Fig.107_320_04]

the original branch line integral (2-14) can be transformed to

式(2-16) [Fig.107_320_05]

Since A is assumed non-zero on the branch line, the integral can be evaluated by the saddle point method for large values of k\infty\r, if(\alpha\/A)Zs is not a rapidly,varying function with respect to \theta\. Thus, we have

式(2-17) [Fig.107_320_06]

3. Edge waves generated by a bottom deformation

The form of edge waves generated by a bottom deformation can be derived from (2-4).
Here, the Green function Ge is given by substituting (2-13) into (2-5) as

式(3-1) [Fig.107_320_07]

Therefore, from (2-4), we have, for x > 0,

式(3-2) [Fig.107_320_08]

where

式(3-3) [Fig.107_320_09]

式(3-4) [Fig.107_320_10]

and

式(3-5) [Fig.107_320_11]

Here * indlcates the complex conjugate and for x ＜ 0, the sign of knx in the right hand side of (3-4) should be changed. Rn (\omega\n) defined by (3-3) may be called the response factor and is written alternatively as

式(3-6) [Fig.107_320_12]

where Nn is the total energy integral and Un is the group velocity of the n-th mode wave (See, Appendix A). It is noted that Rn (\omega\n) is inversely proportional to the group velocity Un.
The velocity components due to edge wave modes, ue, and ve, in the x and y-directions, respectively, can be expressed in parallel to the elevation \zeta\e if we make use of the velocity eigen-functions.
Thus, we heve

式(3-7) [Fig.107_321_01]

where

式(3-8) [Fig.107_321_02]

式(3-9) [Fig.107_321_03]

The evaluation of \zeta\e, ue, and ve, for large values of x can be made by means of the stationary phase method if d^{2}kn/d\omega^{2}\n) \neq\ 0. Near the critical points where d^{2}kn/d\omega^{2}\n \simeq\ 0, we may use the Airy Integral. However, we do not go into details (See, for example, NAKAMURA, 1962,ROSENBAUM.1960).
Here it is only remarked that.Sn (\omega\n) contains all the information of the source characteristics relative to an edge wave mode \eta\, and Rn (\omega\n), which is determined by the shelf topography, indicates what part of the source spectrum is most efficiently converted into edge waves.
The energy flux dEe/dt per unit time due to edge waves passing through a vertical plane parallel to the y-axis at a very long distance x (x > 0) from the source is estimated by

式(3-10) [Fig.107_321_04]

and the total energy Ee transmitted through the plane at x in the time interval -\infty\＜ t＜ \infty\ becomes

式(3-11) [Fig.107_321_05]

The substitution of \zeta\e and ue from (3-2) and (3-7) into (3-11) yields after some manipulations;

式(3-12) [Fig.107_321_06]

Since the energy is transmitted in both positive and negative x-directions, the total energy of the generated edge waves is the summation of Ee in both directions. If the source deformation \omega\B is symmetric with respect to the y-axis, the total energy is simply 2Ee. The expression (3-12) can be obtained, alternatively, by considering the total wave energy, travelling, say, in the x-positive direction at a ffxed time t (t > \tau\;\tau\ is the time of the termination of the bottom movement)in the domain -\infty\ ＜ x ＜ \infty\ and 0 ＜ y ＜ \infty\.
Since the power spectrum F(\omega\) of the water level variation along the coast (y=0) for a very long distance (x ＞ 0) from the source is given by

式(3-13) [Fig.107_321_07]

from (3-2), it follows

式(3-14) [Fig.107_321_08]

In reverse, if the power spectrum is obtained from observation and it is differentiated into edge wave modes by the cross spectral analysis of the records at several stations along the coast (e.g., MUNK et al.,1964), we may write

式(3-15) [Fig.107_321_09]

and

式(3-16) [Fig.107_321_10]

Therefore, the total energy of edge waves travelling, say, in the positive x-direction is estimated by

式(3-17) [Fig.107_321_11]

If the higher mode edge waves make significant contribution to the total energy of edge waves, it is not possible to estimate total energy from a record taken at a single station along a coast. However, for a broad deformation of the bottom covering the whole width of the shelf, it will be seen later that the edge waves of the fundamental mode are most effectively excited, so that the energy may be estimated approximately by considering the fundamental mode only.

4.Waves in deep water for long distances from the source

The Green function Gd for the leaky mode is obtained by substituting (2-17) into (2-5) as follows;

式(4-1) [Fig.107_322_01]

The substitution of (4-1) into (2-4) yields

式(4-2) [Fig.107_322_02]

where

式(4-3) [Fig.107_322_03]

式(4-4) [Fig.107_322_04]

and

[Fig.107_322_05]

In the above expressions, r and \varphi\ are transformed into polar co-ordinates (r*, \varphi\) with the origin at x=0, y=yl, under the assumption that lx0I/r* ＜＜ 1;namely

式(4-5) [Fig.107_322_06]

It is remarked that, if lxI ＜ lx0I, we have

式(4-6) [Fig.107_322_07]

and the expression (4-4) will be modified.
As will be shown later, IRdI corresponds to the amplification factor for plane waves on the shelf and Sd represents the source spectrum in the direction \varphi\.
The radial energy flux dE_{\varphi\}d/dt for long distances from the source per unit time within the angle d\varphi\ is given by

式(4-7) [Fig.107_322_08]

where ud is the radial velocity (〜c\infty\\zeta\d/h\infty\).
Therefore, the total energy flux in the \varphi\-direction E_{\varphi\}d is, from (4-2),

式(4-8) [Fig.107_322_09]

and the total energy Ed radiated into deep water is

式(4-9) [Fig.107_322_10]

This expression corresponds to (3-12) for the normal modes.
The directivity of the energy radiation in half space may be expressed by the directivity coefficient Q defined by,

式(4-10) [Fig.107_322_11]

where E0 is the total tsunami energy generated by the bottom deformation. Q(\varphi\) gives a measure of efficiency of energy radiation in particular direction. Because of the presence of the edge wave energy Ee propagated parallel to the coast, the dlrectivity coefficient for \varphi\= +-\pi\/2 should be modified to include Ee, but in the the later discussion Ee is not included for simplicity.
To discuss the directive difference of energy radiation in deep water, however, it is more convenient to define the modified directivity coefficient Q*(\varphi\) by

式(4-11) [Fig.107_322_12]

excluding the edge wave energy Ee. Since the ratio Ed/E0 varies according to the geometric shape and location of the source relative to the shelf configuration, it is important to keep in mind the difference of the reference energy, E0 or Ed.
Let us examine the energy radiation from a point source on the basis of a geometrical optics approximation. The relationship between the ray directions, measured from the off-shore direction, \theta\0 and \varphi\, at the source position and at the shelf edge, respectively, is simply given by the refraction law:

式(4-12) [Fig.107_323_01]

where h0 and h\infty\ are the depths of water at the source and in deep water, respectively.
At the critical angle \theta\ at the source, \varphi\ becomes \pi\/2 and the total reflection of ray occurs. Thus, the proportion of energy Ed radiated into deep water to the total energy E0 is given by

式(4-13) [Fig.107_323_02]

where the energy radiated toward the coastline is assumed to be completely reflected.
Because of the energy conservation between neighboring rays, we may put

式(4-14) [Fig.107_323_03]

Thus, from (4-10),(4-12) and (4-14), we have

式(4-15) [Fig.107_323_04]

and from (4-11),(4-13) and (4-15), Q* becomes

式(4-16) [Fig.107_323_05]

It is evident that for m = 1, both Q and Q* are unity for all \varphi\, indicating the isotropic radiation. With the decrease of m, Q decreases monotonically for a given \varphi\. However, the directivity expressed by Q*(0) increases with the decrease of m, and for very small m, Q*(\varphi\) becomes

式(4-17) [Fig.107_323_06]

The relationships essentially similar to (4-13) and (4-15) were derived by MIYOSHI (1968) for the case of a uniform shelf slope. (ln his case, the energy radiated toward the coast is assumed to be completely dissipated, so that Ed/E0 is one half of (4-13)).

5. A simple example

As an application of the theory, let us consider a case of the step-type variation of a bottom at y=l;namely,

式(5-1) [Fig.107_323_07]

with

[Fig.107_323_08]

Corresponding to this geometry, we put

式(5-2) [Fig.107_323_09]

The elementary solutions Zs and Zr for k^{2} ＜ \omega\^{2}/c2^{2} are;

式(5-3a,b) [Fig.107_323_10]

and

式(5-4a,b) [Fig.107_323_11]

where

[Fig.107_323_12]

and

式(5-5) [Fig.107_323_13]

The factor A defined by (2-8c) now becomes,

式(5-6) [Fig.107_324_01]

In the range, \omega^{2}\/c2^{2} ＜ k^{2} ＜ \omega^{2}\2/c1^{2}, the normal mode solution is given by

式(5-7) [Fig.107_324_02]

where

式(5-8) [Fig.107_324_03]

and the eigen-values kn are determined by setting A=0 in (5-6); namely The normal mode characteristics in the present shelf geometry are very well known in studies of the wave propagation in various layered medium such as sound propagation in layered water and Love waves in the elastic medium. The general characteristics are briefly reviewed in Appendix B.
As a model of the tsunami source, let us consider the abrupt uniform deformation of a rectangular area of the sea bottom; namely,

式(5-10) [Fig.107_324_04]

Then, the initial potential energy of the tsunami E0 is given by the elevation of the water level equivalent to the bottom deformation such that

式(5-11) [Fig.107_324_05]

where b*=b2-b1. On the other hand, the total energy of edge waves of the n-th mode is

[Fig.107_324_06]

from (3-12). The source spectrum Sn, defined by (3-5) may be written in the present model as

式(5-12) [Fig.107_324_07]

where

式(5-13a) [Fig.107_324_08]

and

式(5-13b) [Fig.107_324_09]

with Zn(y0) defined by (5-7).

Thus, the proportion of the trapped wave energy of the n-th mode to the total tsunami energy is gievn by

式(5-14) [Fig.107_324_10]

In the actual computation, it is convenient to make use of the transformation from\omega\n to \theta\1, where \theta\1 is the directional angle of the elementary wave on the shelf and the wave number kn in the x-direction is given by kn= (\omega\n/c1)sin\theta\1, (See,Appendix B).Then,(5-14) becomes

式(5-15) [Fig.107_324_11]

where \sigma\n=(\omega\n/c1)l,and m=(h1/h2)^{1/2}.
The source spectrum for the leaky waves defined by(4-4) is now written as

式(5-16) [Fig.107_324_12]

where

式(5-17a) [Fig.107_324_13]

式(5-17b) [Fig.107_324_14]

and k=k\infty\sin\varphi\, K*=k\infty\cos\varphi\,k\infty\=\omega\/c2.
From (4-9) the total energy of the leaky waves is

[Fig.107_324_15]

Here, the amplification factor IRdI defined by

(4-3) becomes

式(5-18) [Fig.107_325_01]

Thus, the proportion of the leaky wave Ed to the total energy is

式(5-19) [Fig.107_325_02]

where

式(5-20) [Fig.107_325_03]

with \sigma\=k\infty\l.
Q (\varphi\) gives the directivity coefficient of deep water energy radiation defined by(4-10). The modified directivity coefficient (Q*(\varphi\) is given by

[Fig.107_325_04]

In the upper part of Figure 2, the mode characteristics such as the phase and group velocities, V and U, as well as the response factor R, are shown as a function of nondimensional frequency \omega\l/c1 for the case of h1/h2 = 0.05. The group velocity has the minimum and the response factor has the distinct maximum in the interval \pi\/2 ＜ \omega\l/c1 ＜ \pi\ for the zeroth mode (n=0) and 3\pi\/2 ＜ \omega\l/c1 ＜ 2\pi\ for the first mode (n=1). The response factor decreases rapidly for small frequencies and becomes zero at the cut-off frequency,(\omega\l/c1=0, for n=0, and \omega\l/c1= \pi\/(1-0.05)^{1/2} for n=1).
Furthermore, the minium group velocity is lower and the maximum response factor larger for the higher modes. The generated energy spectra given by (5-14) for the source models of b*=l on the shelf (b1=0,b2=l) and a=0.5l, l,2l,respectively, are shown in the lower figure of Figure 2. The higher mode contribution is verv small in this model. It is seen that the spectral peak lies in the neighborhood of the maximum response, irrespective of the source, and the spectral contribution from very low frequencies is insignificant.
In Figure 3, the case of h1/h2=0.5,n=0, is shown. The difference from the case, of h1/h2 =0.05 is that the group velocity minimum is not so distinct and the response factor do not show the intermediate maximum. Corresponding to these differences of mode characteristics, the generated energy spectra have rather broad peak. The case when the source is in deep water (b1=l, b2=2l) is also shown in the bottom of Figure 3, which shows that the generated energy spectra of edge waves are very small compared with the case when the source is on the shelf.
In Figure 4, the energy partition between the edge waves and the waves radiated into deep water is shown as a function of the lateral length, 2a/l, of the source for h1/h2=0.05.
The source length off-shore is held constant (b*/l=0.5) and the location of the source is either b1/l =0.0 or b1/l=0.5.
In principle, we may say

[Fig.107_326_01]

so that, if the proportion of the edge wave modes increases, the proportion radiated into deep water decreases and vice versa. As shown in Figure 4, the proportion of the edge wave modes decreases with the increase of the lateral length of the source. It should be remembered, however, that the total energy of the generated wave increases linearly with the increase of the lateral length of the source, so that the actual energy contained in the edge wave modes increases gradually with the increase of the lateral length. Comparing the cases of b1/l=0.0 and b1/l=0.5, it can be seen that if the shape of the source is similar and located on the shelf, the edge wave energy of the higher modes is conspicuous for the source located near the edge of the shelf. However,the nearer is the source to the shelf edge, the smaller is the total of the edge wave energy in various modes.
Although it is not shown in the figure, if the source is located in the deep sea, the energy of the edge wave modes is less than 10 percents of the total energy for the present model.
In Figure 5, the similar cases are shown for h1/h2=0.5. Compared with Figure 4, it is found that the edge wave energy decreases with the decrease of the depth difference as is expected from the decreasing reflective efficiency for smaller depth change.
The directivity coefficient Q(0) normal to the coastline for b*/l=0.5 in shown in Figure 6 as a function of the location of the source 2a/l, with the lateral length of the source, 2a/l (=0.5, 1.0, 2.0), and the depth ratio, h1/h2 (=0.05 and 0.5), as parameters. It is evident that for each combination of 2a/l and h1/h2, Q (0) increases with the increase of b1/l, and approaches a stationary value for b1/l > 1.5.
This stationary value is close to the value for the case of a flat bottom without a shelf (See, KAJIURA,1970, Figure 8), indicating that the effect of the shelf is significant only if the source is located in the vicinity of the shelf and the nearer the source to the coastline, the smaller is the energy flux in the direction normal to the coastline. The increase of the lateral length and/or the decrease of the depth difference favor the lncrease of Q(0). It is reminded, however, that the larger Q(0) does not necessarily indicate the larger directive difference of energy radiation in deep water.
If the game situation is shown in terms of the modified directivity coefficient Q*(0), it is found that the value of Q*(0) does not increase monotomcally with the increase of b1/l for small values of 2a/l, as shown in Figure 7.
At 2a/l=0.5, for example, Q*(0) is larger for the source on the shelf (b1/l=0.0,0.5) than that for the source in deep water. In contrast, for larger values of 2a/l, Q*(0) increases monotonically with the increase of b1/l.
To see the situation more clearly, the dependence of Q*(\varphi\) with \varphi\ is shown in Figure 8, for b*/l=0.5 and h1/h2=0.05. The cases of 2a/l=0.5,1.0,2.0 are given separately, with b1/l as a parameter. It is noted that, for the case of b1/l=1.5, the shape of Q*(\varphi\) and its dependence on the lateral length 2a/l are roughly similar to the case of a flat bottom without a shelf. With the approach of the source to the shelf edge from the deep-sea side, Q*(\varphi\) decreases for small values of \varphi\ (consequently, Q*(\varphi\) increases for large values of \varphi\). On the other hand, if the source is on theshelf,Q*(\varphi\) for small values of \varphi\ increases slightly with the approach of the source to the shelf edge, and the dependence of Q*(\varphi\) to the lateral length, 2a/l, is much smaller than for the source in deep water. Thus, for the case of 2a/l=0.5, which is the square source, an almost linear variation of Q*(\varphi\) with \varphi\ exists for the source on the shelf because of the refractive effect of the shelf edge, in contrast to almost uniform Q*(\varphi\) for the source in deep water. For larger lateral scale, when the geometric shape effect is important, the presence of the shelf seems to have an effect to diminish the directivity compared with the case without a shelf.

6. Wave pattern in deep water

To discuss the wave pattern in deep water, let us express Gd given by (4-1) together with (5-4) to (5-6) in series form compatible with the ray theory.
Transforming (5-6) in the form;

式(6-1) [Fig.107_329_01]

with R=(\mu\-1)/(\mu\+1), the reflection coefficient, we may write Gd for y0 ＜ 1 as follows;

式(6-2) [Fig.107_329_02]

Here H(x) is a unit step function, and

式(6-3a) [Fig.107_329_03]

式(6-3b) [Fig.107_329_04]

式(6-3c) [Fig.107_329_05]

式(6-3d) [Fig.107_329_06]

In the derivation of (6-2),the following integral formula is utilized:

式(6-4) [Fig.107_329_07]

In (6-2), it is easily seen that the term involving L2n is the contribution from the mirror image of the source with respect to the coastline at y=0. The effect of the shelf appears in the factor {2/(\mu\+1)}R^{n} as well as in the modification of the length scale in the y-direction by the factor (1-m^{2}sin^{2}\varphi\)^{1/2}/m relative to the deep water length scale. In particular, for the Brewster angle \varphi\* in deep water given by

[Fig.107_329_08]

\mu\ becomes unity so that R=0 and

式(6-5) [Fig.107_329_09]

SRETENSKY and STAVROVSKY (1961) discussed the case without a shelf, which corresponds to m=1,so that his Green function is essentially of the type given by (6-5). It should be noticed that the first term of (6-5) for m=1 is the long-distance approximation of the original Green function in an infinite ocean of constant depth;

式(6-6) [Fig.107_329_10]

(Slightly different definition of G is commonly used; see, for example, Eqs.(2.1) to (2.3) of KAJIURA(1970)).
For y0 > 1, a similar kind of expansion gives

式(6-7) [Fig.107_329_11]

where

式(6-8a) [Fig.107_329_12]

式(6-8b) [Fig.107_329_13]

式(6-8c) [Fig.107_329_14]

In the right hand side of (6-7), the first term is the direct wave, the second term is the reflected wave at the shelf-edge and the third term is the waves multiply reflected on the shelf. It is noticed that 1-R^{2}=TT*, where T and T* are the transmission coefficient from the shelf to deep water and vice versa.
For a rectangular uniform source covering the whole width of the shelf and the lateral length 2a, the wave form for the sudden deformation D0 can be given by

式(6-9) [Fig.107_329_15]

where

式(6-10) [Fig.107_329_16]

with the sign combination in the bracket as follows:

[Fig.107_330_01]

Lon,l*,\mu\, R are given previously and

[Fig.107_330_02]

where \tau\ is the time interval from the wave front arrival. The wave shape in deep water is shown in Figure 9 and Figure 10 for the cases of h1/h2=0.05 and 0.5, respectively. The general characteristics are the same as in the case of a flat bottom. However, the reflection and refraction effects are added and the secondary wave appears. For h1/h2=0.05, the trough depth is larger than the crest height. For h1/h2=0.5, the secondary wave is already very small, because of a small reflection coefficient.
The wave form is very sensive to the source characteristics.

Conclusion

The partition of tsunami energy in trapped and leaky waves due to the presence of the shelf is discussed for a very simple topography of the shelf. The general applicability of the present conclusion depends on the nature of the eigen-function Zn for a particular topographic situation. However, it may be said that the qualitative character of Zn is similar for small n, if the geometry is such that the shelf extends to a certain distance and a steed slope connects the shelf to the deep ocean proper. Thus, the following conclusions derived for a simple model would be applicable to more general cases.
1) The efficiency to generate edge wares decreases with the increase of the lateral dimension of the source. In other words, the generation of the edge wave energy is not sensitive to the lateral dimension of the source.
The nearer is the source to the coastline, the greater is the edge wave generation and if the source is in deep water, the edge wave generation is very small.
2) If the source is on the shelf, the efficiency to generate edge waves of higher modes depends on the location of the source relative to the shelf as well as the width of the source relative to the shelf width. For a source with small width, the higher modes are more effectively generated when the source is located near the edge of the source.
3) If the directivity of radiated waves in deep water is expressed by the directivity coefficient in the direction normal to the coastline, the directivity increases with the increase of the lateral dimension of the source and/or the decrease of the depth difference between the shelf and deep water. In general, the effect of the shelf is confined to the case of source location close to the shelf and the nearer is the source to the coastline, the smaller is the directivity.
4) For a square source on the shelf, the directional difference of wave energy in deep water is caused mainly by the refractive effect of the shelf edge. However, for larger-lateral length of the source, the geometric shape effect of the source is dominant and the shelf acts to diminish this effect.
5) The wave form in deep water reflects the effect of multiple reflections on the shelf. In the direction parpendicular to the shelf orientation, the effect is most significant and the major「period」of the wave pattern corresponds roughly to the period of shelf oscillations. However, in the actual situation in the ocean, the coast and shelf are not so straight and, in deep water, the waves reflected from various coastlines and shelf slopes interfere with each other in the later phases, so that the coherence of the wave patterns observed in mid-ocean stations may not be high.

Acknowledgments

The author wishes to express his sincere thanks to Miss I. KATO for her assistance in various phases of the study.

References

AIDA,I.(1969): Numerical experiments for the tsunami propagation-the 1964 Niigata tsunami and the 1968 Tokachi-oki tsunami. Bull. Earthq. Res. Inst.,47,673-700.
HWANG, Li-San, and D. DIVOKY(1970): Tsunami generation. J. Geophys. Res.,75,6802-6817.
JEFFREYS, H.(1959): The Earth (4th Ed.), Cambridge Univ. Press.
KAJIURA, K.(1970): Tsunami source, energy and the directivity of wave radiation. Bull. Earthq. Res. Inst.,48,835-869.
MIYOSHI, H.(1968):Re-consideration on directivity of the tsunami (1). Zisin, ii,21,121-138 (in Japanese).
MORSE, P.M., and H. FESHBACH (1953): Methods of theoretical physics, Vol.1. McGraw Hill Book Co., New York.
MUNK, W., F. SNODGRASS and F. GILBERT (1964): Long waves on the continental shelf: an experiments to separate trapped and leaky modes. J. Fluid Mech.,20,529-554.
NAKAMURA, K.(1962): The generation of edge waves inpinging from the outer sea. Sci. Rep. Tohoku Univ., Ser.5, Geophys.,14,27-40.
OGAWA, K.(1960): Edge waves induced by a radially spreading long wave and its damping due to the irregularity of coast. Contribution from the Mar. Res. Lab., Hydrographic Office of Japan,1,103-123.
ROSENBAUM, J.H.(1960): The long-time response of a layered elastic medium to explosive sound. J.Geophys. Res.,65,1577-1613.
SEZAWA, K., and K. KANAI (1939): On shallow water waves transmitted in the direction parallel to a sea coast, with special reference to Lovewaves in heterogeneous media. Bull. Earthq. Res. Inst., I7,685-694.
SLATKIN, M.W.(1971): Long waves generated by ground motion. J. Fluid Mech.,48,81-90.
SRETENSKY, L.N., and A.S. STAVROVSKY (1961) Computation of the height of tsunami waves along the coast. Trans, Mar. Hydro. Inst., Acad. Sci. USSR,24,23-43.(English Translation)

Appendix A Energy integrals and the phase and group velocities

The「norm」of the eigen-functions is related to the energy of the mode waves. In the present problem, the equipartition of kinetic and potential energy can be the wave energy is proportional to the「norm」Nn in a generalized sense when the volocity eigen-functions as well as the vertical displacement eigen-function Zn are taken into account.
From the equations of motion, it is straightforward to derive the velocity eigenfunctions in the form given by (3-8)and (3-9), where Hn is replaced by Zn. Therefore, the total energy integral Nn becomes

式(A-1) [Fig.107_332_01]

where

式(A-2) [Fig.107_332_02]

式(A-3) [Fig.107_332_03]

式(A-4) [Fig.107_332_04]

Furthermore, the equipartition of potential and kinetic energy gives

式(A-5) [Fig.107_332_05]

Making use of these lntegrals In(0),In(1),In(2), the phase and group velocities Vn and Un of the normal mode wave can be derived easily by the variational method (JEFFREYS,1959, P.110).
Thus, we have

式(A-6) [Fig.107_332_06]

and, alternatively,

式(A-7) [Fig.107_332_07]

式(A-8) [Fig.107_332_08]

The response factor Rn (\omega\n) defined by (3-3) now becomes

式(A-9) [Fig.107_332_09]

Appendix B The wave characteristics for a shelf with a step type bottom topography

In the formulation given in Section 5, it is instructive to write the wave number k in the x-direction as

式(B-1) [Fig.107_332_10]

where the angles \theta\1 and \theta\2 are the propagation direction of an elementary plane wave with respect to the y-axis. Then, the wave numbers in the y-direction on the shelf and in deep water are,

式(B-2) [Fig.107_332_11]

The refractive index from the shelf to deep water is

式(B-3) [Fig.107_332_12]

and the normal impedance ratio of a wave in deep water to that on the shelf is

式(B-4) [Fig.107_332_13]

The reflection coefficient R of a plane wave from the shelf to deep water is given by

式(B-5) [Fig.107_332_14]

When \theta\1 > Sin^{-1}m, the wave on the shelf is totally reflected at the shelf edge and the critical condition of

[Fig.107_332_15]

corresponds to K*=0, the branch point. In the situation of total reflection, we have normal modes, the characteristic equation of which gives eigen-frequencies for a given value of \theta\1 larger than \theta\1*; namely

式(B-6) [Fig.107_333_01]

where

[Fig.107_333_02]

This equation indicates that the frequency of the higher modes (n > 1) has the lower limit \sigma\n* at

式(B-7) [Fig.107_333_03]

corresponding to the branch point. With the approach of \theta\1 to +-\pi\/2, \sigma\n increases indefinitely.
The energy integrals can be written in terms of \theta\1 as follows;

式(B-8) [Fig.107_333_04]

式(B-9) [Fig.107_333_05]

式(B-10) [Fig.107_333_06]

The phase and group velocities Vn and Un are simply written as

式(B-11) [Fig.107_333_07]

式(B-12) [Fig.107_333_08]

and the response factor Rn is

式(B-13) [Fig.107_333_09]

It should be noticed that from (B-12) and (B-13),

式(B-14) [Fig.107_333_10]

On the other hand, we may write, in general,

式(B-15) [Fig.107_333_11]

Thus, we have, from (B-14) and (B-15),

式(B-16) [Fig.107_333_12]

Since \sigma\n increases with the increase of \theta\1, In(1) given by (B-9) is a monotonically decreasing function with respect to \theta\1 in the range, \theta\1* ＜ \theta\1 ＜ \pi\/2,where In(1)→\infty\ for\theta\1→\theta\1* and In(1)→c1^{2}l/2 for \theta\1→ \pi\/2. Furthermore,In(1) decreases faster with respect to \sigma\n for smaller values of m. Taking this behavior of Inch into account, it can be seen that Rn may have the maximum at the intermediate value of \sigma\n if m is small. The maximum value of Rn increases with the decrease of \sigma\n and, at the same time, the value of \sigma\n, at which the maximum of Rn appears, decreases.
Both the phase and group velocities approach c2 and c1 as \theta\1 approaches \theta\1* and \pi\/2, respectively. The group velocity becomes minimum at the intermediate value of \sigma\n (=\sigma\n**), because of the factor involving cot^{2}\theta\1/ln(1) in (B-12).
With the decrease of m, the minimum value of Un/c1 as well as its location \sigma\n**decreases.
It is interesting to consider a limiting case of m=0. In this case, Rn becomes infinity at \sigma\n=\pi\(n+1/2) and decreases as 1/sin \theta\1 with the increase of \sigma\n. It is zero for \sigma\n ＜ \pi\(n+1/2).
The group velocity is given by Un/c1=sin \theta\1.
Thus, no wave energy propagates beyond the wave front moving with the velocity c1 and the normal mode wave is formed by the reflected waves only. The group velocity is zero at \sigma\n=\pi\(n+1/2) where \theta\1=0 so that the normal mode waves, corresponding to the group velocity minimum, degenerate into the or inary onedimensional oscillations on the shelf with the infinite wave length along the coast. From these considerations, the edge waves with \sigma\n* ＜ \sigma\n ＜ \sigma\n** may be called a refracted wave branch and those with \sigma\n** ＜ \sigma\n a reflected wave branch.

陸棚近傍でおこる津波エネルギー放射の方向性梶浦欣二郎

要旨

陸棚近傍での広範囲な地殼変動によっておこる津波エネルギー放射を,線型,長波近似の範囲内で理論的に取扱い,特に階段状の陸棚の場合について具体的な議論を行なった.
　陸棚の影響が顕著にあらわれるのは,波源が陸棚上にある場合で,発生する津波エネルギーのうち陸棚上にエツヂ波としてとどめられるエネルギーの割合は,波源の岸に平行な方向の長さが増すとともに減少する.波源が岸に近いほど,エツヂ波全体のエネルギーの割合は増すが,幅(岸に直角方向の長さ)の小さい波源が,陸醐の端近くにあると,高調波成分の割合が増してくる.
　外洋側の単位角当りに放射されるエネルギーの津波全エネルギーに対する割合から方向係数をきめると,波源の横方向の長さがますとともに,また水深比がへるほど,陸岸に直角方向の方向係数は増加する.これはまた,波源の位置が陸岸から遠ざかるほど増加し,陸棚の端から波源がやや沖側になると,平坦な海の場合とほぼ等しくなる.
　これに対し,陸棚上の波源から外洋に放射された波だけについて,相対的な方向性を議論すると,波源の幾何学的形状による方向性が小さい間は,陸棚の屈折効果が方向性をきめる.一方,陸棚にそう波源の長さが大きくなると形状による方向性が卓越し,屈折効果はむしろ,方向性を弱めるように働らく.

Journal of the Oceanographical Society of Japan Vol.30, pp.271 to 281,1974 Effect of Stratification on Long Period Trapped Waves on the Shelf* Kinjiro KAJIURA**

Abstract

The effect of stratification on very long-period waves trapped on a straight continental shelf of constant depth is examined for a two-layer model. There are 4 modes in this system.
The characteristics of the mode with the largest phase velocity can be approximated by the barotropic mode. The mode corresponding to the barotropic shelf-wave mode is modified by the baroclinic motions significantly, and in the limit of very narrow shelf width, the mode characteristics are transformed from those of the barotropic shelf-wave to the baroclinic Kelvin wave if the long-shore wave length is larger than the internal deformation radius. In this case, the stratification has an apparent effect of increasing phase velocity of barotropic shelf-waves.
The remaining two modes are dominated by baroclinic motions with significant contribution from barotropic motions: among which the one has a shelf-wave characteristics for small values of the shelf width and approaches the mode corresponding to the baroclinic Kelvin wave in shallower water for large shelf width and the other is a stationary mode. If the long-shore wave length is much shorter than the internal deformation radius, the motions in the upper and lower layers are decoupled: the surface and bottom modes analogous to those discussed by RHNES (1970) appears.
If the interface is deeper than the shelf depth, the stationary mode is absent and the characteristics of the third mode approaches those of the baroclinic double Kelvin wave mode as the shelf width increases.

1. Introduction

In 1971 the abnormal rise of sea level along the pacific coasts of Japan was observed after the passage of a typhoon (YOSHIDA, SHOJI, and Masuzawa,1972). Events of this kind were not uncommon (ISOZAKI,1972) and the subsurface water temperature offshore often showed a marked change accompanying the sea level disturbance (SHOJI, private communication).
Since the sea level disturbances moved slowly to the west with the phase speed of several meters per second, the generation and propagation of a continental shelf-wave (ROBINSON, 1964) were considered to have played an important role. Numerical model experiments indicated that the continental shelf-wave was indeed generated by a moving typhoon (ENDOH, 1973). Even in a two layer model, the generated shelf-wave was hardly affected by the stratification, although the disturbance of the interface remained for a long time behind the shelf-wave (SUGINOHARA,1973).
As to the modification of shelf-wave characteristics by the density stratification, MYSAK (1967) showed that the density stratification in deeper water increases the phase velocity of continental shelf-waves significantly. On the other hand, before the concept of「shelf-wave」was advanced, there had been an attempt to explain coastal sea level disturbances, moving slowly clockwise around the Japanese Islands (SHOJI,1961) in terms of internal Kelvin waves (YOSHIDA,1960; KAJIURA,1962) with the emphasis in the baroclinic motions generated by the long-shore wind stress close to the coast.
From somewhat different point of view, RHIINES (1970) discussed the mode waves on a sloping bottom in a stably stratified ocean and indicated the importance of a non-dimensional parameter B for low frequency waves: (B=ND/(fL);N,Vaisala frequency; f, Coriolis parameter; D,vertical scale; L, horizontal scale). The behavior of waves with small B is barotrogic and the baroclinic motion becomes important for large B. In a two layer model of the ocean, the parameter B can be interpreted as the ratio of the Rossby’s internal deformation radius to the horizontal scale of the wave.
Therefore, the explanation of the slowly moving sea level disturbances in terms of the internal Kelvin wave may be valid only when the width of the varying depth near the coast is narrower than the internal deformation radius.
In the present pager, a simple analysis is made on the mode characteristics of trapped waves in a two layer ocean with a shelf in order to clarify the relative importance of the barotrogic and baroclinic motions in relation to the ratios of various length scales relevant in the problem. The depth on both sides of the shelf break is assumed constant and the interface is assumed either shallower or deeper than the shelf depth. Thus, in this model, the coupling of the barotropic and baroclinic motions is possible only at the shelf break. There are 4 (3) modes of trapped waves if the interface is shallower (deeper) than the shelf depth, and the characteristics of slowly moving shelf waves are expected to be modified significantly.

* Received July 5,1974
** Earthquake Research Institute, University of
Tokyo, Bunkyo-ku, Tokyo,113 Japan

2. Formulation of the eigen-value problem

Let us consider a straight shelf with the step type bottom topography and take the righthanded Cartesian co-ordinates as shown in Figure 1: the x-axis toward the open ocean with the origin at the shelf break and the zaxis positive upwards. The width and depth of the shelf is L and D, respectively. Assuming a two-layer fluid model, the quantities in the upper and lower homogeneous layers are distinguished by subscripts 1 and 2, respectively.
Thus, D1 and D2 are the layer thicknesses on the shelf. In the later discussions, the subscripts 1 and 2 are also used to denote the quantities related to barotropic and baroclinic motions, respectively. The quantities in deeper water are expressed by putting「dash」on the right shoulder. The relative density-difference \mu\ in two layers is assumed very small (\mu\= (\rho\2-\rho\1)/\rho\2 \simeq\ 2×10^{-3} :\rho\ is the density).
The motion is assumed hydrostatic and nondissipative. Then, under the usual linearizing assumptions, the motions of the barotropic and baroclinic modes are independent in the stable stratified ocean with constant depth. They are governed by the equation of the same type:

式(2-1) [Fig.107_336_01]

where t is time, t is time,\Delta\^{2} is the two-dimensionaI (horizontal) Laplacian, Ri is the representative vertical displacement of each mode (i),cl is the representative velocity. For a two-layer fluid model, i=1 or 2 and within the error of O(\mu\) we have

式(2-2) [Fig.107_336_02]

with g the acceleration due to gravity.
The x and y components (Ui, Vi) of the vertual volume transport vector Ui corresponding to Ri satisfy the following equations:

式(2-3a) [Fig.107_336_03]

式(2-3b) [Fig.107_336_04]

The actual surface and interface displacements \zeta\1, \zeta\2 can be expressed by

式(2-4a) [Fig.107_336_05]

式(2-4b) [Fig.107_336_06]

and the volume transport vector Q1 and Q2 in the upper and lower layers are

式(2-5a) [Fig.107_337_01]

式(2-5b) [Fig.107_337_02]

Exactly the same relationships from (2-1) to (2-5b) hold in deep water, provided that all relevant quantities are dashed.
At the coastal boundary, no volume flux across the boundary exists. In terms of mode solutions, this condition requires

式(2-6) [Fig.107_337_03]

The offshore boundary condition for very long distances from the shelf is

式(2-7) [Fig.107_337_04]

since we are concerned with trapped waves only. The boundary conditions at the shelfedge are the continuity of vertical displacements at the surface and at the interface, together with the continuity of the volume fluxes across the shelf break in the upper and the lower layers, respectively. In terms of mode solutions with 0(\mu\) neglected compared with I, these conditions can be transformed to

式(2-8) [Fig.107_337_05]

式(2-9) [Fig.107_337_06]

and

式(2-10) [Fig.107_337_07]

式(2-11) [Fig.107_337_08]

where r= D2/D1, r’=D2’/D1’.
Assuming the sinusoidal wave motion traveling along the shelf

式(2-12) [Fig.107_337_09]

with m the positive wave number in the ydirection and \omega\ the angular frequency, (2-1) is transformed into

式(2-13) [Fig.107_337_10]

with

式(2-14) [Fig.107_337_11]

Since we are concerned with the slowly moving long-period wave only, the wave period (2\pi\/\omega\) is assumed considerably larger than the inertial period (2\pi\/f) and (\omega\/f)^{2} is neglected compared with unity. Thus, Ki is independent of \omega\. It is mentioned that for(mcl/f)^{2} ＜＜ 1, Ki is the reciprocal of the Rossby’s deformation radius LRi (=ci/f).
Now taking real and positive Ki and Ki’, we write the solutions to (2-13) as follows:

式(2-15) [Fig.107_337_12]

式(2-16) [Fig.107_337_13]

(2-16) automatically satisfies the condition (2-7), and the condition (2-6) requires formally that

式(2-17) [Fig.107_337_14]

where

式(2-18) [Fig.107_337_15]

From (2-15) and (2-16), it is possible to classify the wave form in the.x-direction, Zi, according to the values of \alpha\i. For \alpha\i ＜ 0,IZiI has the maximum at the coastal boundary and decreases offshore somewhat like the Kelvin wave. For \alpha\i=0, Zi vanishes at the shell break so that Zi’ is always zero. On the other hand, for\alpha\i > 0, IZiI has local maxima at the coastal boundary and at the shelf break. In particular, for O ＜ \alpha\i ＜ 1, Zi changes sign on the shelf (shelf wave type) and, for \alpha\i > 1, Zi keeps the same sign throughout (double peak type). The case when \alpha\i=1 corresponds to a so-called「double Kelvin wave」for which the vertical displacement decreases exponentially on both sides of the shelf break.
Since \alpha\i is determined by (2-17), the wave form can be found from the relative magnitude of Xi and Ti as follows:
double peak type (\alpha\i > 1) for Xi > 1/Ti
Kelvin wave type (\alpha\i ＜ 0) for 1/Ti > Xi > Ti
shelf wave type (0 ＜ \alpha\i ＜ 1) for Ti > Xi
double Kelvin wave type (\alpha\i=1) for Ti =1 and Xi \neq\ 1 For the case when Ti=Xi=1,\alpha\i is indeterminate in (2-17). However, this case corresponds to the ordinary Kelvin waves in shallower water.
The boundary conditions from (2-8) to (2-11) at the shelf break give after some manipuladons:

式(2-19) [Fig.107_338_01]

and

式(2-20) [Fig.107_338_02]

where

式(2-21a) [Fig.107_338_03]

式(2-21b) [Fig.107_338_04]

式(2-21c) [Fig.107_338_05]

式(2-21d) [Fig.107_338_06]

In these equations, the following abbreviations are used.

[Fig.107_338_07]

with

[Fig.107_338_08]

From (2-19) and (2-20), the characteristic equation is derived by putting

式(2-22) [Fig.107_338_09]

Neglecting the term of O(\mu\^{1/2}) compared with 1, the dominant term in the mode coupling can be approximated by

式(2-23) [Fig.107_338_10]

If the coupling effect is completely neglected, all and a22 become the characteristic equations for the barotropic and baroclinic modes, respectively. It is noticed that a22 can be converted to all by changing the parameters:

[Fig.107_338_11]

Thus, the mode characteristics are similar to each other. The main point in the present paper is the modification of these mode characteristics by the presence of coupling.
By the substitution of (2-17) into (2-21a), (2-21b) and (2-23),(2-22) may be written in an explicit form:

式(2-24) [Fig.107_338_12]

where P1=D/D’, and P2=r/r’.
It is immediately found that one root of (2-24) is zero. This shows the generation of astationary wave mode by the coupling of barotropic and baroclinic motions. On the other hand, if (mc2’/f)^{2} ＜＜ 1, we have k2^{2} > (k2’)^{2} >> 1 \simeq\ (k1’)^{2} : (k2^{2},(k2’)^{2}〜\mu\^{-1}) and the interaction term in (2-24) may be neglected within the error of O(k2^{-1}) for a solution X1 of O(1).
Thus, the mode characteristics for X1 > O(1) can be approximated by those of the barotropic mode.
From (2-4a), the ratio \gamma\ of the surface displacement at the shelf break due to the baroclinic motion to that of the barotropic motion is

[Fig.107_338_13]

This can be transformed to

式(2-25) [Fig.107_338_14]

It is noticed that \gamma\ is O(k2^{-}1) at most for the mode with X > O(1), so that the baroclinic motion may be neglected. However, the interface displacement near the shelf break may be governed by the baroclinic motion because R2/R1〜O(\mu\^{-1/2}). For small values of X1, \gamma\ is of O(1) in general so that barotropic and baroclinic motions are equally important and \gamma\ ＜ 0 for X1 > Ti and \gamma\ > 0 forX1 ＜ T1.

3. Mode characteristics for a homogeneous fluid

Before going into discussions of mode characteristics for a two-layer fluid system, let us review the modes for a homogeneous fluid (see, for example, LARSEN,1969). The characteristic equation is given by a11=0 or from (2-24)

式(3-1a) [Fig.107_339_01]

where

式(3-1b) [Fig.107_339_02]

式(3-1c) [Fig.107_339_03]

式(3-1d) [Fig.107_339_04]

a. Very large shelf width; K1L >> 1 and
T1 \simeq\ 1
Two roots of (3-1a) are

式(3-2a,b) [Fig.107_339_05]

Now, for the very large long-shore wave length,(c1’,m/f)^{2} ＜＜ 1, we have (k1’)^{2}=P1 and X1 becomes the phase velocity cp of the mode relative to c1. Thus, we have

式(3-3a,b) [Fig.107_339_06]

The former corresponds to the Kelvin wave on the shelf and the latter corresponds to the double Kelvin wave near the edge of the shelf (LONGUET-HIGGINS,1968). The phase speed of the double Kelvin wave is faster or slower than the shallow-water Kelvin wave depending on the depth ratio P1 smaller or larger than 1/4.
For the very small long-shore wave length, (c1m/f)^{2} >> 1, we have k1’ \simeq\ 1, so that

式(3-4a,b) [Fig.107_339_07]

The former is an apparent root which does not correspond to the reality unless the condition of Kelvin wave is satisfied. The latter corresponds to the non-divergent short wave length limit of the double Kelvin wave near the edge of the shelf (RHINES,1969).
b. Small shelf width; T1~K1L ＜＜ 1
In this approximation,(3-1a) may be reduced to

式(3-5) [Fig.107_339_08]

and the two roots are approximated by

式(3-6a,b) [Fig.107_339_09]

If (mc1’/f)^{2} ＜＜ 1, the first root corresponds to the deep-water Kelvin wave mode, and the second root is the shelf-wave derived by LARSEN (1969). Phase velocities of these waves are approximately

式(3-7a,b) [Fig.107_339_10]

If (mc1/f)^{2} >> 1, the former root reduces to \omega\ \simeq\ f but the latter is unchanged.
c. Very large wave length; (mc1’/f)^{2} ＜＜ 1
In this case k1’^{2}=P1 and the variation of phase velocity of each mode with respect to K1L can be studied by tracing the two roots of (3-1a) as shown in Figure 2. If P1 ＜ 1/4, the deep-water Kelvin wave mode and the shelfwave mode for small K1L are transformed into double Kelvin wave mode and shallow-water Kelvin wave mode, respectively, as K1L increases. If P1 > 1/4, the opposite is true, because the phase velocity of the double Kelvin wave mode is smaller than that of the shallow-water Kelvin wave mode in this case.
Completely analogous results are expected for the internal wave modes if the coupling of the barotropic and baroclinic motions is neglected.

4. Mode characteristics in a two-layer fluid with the thickness of the upper layer shallower than the shelf depth

Taking the mode characteristics for a barotropic fluid into consideration, let us examine the modes for a two-layer fluid given by (2-24).
At first,let us consider the case of the very large shelf width. If T2\simeq\1 , X2=1 is a root and, furthermore, if T1\simeq\1, X1=1 is also a root. These roots correspond to the baroclinic and barotropic Kelvin wave modes in shallower water if (mc2/f)^{2} ＜＜ 1 and (mc1/f)^{2} ＜＜ 1, respectively. In this case (T1→1), the last root is given by

式(4-1) [Fig.107_340_01]

which is reduced to (3-2b) when (mc2’/f)^{2} ＜＜ 1.
Since in general (2-24) is complicated, simplified equations are derived for two cases of the long-shore wave length; (a) mc2/f)^{2} >> 1 and (b) (mc2/f)^{2} ＜＜ 1. The case (a) corresponds to the assumption of non-divergence of the baroclinic motion as well as the barotropic motion.
The case (b) gives the strong divergence of the baroclinic motion, so that the coupling of the motions in the upper and lower layers is large.
(a)(mc2/f)^{2} >> 1
In this case,k1’=k2’=k2=1 and X1’=X2’=X2=X1=\omega\/f. Therefore,(2-24) can be simplified to yield

式(4-2) [Fig.107_340_02]

with

[Fig.107_340_03]

The roots of (4-2) are X1=0,1, and W among which X1=0 corresponds to the stationary mode and X1=0 is the inertial oscillation and only valid when the condition of Kelvin wave in shallower water is satisfied. The most intersting root is X1=W.
If T1, T2→1, this root becomes

式(4-3) [Fig.107_340_04]

and the wave forms of both the barotropic and barochnic motions are of the double Kelvin wave type (\alpha\l=1). Compared with (3-4b) for the purely barotropic case, it is seen that the frequency is increased by the presence of the stratification. Furthermore, the volume transport vector Q1 in the upper layer given by (2-5a) reduces to zero since (1-r)U1=-rU2 and the volume transport vector in the lower layer given by (2-5b) becomes Q2=U1. Thus, the wave is analogous to the bottom wave discussed by RHINES (1970). This is in contrast to the stationary mode (X1=0), in which the motion is confined in the upper layer only in the present approximation.
If T1 ＜＜ 1 and T2 \simeq\ 1, we have approximately

式(4-4) [Fig.107_340_05]

The wave forms of the barotropic and baroclinic motions are the Kelvin wave type and the double Kelvin wave type, respectively. Compared with (3-4b), it is found that P1 is replaced by P2, showing that the baroclinic motion is dominant.
(b) (mc2/f)^{2} ＜＜ 1
In this case, we may safely assume the inequality:

[Fig.107_340_06]

Therefore, the three roots besides zero can be determined approximately by the following characteristic equation within the error of O(k2^{-1});

式(4-5) [Fig.107_340_07]

where a and b are the same as (3-1b),(3-1c)

and

式(4-6a) [Fig.107_341_01]

式(4-6b) [Fig.107_341_02]

with

式(4-6c) [Fig.107_341_03]

式(4-6d) [Fig.107_341_04]

For the Iarge shelf width (K1L >> 1), we may put T1,T2 \simeq\ 1 and the relative orders of the coefficients in (4.5) are

[Fig.107_341_05]

Therefore, the smallest root is approximated by

[Fig.107_341_06]

This corresponds to the baroclinic Kelvin wave mode in shallower water. The remaining two roots are approximately the same as for the homogeneous fluid (3.2a, b).
On the other hand, for the small shelf width (K1L ＜＜ 1), we have

[Fig.107_341_07]

so that

[Fig.107_341_08]

In this case, the largest root is approximately the same as for the homogeneous fluid (3.6a): namely the deep-water Kelvin wave mode if (mc1’/f)^{2} ＜＜ 1. The remaining two roots can be determined approximately by

式(4-7) [Fig.107_341_09]

where

式(4-8a) [Fig.107_341_10]

式(4-8b) [Fig.107_341_11]

In particular, if K2L ＜＜ 1, two roots are approximated by

式(4-9a,b) [Fig.107_341_12]

The former corresponds to the baroclinic Kelvin wave mode in deep-water (\alpha\2 ＜ 0) if (mc2’/f)^{2} ＜＜ 1 and the latter to the shelf-wave (0 ＜ \alpha\2 ＜ 1) which is different from the barotropic shelfwave mode (3-6b).
The variation of phase velocity of each mode with K2L is shown in Figure 3 by solving (4-5) numerically with (mc1’/f)^{2} ＜＜ 1. Let us denote the modes given by three roots of (4-5) as the first, second and third modes in the decreasing order of the roots. The first mode is then found to be almost the same as the barotropic mode provided that the baroclinic motion is also significant near the shelf break.
For P1 > 1/4, the wave form of the barotropic motion varies from the Kelvin wave type to the double peak type and finally to the double Kelvin wave type as K1L increases from zero to infinity. The accompanying wave form of the baroclinic motion is of the double Kelvin wave type for K2L >> 1. Near the shelf break, the interface disturbance is larger than the surface disturbance by O(k2).
The second mode with the intermediate phase velocity is the most interesting. The barotropic wave-form changes from the Kelvin wave type tothe shelf-wave type at X1=T1 as K1L increases from zero and approaches to the Kelvin wave in shallower water as K1L >> 1. On the other hand, the wave-form of the baroclinic motion changes from the Kelvin wave type to the double peak type at k2X1T2=1 as K2L increases and finally approaches the double Kelvin wave type for K2L >>1. For a moderate value of K1L when the phase velocity of the second mode is close to that of the barotropic shelfwave mode, the baroclinic motions are still important near the edge of the shelf. It may be noticed that the characteristics of the second mode shows an apparent effect of increasing phase velocity of the barotropic shelf-wave due to the presence of stratification.
The third mode with the smallest phase velocity (X2 ＜ 1) is predominantly of the baroclinic nature with significant contribution from barotropic motions. The wave-forms of both the barotropic and baroclinic motions are of the shelf-wave type. With K2L→\infty\, however, the baroclinic motion is confined near the coastal boundary as a Kelvin gave and the barotropic motion disappears.
(c) Stationary wave mode: (X1=0)
If we take the solution of the form (2.12) together with (2.15) and (2.16),it is straightforward to derive the stream functions \Psi\1 and \Psi\2 in the upper and lower layers satisfying (2-6) and (2-7) in the form:

式(4-10a) [Fig.107_342_01]

式(4-10b) [Fig.107_342_02]

with

式(4-11a) [Fig.107_342_03]

where C is a constant. In deep water, all quantities in (4-10a) and (4-10b) are dashed and

式(4-11b) [Fig.107_342_04]

It is easily seen that, if (mc2/f)^{2} >> 1, namely when the divergence of the baroclinic motion is negligible, we have K1=K2=m. Therefore, under the condition of mL >> 1, we have Ti=1, \psi\1=\psi\2,and from (4-10b), the motion in the lower layer vanishes. Thus, we may call this mode a surface mode in contrast to a bottom made discussed in (a) of this section.
It is mentioned that, since \omega\=0, the flow is purely geostrophic and the solution is not ristricted to the form (4.11a, b). More general form of a stationary vortex can satisfy the boundary conditions (2.6) to (2.11).

5. Mode characteristics in a fluid with two layers in deep water only

Parallel arguments to Section 2 lead to the characteristic equation for a model with the two-layer in deeper water only:

式(5-1) [Fig.107_342_05]

(a) (mc2’/f)^{2} >> 1
In this case, we have k1’,k2’=1 and (5-1) reduces to

式(5-2) [Fig.107_342_06]

where

[Fig.107_342_07]

The root X1=1 is trivial and the important root is

式(5-3) [Fig.107_342_08]

Since X1 ＜ T1, the motion on the shelf is of the shelf-wave type. For T1→1, we arrive at the double Kelvin wave. The difference of this wave from the barotropic case (3-4b) is the replacement of P1 by P3: namely the actual depth of deeper water is replaced by the thickness of the upper layer.
(b) (mc2’/f)^{2} ＜＜ 1
In this case, we have k2’^{2} >> 1 \simeq\k1’^{2},T1,and

(5-1) may be transformed to

式(5-4) [Fig.107_343_01]

within the error of O(k2’^{-1}). Here, a and b are the same as (3-1b) and (3-1c), and

式(5-5a) [Fig.107_343_02]

式(5-5b) [Fig.107_343_03]

For a very large shelf when K1L >> 1, we have T1=1 and the larger two roots can be approximated by the roots for the homogeneous fluid,(3-3a, b). The smallest root is approximated by

式(5-6) [Fig.107_343_04]

This root corresponds to the double wave mode (\alpha\1=1) with the internal wave type in deeper water.
For a small shelf width when K1L ＜＜ 1 , we have T1 ＜＜ 1 and the largest root can be approximated by the corresponding root for the homogeneous fluid (3-6a). The smaller two roots are given by

式(5-7) [Fig.107_343_05]

If K2’L ＜＜ 1, we have

式(5-8a,b) [Fig.107_343_06]

(5-8a,b) is similar to (3-6a,b) and the larger root corresponds to the deep-water baroclinic Kelvin wave mode and the smaller root is the shelf-wave mode with the baroclinic character dominating in deep water.
If we put formally K2L >> 1, the larger root of (5-7) becomes

式(5-9) [Fig.107_343_07]

Namely, the mode corresponding to the larger root resembles the barotropic shelf-wave mode (3-6b). The increase of the phase velocity of a shelf-wave due to the effect of stratification discussed by MYSAK (1967) is thus explained by the transition of the predominant motion of this mode from the barotropic shelf-wave mode to the baroclinic Kelvin wave mode in deeper water as the shelf width decreases (K2’L ＜ 1).
He apparently missed the baroclinic shelf-wave mode of much slower phase velocity(5-8b}.
It should be noticed that, as P3→1, the smallest root approaches zero; in other words the stationary mode is a limiting case of the mode corresponding to the smallest root in the present model as D/D1’→1.
The variation of phase velocity of each mode with K2’L is shown in Figure 4, by solving (5-4) numerically. It is seen that the mode with the smallest phase velocity resembles the shelf-wave mode for small K2’L and the baroclinic double Kelvin wave mode as K2’L increases. On the other hand, the mode corresponding to the deep-water baroclinic Kelvin wave mode for small K2’L is transformed to the mode corresponding to the barotropic shelfwave mode and finally to the shallow-water Kelvin wave mode (or barotropic double Kelvin mode if [Fig.107_344_01] increases.

6. Conclusions

The characteristics of the trapped modes in a two-layer shelf model are examined. The relative order of various scales: the long-shore wave length, the Rossby’s deformation radius and the shelf width, are crucial to determine the characteristics of various modes. The variation of the phase velocity of each mode with the increase of the shelf width is shown in Figures 3 and 4, for a very large long-shore wave-length. The shelf-wave mode splits into two parts near fL/c2〜1 or 2 by the presence of stratification. With the decrease of the shelf width the barotropic shelf-wave mode is transformed into the deep-water baroclinic Kelvin wave mode and the shelf-wave mode formed by the combination of the barotropic and baroclinic motions with much smaller phase velocity appears.
If the interface of the two fluid layers is shallower than the shelf depth, a stationarv mode is possible, and for large shelf width the double Kelvin wave mode of baroclinic nature does not exist. If (mc2/f)^{2} >> 1 and (mL)^{2} >> 1 , the stationary wave mode and the shelf-wave mode become the surface mode and the bottom mode, respectively, where the motion is confined either in the surface or bottom layer only.
If the interface is deeper than the shelf depth, the shelf-wave of the baroclinic character in deeper water exists for small shelf width. In contrast, the barotropic shelf-wave mode is transformed with the decrease of the shelf width into deep water baroclinic Kelvin wave mode near fL/c2’〜1 or 2, so that the phase velocity of this mode apparently increases by the presence of stratification. For large shelf width, two double Kelvin wave modes appear with barotropic or baroclinic characters in deeper water,respectively.
The present model is admittedly too simple to understand fully the mode characteristics of trapped waves in a real ocean. The inclusion of the depth variation on the shelf would complicate the coupling of the barotropic and baroclinic motions, because of the presence of shelf-waves of higher modes. However, if the slope on the shelf is small, the coupling of two kinds of motions on the shelf may be safely neglected if the Rossby’s internal deformation radius is much smaller than the horizontal scale of the wave. The existence of the rapid variation of the depth at the shelf break is considered to be the essential factor causing the coupling, so that the present model may be justfied with respect to the qualitative nature of mode characteristics.

Acknowledgment

The author is grateful to Miss KATO for her help in numerical computation.

References

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ISOZAKI,I.(1972): Unusually high mean sea level in September 1971 along the south coast of Japan -1.Some aspects of high sea level with time scale more than one week-. Pap. Meteorol. and Geophys., 23, 243-257.
KAJIURA, K.(1962): A note on the generation of boundary waves of Kelvin type. J. Oceanog. Soc. Japan,18,49-58.
LARSEN, J.C.(1969): Long waves along a singlestep topography in a semi-infinite uniformly rotating ocean. J. Marine Res.,27,1-6.
LONGUET-HIGGINS, M.S.(1968): On the trapping of wares along a discontinuity of depth in a rotating ocean. J. Fluid Mech.,31,417-434.
MYSAK,L.A.(1967): On the theory of continental shelf waves. J. Marine Res.,25,205-227.
RHINES, P,B.(1969): Slow oscillations in an ocean of varying depth,Part 1.Abrupt topography. J.Fluid Vlech.,37,161-189.
RHINES, P.B.(1970) Edge-, bottom-, and Rossby waves in a rotating stratified fluid. Geophys. Fluid Dynamics,1,273-302.
ROBINDON, A.R.(1964): Continental shelf waves and the response of the sea surface to weather system. J. Geophys. Res.,69,367-368.
SHOJI, D. (1961): On the variation of the daily mean sーa levels along the Japanese Islands. J. Oceanog. Soc. Japan,17,141-152.
SUGINOHARA, N. (1973): Response of a two-layer ocean to typhoon passage in the western boundary region. J. Oceanog. Soc. Japan,29,236-250.
YOSHIDA, K.(1960): The oceanic waves of days to months periods. Rec. Oceanog. Wks. Japan,5, 11-24.
YOSHIDA, K., D. SHOJI and J. MASUZAWA (1972): A possible interaction between the storm tides and the Kuroshio- a speculation on the recent floods. Rec. Oceanog. Wks. Japan,17,47-51.

陸棚上の超長周期波に対する成層の効果梶浦欣二郎

要旨

二層モデルによって,一様水深の真直な陸棚上に生ずる超長周期波の特性を調べ,陸棚幅,岸沿いの波の波長等の違いによって,特性がどうなるかを明らかにした.
このモデルでは四つのモード波が出来るが,そのうち位相速度が最大のモード波は順圧モードで近似出来る.
順圧の陸棚波に対応する二番目のモード波は傾圧的な海水運動によってかなりの影響をうけ,陸棚幅が狭い極限では外海側の内部ケルビン波に対応するモードに移行する(岸沿いの波の波長が大きい場合).従って,この場合はみかけ上陸棚波速度が成層の影響によって増加するようにみえる.残りの二つのモード波は内部波的であるが,順圧的な海水運動も重要な効果を及ぼしている.そのうち,一つは陸棚幅が狭い場合には陸棚波的な振舞をし,陸棚幅が大きい場合には陸棚上の内部ケルビン波になる.もう一つは定常波モードである.岸沿いの波の波長が内部ロスビー変形半径より短いときには上,下両層の運動は無関係となり,表面モードと海底モードとが出来る.
もし,二層の境界面の深さが陸棚水深よりも大きいときには定常波モードはなくなり,第三番目のモード波の特性は陸棚幅が扶い場合には陸棚波的で,陸棚幅が広い場合には内部波の特性をもった二重ケルビン波に近づく.位相速度の大きい初めの二つのモード波の特性は境界面の深さが陸棚水深より小さい場合と本質的に変わりはない.

BULLETIN OF THE EARTHQUAKE RESEARCH INSTITUTE Vol.56 (1981), pp.415-440 20.Tsunacmi Energy in Relation to Pacrameters of the Earthquake Fault Model. By Kinjiro KAJIURA, Earthquake Research Institute. (Received July 30,1981)

Abstract

Tsunami energy generated by an earthquake is estimated on the basis of a simple fault origin model of the earthquake. Tsunami energy Et is given by

[Fig.107_346_01]

where Mw is the moment-magnitude of earthquake and F is a function of fault parameters (maximum F is about 0.1), such as the dip angle \delta\, slip angle \almbda\ and the relative depth h*(=H*/L ; where H* is the mean depth of the fault plane with the length L and width W). The aspect ratio (=W/L) is assumed to be 1/2.
The variation of F with respect to the full range of \delta\,\lambda\,or h* (＜ 1.0) is about a factor of 10. In particular, the difference of tsunami energy between the vertical faults with the dip and strike slips is conspicuous. Since the depth dependence of the tsunami energy is given in terms of the relative depth h*,the decrease of energy with the increase of the fault depth H* is more significant for smaller earthquakes.
The results are compared with empirical values of tsunanmi energy published so far. The general trend of logEt with respect to Mw, is consistent with the above formula. However, it is noted that the values of tsunami energy derived in the past on the basis of the energy flux method were systematically overestimated by a factor of 10 or more. On the other hand, the maximum tsunami energy (Chilean earthquake of 1960) would be around 10^{23} ergs and somewhat lower than the value expected from the formula.

1. Introduction

It is known that major tsunamis are generated at thc time of large shallow earthquakes occurring undcr the sea. More than 20 years ago IIDA (1958) showed a relationship between the physical sizes of tsunamis and earthquakes by correlating the so-called Imamura-lida magnitude m of tsunamis with the earthquake magnitude M (JMA magnitude Mjdetermined by the Japan Meteorological Agency was used) of the corresponding earthquakes, though the correlation was not so good. To appreciate this kind of empirical relation between the physical sizes of tsunamis and earthquakes, we need to understand several different problems:1) What is the most appropriate measure to represent the overall size of a tsunami?
2) is the Gutenberg-Richter surface wave magnitude M, adequate as a measure of the「size」of a very large earthquake? 3) What is the cause of the essential scatter in the relation of the tsunami magnitude (a single parameter representation of the「size」of a tsunami, whatever the definition may be) to the earthquake magnitude (or the total seismic moments M0)?
The first problem has been discussed by many researchers, each of whom criticized the deficiencies of the Imamura-lida scale and proposed a new definition of the tsunami magnitude scale (WATANABE,1964: IIDA et al.,1967;SOLOVIEV,1970;HATORI,1973;ADAMS,1974;ABE,1979; MURTY and LOOMIS,1980). After all, it seems to be the consensus from the beginning of the use of a tsunami magnitude scale by IMAMURA (1942, 1949) that the scale should in someway be related to the total physical size (total energy) of the tsunami and not merely an index to represent the intensity of a tsunami at a specific location. MURTY and LOOMIS (1980) proposed to define the magnitude scale directly on the basis of the total tsunami energy. However, to determine the tsunami energy from observed wave data is not an easy task in reality and only a few reliable estimates are available so far. HATORI (1970) summarized the data published before 1970. The second problem has been argued the past 10 years or so on the basis of physical models of the earthquake source and a new definition of the earthquake magnitude scale Mw, namely a moment-magnitude scale based on the total earthquahe moment M0 was proposed by KANAMORI (1977) to overcome the saturation problem of the seismic wave amplitude for great earthquakes (Mw >= 8) and, at the same time, to guarantee the smooth continuation to the surface wave magnitude Ms for earthquakcs of moderate sizes. This continuation was examined later by PURCARU and BERCKHEMER (1978) and SINGH and HAVSKOV (1980). As for the third problem, it has been empirically known that 1) the tsunami generation is influenced by the depth of the earthquake hypocenter: deep-focus earthquakes with the focal depth larger than, say, 60km hardly ever generate a very large tsunami (IIDA,1958,1963a), and 2) tsunamis seem to be generated by earthquakes having predominantly a large dip-slip component of the fault dislocation and earthquakes of the strike slip type ordinarily result in only minor tsunamis (IIDA,1970; WATANABE,1970).
With advances in the physical understanding of the earthquake mechanism (see, for example, AKI,1972; KANAMORI and ANDERSON,1975) these problems can be discussed from a unified point of view. The generation of tsunamis has been discussed directly on the basis of a simple source model of earthquakes in the solid-liquid coupled system (PODYAPOLSKY,1970; YAMASHITA and SATO,1974; ALEXEEV and GUSIAKOV, 1974;WARD,1980). This is considered to be the most straightforward approach to the tsunami generation problem. However, to understand the gross characteristics of tsunamis, such as the total energy, in relation to an assumed fault model of large earthquake, it may be advantageous to use rough estimate method. Since the major part of the permanent crustal deformation at the time of an ordinary large earthquake is believed to be completed in a relatively short time interval (within a minute or so) compared with the generated tsunami「period」,the total tsunami energy may be roughly estimated from the initial potential energy of the water surface disturbance computed on the basis of the instantaneous deformation of the ocean bottom, which is equivalent to the static deformation caused by the earthquake fault.
If this assumption is accepted, the relation of the tsunami energy to various earthquake parameters can be found by analyzing the dependence of the static deformation of the sea bottom on the fault parameters of an earthquake. It is significant that the recent advance of numerical methods to solve thc shallow water hydrodynamical equations enable us to reproduce the coastal height distribution of the tsunami generated by a given large scale bottom deformation with considerable accuracy and, in fact, AIDA (1977,1979a) confirmed that, in many cases of large tsunamis, the distribution of tsunami heights observed along the coast can be explained to a reeasonable degree of accuracy by a suitably assumed earthquake fault model consistent with seismological data. Thus, in the present paper, we focus attention on the dependence of tsunami energy on various parameters of an earthquahc fault. At the same time, the reliability of various estimates of tsunami energy published so far are critically reviewcd and compared with the estimate bascd on the fault model.
Up to the present, the operational tsunami warning practice depends heavily on the empirical relation analogous to Iida’s relation because the information obtaincd immediately after the occurrence of a large earthquake is limited to something like the Gutenberg一Richter earthquake magnitude M and the location of the epicenter. For tsunamis propagating from a distant source region, it is possible to supplement waterwave data at various places later to improve the prediction, but for tsunamis generated in the vicinity of the coastal area of interest, or where no waterwave data is available, it is advantageous to use the earthquake moment M0 and possibly the kinematic parameters of the fault model, if they can be estimated quickly after the earthquake occurs. It appears that the use of these parameters in operational tsunami warning is possible (KANAMORI, personal communication,1981). In this respect, the present study may be regarded as a kind of sensitivity study of various fault parameters influencing the total tsunami energy. Of course, there are other dynamical problems in connecting the total tsunami energy to the wave heights realized along the coast. However, these problems are outside the scope of the present study.

2. Relation between tsunami energy and earthquake magnitude

It is instructive to first review some relations concerning the earthquake fault model. Take a simple fault plane of a rectangular shape as shown in Fig.1, where L is the length, W the width, S the area (=LW), and D the mean dislocation on the fault. The dip angle of the fault plane is \delta\, and \lambda\ is the slip angle of the dislocation on the fault plane measured from the horizontal axis. Then, the dislocation theory in elasticity applied to the earthquake fault model predicts the following approximate relations for physical quantities such as the total seismic morment M0, the mean stress drop \Delta\\sigma\, and the change of total strain energy W (see AKI,1972):

式(1) [Fig.107_350_01]

式(2) [Fig.107_350_02]

and

式(3) [Fig.107_350_03]

where \mu\ is the rigidity with the value between 3〜7×10^{11} dyne cm^{-2} depending on the crustal condition in the earth, and C (2.4〜5) is the shape factor depending on the fault geometry and slip direction. For a buried dip-slip faulting on a rectangular fault, [Fig.107_350_04]. \sigma\ is the average stress given by \sigma\= (\sigma\1十\sigma\0)/2 with \sigma\0, and \sigma\1 the average stresses on the fault plane immediately before and after the earthquake occurrence and the stress drop \Delta\\sigma\ is equal to (\sigma\0-\sigma\1). The total seismic energy Es may be formally related to the change of strain energy W due to dislocation in the form E=\eta\W where \eta\ is the efficiency related to \sigma\1. From (1) and (3), the minimum estimate of strain energy release is Wmin=(\Delta\\sigma\/2\mu\)M0 and it seems reasonable to assume Es〜Wmin.
One basic empirical relation in the simple fault model of shallow earthquakes is that the fault area S and the earthquake moment.M0 are closely related (ABE,1975;KANAMORI and ANDERSON,1975); namely,

式(4) [Fig.107_350_05]

The constant of proportionality\alpha\ appears to vary by a factor of 3 depending on the tectonic setting of the region in which the earthquake takes place: \alpha\〜1.23×10^{7} dyne cm^{-2} for interplate earthquakes and about 3×10^{7} dyne cm^{-2} for intraplate earthquakes (S and M0 are expressed in c.g.s. units). Substitution of (4) into (1) yields D/S^{1/2}=\alpha\/\mu\ and (2) becomes \Delta\\sigma\ =C\alpha\. Roughly speaking, the length to width ratio of the fault plane L/W is about 2 (ABE,1975;GELLER,1976), so that the former relation suggests the geometrical similarity of the earthquake fault, and the latter relation indicates that the difference of \alpha\ is the reflection of the variation of the stress drop \Delta\\sigma\. It is very often said that the stregs drop \Delta\\sigma\ is about 30 bars for earthquakes on interplate boundaries. ABE (1975) actually estimated \alpha\ from \Delta\\sigma\/C with \Delta\\sigma\=30 bars and C=2.44.
Now extending the earthquake energy-magnitude relationsllip

式(5) [Fig.107_350_06]

to the range where the surface wave magnitude Ms suffers a saturation problem, a new magnitude scale Mw may be defined by the substitution of the minimum strain energy release Wmin for Es (KANAMORI,1977):

式(6) [Fig.107_351_01]

Kanamori took a constant strain drop \Delta\\sigma\/\mu\〜10^{-4} for major earthquakes occurring along interplate boundaries and gave

式(7) [Fig.107_351_02]

The usefulness of this magnitude scale for earthquakes of moderate sizes was examined by PURCARU and BERCKHEMER(1978) and SINGH and HAVSKOV (1980). Since the strain drop \Delta\\sigma\/ \mu\ is fixed in (7), this magnitude scale is a function of the total moment M0 only and is called the momentmagnitude.
In the fault model of earthquakes there is one important dynamic pararneter called the rise time \tau_{s}\ which is related to the movement of the fault dislocation. The mean velocity of dislocation D/ \tau\s is proportional to the initial effective stress \sigma_{e}\ and if \sigma_{e}\ is approximated by \Delta\\sigma\, the rise time \tau_{s}\ is given by \tau_{s}\=\mu\D/(\beta\\Delta\\sigma\) (GELLER,1976) where \beta\ is the shear wave velocity. Substituting relevant numerical values, the velocity of dislocation is found to be around 1m sec^{-1}. Invoking the geometrical similarity of the fault and the constancy of \Delta\\sigma\, it follows that

式(8) [Fig.107_351_03]

On the other hand, the time scale corresponding to the rupture propagation with the rupture velocity v(v〜0.7\beta\) over the whole length L of a fault is L/v〜2S^{1/2}/\beta\ and about 5 to 10 times greater than the rise time \tau_{s}\. Therefore, we assume the representative time scale related to the permanent deformation of the crust to be \tau\*s^{1/2}/\beta\ .
Now the representative time scale \tau\ of the tsunalni generated by the crustal deformation of the ocean bottom may be expressed by \tau\=S^{1/2}/c where c(=(gd)^{1/2}: d the depth of water) is the velocity of a shallow water gravity wave. Thus, the ratio \tau\*/ \tau\(=c/\beta\) plays an important role in the efficiency of tsunami generation due to the bottom deformation. If \tau\*/ \tau\ ＜＜ 1,it is known that the generated tsunami energy.Et is almost independent of \tau\* and the same as the energy Et0 in the case of the instantaneous deformation of the bottom (for a model of uniform uplift, Et/Et0 is about 0.9 for \tau\*/ \tau\〜0.2: KAJIURA,1970). In realistic situations, c/\beta\=＜ 1/20 and the condition of \tau\*/\tau\ ＜＜ 1 always holds. If \tau\*/ \tau\ > 1,the ratio Et/Et0 is generally proportional to (\tau\*/\tau\)^{-n} (n is unity for the case of one-dimensional propagation and n is somewhat larger for the case of two-dimensional propagation). It is known that in the case of an instantaneous deformation of a very large portion of the sea bottom (S^{1/2}/d >>1), the initial disturbance at the sea surface is approximately equal to the bottom displacement at least where the gravity wave part is concerned. Of course, smaller scale components of the bottom displacement are attenuated by the hydrodynamic effect in water, so that the surface disturbance is a smoothed version of the displacement with a filter function of 1/cosh (kd) where k is a wave number. That is, the bottom deformation on a scale less than about 2d is very strongly attenuated in the water surface disturbance. The oscillatory time dependence of the bottom displacement does not contribute significantly to the final gravity wave generation, if the angular frequency of oscillation \omega\ is in the range of \omega^{2}\d/g > 1.
Thus, we may obtain a rough estimate of the total tsunami energy Et from the initial potential energy corresponding to a static mound of water generated at the sea surface by the sudden displacement z of the sea bottom:

式(9) [Fig.107_352_01]

where \rho\ is the density of water and S’ is the surface area in which the vertical static displacement z takes place at the time of the earthquake occurrence. Now, scaling the bottom displacement in terms of the dimensions of a plane fault, we get

[Fig.107_352_02]

where H is the depth of the upper rim (parallel to the free surface) of a fault plane, and (9) can be formally written as

式(10) [Fig.107_352_03]

where

式(11) [Fig.107_352_04]

Note that F(\delta\,\lambda\, h,r) is a non-dimensional function apparently independent of the absolute values of D and S (namely independent of Mw).
Apart from the factor F,(10) shows that Et is proportional to D^{2}S and, taking the relations (1) and (4), we have

式(12) [Fig.107_352_05]

The proportionality of Et to M0^{4/3} was mentioned by AIDA (1977) but WARD (1980) suggested M0^{2} dependency by assuming a moment tensor point source.
Now, adopting the relation (7) and using the value \alpha\=1.23×10^{7} dyne cm^{-2}, \mu\=5×10^{11} dyne cm^{-2}, and \rho\g=10^{3} c.g.s. units it follows that

式(13) [Fig.107_353_01]

Because of the range of variation of \alpha\ and \mu\, the value of the constant term in (13) is somewhat uncertain and the value of Et may vary by a factor of about 2. Since the maximum value of F is about 0.11 as will be introduced later, the gross upper limit of the total tsunami energy may be formally shown as

式(13) [Fig.107_353_02]

Comparing (14) with (6), we have

式(15) [Fig.107_353_03]

the ratio、Et/Es increases with the increase of earthquake magnitude, but even for maximum earthquake (Mw〜9.5), the ratio is less than 10^{-2}. The relation suggested by (13) will be checked later by the examination of an empirical relationship between the estimated tsunami energy by various methods and the moment-magnitude Mw of the corresponding earthquake.
The difference between the tsunami energy and seismic energy in their dependence to the earthquake moment M0 or magnitude Mw may be interpreted as the effect of the rise time \tau_{s}\ of the fault dislocation. For seismic wave energy the rise time \tau_{s}\ plays an essential role (HASKELL, 1964) but for tsunami energy, the time scale of tsunami generation is much larger than the time scale of the crustal deformation (\tau\*/ \tau\ ＜＜ 1) so that the total energy of gravity waves is almost independent of the time scale \tau\* related to the permanent deformation of the crust (a kind of saturation problem).

3. Dependence of tsunami energy on earthquake fault parameters

For a given earthquake magnitude, F defined by (11) shows the dependence of tsunami energy on geometric parameters of the fault. To find the factor F, we asgulne a uniform rectangular plane fault with given parameters \delta\, \lambda\,h and r in a semi-infinite elastic solid with equal Lame constants. Only cases of r=1/2 are considered, though the aspect ratio varies between 2 >= r >= 1/4 (ABE,1975; GELLER,1976). The static displacernent z* of the free surface of a solid space is computed within an area 2L×2L by the analytical formula given by MANSHINHA and SMYLIE (1971). And z^{2}* is integrated in a limited region delineated by lz*/z*maxI >= 0.1 with z*max representing the absolute value of the maximum computed displacement. Actual computations are carried out for 0°=＜ \delta\, \lambda\ =＜ 900,and 0.04 =＜ h =＜ 0.44 with the intervals 15°,30°, and 0.1 respectively.
In addition, the computation for h*=1.0 is also made. It should be noticed that the relative depth h of the upper rim of the fault is related to the relative mean depth h*of the fault by

式(16) [Fig.107_354_01]

so that the apparent conclusion as to the dependence of F on \delta\ and \lambda\ is affected by the choice of the representative depth.
Figure 2 shows the dependence of F on \delta\ and \lambda\ for h=0.04 which is the case when the upper rim of the fault plane is close to the surface. The maximum value of F (〜0.11) is found when \lambda\=90° and \delta\ is 45°〜60° and Fdecreases sharply for small values of o. In Fig. tea, b, c, the mean depth h* is fixed (h*=0.29, 0.44 and 1.0) instead of the depth of the upper rim. The variation of F with respect to \delta\ is not so conspicuous and within a factor of 2, except for cases of strike slip faults (\lambda\=0°) in which the values of F vary by a factor of 10 between \delta\=0° and 90°. That is, F is smallest in the case of a vertical strike slip fault. It is seen in Fig.3a that the variations of F have an opposite trend with respect to \delta\ for\lambda\ larger or smaller than about 45°. The variation of F with respect to \lambda\ depends on \delta\ and the largest change occurs for a vertical fault \delta\=90° in which the variation amounts to a factor of 10 or more between the dip slip and strike slip faults but for small \delta\ the variation is rather small.
The depth dependence of F can be clearly seen in Fig.4 which shows the variation of F as a function of the relative depth h*(\delta\=30°,90°and \lambda\=0°,90°). Although not shown in the figure, the maximum values for small dip angles (\delta\ =＜ 15°) generally occur at an intermediate depth (h*〜 0.2).This feature is qualitatively consistent with other studies (PODYA- POLSKY,1970;YAMASHITA and SATO,1974). In contrast, for large dip angles (\delta\≧45°) the maximum values appear at the shallowest depth h*, except when the slip angle \lambda\ is small (\lambda\ ＝＜ 15°), and the effect of the optimum depth does not show up clearly, probably because the fault ex-tends deeper than the optimum depth even in the case when the upper rim of the fault approaches the surface.
For h* larger than, say,0.5, the decrease of F with h* may be regarded as exponential with the decay constant of about 2.4. Since the mean relative depth h* is given by H*/L with H*the mean depth and L the length of the fault respectively, the decrease of F with respect to the increase of the mean depth H* is relatively faster for smaller earth- quake magnitude Mw because L decreases with the decrease of Mw (if we assume W/L=1/2, \alpha\=1.23×10^{7} dyne cm^{-2}, it follows from (4) and (7): log L (km)=一0.5Mw-1.85). This indicates that, for a fixed mean depth H* larger than the optimum depth, the relation between Et and Mw deviates considerably from the relations log Et〜2Mw given by (14). In Fig.5, the Et-Mw relation in cases of vertical dip slip faults (\delta\=90°, \lambda\=90°) with h*=0.29, F=0.103 (the upper rim of the fault very close to the surface) and H*=50 km are shown. For example, at Mw=7.0, the tsunami energy when H*=50km is reduced by a factor of 10 from the value for h*=0.29. In the figure, a case of vertical strike slip (\delta\=90°,\lambda\=90°) with h*=0.29,F=0.002, is also shown for comparison.

4. Empirical methods to estimate tsunami energy

a) Relative energy method
Because of a close relationship between the tsunami magnitude scale m and the total tsunami energy, it may be expedient to explain the def- inition of the so-called Imamura-lida magnitude scale. The original definition of the tsunami magnitude scale proposed by IMAMURA (1942) was of a rather qualitative nature, because he intended only to assign the size for old Japanese tsunamis described very briefly in historical docu- ments. An important point in his classification is that the 1896 Sanriku tsunami was assigned grade 4 and the 1933 Sanriku tsunami grade 3. Other tsunamis are classified by comparison with the above two tsunamis. TAKAHASI (1951) translated the qualitative definition of the Imamura's tsunami magnitude in an explicit form:

式(17) [Fig.107_357_01]

where H is the「maximum」height and the suffix 0 indicates reference values.(He took m0=5 for H0=30 meters.) Although he stated that H is the maximum height, he apparently used some kind of a mean maximum value for H in practice. This is evident because he assigned the magni- tude m=3 for the 1933 Sanriku tsunami, which corresponds to H=7.5 meters, despite the fact that the maximum oboerved runup was 28 meters. Later, IIDA (1963) took the reference values in (17) as m0=0 for H0=1 meter. The difference of the magnitude scales defined by Takahasi and Iida is actually small if the definition of H is the same. SOLOVIEV (1970) called this kind of tsunami scale m as an intensity scale i and explicitly defined H as H=√2H with H the mean of inundation heights over the coast where the tsunami activity was significant (He took i0=0 for H0=1 meter). Originally, the definition of the so-called Imamura-lida magnitude scale for large tsunamis way stated in terms of the observed maximum height, so that the ambiguity in the definition of H in (17) caused con- fusion among researchers. The confusion was accelerated when IIDA et al. (1967) defined the magnitude scale by (17) in their tsunami catalogue with the statement that H is the actual observed maximum height.
With the accumulation of data on smaller tsunamis, it was felt that a clear distinction between the magnitude scale and the intensity scale was desirable. Various proposals were made to define the magnitude scale based on the wave height estimated at a fixed distance from the tsunami source (WATANABE,1964; HATORI,1973; ADAMS,1974). Discussions of the magnitude scale of the tsunami itself, however, are not the subjectof the present paper.
Since the shortest distance between the coast and the tsunami source for most large tsunamis experienced in Japan do not differ significantly, TAKAHASI (1951) considered that the total tsunami energy Et is propor- tional only to the square of the representative tsunami height H observed on the coast. Thus, he proposed the formula,

式(18) [Fig.107_358_01]

where H0 is the reference tsunami height corresponding to Et0.
Changing (17) to (18), he gave

式(19) [Fig.107_358_02]

In this case, he took the 1933 Sanriku tsunami as a standard, for which m0=3 and Et0 =1.6×10^{23} ergs (This value of Et0 was estimated by the energy flux method, but was too large by a factor of 50, which will be explained later).
Taking these numerical values into account,(19) becomes

式(20) [Fig.107_358_03]

IIDA (1963b) extended Takahasi's argument by combining (20) with the formula he derived between m and M (JMA magnitude of the earthquake), namely

式(21) [Fig.107_358_04]

to show roughly

式(22) [Fig.107_358_05]

Recalling the Gutenberg-Richter seismic energy-magnitude relation (5), he concluded that Et/Es〜1/10. The numerical value of this ratio depends on the choice of the Et0 value. WATANABE (1964) and WILSON (1964) also stated that the ratio of the tsunami energy to the seismic energy is constant.
It should be noted that Takahasi's conjecture leading to (18), which is followed by several investigators, neglected one important factor in the consideration of the relative tsunami energy, even if the assumption of the equal distance between the coast and the tsunami source is accepted. Namely, the「period」(or the「wave length」) of a tsunami which is also a function of the earthquake magnitude (TAKAHASI,1963). Furthermore, the relation given by (21) greatly lacks certainty. Thus, the formula (22) and its consequences such as the constant energy ratio between the tsu- nami and earthquake are not well founded.
b) Energy flux method
For a free, progressive, plane, shallow waterwave packet of finite duration, the total flux of energy Ef(\theta\) across a section of unit width perpendicular to the direction of the wave propagation \theta\ may be written as

式(23) [Fig.107_359_01]

where An is the successive amplitudes (the crest height or trough depth from the mean level),\Delta\\tau_{n}\ is the duration of the elevation or depression (half the wave「period」of successive waves), and N is the number of half waves radiated from the source. And d is the water depth where An is determined, but the variation of d in one wave length should be small.
Assuming the isotropic radiation of energy from a point source, TA- KAHASI (1951) proposed to compute the total energy Et of the tsunami in the form

式(24) [Fig.107_359_02]

where R is the distance from the epicenter of the earthquake to the loca- tion where An is estimated.
The refractive effect of the depth variation during the long distance travel of tsunami waves radiated from a point source may be expressed by the refraction coefficient P(\theta\) defined by

式(25) [Fig.107_359_03]

where \theta\ is the angle of the observing Iocation from a reference direction at the source,\theta\0 is the angle in radian between the neighboring wave rays at the source and S(\theta\) is the separation of the same neighboring rays at the distance R from the source. Applying this coefficient,(24) is modified to

式(26) [Fig.107_359_04]

The geometric optics approximation is valid only if the relative depth change in one wave length is small. Because of the long wave length of ordinary tsunami waves, this approximation is probably not satisfactory near the shelf region and the coast.
In reality, the radiation characteristics of the tsunami energy depend on the characteristics of the earthquake source mechanism and, for an elongated dipole type tsunami source, this effect is very important. The effect may be conveniently expressed by the source directivity coefficient Q(\theta\) defined by

式(27) [Fig.107_360_01]

where Ef*(\theta\) is the energy flux of radiated waves per unit width at a distance of R when the depth is uniform. For a simple uniform uplift of the rectangular portion (L×W) of the ocean bottom, Q (\theta\) depends on the ratio W/L and if the reference direction is along the major axis of the source, Q (90°) 〜L/W (KAJIURA,1970).
When the depth variation in the vicinity of the tsunami source is significant, it is impossible to separate the topographic effect from the source directivity defined by (27). However, for a rough approximation, we may replace Ef*(\theta\) by Ef(\theta\)/P(\theta\) and write

式(28) [Fig.107_360_02]

The difficulty in applying this method is, apart from the assumptions on P(\theta\) and Q(\theta\), the estimate of the incoming wave amplitude An,at the depth d and the duration \tau\=\Sigma\\tau_{n}\ of waves from data taken along the coast where observed wave signatures are strongly modified by such effects as the shoaling, coastal reflection, refraction, diffraction, scattering and wave trapping on the shelf and inside the bay. Despite these diffi- culties, TAKAHASI (1951), IIDA (1963b) and HATORI (1966) estimated values of the total energy of many tsunamis in Japan on the basis of (24). Although they did not give the details of their computations, it appears that the incoming wave amplitude An, and the duration of waves \tau\ (they counted 5 half waves) were overestimated roughly 2 to 3 times, respectively. If the effect of the directivity is also taken into account, their values of energy would be systematically biased by a factor of 10 to 20. TAKAHASI (1951) and IIDA (1963b) appear to have taken d as the mean depth between the epicenter of earthquake and the observing loca- tion which is about 10 times larger than the depth where An was esti- mated in the case of the 1933 Sanriku tsunami. The values of Et in the 1933 Sanriku tsunami estimated by Takahasi and Iida are 1.6×10^{23} and 1.8×10^{23} ergs, respectively, so that the probable value after removing the systematic bias of about 50 wouid be 3×10^{23} ergs.
A modified version of the Takahasi method was used by SOLOVIEV (1970) to compute the tsunami energy from tide gauge data. He replaced [Fig.107_360_03] in (23) by A^{2}\tau\ where A is the mean amplitude within the dura- tion \tau\ of the tsunami activity. When the maximum amplitude Amax is used instead of A, \tau\ is replaced by \tau\/10. Values of individual tsunami energy were not presented in his paper, but his Fig.2 suggests the mean relation between his intensity i and Et:

式(29) [Fig.107_361_01]

Although the intensity i is found to scatter around a_mean i for a given earthquake magnitude M, SOLOVIEV gave the relation between the mean value i and the earthquake magnitude M by

式(30) [Fig.107_361_02]

If (30) is substituted for(29), it follows that

式(31) [Fig.107_361_03]

As already mentioned, his values of A and \tau\ are overestimated and, be- cause of the very long duration \tau\ he took, the reduction factor may be about 1/20〜1/30.
HIRONO (1961) used the tide gauge data obtained at small islands in the Pacific to estimate the energy of the Chilean tsunami in 1960 based on (26). The values of S(\theta\) in the deep sea are estimated by a refraction diagram drawn from a point source lying on the Chilean coast (assuming the total reflection of waves at the coast, the factor 2\pi\ in (26) is replaced by \pi\). The energy fluxes seem to be estimated by taking tide gauge records on the islands without modification. The values of energy thus estimated are (1.3 and 3.8)×10^{23} ergs from data at Christmas Island and Johnston Island, respectively. Because of a very long extension of the source region of this tsunami, the effect of directivity must be significant and the probable value of energy may be close to 1〜2×10^{23} ergs. Hirono also mentioned that a silnilar method yielded Et〜4×10^{23} ergs for the Kamchatska tsunami in 1952, but details were not available.
WATANABE (1964) also estimated the values of tsunami energy for several earthquakes taking the refractive effect into consideration. In his case, the tsunami wave height at the source region was computed from tide gauge data at several coastal stations and the energy was estimated by applying (24) assuming、R=50km. Although he gave numerical values corresponding to An,\Delta\\tau_{n}\, and N, the values of Et given by him are not reproducible. GRIGORASH and KORNEVA (1972) also stressed the impor- tance of refractive effects in the estimation of tsunami energy.

c) Reverberation method
Another way to estimate the total energy of a very large tsunami is to make use of the reverberation of a tsunami throughout the whole Pacific Ocean (MILLER et al.,1962). If the energy density in the later stage of a very large tsunami is uniformly distributed over the whole Pacific Ocean and if the decay rate is uniform in space and time, it is possible to estimate the initial total energy by extrapolation from the knowledge of the energy density in the later stage and the decay rate together with the area of the Pacific Ocean. However, there are indica- tions that the decay time to (the time in which the energy decreases by a factor of 1/e) of the tsunami energy observed at a coastal station in- creases with decreasing energy density. At the beginning,t0〜0.5 day and becomes larger as time elapses. On the other hand, the state of complete 「diffusion」of the tsunami energy over the entire Pacific Ocean would take many days. Here we have the problem that by the time the ideal state of the diffusion of tsunami energy is attained the energy level would be already so small that it would be difficult to distinguish the tsunami energy from background wave activity (LOOMIS,1966).
MILLER et al. speculated the total energy of the Chilean tsunami in 1960 to be on the order of 3×10^{23} ergs by inferring the initial energy density (12 hours after the occurrence of the earthquake) in the open sea from data obtained at La Jolla on the California coast by taking the local enhancement factor into account. They apparently used the decay time of the order of 1.5 days(Fig.7, MILLER et al.,1962) in the interval of about 5 days.
VAN DORN (1963) also applied the same idea to estimate the tsunami energy of the 1957 Aleutian earthquake on the basis of data obtained at Wake Island. He used the decay time on the order of 4 days (Fig.7, VAN DORN,1963) estimated from data in several days after 40 hours from the beginning of the record and obtained the energy Et as 2.5×10^{22} ergs.
It should be noted that the decay rates used in these two studies are different, suggesting that the estimated decay rates were of the regional feature. Furthermore, it is extremely difficult to estimate the mean energy density for the entire Pacific Ocean from data obtained at a single station. As already noted by MILLER et al. (1962), there exist many am- biguities in the chain of reasoning to arrive at the value of total energy and the value should be trusted by the order of magnitude only.

d) Spectral inversion method
VAN DORN (1963) also attempted to eatimate the initial wave form of the 1957 Aleutian tsunami by a spectral inversion method of the ob- served dispersive wave pattern at Wake Island. To make the inversion unique, he assumed an axially symmetric initial disturbance and obtained the bell shaped depression with a slight elevation around the fringe of the depression. The potential energy was computed from this initial distur- bance to be 2.7×10^{22} ergs. The deficiency of this method when applied to a tsunami problem is that the assumed axisymmetric original disturb- ance may not be valid because the effect of directivity is significant. In this respect, a deduction from a single station data is quite questionable because the wave patterns are expected to differ considerably depending on the azimuth of the observing site with respect to the tsunami source. The estimated values of tsunami energy by the reverberation method and the spectral inversion method for the 1957 Aleutian tsunami turned out to be very close, but this coincidence seems to be fortuitous.

e) Potential energy method
This method of estimating energy was already discussed in relation to the earthquake fault model. Instead of estimating the crustal deformation from a fault model, the tsunami energy may be computed from empirical data of crustal uplift estimated geodetically. When a rough idea about the vertical displacement D(x,y) and the total area S are obtained by a field survey, it is possible to estimate、Et by the formula [Fig.107_363_01] with [Fig.107_363_02]. This kind of computation was carried out for the 1964 Alaska tsunami by several investigators. The derived values of energy are (in ergs):2.3×10^{21} (VAN DORN,1964),5.88×10^{21}(PARARAS- CARAYANNIS,1967),1.4×10^{22} (HATORI,1970), and 2.2×10^{22} (BERG et al., 1972). It is noticed that the differences between the earliest and the latest values amount to a factor of 10. Since the last two figures are based on detailed data of crustal uplift, the true energy would be some- where around 2×10^{22} ergs. HATORI (1970) also computed the tsunami energy for several Japanese earthquakes by this method. The difficulty in applying this method is that the crustal uplift oc- curring underwater is difficult to estimate. As seen in the example of the 1964 Alaska earthquake, the uncertainty may amount to a factor of 10 if no reliable data is available.
Now let us apply the present method to the 1960 Chilean tsunami. According to PLAFKERS's (1972) investigation on the crustal deformation and also from data of tsunami height distribution along the Chilean coast (SIEVERS et al.,1963) the source area of the tsunami may be roughly estimated to be confined between the coast and the trench axis and to be of the dimensions 900km×120km with the longer axis parallel to the coast. In view of the maximum uplift of 6m observed at Isla Guanbin, and also that the mean inundation height of the tsunami was about 10m, it may be reasonable to assume that the mean maximum uplift was about 7m along the offshore side of the source area and decreased to zero in cosine form toward the Chilean coastline. With this geometry of the uplift, the total potential energy Et was

式(32) [Fig.107_364_01]

CARRIER and GREENSPAN (1958) showed that for a special type of an initial mound of water with the maximum height Hm the runup height on a linearly sloping bottom is 1.45 Hm. Therefore, the maximum uplift of 7m for the inundation height of 10m may be acceptable,but the mean maximum uplift of 7m all along the length of the source region seems to be too large. The values of tsunami energy for the Chilean earthquake estimated by various methods are (in ergs):4.5×10^{23} (IIDA,1963b),1.3〜 3.8)×10^{23} (HIRONO,1961),3×10^{23} (MILLER et al., 1962), and 10^{23} (present study). All estimates involve uncertainty but the most probable value would be about 10^{23} ergs.

f) Numerical inversion method
Presently the most reliable energy estimate may be obtained by means of numerical experiments of the tsunami generation and propagation on the basis of shallow water equations. By the trial and error method, starting from a known earthquake fault model, it is possible to find a probable model of a tsunami source which can best explain the observed coastal tsunami height distribution (AIDA,1972,1974,1978a,1978b,1979a, 1979b). In most of these studies, Aida gave numerical values of tsunami energy computed by the potential energy method based on the derived tsunami source models. Since the uncertainty in the source estimation of tsunamis expressed by the relative standard deviation between the computed and observed tivave amplitudes along the coast is about 0.4 (AIDA,1979a), the accuracy of energy estimation is probably better than a factor of 2. He extended the numerical method to several tsunamis in the past and estimated the total tsunami energy as well as the earthquake moment which generated the tsunami (AIDA,1977). The values of tsuna- mi energy for the Izu-Oshima-Kinkai earthquake of 1978(AIDA,1978a) and the Tonankai earthquake of 1944 (AIDA,1979b) are not stated explic- itly in Aida's papers, but they are 10^{17} and 2.3×10^{20} ergs, respectively (AIDA, personal communication). There results are considered, at present, to be the best estimate of tsunami energy.

g) Empirical tsunami energy as a function of the moment-magnitude of earthquakes
The tsunami energy Et estimated by various methods is plotted as a function of the moment-magnitude Mw as far as possible. For most of large earthquakes considered here, the earthquake moment M0 is already known (see, for example, OHNAKA,1978), and for smaller earthquakes (M ＝＜ 7.5) the difference of MJ (JMA magnitude) and Mw is usually small so that MJ is used for Mw if M0 is not available. Figure 6 shows the total energy of tsunamis estimated by the energy flux method in the vicinity of Japan (IIDA,1963b; HATORI,1966,1971,1972). The dashed line cor- responds to (31) derived from Soloviev's data (1970) with M substituted for Mw. In Fig.7 the total energy estimated by the potential energy method (BERG et al.,1972; HATORI,1970, present paper) and the numeri- cal inverslon method (AIDA,1972,1977,1978b,1979a, unpublished data) are plotted as a function of Mw. The solid lines in these figures corre- spond to (14). It is seen that the general trend of logEt with respect to Mw, in Fig.6 is well represented by (14), provided that the values estimated by the energy flux method are roughly overestimated by a factor 10 to 20. It may be natural that the tsunami energy estimated by the numeri- cal inversion method follows closely the relation given by (14) as seen in Fig.7, because the tsunami generation by ordinary earthquakes and the observed tsunami behaviors are found to be explained, at leapt as a first approximation, by fault models seismically determined, and (14) is based on the relation of idealized fault models to the moment-magnitude Mw of earthquakes. For very large earthquakes (M>=9.0), however, the empirical values are somewhat lower than those expected from (14).

5.Summary and discussion

On the basis of a simple kinematic similarity model of the earthquake fault, tsunami energy is estimated as the potential energy of the initial sea-surface distubance which is estimated to be equivalent to the static displacement of the sea-bottom due to the fault dislocation. The relation between the tsunami energy Et and parameters(Mw,\delta\,\lambda\,h*) of the earth- quake fault is expressed by (13). The dependence of Et on the molnent- magnitude Mw is logEt〜2Mw if \delta\,\lambda\,h* are held constant, so that log(Et/Es)〜0.5Mw.
The dependence of Et on \delta\,\lambda\, and h* is given by a function F(\delta\,\lambda\,h*) which is independent of Mw. Behaviors of F with respect to \delta\,\lambda\, and h* are shown in Figs.2 to 4. Roughly speaking, the variation of F is within one order of magnitude for each change of parameters in the intervals 0°＝＜ \delta\ ＝＜ 90°,0°＝＜ \lambda\ ＝＜ 90°, and h*＜ 1.0. The largest value of F (=0.11) is found for \delta\〜60°,\lambda\〜90°and h〜0.04 (the computed shallowest depth of the upper rim of the fault). In general, for any fixed values of h* and \delta\,Et decreases with the decrease of \lambda\ and the range of variation is the largest when \delta\=90°(vertical fault). The dependence of Et on \delta\ shows an opposite trend with respect to being larger or smaller than about 45°:for small \lambda\, Et decreases as \delta\ increases, and for large \lambda\, Et increases as \delta\ increases (if the relative depth h* is larger than, say,0.5, Et in- creases with the increase of \delta\ up to about 45° and then slightly decreases as \delta\ approaches 90°). The range of variation of Et with respect to \delta\ is largest when \lambda\=0°(strike slip).
For small values of h*, the depth dependence of Et varies according to the values of \delta\ and \lambda\ :for small \delta\ and \lambda\, there exists an optimum depth h*at 0.2〜0.3 when Et is the maximum, but for large \delta\ and \lambda\, there is no such optimum depth and the maximum value of Et occurs for shallowest faults. For large h*(say h*>= 0.5), the decrease of Et with the increase of h* is roughly exponential with decay constant of about 2.4. Since the depth dependence of Et is expressed in terms of the relative depth ( [Fig.107_367_01] ), the decrease of Et with respect to the increase of the actual depth H* is more rapid for earthquakes of smaller magni- tude Mw.
The empirically estimated values of tsunami energy, except those deduced by the numerical inversion method, can be trusted by the order of magnitude only because of many ambiguities in the derivation of final values. Even so, the general feature of the、Et〜Mw relation is consistent with the present study, but there seems to be a systematic bias by a factor of 10 or more in the tsunami energy deduced by the conventional energy flux method.
There are many problems concerning the source parameters of earth- quakes, even if we admit a purely elastic dislocation model as the earth- quake mechanism. The aspect ratio W/L may differ considerably from 1/2 and there seems to be a tendency that this ratio decreases for a very large Mw. The rigidity \mu\ may vary by a factor of 2 to 3, and the con- stan t\alpha\ in (4) may be about 3 times larger than that adopted (\alpha\=1.23× 10^{23} dyne cm^{-2}. Taking these uncertainties into account, the constant term in (13) may vary considerably and the estimate of energy Et may vary be a factor of 2. To be more realistic in a model of the earthquake fault, we may take the layering of the crust and/or the non-uniform dis- tribution of the stress drop into consideration. Furthermore, the rectan- gular plane fault is also an idealization. For very large earthquakes, there may exist an imbricate fault which has a large dip angle \delta\. In view of these features of very large shallow earthquakes along interplate boundaries, the actual static displacement of the ocean bottom may deviate considerably from that deduced from a simple rectangular plane fault with uniform rigidity and dislocation. For example, according to (14) the possible maximum value of Et is 3.5×10^{23} ergs for Mw=9.5. In reality, however, the most probable value of the tsunami energy of the Chilean earthquake in 1960 (Mw=9.5) is estimated to be on the order of 10^{23} ergs from empirical data. In the present state of the art concerning the under- standing of earthquake mechanism, however, it may not be fruitful in the tsunami problem to take a complicated fault model into consideration.
Another problem is the assumption that the initial water surface disturbance is equivalent to the static vertical displacement of the ocean bottom. This assumption is generally valid only for large scale bottom deformations. For moderate to small earthquakes (M ＝＜ 7.0) with the epicenter located in deep ocean (d〜5km), this assumption may become a fair approximation. Because, in this case, the width of the fault is about 20km forMw〜7.0) and a considerable amount of power in the static bot- tom deformation may be present in small scale components which would be filtered out in the actual sea-surface disturbance by a factor of 1/cosh (kd). From this point of view, tsunami energy for small values of Mw, (＝＜ 7.0) given by (13) may be overestimated. Values of tsunami energy for small earthquakes are expected to be more sensitive than for large earthquakes to the depth of water d and tbe mean depth H* of the fault so that the empirically determined values of enemy would be scattered over a wide range.
Taking all these uncertainties into consideration, it seems to be very difficult to determine the tsunami energy beyond the accuracy of a factor of 2 to 3 when we use the parameters of a simple earthquake fault. It is also mentioned that the accuracy of the seismic moment determination, which is said to be better than a factor of (AKI,1972), limits the relia- bility of the tsunami energy estimation by a factor of about 2.

Acknowledgements

I thank Miss I. Kato for her assistance in numerical computations
and for typing the manuscript.

References

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20. 津波エネルギーと地震断層モデルのパラメ一タとの関係 地震研究所　梶浦欣二郎

要旨

　地震によっておこる津波のエネルギーを,幾何学的相似を仮定した簡単な地震断層モデルにもとず いて推定した.その結果によると津波エネルギー Et（エルク）は,

[Fig.107_371_01]

で与えられる. ここで，Mwは地震のモーメント・マグニチュードであり, Fは地震モーメントと 直接関係しない断層パラメータの閏数である.長ざ L,幅 W の地震断層を考えると, Fに含まれ るパラメータは断層面の傾斜角 \delta\ ,断層面上のすべり方向 \lambda\,断層面の相対深さ h*(h*=H*/L; H* は断層面の平均深さ, Lは断層の長さ),およひ断層面のたてよこ比 W/L であるが,このうちW/ L=1/2 を仮定した．
これらのパラメータの変化により,Fは最大1桁くらい変るが,最大の Fの値は約0.1である. 鉛直の断層で,たてずれのときとよこずれのときの津波エネルギーの違いが最も大きい. 　断層面の深さによる津波エネルギーの違いは，相対深さh* の関数であり，やや深い地震では Et 〜exp(-h*/2.4) の程度にエネルギーは深さとともに減少する．h*はもともと断層の長さLで無 次元化されているので,Mwが小さくなうと L が減少するため，小さな地震ほど波エネルキーの 深さH*による減少率は大きくなる.
これらの結果を，今までの津波エネルギーの推測値と比較する.まず,いろいろの方法によって推 定した津波エネルギーの信頼度を検討し，エネルギー推定の不確かさを明らかにした．最も普通に 行われる，エネルギー・フラックスを利用する方法では，今迄の値は系統的に10倍以上の過大評 価であるがMwに対する依存性はLogEt〜2Mwと矛盾しない．津波数値計算と実測波高との比 較から求めた津波エネルギーは，もともと断層モデルから出発していることもあって上式でよくあら わされる．最大級の地震津波であるチリ津波について，地殻変動の推定から求めた津波エネルギーは 10^{23} エルクの程度と思われ，上式の方が過大評価ぎみである．これは，断層モデルを簡単化しすぎて いることにも原因があるであろう．

Some Statistics Related to Observed Tsunami Heights along the Coast of Japan Kinjiro KAJIURA Earthquake Research Institute. University of Tokyo,Tokyo,Japan (Received October 20.1981 :Revised December 10,1981)

Some spatial statistics related to tsunami heights observed along the coast or Japan are made on the basis of data collected during field surveys taken after the occurrence of large tsunamis. For a regional frequency histogram of tsunami heights along a coast of about 300 km,the long-normal distribution is found to be a reasonable first approximation, supporting VAN DORN's result (1965). However, the standard deviation seems to vary by a factor of about 1.5 around a central value of 0.24 (logarithmic scale) depending on the characteristic 「period」of the tsunami as well as the topography of the coast.The height ratio of the observed maximum Hmax, to the mean, H, is about 4. On the other hand. for local statistics in the range of about 20 km of the coast, the height ratio of the Maximum, Hn.max, to the mean, Hn, is about 2. Statistically speaking, the maximum double amplitude of tide gauge rccords is close to the run-up height observed in the vicinity, although the indivldual ratio varies by a factor of two.
The maximum local-mean tsunami height,Hn,max, is related to the moment- magnitude of earthquakes, Mw,by [Fig.107_372_01] .The uncertainty factor of estimated Hn.max for individual tsunami is about 1.5 and the formula fails for very great earthquakes (Mw > 8.5).

1. Introduction

In Japan. therc are a considerable amount of coast tsunami data (inundation or run-up heights) collected by field survey teams after the occurrence of large tsunamis. The observed values of tsunami height for a single event vary not only regionally but also locally. lt is noted that the variation of heights is very large even within a short stretch of the coast、 say 10 to 20 km. Besides errors involved in measurements, the diversity of tsunami heights observed along the coast is customarily attributed to various factors such as wave refraction. scattering resonance and non-linear effects due to topographic irregularitics of various scales. instead of trying a deterministic approach to this problem, a simple statistical description of the wave height diversity is attempted here to express the gross features of tsunamis. A similar study was made by VAN DORN (1965), who proposed a log-normal distribution for the occurrence frequency of observed tsunami heights along a coast.
On the basis of this kind of statistical knowledge we may be able to attain a better understanding of the relationship between the.「maximum」 tsunami height observed along the coast and the earthquake magnitude (the moment-magnitude) and also establish better criteria for designing protective measures against tsunamis. Local tsunami risk analysis, which has been up to the present mainly concerned with probabilistic statements of the maximum tsunami height with respect to time at a specific location, may also be improved by a knowledge of the spatial statistics.

2. Sampling Problems Related to Run-Up Heights

When undertaking a statistical analysis of run-up data、we are immediately faced with the problems of data quality and sampling. In most field surveys of tsunamis, the coastal areas covered by survey teams are limited within the region where tsunami heights are significantly large: the peripheral regions with an average tsunami height less than about 0.5m are seldom covered unless some special interest exists. Even in the vicinity of the central area of tsunami attack data sampling is very dense in the region of populated and relatively flat lands of bay coasts, whereas remote locations posing difficult access such as a rocky open sea coast are poorly sampled.The nature of this data sampling causes difficulty in the interpretation of statistics.
Other problems are: 1) different critcria for the height measurements. and 2) accuracy of the measurements. Some survey groups appear to choose water marks which represent more or less the mean height in the locality while rejecting exceptional values.Other groups prefer to choose the highest mark in the locality even if it might be a result of water-splash. Furthermore. because of the difficulty of finding a reference level, say,the position of a nearby tidal level at the time of measurement or a geodetic bench mark,the accuracy of height measurements varies considerably depending on the local situation as well as thc survey team. For some old records. even the reference levels relative to which the observed height values were reduced were not clearly stated. For more recent tsunamis, the height values are usually referred to T.P. (the reference level of land maps in Japan which is close to the local mean sea level at Tokyo: Tokyo Peil).
If the data taken by various survey teams are combined the problem of redundant data arises (the same water marks are likely to be measurcd repeatedly by differellt survey teams when water marks are very distinct). There may be other problems to be considered in the interpretation of a statistical descripttion.
Examples of the variation in data sets taken by different survey teams in the same area for the same tsunami are shown in Fig.1. For the 1896 Sanriku tsunami, the overlapping area of interest is the coast of iwate Prefecture about 200km Iong from Hachinohe to Hirota Bay, wherethe data listed in the「Records of Tsunami Disasters」 (Civil Engineering Section of lwate Prefecture. 1936: nummber of samples 84) are compared with those listed in the report by KUNITOMI (1933). Kunitomi used a combination of data by IKI (1897) and those obtained by the local weather stations (number of samples 91). For the 1933 Sanriku tsunami. the data collected by survey teams of the Central Meteorological Observatory and local weather stations (KUNITOMI 1933:number of samples 201) and those collectcd by a survey team of the Earthquake Research Institute (EARTHQ. RES. INST.,1934;number of samples 523) are compared for the whole Sanriku coast of about 300 km length south of Hachinohe.
The presentation is in the form of box plots, where the box gives values between upper and lower quartiles of data with the horizontal bar as a median value. The maximum and minimum values are shown by circles with dots and the second maximum and minimum by black circles. As seen in Fig.1,the median values differ considerably indicating the bias of different sampling, and this difference can not be attributed to the statistical variability of random sampling from the same population.

3. Statistics of Run-Up Measurements

VAN DORN (1965) presented a spatial distrihution of observed tsunami run-up heights around the Hawaiian Islands. He placed together data of two different tsunamis of 1946 and 1957 observed around the islands of Kauai, Oahu, and Hawaii, by representing the coastal locations in terms of the azimuth measured from the local meridian passing through the center of each island. From the frequency histogram of these data (samples 254), he proposed a log-normal distribution with the standard deviation、\sigma\*, found equal to 1.8 (log\sigma\*=0.255). A similar analysis for the 1933 Sanriku tsunami also yielded the identical standard deviation.
In the present paper, six tsunamis observed in Japan with sufficient data available are discussed; they are the l896 Sanriku.1933 Sanriku,1946 Nankaido,1960 Chile, 1964 Niigata, and l968 Tokachi-oki tsunamis. Except for l946 and l964 tsunamis. run-updata determined are all for the Sanrikucoast in the north-eastern part ofJapan. The 1946 tsunami attacked the region from Kii to Shikoku in the southwestern part of Japan, and the l964 tsunami occurred off the Niigata coast in the,Japan Sea. The Some Statistics Related to Observed Tsunami Heights along the Coast of Japan

height values for the 1968 tsunamis are corrected by adding 75 cm to give the height from the low tide level, because this tsunami occurred at low tide. For all other tsunamis, no correction is made. The relevant information concerning earthquakes and tsunamis to be discussed is listed in Table 1.

3.1 Regional statistics

For each tsunami data obtained by different survey teams are lumped together for the total range of about 300 km of the coastal stretch where major tsunamis were experienced: for the 1896 1933 1960 1968 tsunamis, the region is from Hachinohe in Aomori Prefecture to Ishinomaki in Miyagi Prefecture, for the 1946 tsunami,the region is from Shingu in Wakayama Prefecture to Sukumo in Kochi Prefecture,and for the 1965 tsunami. the region is from Funakawa in Akita Prefecture to Naoetsu in Niigata Prefecture, including Sado Island.
Frequency histograms of run-up or inundation heights, H are presented in Fig.2, where the abscissa is expressed in terms of percentage of occurrence,and the ordinate is in the logarithmic scale of the observed value, H normalized by the logarithmic mean value,H. The histograms for the 1896 and 1933 tsunamis are flatter than the normal distribution、for the 1960 and 1968 tsunamis they are skewed to the right considerably. and for the 1964 tsunami a small secondary peak is seen in the histogram. It is quite clear from these frequency histograms that the difference in the sample distributions can not be explained simply as a result of random sampling from the same normal population.
However, if we combine all these frequency distributions with equal weight, the resultant distribution is closely approximated by a log-normal distribution only slightly skewed to the right as shown in Fig.3. suggesting that the type of frequency distribution for an individual tsunami deviates considerably from the log-normal one, but that there is no systematic deviation of paramcters as a whole.
As shown in Table 1, the standard deviations,SDT (logarithmic scale in height) for the six tsunamis are within a range of 0.14 to 0.32 and the average value,＜SDT＞,for all data combined is 0.24. This average value favorably compares with the Van Dorn's estimate (\simeq\ 0.26).
The skewncss, c3,varies between -0.66 to 0.07. and the kurtosis,c4, between 2.2 to 3.6 (logarithmic scale in height). There appears to be a tendency for c4 to increase with decrease of SDT. The parameter values for all data combined are c3=-0.08 and c4 = 3.08. These values are very close to the normal distribution for which c3=0 and c4=3.0.
Even taking uncertainty of parameter values due to biased sampling into consideration, we can recognize a distinct difference of the values of SDT, c3 and c4 between the data for the 1896 and 1933 tsunamis and for the 1960 and 1968 tsunamis, although the data were sampled from the game geographical region of the Sanriku coast.The former group has larger SDT values than the latter. This difference may be attributed to the physical difference of the tsunami characteristics such that the former group had shorter predominant periods of incoming waves (10 to 15 minutes) than those of the latter group (more than 25 minutes), and that the bay responses to incoming tsunamis were quite different.
Regional maximum and minimum values Since the frequency distribution for an individual tsunami deviates from log-normal, it is difficult to discuss the statistical characteristics of extreme height such as the maximum. Hmax, and the minimum,Hmin. As a tentative attempt,the values of log(Hmax/H) SDT and log(Hmax/Hmin)/(2SDT) are plotted in Fig.4 as a function of the number of total samples. For reference the thcoretical probabilities of standardized normal extremes are also shown in Fig.4, where the thick line gives the mode and thin lines represent lines below which 10 percent and 90 percent of sample values of the extremes fall respectively (GUMBEL. 1958: Graph 4.2.2.(2)). ln the figure, data for the 1923 Kanto, 1944 Tonankai,1952 Tokachi-oki and 1968 Hiuganada are also included although the number of sampled data is limited. Relevant data are listed in Table 2.
lt is seen that all but one of the maximum values of the tsunami data considered fall below the median line of the normal extreme because of relatively low values of kurtosis or large values of skewness. If the standard deviation is recduced to about 0.8 to 0.9 of the actual vaiue. all the extreme values might fall in the range expected from the log- normal distribution reasonably well. On the other hand, all the plotted polnts of log (Hmax/Hmin)/(2SDT) fall within the 10 and 90 percent lines, although the statistics of this variate obey a slight different law from the former variate.
If we adopt the normal distribution,the mode of the normalized extreme values for 100 and 500 samples are about2.4 and 2.9 respectively.Thus, if we assume the standard deviation,\sigma\,to be 0.9×SDT with SDT \simeq\ 0.24, we have; Hmax/H〜 3.3 and 4.2 for 100 and 500 samples, respectively.These values may be favorably compared with the range of Hmax/H directly computed from data listed in Tables 1 and 2, where the ratio is from 2.0 to 4.9 with a mean (logarithmic) value of 3.6.

3.2 Local statistics

The height statistics for all observed data of a given tsunami include various physical processes related to the regional as well as the local character. To separate local characteristics from regional ones, we have divided the data for four tsunamis experienced in the Sanriku coast along a stretch of about 300 km from Hachinohe to Ishinomaki into 17 local sets in intervals of about 20 km each as shown in Fig.5. This smaller length of coast includes roughly one or two units of topographic features such as coastal indentations and embayments. In most,segments, the number of height samples for each tsunami varies from about 5 to 70, depending on the topographic features as well as the interest of the survey teams.
For each tsunami, the distribution of logarithmic mean values.Mn( =log Hn), and the standard deviation, SDn. therefrom with respect to their locations is shown in Fig.6. The geographical variations of the mean values. Hn should be better understood by the deterministic approach because they are mainly dictated by the gross coastal topographic features of the location in relation to the tsunami source characteristics (this is evident in view of successful tsunami numerical simulations). The 1896 and 1933 tsunamis show similar variations of Hn in the southern half of the Sanriku coast (segment numbers greater than 5). In the northern region,the mean heights of the 1896 tsunami are larger than those of the 1933 tsunami, suggesting the location of the source area of the 1896 tsunami to be farther north of the source of the 1933 tsunami. ln contrast, the 1968 and 1960 tsunamis exhibit quite different distributions. In particular, the geographic variation is small for the 1960 Chilean tsunami because this tsunami propagated across the Pacific Ocean, and the incident waves along the Sanriku coast were believed to be relatively uniform with Iong periods. These differenccs of mean height distribution are also reflected in the frequency histograms already shown in Fig.2.
If we calculate the standard deviations, MSD. of these Iocal mean values,Mn, within the region considered. numerical values fall between 0.08 and 0.26 with the average value ＜MSD＞ for all tsunamis being 0.17 as shown in Table 3. Recalling the average standard deviation. ＜SDT＞. of all data which was 0.24. it is found that about half of the variance comes from the regional variability of the mean values.The ratios of the maximum. Hmm,max, of the local mean value Hn to the total mean value H, i.e., Hn,max/H are in the range of 1.4〜2.5 with the mean (logarithmic) value of about 2.0.
In Fig.6, it is seen that the regional variation of the standard deviation,SDn, is considerable (see also Table 3). The mean value.SDn over the whole region varies in the range of 0.1 and O.2 depending on the particular tsunami with an average value over all tsunamis,＜SDn＞. of 0.16. Here again SDn is larger for the l896 and 1933 tsunamis than for the 1960 and 1968 tsunamis. reflecting difrerences in characteristics between these two groups of tsunamis.
If we normalize individual SDn by the SDn of each tsunami SDn/SDn varies from place to place in the range from 0.6 to 1.5. reflecting the local topographic effects. The variation of SDn/SDn for various tsunamis at the same Iocation is mostly within 30 percent,but at some specific sites it becomes very large, suggesting different responses of the locations to the incoming tsunamis.
Local maximum and minimum values The ratio of the maximum Hn.max to the mean. Hn of run-up heights (logarithmic scale) within each segment is normalized by the individual standdard deviation,SDn and is shown in Fig,7 as a function of a number of samples. The thick and thin lines drawn in the figure are the same as in Fig.4. It is noted that the plotted points scatter over a wide range, although centered very roughly on the mode of extremes for the normal distribution. On the other hand,the observed ratios of maximum Hn.max and minimum.Hn.min. (logarithmic scale) normalized by 2SD) (not shown here) lie within the range of the 10 and 90 percent lines for the normalized extreme deviate based on the log-normal distribution. This character is similar to the case of regional statistics (see Fig.4).
Although the values of SDn vary over a wide range, let us tentatively assume the log-normal distribution with the local standard deviationSDn to be 0.16 as a model of the local run-up height frequency. Then, since the mode of the normalized extreme values for about 30 samples is 2.0, we have Hn.max/Hn 〜2.0. On the other hand,Since Hn.max/H 〜2.0 as already mentioned, we have Hmax/Hn〜4.0. for the total regional data, which is consistent with the result of the regional statistics. In passing, it is noted that the ratio, Hn.max/Hn (〜2.0), here derived is considerably larger than that suggested by SOLOVIEV (1978) who indicated that Hmax/Hav 〜√2, with Hav representing average values of wave run-up over a certain

4. Relation of Field Data and Tide Gauge Data

In an ideal case when a tide gauge registers the water level outside of the tide-well faithfully without any filtering effects, the maximum water level at the site should be equal to the maximum crest height recorded on the gauge relative to the same reference level. In reality, the maximum crest heights of tide records (very often represented by the maximum value of the crest elevation relative to the ordinary tide level) are almost always smaller than the observed inundation heights measured in the vicinity. In practice, the maximum wave height ,A,(crest to trough distance) measured by a tide gauge is often used to represent a tsunami height at the coast. However, the relation between A and the inundation heights,H, measured by field survey has not been studied systematically because data sets are scarce. WILSON (1972) quoted an opinion of Cox that H would be from 1 to 1.5 times the maximum height,A,recorded on a tide gauge in the vicinity, this factor being a function of location, period of waves, and damping characteristics of the tide gauge.
This relation is examined here on the basis of available data for four tsunamis: the l933,1952,1960, and 1968 tsunamis observed on the Hokkaido and Sanriku coasts, although the compared run-up heights were not measured at the exact site of the tide gauge. In Fig.8,we can find that A/H lies between 0.5 and 2 with a mean (logarithmic) value of approximately 1.0. Thus, the wave height, A,measured by a tide gauge can be considered to represent the run-up height, H, measured in the vicinity. Furthermore, the value of A might be used in place of Hn in a rough estimation as shown also in Fig.8.

5.Dependence of Coastal Tsunami Heights on the Moment-Magnitude of Earthquakes

The relationship between the「maximum」tsunami height, Hmax,(or the tsunami 「magnitude」) and the earthquake magnitude,M, has been studied by various investigators (see, for example, IIDA,1963; WiLSON,1972). Since it is believed that the moment-magnitude, Mw, of earthquakes (KANAMORI.1977) is better suited for correlation with the size of the tsunami than M, the correlations of the observed maximum, Hmax, maximum local mean,Hn.max,or regional mean,H, of run-up heights to the moment-magnitude,Mw, of earthquakes are shown in Fig.9. In this figure,some data are added (since 1920) for the observed maximum run-up heights of large tsunamis observed outside of Japan (IIDA et al.,1967).
The full line in the figure is the relationship given by

式(1) [Fig.107_383_02]

It is interesting to notice that the above formula is nothing but the relation assumed by ABE (1981) in his Fig.8 between the upper limit of the tsunami height (the maximum double amplitude of the tide gauge) and the moment-magnitude Mw. In other words, the limiting tsunami height assumed by ABE (1981) corresponds to Hn.max. Dashed lines in the figure are drawn by taking the empirical relations such that Hmax/Hn.max 〜2.0 and Hn.max/H 〜2.0 into account (the constant term in (1) becomes 3.00 or 3.60, respectively).
As already discussed,the observed maximum run-up height Hmax, for a given number of samples scatter statistically over a considerable range, so that application of the formula should be made with caution. A rule of thumb may be all uncertainty by a factor of 1.5 for the estimation of maximum run-up height, Hmax by this formula. Another notable point is that for great eathquakes with Mw larger than, say,8.5, the
observed maximum values do not follow the trend of the formula and seem to be　saturated at about 20 to 30 m. Since the observed maximum heights are rather isolated values and the maximum local mean values are about one half of the observed　maximum, it seems that the maximum local mean tsunami height will never exceed 15 m for ordinary tsunamis generated by the tectonic movements due to earthquakes.　 lt is remarked that the mean slip displacement D of the idealized earthquake fault　for a given earthquake moment (ABE.1975) may be written in terms of Mw as

式(2) [Fig.107_384_02]

Compared with (1), it may be said very roughly that Hn.max is about two times D, or　that D corresponds to the overall mean run-up height H of the tsunami observed along　a coastal stretch of ahout 300km, at least for data in Japan.

6. Conclusion

Because of the strong dependence of tsunami behavior along the coast on both the　characteristics of the incoming tsunami wave train and the topographic conditions of　local and regional scales, it is very difficult to draw general conclusions concerning the　spatial statistics of coastal wave run-up heights applicable to all regions and tsunamis.　However,from the analysis of data in Jap, an idealized statistical model, although　very rough has been constructed. In summary,it was found that:1) Regional (300 km　range) and local (20 km range) frequency distributions of run-up heights are log-　normal with the standard deviations 0.24 and 0.16 (logarithmic scale in normalized　height),respectively. 2) As a rule or thumb,relations among various height values, such　as the observed maximum Hmax,the local maximum,Hn.max, the local mean, Hn, the 　maximum local mean.Hn.max, and the regional mean H are Hn.max/Hn〜2.0,　Hn.max/H〜2.0, and Hmax/H 〜4.0. 3) The relationship between Hn.max and the
moment-magnitude Mw,of earthquakes for [Fig.107_385_01] is roughly represented by Eq.　
(1). 4) For an individual tsunami,the deviations of observed values from those deduced　from this idealized model may lie within a factor of about 1.5.

l wish to thank Miss Kato for her valuable assistance in computation of statistics.

REFERENCES

ABE, K., Reliable estimation of the seismic moment of large earthquakes.J.Phys.Earth,23,381-390,1975.
ABE. K., Physical size of tsunamigenic earthquakes of the northwestern Paicific.Phys.Earth Planet.Inter., 27.194-205.1981.
AlDA. I., K.KAJIURA.T. HATORl.and T. MOMOI, A tsunami accompanying the Niigata earthquake of June 16.1964,Bull.Earthq.Res.Inst.,42.741-780.1964 (in Japanese).
CENTRAL METEOROLOGICAL OBSERVATORY, General survey of the great Tonankai earthquake on December 7,1944,Special Pub.. CMO, pp.1-94.1945 (in Japanese).
CENTRAL METEOROLOGICAL OBSERVATORY, General survey of the great Nankaido earthquake on December 21.1946. Special Pub., CMO. pp.1-84,1947 (in Japanese).
CENTRAL METEOROLOGICAL OBSERVATORY, General survey of the Tokachi-oki earthguake in 1952 Hokkaido.Quart.J Seismol.,17,50-87.1953 (in Japanese).
COMMITTEE FOR FIELD INVESTIGATION OF THE CHILEAN TSUNAMI OF 1960. Report on the Chilean Tsunami of Mar.24.1960.as Obserced along the Coast of Japan.397.pp.,Maruzen. Japan.1961.
EARTHQUAKE RESEARCH INSTITUTE,Reports on the Sanriku Tsunami of1933, Bull. Earthq. Res. Inst. Supplementary volume 1. Figs.98-209.1934.
EARTHQUAKE RESEARCH INSTITUTE,Reports of the investigation of the Nankaido earthquake on December7. 1946.,Spcial Rep., Earthq.Res. Inst.,5. 1-195. 1947 (in Japanese).
GUMBEL.E.J.,Statistics of Extremes.375 pp., Columbia Unic.Press.New York.1958.
HATORI.T., and K.KAJIURA. Tsunamis in the south Kanto district.Pub.for the 50th Anniv. Great Kanto Earthq.,1923,Res.Inst., 57-66.1973 (in Japanese).
HYDROGRAPHIC DEPARTMENT OF JAPAN, Report on tsunamis at the time of the Nankaido earthquake in 1946.Special issue,Hydrographic Bulletin,201,1-39,1948 (in Japanese)
llDA. K., Magnitude,energy and generation mechanisms of tsunamis and a catologue of earthquakcs asstociated with tsunamis.in Proc.Tsunami Meetings Associated 10th Pac.Sci Congr.,Honolulu,edited by D.C.Cox. I.U.G.G. Monogr.24.pp.7-18,1963.
llDA. K., Geographical distribution of seismicity and damage caused by the Tonankai earthquake on Deccmber 7,1944. Report prepared for the commettee for disaster prevention. Aichi Prefecture. pp. 1-120.1977 (in Japanese).
llDA. K., D.C.Cox.and G.PARARAS-CARAYANNIS. Preliminary catalog of tsunamis occurring in thc Pacihc Ocean. Data Rep, No.5. HIG 67 10. Hawaii lnst. Geophys., Univ. of Hawaii.1967.
IKEDA. T., Tsunamis in the lzu and Awa districts.Pub.Earthq.Inv.Committee-B,100,97-115,1925 (in Japanese).
lKI,T., Report of the great Sanriku tsunami,Pub.Earthq.Inv.Committee-B,11,5-34,1897 (in Japanese).
KAJIURA, K.,I. AIDA, and T. HATORI, An investigation of the tsunami which accompanied the Hiuganada earthquake of April 1,1968,Bull.Earthq.Res.Inst.,46,1149-1168,1968 (in Japanese).
KANAMORI, H., The energy release in great earthquakes,J.Gep@hys.Res.,82,2981-2987,1977.
KlSHI, T., Tsunami on May 16,1968,on the coast of Hokkaido and Tohoku, in General Report on the Tokachi-Oki Earthquake.1968, pp.207-256, edited and published by the lnv. Committee of the Tokachi-oki earthquake 1968, Hokkaido University,1969 (in Japanese).
KUNITOMI, S., On the Sanriku earthquake and tsunami, Quart.J. Seis.,7,1-43,1933 (in Japanese).
OMOTE, S., The tsunami, earthquake sea waves, that accompanied the great earthquake on December 7, 1944,Bull. Earthq. Res. Inst.,24,31-57,1946 (in Japanese).
SECTION OF CIVIL ENGINEERINGS, IWATE PREFECTURE, Records of Tsunami Disasters, pp,24-30, published by Iwate Pref.,190 pp.,1936 (in Japanese).
SOLOVIEV, S. L,Tsunamis, in The Assessment and Mitigation of Earthquake Risk,(Natural Hazards,I), pp. 118-139,UNESCO,1978.
VAN DORN W. G., Tsunamis, in Advances in Advances in Hydroscience , edited by Ven Te Chow,2,1-48, Acad. Press, New York and London,1965.
WILSON. B. and TORUM, Run-up heights of the major tsunami on North American coasts, in The Great Alaska Earthquake of 1964: Oceanogr. and Coast.Engineering, pp.158-180. NAS Pub.1605, Washington, Nat. Acad. Sci.,1972.
YOSHIDA, K., T. YAMAGIWA, K. KAJlURA, and T. SUZUKI, On the tsunami accompanying the Nankai earthquake, Part 1,Geophysical Note, Geophys. lnst., Univ. of Tokyo,1,1-20,1947 (in Japanese).
YOSHIDA, K., On the tsunami accompanying the Nankai earthquake, Part 2, Geophysical Note, Geophys. Inst., Univ. of Tokyo,1,1-15,1947 (in Japanese).

孤立波の陸上遡上について 一砕波と海底摩擦効果の検討一 梶浦欣二郎*

[要旨]

　津波の陸上遡上の数値シミュレーション手法の信頼性向上のため,砕波を含む孤立波の遡上を数値的に計算し,これを既応の実験結果と対比し,摩擦係数として,f=0.01が適当であることを示した。

1.　はじめに

　津波陸上遡上の数値シミュレーション手法については,最近かなり多数の論文が発表されており(相田,1977;首藤・後藤,1977;岩崎・真野,1979;Pedersen　andGjevik,1983等),それぞれに成果をあげている。多くの方法は波が非砕波領域にあるときには,格子間隔その他のパラメータを適当にえらぶかぎり,ほぼ同様な精度で計算結果が得られるようであり(後藤・首藤,1980),水理実験結果との比較もなされているし(岩崎他,1981),また,過去の津波浸水高分布との比較もなされている(例えば,相田他,1983)。　しかしながら,もし波が海岸近くで砕波する場合も含めると,まだ計算手法に問題がのこっていろ。
　波が砕波する場合の陸上遡上機構に関しては,有限振幅浅海波の方程式の特性曲線法による解法を基礎として多くの試みがなされている (例えば, [Fig.107_387_01] et al.,1968;Meyer＆Taylor,1972;椹木,1973参照),,そこでは,底面摩擦を導入することも試みられているが実測値との比較ではそれほど満足すべきものではないようである。遡上する波の先端の波形,即ちフロントの境界条件に関しては,摩擦力の存在と関連してダム破壊後の流れに関するWhitham (1955) の研究以来いろいろの議論 (岩崎・富樫,1969;Shlmada,1981;松富,1981; [Fig.107_387_01]1979)があるが,まだ遡上する波のなかのエネルギー逸散に関してその機構がよくわかっていないのであまり詳しいことはいえない。
　一方で,一様段波を対象とした遡上の理論や,その数値計算 (Hibberd＆Peregrine,1979) の結果を,Miller (1968) による実験値と比較すると大きなくいちがいがあり,海底勾配が小さくなるほどその差が大きくなることがわかっている。その原因が摩擦にあるとの想定のもとで,Pachham＆Peregrine(1980) は速度の2乗に比例する形の海底摩擦を導入して数値計算を行ない,かなり大きな摩擦係数をつかうと実験結果を説明出来るとのべている。
　津波の遡上に関連していえぱ,一様段波よりは孤立波的な波の遡上をしらべる方がより利用し易いと思われるので,ここでは孤立波が一様斜面上を進行し陸上に遡上するとき,遡上高が入射孤立波のパラメータにどう依存するかを数値計算し,それを既存の多数の実験結果と比較して,最も適当と思われる海底摩擦係数を推定することを試みる。このとき,広いパラメータ範囲を考えるので,孤立波は非砕波のときのみならず,汀線近傍で砕波する (段波タイプ) 場合も含んでいる。一般にアーセル数の大きな波の浅水変形では,段波のほか,波の先端部のソリトン分裂をも説明出来る方程式系を基礎とすべきであろうが,さしあたっては分散性の波は考えないことにする。
　数値計算との比較を統一的に行なうため,既応の多数の研究者による孤立的な波の遡上に関する実験結果を検討しなおす。

2.　浅海波方程式系

　波形曲率の効果 (波数分散性) を無視すれ
ば,非線形の一次元浅海波方程式系は,

式(1) [Fig.107_388_01]

式(2) [Fig.107_388_02]

とかける。ここで,xは水平座標であり,原点を斜面の初まりの点にとり,岸向きに正とする。tは時間,　gは重力加速度,　uは,水平流速,\eta\ は平水面からの水位上昇,dは平常時の水深でありfは海底摩擦 \tau\ を\tau\=\rho\fIuIu(\rho\:水の密度)とおいたときの摩擦係数である。摩擦力の表現にはこのような単純なものばかりでなく,マンニングの粗度係数nを使うものなどがあるが,ここでは立入らず,水深にかかわらず f は一定とする。
　上述の(1),(2)に含まれる量を,一様勾配sの斜面起点の水深d0,斜面起点から汀線までの長さ l 支び重力加速度 g をつかって無次元化する。いま無次元量に*をつけると,

式(3) [Fig.107_388_03]

であり,(1),(2)は

式(4) [Fig.107_388_04]

式(5) [Fig.107_388_05]

とかける。
　無次元化された与程式系(4),(5)をみれば明らかなように,摩擦項を除いて,みかけ上,運動は海底勾配sに無関係となっている。これは長波に関するフルードの相似則にあたる。間接的には時間の無次元化の関係式をみてもわかる通り,代表的な時間スケールTについての相似条件は,パラメータ [Fig.107_388_06]の等しいことである。波の波長 [Fig.107_388_07]をつかうと,パラメータは\lambda\/l ともかける　いま,角周波数 \omega\(=2\pi\/T) を無次元化して \sigma\ とかくと

式(6) [Fig.107_388_08]

であるから,相似条件は \sigma\/s の等しいことてある.したがって,同じフルード数 (代表的な流速をU,波高をHとすると,U√gd0又はH/d0 の等しいとき),及び \sigma\/s に対し,摩擦効果を無視すれば現象は相似となる.例えば,波のはい上がり高さRと入射波の波商Hとの関係は

式(7) [Fig.107_388_09]

とかける。もし摩擦の効果が大きければ

式(8) [Fig.107_388_10]

のタイプと予想される。実際の現象がこのタイプの実験式で記述出来ないときには,方程式系 (1),(2)が不十分であって,例えば波数分散性を導入するとか,エネルギーの消耗についての表現式を考えなおすとかをする必要があるであろう。

3.周期波の遡上高

　まず,周期波の遡上高Rに関し,後の議論に関連することを復習する。今までの多くの研究では,風浪やうねりのような短周期波を対象とすることが多かったのて,実験波のパラメータを沖彼のものに換算するのが普通であるが,津波のような長周期波を対象とするときは浅海波のパラメータの方が便利である.
　傾斜sの斜面が無限遠方まで続くとき,非砕波(線形)で摩擦効果を無視したときの周期波の遡上高Rは,深海波の波高H_{\infty\}との間に

式(9) [Fig.107_389_01]

の関係がある。
　一方で,周期的な深海波が浅晦に進んできて,水深dで完全に長波となったときの波高Hdは,浅水変形の式から

式(10) [Fig.107_389_02]

てあり,長波の条件から \sigma\~0.3 が要求される。ここで \sigma\ は (6) で定義された無次元角周波数であり,水深dでの長波の波数をkとすると

[Fig.107_389_03]

ともかける。ついでにのべれば,波形勾配は

式(11) [Fig.107_389_04]

である。
　ここで,(9)と(10)とを組含せると

式(12) [Fig.107_389_05]

となる.
　よく知られているように,一様水深d0の水路から,一様傾斜sの斜面上に入射する周期波の遡上は (Keller and Keller,1964),

式(13) [Fig.107_389_06]

とかかれ,J0(\xi\),J1(\xi\) はべッセル関数である。この式のうち斜面上での固有振動の部分を除くと(12)と等しくなる。後述するように.,過渡的な正弦彼の入射を考えると,第一波は(12)の形の遡上となり,第二波以降は(13)の形に近づくことが判っている。
　さて,周期波の砕波限界の条件として,Miche (1951) は深海波の限界波形勾配を海底勾配の関数として与えたが,海底勾配sの小さいとき,その形は長波についてのCarrler＆Greenspan (1966) のものと同形であり,次式であらわされる。

式(14) [Fig.107_389_07]

ここで係数\beta\は

[Fig.107_389_08]

である。
　周期波の場合,\beta\=1の方が理論上は正確であると思われるが,Van Dorn (1978)は砕波限界として \beta\=2 が実験と合うとしている。また,主として孤立波タイプの波について富樫・中村(1975)は首藤(1972)の方式で砕波の限界をしらべ,\beta\\simeq\2 を得ているが,一方で引波を含む津波の実験で岩崎その他 (1982) は,\beta\\simeq\1とのべている.
　ここで(12)と(14)とを組合せると,砕波限界浅高として

式(15) [Fig.107_389_09]

が遅られる。いまかりに (\beta\/\alpha\) \simeq\ 1として,実験波の波高が 0.5 ＞H/d ＞0.01のときを考えると,砕波限界は1.3＜\sigma\/s ＜ 6.3の範囲ということになり,\sigma\/s の割合せまい範囲になる。
　完全に砕波する周期波の遡上高については,Huntの実験式 (Hunt,1959) が育名であろが,Van Dorn (1966) はこれを変形して,

式(16) [Fig.107_389_10]

とかきあらためた。この式は海底勾配がs ＞ 1/10の程度,沖波波形勾配がそれほど小さくない範囲の実験からきめられたものであり,海底勾配及び波形勾配がともにきわめて小さいときに適用出来るかどうかには問題がある。ここで,沖波波高H_{\infty\} を(10)をつかって浅海での波高Hdにかきなおすと

式(17) [Fig.107_390_01]

となる。浅海での波高Hdをつかった式には(2\sigma\)^{1/4} の項が加わっていることに注意されたい。Togashi and Fuhrboter (1981) の実験結果をみるとこの項の効果があらわれているようである。また,引波初動の砕波タイプの波の遡上を実験的に調べた岩崎その他(1982) の場合,(2\sigma\)^{1/4}が平均0.72でその変動幅がせまいので (17) によると(2\pi\)^{1/2}(2\sigma\)^{1/4}という係数は1.80となり,実験的に与えられた1.76とよく合っている。このことからみると,Huntの式は海底勾配がかなり小さいところでも,それに応じて波形勾配が小さければ使えるようである。
　砕波の遡上高をあつかうHuntの式に関しては,いろいろの議論があり,R/H_{\infty\}〜S^{2/3}という実験(高田,1970)もある。ここで,もし(17)のが(8)のタイプの実験式になると考えれば,一つの可能性は

式(18) [Fig.107_390_02]

であり,摩擦係数fに対する遡上高Rの依存性がf^{-1/4} であるのがもっともらしい。
　このほか,周期波の遡上高Rのなかには,放射圧力 (Radiation Stress) による定常的な水位上昇 (Set Up) が含まれており,海底勾配が小さくなると (\sigma\/sが大きい),水ぎわでの周期的な前後運動による遡上はほとんどなくなり,大部分が定常的な水位上昇になるということも判っている(例えば三井,1973;Van Dorn,1976をみよ)。逆に波が水ぎわ近くで砕波し,周期的な前後運動の大きいときには,周期波に特有な遡上のもどり流れと次の砕波との相互作用が遡上高を決定するのに大きな役割を占めると思われるので,これを理論的あるいは数値実験的に再現するには砕波によるエネルギー逸散の定式化とも関連してかなりむつかしい問題があると思われる。
　以上のような砕波する周期波の遡上に関する知識を参考としながら,次節以下では孤立的な波(遡上にあたってそのまえの波によるもどり流れの影響がない)の遡上を議論する。

4.　斜面上の孤立的な波

　まず初めに,一様水深の水路を,押し波を初動とする過渡的な正弦長波が進行し,一様勾配の斜面上に入射するとき,非砕波第1波の遡上高が後続の波の遡上高とどう違うかを調べる。
　このような問題は,線形の範囲内で摩擦を無視すると解析解が求められる(非線形性をとり入れても,適当な変換をほどこせば線形化できる)。しかし,実際上,一般的な入射波形に対しては,フーリエ逆変換を数値的に行う必要がある。ここでは,(x,t)の座標系のかわりに [Fig.107_390_03] を座標系とした階差法で線形の方程式系を数値計算し,R/Hを入射波のパラメータ \sigma\/s に対して示したものが第1図である。
　いま,一様水深d0の水路からの入射波\eta\iを

式(19) [Fig.107_390_04]

とおくと(13)中の波高Hd0 は Hd0=2A であり,\sigma\は(6)で定義される。図中(13)の関係は実線で示してある。これに対し,第1波に関しては,これを孤立的な波とみなし,Aを波高,半周期 T/2をその時間スケールと考えてH及び \sigma\ を定義したときの関係が白丸で示してある。また,x=0の点,即ち斜面の初まる点での第1波の最大水位(入射波と反射波の合成)を波高Hと考えたときの関係が黒丸で示したものである。
　この図をみて明らかなように,\sigma\/s ＜ 3では,基準の高さをx=0の点での最大水位にとるか(黒丸),入射波の高さAにとるか(白丸)で大きな違いが出るが,\sigma\/s> 3 ではその差がほとんどない。これはx=0における入射第1波の高さに波の反射の効果が含まれているかどうかによるものであり,\sigma\/sの小さいとき,白丸であらわしたR/Hは2に近づき,黒丸であらわしたR/Hは1に近づく。一方で\sigma\/s > 3では

式(20) [Fig.107_391_02]

と近似され,(12)と比較すると係数 \alpha\ は約1.24倍である。
　また,(19)で与えられるような入射波の場合,非砕波領域では第1波の遡上高R1は第2波以下の遡上高R2よりも小さい。これは砕波する波の場合に第1波の方が後続の波よりも遡上高が大きいという経験的事実と対比して興味のあることである。
　ここで,後述する孤立波の遡上実験とも関連して,一様水深d0の水路からの入射波が,理論的な孤立波である場合のパラメータ \sigma\ について検討する。
　孤立波の水位 \eta\,は,最大値をHとすると,

式(21) [Fig.107_391_03]

とかける。ここでXは波の波速をcとして,

[Fig.107_391_04]

であり,X=0は波峯の位置にとってある。
　簡単な近似として,見かけの半波長 \lambda\/2を,波高比 \eta\/H =0.1の点までの距離Xで定義すると

式(22) [Fig.107_391_05]

となり,アーセル数 Ur=(H/d)(\lambda\/d)^{2}\simeq\4.2である。狐立波の代表的長さ \lambda\ については,種々の定義ができ,例えば Munk (1948)のものは(22)とやや異なる。孤立波の波長(22)をつかうと,平均阻度(Steepness)は,

式(23) [Fig.107_392_01]

最大傾斜角は,

式(24) [Fig.107_392_02]

である。比較のために正弦波の第一波の正の部分の体積と今定義した孤立波の一波長分の体積,及び波高をそれぞれ等しいとおくと,波長の間の関係は

式(25) [Fig.107_392_03]

ときめられる。ここで \lambda\s は正弦波の半波長である。
　一方で,波速cは

式(26) [Fig.107_392_04]

とおけるから,無次元化した代表的時間スケ一ルT*は

式(27) [Fig.107_392_05]

であり,無次元角周波数 \sigma\ は

式(28) [Fig.107_392_06]

とかける。

5.孤立波遡上に関する実験結果の検討

　後述する遡上数値実験結果と比較するため,既応の水理実験結果を再検討する。孤立波の一様傾斜面はい上り高さRと,一様水深部での波高Hとの関係を調べる水理実験は過去においていろいろと試みられ,それぞれに適当なパラメータ範囲で実験式が得られている。このとき,砕波の有無はエネルギーの逸散とからんで遡上高に大きな影響を与えるので,孤立波の砕波,非砕波に関する特性を知っておくことが大切である。
　Street and Camfield (1966)は，やや定性的であるが,第2図に示すように,相対波高 H/h0 及び底面傾斜 s をパラメータとして孤立波の岸近くでの特性を分類した。これによると,ほぼ傾斜角8°〜10°(s:0.14〜0.17)を境として,それより急な傾斜面では孤立波は砕けないで遡上する。これに反し,ゆるい傾斜面では必ず波が砕けるが,sの減少又はH/h0 の増加にともなって,巻き波 (plunging)領域からくづれ波 (spilling) 領域に移行する。例えば s〜0.04 のとき,H/d0＜ 0.3で巻き波領域であるが,s〜0.01のときはほぼすべての波がくづれ波となる。定性的には似た結果が花安その他(1970)にものべられており,sがやや大きく,H/h0が小さいときには孤立波の波形の前傾が大きくなって先端附近が巻き波的となる。このことは,砕波高HBと砕波水深hBの比 HB/hB が sとともに増加し1以上になることからもうかがえる。
　第2図ではH/h0がきわめて小さいとき,すなわち波長の長い孤立波の場合にどうなるかはっきりしたことは判らないが,浅海波方程式系の相似パラメータ \sigma\/s がある値以下で非砕波になると考えると,孤立波の場合は\sigma\/s〜(H/d0)^{1/2}s^{-1} であるから,(H/d0)＜＜ 1 ではsのかなり小さいところまで非砕波領域がのびることが期待される。
　さて,途中の浅水変形や砕波形態の相異には目をつぶって,孤立波タイプの波の遡上高Rと,入射孤立波の波高Hとに関係についての実験式を次にあげる。
　Hall and Watts (1953)は,0.05 > H/d0> 0.5, 5°＝＜ s ＝＜ 45°の範囲の実験値を

式(29) [Fig.107_393_01]

の式で整理し,Ks及びmをいろいろのsについてきめた。論文では斜面先端での波高をHとするとのべているが,実際の測定は斜面の影響のない点で行なわれているらしく,Hは入射波波高とみなせるようである。また,sの小さいとき,係放が変化するのは摩樺がきいているからではないがとのべているが,第2図からもわかるように,この実験は大部分が非砕波領域で行なわれており,sの小さいところでやっと砕波領域に入るので,係数に変化があるのは砕波の有無と関連づけられよう。
　Kaplan (1955) は,厳密には孤立波ではないが,過渡的な波の第1波を使って,ほぼ,10^{-3} ＜ H/\lambda\ ＜ 10^{-1}, s=1/30,1/60の範囲の実験値を

式(30) [Fig.107_393_02]

の式で整理し,それぞれのsについてKe,nをきめた。
　Koshi and Saeki (1966) 及び花安その他(1970) は,それぞれ 1/10 >= s >= 1/30,及び1/50 >= S>= 1/150の傾斜,また,3×10^{-3} ＜ H/d0 ＜ 8×10^{-1},及び10^{-2} ＜ H/d0 ＜ 7×10^{-1}の波高範囲で実験を行ない,これを

式(31) [Fig.107_393_03]

の実験式として整理した。Kishi and Saekiの実験では底面が粗な斜面を使っている。この北海道大学グループの実験では,水路端の空気室内に初期火面上昇を与え,これが水路中を伝播して孤立波を形成する方式をとっているので,初期アーセル数の制限から,きれいな孤立波が与えられた水路長のなかでつくれるH/d0にはある限界があると思われる。空気室の水平スケールを \lambda\0,水位上昇量をH0,水深をd0とすると,初期アーセル数Ur0は

式(32) [Fig.107_393_04]

であり,Prins (1958) の実験結果などから類推すると,きれいな孤立波のできるのは

式(33) [Fig.107_393_05]

の範囲である。このことから考えると,北海道大学グループの実験のうちH/d0の小さいものでは,分散性の波になり理論的な孤立波に近い波ができていないと思われる。したがって,以後は,H/d0 ＜ 0.04のデータは使用しない。実際,Kishi and Saeki (1966)の論文でも,H/d0 ＜ 0.04のデータは使われているようにみえない。
　さて,(29),(30),(31)の実験式は,花安その他(1970)で示されているように,孤立波についての (H/d0)と (\lambda\/d0) との関係式をつかって換算が可能とすると,すべて(31)と同形となる。彼等は,こうして全実験式を(31)のかたちとしたときの係数Kh及びaの値を傾斜sに対して図示した。これによると，KhもaもSに対してきわめて規則的に変化し,異なった実験から得た結果が相補的であることがうかがえる。
　このことをもう少し詳細に検討し,また,周期波の遡上高との関連も見易くするため,相対遡上高 R/Hを \sigma\/s の関数として表示したのが第3図である。ここで \sigma\ は(28)で定義されたものである。図中,SAEKI-I,SAEKI-II,はそれぞれKishi and Saeki(1966) 及び花安その他(1970)のデータのうちH/d0 ＞ 0.04のものであり,H＆Wは Hall and Watts(1953) のデータである。Kaplan (1955) のものは,個々のデータが不明のため,実験式を書きなおし,S=1/30,1/60の両方の平均をとって,

式(34) [Fig.107_394_01]

とした直線が記入してある。上式とKaplanの与えた個々の実験式との差は数パーセント以内である。また直線の範囲は,図をみやすくするため,実際の実験範囲 10 ＜ \sigma\/s ＜ 10^{2}よりやや広くとってある。
　第3図によると,期待されるように異なった実験結果が第一近似としては一本の曲線上にのっているようにみえる。また,孤立波が砕波するか否かの区別は,ほぼ \sigma\/s〜5のあたりできまるようであり,R/Hの最大値は3.5の程度と思われる。周期波の場合の砕波限界の関係式 (15) からみても,この \sigma\/s〜5という値はそれほど不自然ではない。非砕波領域では,近似的に

式(35) [Fig.107_395_01]

砕波領域では,近似的に

式(36) [Fig.107_395_02]

である。特に図からもわかるように s ＜ 1/50では値がかなりばらついており,これは砕波が主としてくづれ波 (spilling wave) になること,摩擦効果が大きくきくことなど,種々の要因に関係しているようである。
　純粋の孤立波実験に対し,首藤(1966)や富樫・中村(1975)の津波遡上実験では,波のアーセル数が100以上と大きく,時間がたつとソリトン分裂形に移行する波をつかっている。したがって,実験範囲で H/d と時間スケールTとは独立である (この点が孤立波実験と異なる)。首藤(1966)の実験は主として非砕波領域のもので,結果は理論的予測に近いとのべている。富樫・中村(1975)は R/H を l/\lambda\ の関数とする実験式を提出している。ここで,Hは水平の長さlの斜面直下での波高であり,入射波高ではない。また,2\pi\l/\lambda\=\sigma\/s の関係がある。
　首藤(1966)及び富樫(1976)の実験データを,第3図と同様にR/Hと \sigma\/s の関係として表示したのが第4図である。但しHは斜面直下での最大水位である。この図は富樫(1976)の図3・66と本質的に同じものであり,底面傾斜の範囲は1/20〜1/60である。違うところは,最近岩崎その他(1982)の発表している実験結果のうち,押し波初動の第一波のデータ(s=1/30)をつけ加えてある。すべてのデータは第4節の孤立的な波の波高H,時間スケールTの定義にしたがっている。
　図中,異なった記号は,それぞれ相対的入射波高A/h0について,△＝＜ 0.05＜ □ ＝＜ 0.10 ＜ ○ ＝＜ 0.20 ＜ ×ととってある。富樫(1976)のデータのうち,\sigma\/s＜ 2.0のものについてはH/h0 からA/h0 を推定し,\sigma\/s>= 2.0ではH/h0をA/h0 と等しいとおいた。　また,図中の直線及び波線は第1図に示した正弦波の第一波に関するものであり,ここで用いた実験波形の方は,

式(37) [Fig.107_396_01]

と仮定し,正弦第一波の \sigma\s を,これと体積を等しくする実験波の \sigma\ に換算 (\sigma\=(\pi\/4)\sigma\s)したもので記入してある。
　第4図を第3図と比較すると,Hの定義に違いがあるため\sigma\/s ＜ 1.0では異なっているが, \sigma\/s が1から2のあたりでは割合によく合っている。第4図では\sigma\/s \simeq\ 2のあたりが砕波・非砕波の限界にあたるので,それより大きな \sigma\/s でのR/Hは,入射波高A/h0 の大小によりやや異なった傾向を示し,第3図の孤立波のものと異なる。同じ \sigma\/sに対し,A/h0 の大きなデータほど早く砕波するので,R/Hが小さく出ているのはもっともらしい。特に岩崎その他(1982)のデータはA/h0〜0.02,\sigma\/s>=5の範囲に入るので,まだくだけ寄せ波(surging) の領域にあり,R/Hが4以上にも達している。これからみると,\sigma\/s>= 10で,A/h0 ＝＜ 0.01程度の孤立性の波による遡上実験を行なうと,摩擦の効果がそれほど大きくなければR/Hは5以上にもなり得るであろう。

6.孤立波遡上の数値シミュレーション

　孤立波が一様斜面上を変形しながら岸向きに進行し,陸上に遡上する有様を数値的にシミュレーションするには,ブシネスク方程式(波形曲率項を含む)系をつかうのがよいであろうが,ここでは \sigma\/s の範囲がほぼ10^{2}以下,すなわち l/\lambda\ ＝＜ 16で,斜面上では波形の前傾が大きいところを中心と考えるため,オイラーの方程式系(4),(5) を使用する。砕波する孤立波に対し,波形曲率を考慮しない方程式系が適当かどうかや、疑問の余地があるが,ここでは砕波はすべて段波発生でおきかえられるものと仮定する。
　オイラー方程式(4),(5)の数値計算には,二段階Lax-Wendroffの方法をつかう(Richtmyer and Morton,1957)。陸上に遡上する波の先端条件は,運動する水粒子の先端の水深が0となることで与え,また,先端の進行速度は先端水粒子の速度を内側からの線形外挿値できめる。したがって，波先端の位置は各時間ステップ毎に決定しており,あらかじめ与える空間の格子点とは一致していない。
　水底傾斜面の沖側境界では,理論的な孤立波の波形をもった浅海波を入射波とし,その斜面上の変形を追う方法をとった。沖側境界に内側から進む反射波については,簡単のために,一様水深の場合の線形波自由透過条件を傾斜水底の端でそのまま使ったが,ごく僅かだけ反射がおこっている。
　孤立波から段波が発生するところでのギプス振動を除くため,人為的な拡散項が入れてあるが,段波面近く以外ではその影響は無視出来る。
　このようにして計算した遡上高Rと入射波高Hとの関係が第5図に示してある。摩擦係数f (図にはFRと表示) を0,0.002,0.01,0.04とし,それぞれいくつかのsに対し,H/d0 を変化させてて計算を行ない,R/Hを \sigma\/s に対して示したものである。したがって,同じ \sigma\/s でも,フルード数にあたるH/d0 は異なったものを含んでおり,計算結果でみるかぎりフルード数の影響も多少あるようである (図において同じ摩擦係数の場合に,R/Hの \sigma\/s への依存性が直線的になっておらず,やや点がバラついているのはこのせいである)。しかし,大局的にみると,R/H の \sigma\/s への依存性は,フルード数の変化よりも摩擦係数の変化によって大きく変り,特に \sigma\/s の大きな砕波領域では,摩擦係数の大きさではっきり R/H が変化する。図中の実線は(34)で与えられるKaplan(1955)の結果に対応するものであり,第3図ともあわせ考えると,摩擦係数 f \simeq\ 0.01のあたりが最もよく実験値を再現しているようにみえる。
　非砕波領域では,砕波領域ほど摩擦効果は顕著ではなく,摩擦を省略した計算値もかなり実験結果に近い。しかし砕波領域で摩擦を省略すると今までの水理実験で得られた相対遡上高 R/H の約2.5倍大きな遡上高が得られ,この違いは特に \sigma\/s が増大するほど大きくなるようである。

7.おわりに

　いままでのべてきたのは,実験室内での孤立波遡上高と数値実験結果の比較であり,現実の海浜における津波遡上を考えるときに,f=0.01の程度の摩擦係数が適当なものかどうかには問題も残る。
　また,現実の津波の場合には,第一波を理論的な孤立波で近似するのは適当でないことが多い。例えば昭和58年日本海中部地震津波の場合，能代沖の陸棚端近く,水深100mのところで,第一波の峯高Hが1m程度,時間スケールTが4分程度と考えると,陸棚の傾斜sが1/200の程度であるから [Fig.107_398_01],Ur〜56.5 であり,孤立波に関する第3図よりも,Urの大きいときの孤立的な波にあてはまる第4図を参考にすぺきである。しかし,[Fig.107_398_02]の程度であるから実験値の範囲をこえており,きわめてあらく見積ると,波は多少砕けて surging の領域にあるとしてR/H が5〜6の程度となろう。
　現実の津波遡上について考えるべきもう一つの要素は,水ぎわ近くで海底勾配が急に大きくなるときである。これは,海に面して砂丘の急斜面が存在する場合などにもあてはまるが,このときは,勾配の変化のないときにくらべて遡上高が大きくなることが期待される。
　Kaplan (1955) の実験でも,水ぎわ近くに堤防があって急傾斜になっているとき,堤防のないときにくらべて他は同じ条件でもR/Hが約2倍になっている。短周期の波について水ぎわ近くに堤防の存在するときは,ないときにくらべて砕波の遡上高をたかめ,最大遡上高 R/H は,堤脚水深のあたりで波が砕けるときにあらわれることが知られているが,これらのことを考え合せると能代北方の砂丘地帯で特に遡上高の大きかったことを説明できるかも知れない。
　最後に,第2,3図に用いた孤立的な波の遡上に関する実験データを提供していただいた佐伯・富樫両博士に御礼申します。また,数値計算,作図を手伝ってもらった加藤氏にも感謝いたします。

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