Numerical Simulation of Historical Tsunamis Generated off the Tokai District in Central Japan

The l944 Tonankai and the l854 Ansei−Tokai tsunamis,both of which were generated off the Tokai district in central Japan,are numerically simulated by making use of bottom displacement fields inferred from seismic fault models proposed by various investigators.The reliability of these simulations is examined by comparing simulated wave heights with observed values,and it is found that the simulations based on the seismic fault model are satisfactory.
The source models for the l707,1605,and 1498 tsunamis,which caused severe damage in this district,are next constructed by the trial and error method to be consistent with the distribution of tsunami inundation heights described in old records,although information is rather vague.Among the five tsunamis investigated here,the 1498 Meio tsunami seems ■o have been the worst event and should be noted from the viewpoint of hazard prevention.
Finally,taking the Meio tsunami as an example,a zoning map of destructive forces is prepared for the inundation at the Shimizu Harbor area.To do this,a tsunami and possible run‐up is simulated for Shimizu Harbor with a local simulation model.It is found that breakwaters and reclaimed land which have been constructed recently are effective in reducing tsunami heights within the harbor for tsunamis of the Meio type.

1. Introduction

Several years ago,the possibility was suggested that a destructive earthquake could occur in the near future in the vicinity of the Pacific coast of the Tokai district,central Japan.According to this suggestion,the postulated seismic source area of this earthquake covers the whole of Suruga Bay.Since the seismic source is located very close to land,seismic intensities in the Tokai region would be among the strongest of past earthquakes.
A destructive tsunami might be generated accompanying this earthquake.The prediction of the expected tsunami behavior has become a subject of intense research effort.However,much larger tsunamis than that expected for this earthquake were experienced in this region according to old records.Therefore,it is very important to investigate not only the behavior of the tsunami expected in the near future,but also the strongest tsunami experienced in the past.
First,numerical simulations for tsunamis which have occurred in recent years （1944 and l854）are carried out and compared with available data to verify the reliability of the use of a seismic fault model as a tsunami source.Second,source models for older tsunamis are searched for by simulating tsunamis which reasonably well explain coastal inundation heights described in old records.This enables an estimation of inundation heights along the coast where the behavior of the tsunami was not described in records,and offers a useful guide for the planning of preventive measures against tsunami disaster.In addition,the estimated earthquake source model may give important clues of geophysical interest concerning sizes and geographic locations of earthquakes occurring in this region.
Furthermore,tsunami inundations on land are simulated by a local model,the input of which is obtained by the wider model for which the estimated tsunami source is used.The inundated area and the inundation heights on land can be examined by this simulation.The distribution of hydraulic pressure due to the inundation water current on land is known to be well correlated with the tsunami destructive power.Therefore,a distribution map of the hydraulic pressure may provide useful data for microzoning of tsunami hazards.

2. Numerical Method

A tsunami is described by shallow water equations,since the wave length is very large comparing with the water depth.
【式】：（1）・（2）・（3）
Here,qx and qy are the x-and y-components of volume transports integrated to the bottom,ζis the water surface elevation,is the vertical displacement of the bottom,h is the water depth,ｆis the quadratic friction coefficient of the bottom,D＝h＋ζ−ξ and Q=（q^2x＋q^2y）^1/2.The second and third terms on the right-hand side in Eqs.（1） and （2）were neglected in the wide models of simulations including the tsunami source area.
A vertical displacement of the sea bottom at the time of tsunami generation is assumed to be approximated by the permanent elastic displacement of an idealized seismic fault model（rectangular faulting with uniform slip）,which can be computed by the formulation of MANSINHA and SMYLIE（1971）.The duration time required to complete the bottom displacement is assumed to be much smaller than the propagating time of a shallow water wave across the tsunami source area,so the initial elevation of the water surface is identical to the bottom displacement ξ.
The generated tsunami is simulated by means of Eqs.（1）to（3）which are solved by a space staggered finite difference scheme with a regular grid size of 5km over most of the open sea area.At selected coastal regions shallower than about 200meters,the grid size is decreased in four steps down to 1／16 of the regular grid size.The adoption of a telescopic grid scheme improves the high frequency wave characteristics which critically depend on the grid size,and improves the accuracy of geometrical modelling of the coastal area.The computing time step is 0.05minutes which satisfies the numerical stability condition over the whole area.
The boundary condition at the coast is qx or qy＝0.At the open boundary appearing artificially due to the restriction of the computational area,the following formula related to long progressive waves is assumed to hold:
qx＝±（c^2ζ−q^2y）^1／2
or （4）
qy＝±（c^2ζ−q^2x）^1／2
Here c is a long‐wave velocity and signs at the right hand side are taken to satisfy the condition that qx or qy are in the outward direction from the computing area when ζis plus.

3. Verification of the NumericalSimulations

3.1 The 1944 Tonankai tsunami

The 1944 Tonankai tsunami attacked the Tokai district shown by hatching in the inset of Fig.1.Tide‐gauge records of thetsunami were obtained at Mera,Ito,Uchiura,Matsuzaka,Shimotsu and Tosa-Shimizu.The computation area is 965×400km^2,shown in Fig.1,which includes small areas surrounded by dotted lines near the tide‐gauge stations where telescopic variations are introduced.
For thise earthquake.simulations are carried out making use of the vertical bottom displacment fields inferred from parameters of four different fault models published so far（AIDA,1979）.Figure2 shows these fault models（KANAMORI,1972;ANDO,1975;INOUCHI and SATO,1975;ISHIBASHI,1976,1981）in which rectangles indicate the horizontal projection of the fault plane and arrows indicate the dislocation of the hanging wall side of the fault.Dip angles of fault planes are 10 to 30 degrees in the north‐west direction.
Two parameters K and K are used to express the reliability of a model（AIDA,1978）.Let xi and yi be the respective amplitudes of the observed tsunami and the simulated wave at the i'th station,and let the ratios Ki＝xi/yi be calculated for n stations.The first parameter Kis the geometrical mean of Ki,which is defined as log K ＝（1／n）Σni＝1 log Ki.The factor K represents a correction to be applied to the slip displacement of a fault model.The second parameter K is a logarithmic standard deviation defined by the following formula,which is a measure of the variation of Ki,namely,the accuracy of the height simulation.
【式】：（5）
The reliability of the present simulations is examined for the amplitudes of the first and sceond half cycles of the wave‐time histories,a1 and a2.
Taking the locations of tide‐gauge stations on the abscissa,the ratios Ki／K are plotted in Fig.3.Since Models I and II give large variation of Ki and Model IV gives a systematic diffbrence between three eastern stations and three western stations,these models are not adequate as tsunami sources.Model III seems to be the best model since K is the smallest;K＝1.27 in average for a1 and a2.But the K factor of this model is 0.45.This shows that the dislocation assumed in this fault is too large by a factor of about 2.
The simulation applying the factor K to the fault dislocation resulted in the very good agreement between the computed elevation‐time histories and actual tsunami records as shown in Fig.4.

3.2 The 1854 Ansei‐Tokai tsunami

A fault system for the 1854 Ansei‐Tokai earthquake has been published by ISHIBASHI（1976,1981）as shown in Fig.5.It is composed of two parts.The southwestern part experienced the Tonankai earthquake 90 years after the 1854 Ansei‐Tokai earthquake and thus the strain energy was believed to have been released again.The northeastern part has experienced no event in the 127 years since the 1854 earthquake.Therefore,it is speculated that the next earthquake should occur in the northeastern region.
Vertical bottom displacement fields inferred from this fault dislocation are displayed by solid（uplift）and dashed（subsidence）contour lines in Fig.5.This pattern will be used as the source model for the 1854 Ansei‐Tokai tsunami.
Actual tsunami inundation heights along the coast have been estimated from tsunami behavior described in old documents（HATORI,1976,1977）,which are called the observed values hereafter.For comparison with the observed tsunami heights,the simulated wave heights are determined by either of the following two methods depending on the location.
A）In places covered by the telescoped grid system,the computed maximum elevation above the still water level was used without modification.Shimizu and Omaezaki were added to the telescoped system and Shimotsu and Tosa‐Shimizu were removed.
B）On coastal locations covered only by the coarse grid,the maximum elevation was estimated from the wave height（double amplitude）H0 computed on the 200 meters isobath by applying an amplification factor.The amplification factor in shallow water is determined from a simulation for the 1944 tsunami such that the ratio of surveyed inundation heights on the coast and H0 computed on the 200 meters isobath represent the amplification factor for that location.
Figure6 shows the observed heights（short bars）and the heights calculated by two methods,A（double circle） and B（single open circle）.Simulated values have been corrected for the tide level and vertical ground displacement at the particular time and location.The agreement is fairly good,with ratios of observed and simulated heights in the range of 0.8 to 1.2.
The reliability parameters become K＝0.99 and K＝1.16 for locations where method A was applied,and K＝0.92 and K＝1.2 for methods A and B.

3.3 Discussion of reliability

Many numerical simulations have been carried out for tsunamis off Hokkaido and Tohoku districts（AIDA,1977a,1978）.Table 1a shows the reliability parameters of these tsunami simulations including those of the 1944 Tonankai tsunami.In this group,the verification of the simulation was made using the amplitudes of the actual tsunami records for the first and second half cycles of waves,al and a2.
Reliability parameters which were obtained from the measured or estimated run‐up heights and the simulated heights by method B are shown in Table 1b which includes results from the 1854 Ansei‐Tokai tsunami.
According to these tables,a measure of variation of simulated values K is about 1.4 on the average,without a significant difference between the two tables.In the present simulations,we obtained K＝1.17（a1）and 1.39（a2）for the 1944 tsunami,and k＝1.2 for the 1854 tsunami.These values seem to be quite satisfactory since they are considerably smaller than the average.That is,it has been verified that the tsunami source derived from a seismic fault model can be used in the simulation of tsunami originating in the Tokai district just as well as in the Tohoku district.

4. Source Models for Other Historical Tsunamis

Besides the two tsunamis already stated,very large tsunamis attacked the Tokai district in 1498,1605,and 1707.
There are a considerable number of old records relating to the 1707 Hoei tsunami（HATORI,1977）.Coastal inundation heights of this tsunami inferred from these records are almost the same as that of the 1854 Ansei‐Tokai tsunami.The question as to whether or not the seismic fault plane extended to the head of Suruga Bay has not been resolved yet（HATORI,1977;MOGI,1977;ISHIBASHI,1977）.Results of several simulations in which different locations were assumed for the fault in Suruga Bay showed clearly that a model used for the 1854 tsunami could explain quite well the behavior of the 1707 tsunami on the coast in the vicinity of Suruga Bay（AIDA,1981）.Therefore,the source model for the 1707 tsunami is presumed here to be the same as the model for the 1854 tsunami.
On the other hand,records describing the 1498 and the 1605 tsunamis are so scarce that an estimation of inundation heights based on these is somewhat uncertain.For these tsunamis,several probable models are assumed,taking into account the geographical arrangements of the Suruga and Nankai troughs and also the speculations（HATORI,1975;ISHIBASHI,1978）of the source area.The most reasonable source parameters are selected on the basis of a comparison between computed inundation heights and descriptions in old records.
In summary,seismic faults are shown in Fig.7 which are adequate as source models for five tsunamis in this region.Rectangles denote the horizontal projections of fault planes of the fault locations.Other fault parameters are tabulated in Table2.In Table2,seismic moments are also calculated from fault areas and dislocations assuming a rigidity of 5×10^11 dyne・cm^−2.These historical earthquakes are found to be of the largest class in and near Japan.
Using the fault parameters listed in Table2,reliability parameters for these tsunamis are as follows,K＝1.04 and k＝1.36 for the 1605 tsunami,and K＝1.12 and k＝1.42 for the 1498 tsunami.

5. Regional Distribution of Tsunami Heights

To determine the general characteristics of the various tsunamis,the distribution of wave heights,H0,along the 200 meters isobath is plotted in Fig.8.On the western half,namely in the Kumanonada region,wave heights H0 are almost the same for all tsunamis.However,on the eastern half,namely in the Enshunada region,the differences of H0 from tsunami to tsunami are considerable.Among these tsunamis,the 1498 Meio tsunami shows the highest values which are about 1.5 times that of the 1854 tsunami in many places.
Tsunami heights（maximum water elevation）at several places in four typical bays are shown in Fig.9.At Owase Bay in the western part,the height differences among various tsunamis are very small.However,at Shimizu,Uchiura,and Shimoda Bays in the eastern part,coastal tsunami heights depend strongly on the particular model.At Shimizu and Shimoda Bays,the heights of the 1498 tsunami are about 1.5 times that of the 1854 tsunami.According to these simulations,it appears that the 1498 Meio tsunami was the highest tsunami ever experienced in this region.

6. Simulation of Tsunami Inundation

Simulations of tsunamis inundating over land have been carried out by the author on the basis of the shallow water equations,Eqs.（1）to（3）（AIDA,1977b）.
The water front is determined by checking the boundary between a wet and dry meshes,where the water velocity is related to the square root of the height of water front with a coefficient of 0.5.Energy losses caused by a step change at the original shoreline are replaced by effective energy losses due to bottom friction in one grid interval,in which the effective friction coefficient is assumed to be 0.03.In an urban area consisting of houses,energy losses due to obstacles on land are represented by an effective friction coefficient of 0.02.
The friction coefficient in a bay proper is taken to be 0.005 except where the water depth is less than 5 meters and on land where the coefficient is taken as 0.01.
In the present paper,the inundation corresponding to the Meio tsunami is simulated for Shimizu Harbor.The input wave form is given at the boundary outside of the harbor,for which the results of the wide model discussed in the previous section are utilized.
Figure 10 shows the envelope of the maximum water surface elevation of the tsunami invading Shimizu Harbor with the old topography.The natural shoreline in the harbor is shown by thick solid lines and the ultimate inundated water front is shown by chain lines.Water surface elevations are 3 meters high near the mouth of the harbor and about 5 meters on the ground at the bay head.The value at the bay head is somewhat smaller than the case of non‐inundated model,but at other places,water surface levels are nearly equal to that of non‐inundated model.
Recently,a breakwater has been constructed near the harbor entrance and the topography of the inner harbor has been greatly changed by reclamation works.The behavior of tsunamis of the same type for the present topography simulated under identical input condition as for the old topography is shown in Fig.11.The result shows clearly that the water elevation is reduced by 50 cm at the bay head and 80 cm at most other places.In particular,the decrease of 150 cm is found at Shimizu,in the uppermost portion of the figure.Thus,the construction works of breakwaters and land reclamation in the harbor appear to be effective for reducing inundation heights in the present case.
It has been shown that the percentage of destroyed houses at local area of a town is well correlated to the hydraulic pressure due to the inundation water current calculated from simulations（AIDA,1977b）.The hydraulic pressure is represented by the product of the square of the water current velocity and the water elevation above the ground surface.
The distribution of the maximum representative pressure in m^3／sec^2 is indicated by contour lines in Fig.12.Taking the experience of previous investigations into account,more than 50 ％ of the wooden houses in the area shown by hatching would be destroyed.It is clearly seen that in this region,the riverside,bay head and tip of the peninsula are locations of high risk.

7. Conclusions

A simple earthquake fault model is determined as a first approximation in which for simplicity a rectangular fault and a uniform distribution of a fault slip are assumed.In a simulation of a tsunami based on a simple fault model with a uniform slip over a rectangular fault area,the uncertainty of the result amounts to a factor of 1.2 to 1.5.Some improvements of the approximation may be possible by selection of source models such as the 1854 Ansei‐Tokai and the 1944 Tonankai tsunamis.
Source models of older tsunamis were determined by the same procedure.The most destructive tsunami among them was the 1498 Meio tsunami.It is estimated that this tsunami would have caused run‐up as high as 10 meters at Shimoda,but unfortunately,a description of this tsunami in old records is missing.
Using the tsunami source model obtained here,a local inundation simulation was carried out and a zonation of tsunami hazards attempted through use of the distribution map of hydraulic pressure of the water current invading overland.
I would like to thank Prof.Kajiura,Earthquake Research Institute,University of Tokyo,for helpful suggestions.The computations were carried out by the Computer Center,University of Tokyo and this research was supported by the scientific research fund of the Ministry of Education.

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