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Abstract

 Tsunami energy is determined by means of the following methods,and the results obtained are discussed:(1)Method of using the tsunami waves directly propagated from the origin.(2)Method of using the generation energy of tsunamis at the epicentral area.(3)Method of using the relationship between the magnitude of an earthquake and a tsunami.   (4)Method of using the maximum predominant period of a tsunami.
 Tsunami energy was of the order of 10^24 ergs for the largest tsunami and of 10^20 ergs for the smallest tsunami.Tsunami energy ranged from about one-tenth to about one-hundredth of the total energy of seismic waves.

Discussion

Array As is generally known,tsunamis accompany only those earthquakes whose epicenters are at shallow depths beneath the ocean or close to shore;most tsunamis are generated by tectonic displacement of the ocean bottom,either fault displacements or more general deformations caused by earthquakes.Some tsunamis are also generated by landslides or by submarine volcanic explosions,or even by artificial submarine explosions.Any tsunami warning system should therefore take into consideration the dynamic nature of tsunamis with respect to potential danger.An estimation of tsunami energy is significant in connection with this dynamic nature and with the generation mechanism of tsunamis.The assumed minimum magnitude of earthquakes which are associated with the generation of tsunamis is also important in tsunami waiting,from a seismological viewpoint.
 As already pointed out(2,3,4),tsunami magnitude depend not only on the magnitude of the earthquake but also on certain conditions of the seismic origin,such as,focal depth,water depth at the epicenter,and area of crustal deformation caused by earthquakes.In addition to this,the speed and amount of the displacement may generally affect the generation of tsunamis,but these factors have not yet been established.
However,since the energy of a tsunami caused by tectonic displacement of an ocean bottom results from the energy transferred from the earth’s crust to the sea water,some portions of which disappear changing into heat or sound,the amount of such displacement may be found through a knowledge of tsunami energy.
 In the present paper,tsunami energies are obtained on the basis of practicable methods.The energies thus obtained are compared with each other with respect to the same tsunami.
 An estimation of tsunami energy may be made by means of the following methods:
 (1)Method of using the tsunami waves directly propagated from the origin.
   A tsunami is a train of long waves which originate from the periphery of a submarine epicentral dislocated area.The velocity of wave propagation is
  V = √gh
where g is the acceleration of gravity and h the depth of the sea.Assuming that the spread of energy from a source area is uniform in all directions,the total energy E is given by(7)
  V=π ρgVRΣa^2T  [1]
where ρ is the density of sea water,R the distance from the origin,a the amplitude,and T the half-period of tsunami waves.The summation is made with respect to all half-periods.According to [1],Takahasi(1951)estimated the energy of the 1933 Sanriku tsunami as 16 x 10^22 ergs,and Watanabe(1956)also obtained the energy of the same tsunami as 15 x 10^22 ergs.In addition to this,they also estimated the energies of some of the other large tsunamis which have occurred in Japan;these figures are given in Table1.
 It was found that the total energy thus calculated was of the order of 10^23 ergs for a large tsunami and of 10^20 ergs for a small tsunami.This is approximately equal to from one-tenth to one-hundredth of the total energy of seismic waves,and it may be concluded that the larger the tsunami,the smaller the ratio of tsunami energy to the energy of seismic waves.
 In calculating the tsunami energy,the values of a,T,and V should be taken preferably from observation at a place nearest the origin.When using data obtained at long distances,the convergence and divergence of wave energy caused by refraction and reflection of waves,as well as the scattering or spreading of the energy,must then be taken into considera-tion.To determine accurately the value of V,the submarine topography must be known in detail.
(2)Method of using the generation energy of a tsunami which has originated at the epicentral area.Since a tsunami is considered to originate at the epicentral area of a dislocation,such as sudden rising,subsiding,or faulting on the ocean floor,the tsunami energy(6)is derived from the energy transferred from the sea floor to the water.
 Assuming the sudden vertical displacement at the dislocated area S of the ocean floor,the work done on the water by the sea bottom dislocation due to the earthquake is approximately expressed as
【数式・1】
where b is the vertical displacement at the small portion dS of the dislocated area and h the water depth at this portion.This work should be consumed in increasing the potential energy of the sea water and in the energy transformed into the tsunami waves.Since the former portion of the work expressed as
【数式・2】
the latter portion,i.e.,the energy transformed into the tsunami waves is therefore equal to
  【数式】   [2]
Taking the average vertical displacement ^am and the area of tsunamiorigin A’ at the dislocation area S,[2] becomes approximately
  【数式】   [3]
From [3] it is found that the tsunami energy can be approximately estimated from the area of tsunami origin and the vertical displacement of sea bottom.Since the area of tsunami origin A is closely related to the area of aftershock activity,we may take A for A’.The relationship [4] between the area of aftershock activity A and the earthquake magnitude M is written as
  log A = 0.9M + 7   [4]
Consequently,the tsunami energy is approximately expressed by
  【数式】   [5]
From[5] it is found that the energy of a tsunami is related not only to the earthquake magnitude but also to the vertical displacement at the epicentral area.If the amount of the vertical displacement of the sea bottom is not determined,the tsunami energy cannot be estimated.It has been found,however,that the degree of crustal deformation may be estimated from the energy of tsunamis.Taking the data relating to several large tsunamis,we may find the values of am,as given in Table2.It can be noticed that the amount of the vertical displacement of the sea bottom is of the order of from several meters to about ten meters.
 (3)Method of using the relationship between the magnitude of an earthquake and a tsunami.
 As already pointed out(2,3,4),since the magnitudes of tsunamis(m)are related to those of earthquakes(M),the computation of tsunami energy may be performed by using the following relationships:
  m = 2.61M − 18.44   [6]
  log E = log Eo + 0.6m  [7]
  log E = 10.34 + 1.56M  [8]
together with the relationship,developed by Gutenberg and Richter:
  log Es = 11.8 + 1.5M  [9]
in which E_s is the earthquake magnitude,and E_o = 2.5 x 10^21 ergs.Thus,we can determine the energy of a tsunami from earthquake magnitude or from seismic wave energy.
 Table3 gives the values for the energy of several tsunamis;from these it can be seen that the smaller the tsunami,the greater becomes the difference between tsunami energy and earthquake energy.The accuracy of the values thus obtained depends on accurate seismological data.Most values obtained by method(3)are comparable to those obtained by method(1).
 (4)Method of using the maximum predominant period of a tsunami.
 It is inferred that the difference in the period of tsunamis can be regarded as due to the magnitude of the earthquakes.Using several waves from the beginning in the mareograms of various stations and of various tsunamis,a Fourier analysis was carried out according to the Filon method by the electronic computing machine NEAC-2203 in order to find the predominant period of tsunami waves.Twenty-two tsunamis and 48records were selected for this analysis,as given in Table4.Figure 1 shows the relationship between the predominant period thus obtained and earthquake magnitude,from which the maximum period was found.From this relationship it is found that the greater the earthquake,the larger becomes the period of a tsunami.Taking the maximum period of tsunami Tt,the relationship between Tt and M,as shown in Figure2,may be given as
  log Tt = 0.20 + 0.22 M,  [10]
 or log Tt = 0.15 logE − 1.82  [11]
From [11] we can estimate the energy of a tsunami.Table4 gives the data of various tsunamis thus obtained and,except for the Tokachi tsunami of 1952,the values are in good agreement with those obtainedby methods(1)and(3).

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Table1 Estimations of tsunami energy,as obtained by various investigators
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【数式・1】
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【数式・2】
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【数式】 [2]
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【数式】 [3]
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【数式】 [5]
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Table2.Value of ^am inferred from tsunami energy
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Table3 Tsunami energy derived from tsunami magnitude(formula(7))and earthquake magnitude(formula(8))
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Fig.1−Relationship between the predominant periods of tsunamis,analyzed from observations at various tide gauge stations and earthquake magnitudes.
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Table4.Maximum period obtained by Fourier analysis and energy of tsunamis.
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Fig.2−Relationship between the maximum period of a tsunami and earthquake magnitude.
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Table5 Comparison of energy of tsunamis

Conclusion

 Tsunami energies thus estimated have almost the same value,as given in Table5,but some differences were found in small tsunamis which occurred beneath the sea close to the shore.These differences may be due mostly to the uncertainty of the magnitudes and periods of the tsunamis.At any rate,tsunami energy was of the order of 10^24 ergs for the largest tsunami and 10^20 ergs for the smallest tsunami and was also about from one-tenth to one-hundredth of the total energy of seismic waves.If an increase in the accuracy of the determination of tsunami energy is to be attained,seismological data will have to be determined more accurately.

References

(1)Hirono,T.1953.Report on the investigation of the Kamchatka Earthquake of 1952.Quart.J.Seismol.18:36-39.
(2)Iida,K.1956.Earthquakes accompanied by tsunamis occurring in the sea Off the Islands of Japan.J.Earth Sciences,Nagoya Univ.,4:1-43.
(3)Iida,K.1958.Magnitude and energy of earthquakes accompanied by tsunami,and tsunami energy.J.Earth Sciences,Nagoya Univ.,6:101-112.
(4)Iida,K.1961.Magnitude,energy,and generation mechanism of tsunamis,and catalogue of earthquakes associated with tsunamis.Tenth Pacific Sci.Congr.,Tsunami Symposium No.1.
(5)Matsuo,H.1934.Estimation of energy of tsunami and protection of coasts.Bull.Earthq.Res.Inst.,Tokyo Univ.,Suppl.Vol.1:55-64.
(6)Miyoshi,H.1960.The tsunamis caused by the Myojin volcanic explosions(II).J.Tokyo Univ.of Fisheries 47:1-40.
(7)Takahasi,R.1951.An estimate of future tsunami damage along the Pacific coast of Japan.Bull.Earthq.Res.Inst.,Tokyo Univ.29:71-95.
(8)Yoshida,K.,Kajiura,K.,and Miyoshi,H.1952.On the tsunami of March 4,1952.Geophys.Notes 6:1-6.