Wave Motion 46 (2009) 420–426

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Wave Motion

journal homepage: www.elsevier .com/locate /wavemoti

On the relevance of soliton theory to tsunami modelling

Adrian ConstantinUniversity of Vienna, Fakultät für Mathematik, Nordbergstraße 15, 1090 Vienna, Austria

a r t i c l e i n f o

Article history:Received 12 March 2009Received in revised form 11 May 2009Accepted 26 May 2009Available online 7 June 2009

Keywords:TsunamiSolitonNon-linear waves

0165-2125/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.wavemoti.2009.05.002

E-mail address: adrian.constantin@univie.ac.at

a b s t r a c t

We discuss the relevance of soliton theory to the modelling of tsunamis in the context ofthe two largest tsunamis for which records are available—the December 2004 and the May1960 tsunami. Our contention is that in both cases the scales involved do not permit a bal-ancing effect of dispersion and nonlinearity, and therefore soliton theory is not applicable.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

The word ‘‘tsunami” means ‘‘harbour wave” in Japanese (tsu – harbour, nami – wave). The name captures the essentialfeature of these waves that are nearly impossible to detect at sea due to their deceitable small amplitude (usually with wavesraising almost uniformly over a wavelength of several hundreds of km to less than 1 m above the usual sea level); the tsu-nami waves rise to large amplitudes (up to 10–15 m) is triggered by the diminishing depth effect as they approach the shoreline. Earthquakes, volcanic eruptions, great landslides [45,57,58], and even more localized impulses, such as meteoritestrikes into the sea [44], sometimes cause a large volume of water to be rapidly displaced from its original position withan almost flat surface to form a new pattern featuring more pronounced rises and/or depressions, with the resulting wavemoving away from the location of its creation in the form of a tsunami wave. The catastrophic tsunami of December 26,2004, was one of the deadliest natural disasters in history, devastating many coastal communities across the IndianOcean/Bay of Bengal by killing more than 275,000 people in 11 countries [45,63] and causing enormous material damage.This event prompted the need for the implementation of a tsunami warning system in the Indian Ocean, just as the devas-tating Chile tsunami of May 22, 1960, led to the establishment of the modern Pacific Tsunami Warning System [11]. To pre-dict accurately the appearance of a tsunami it is of paramount importance to understand how these exceptionally powerfulwaves, once initiated, evolve from a small-amplitude disturbance of the sea level (albeit one of large wavelengths, measuredin hundreds km) to become such devastating forces of nature as they crash upon the shore. We are especially interested inwhether a balance between dispersion and nonlinearity is likely to occur, thus providing a setting appropriate for the appli-cation of soliton theory to the modeling the tsunamis. This issue is decided by analyzing the geophysical scales involved. Wewill look at the evidence gathered in the case of the December 2004 tsunami and will also investigate the case of the 1960Chile tsunami. The latter is rightfully considered [45,58] as the most likely candidate to illustrate the relevance of solitontheory to tsunamis since it involved the largest ever recorded earthquake and the tsunami waves propagated all acrossthe Pacific Ocean over distances in excess of 17.000 km. However, we will show that the geophysical scales involved in bothtsunamis lead to time- and space-scales that are orders of magnitude smaller than those required for the applicability of

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A. Constantin / Wave Motion 46 (2009) 420–426 421

nonlinearly dispersive water wave models from soliton theory, like the Korteweg–de Vries or Boussinesq equations [34,43].With dispersion deprived of a balancing effect on the nonlinearity, the appropriate equations for tsunami waves are there-fore those modeling the propagation of very long waves over variable depth—a direction of research that is distinct fromsoliton theory but nevertheless of great current interest (see the discussion in [22]).

2. The governing equations

We assume an initial disturbance in the form of a two-dimensional wave and we are interested in understanding thedynamics of the wave as it propagates across the ocean. This assumption is realistic in the two case studies we undertakesince both the 2004 and the 1960 tsunami were caused by a sequence of earthquakes that occured along approximatelystraight fault lines of length in excess of 1000 km. The initial disturbance taking place along an approximate line, the gen-erated tsunami waves were approximately two-dimensional. In a two-dimensional wave the motion is identical in any direc-tion parallel to the crest line. Therefore, to describe such waves it suffices to consider a cross section of the flow that isperpendicular to the crest line. Relevant for our considerations are also the following facts: the depth of the central Bayof Bengal is relatively uniform with average depth h0 ¼ 4 km, while the bathymetry of the Pacific Ocean between Chileand Hawaii, and between Hawaii and Japan, is relatively flat with average depth h0 ¼ 4:3 km, respectively, 6 km. While inthe depression of the Japan Trench, east of Japan, the depth reaches 9 km, and the Hawaiian Islands are surrounded by aregion in which the depth of the ocean is about 5 km, these deviations from the average relatively uniform depth are verylocalized and occur near the shore, having little influence on the wave dynamics at sea. In studying the propagation of thetsunami across the ocean we will therefore make the convenient assumption of a flat bed.

Choose Cartesian coordinates ðX;YÞ with the Y-axis pointing vertically upwards, the X-axis being the direction of wavepropagation, and with the origin located on the mean water level Y ¼ 0. Let ðUðX; Y; TÞ; VðX;Y ; TÞÞ be the velocity field ofthe two-dimensional flow propagating in the X-direction over the flat bed Y ¼ �h0, and let Y ¼ HðX; TÞ be the water’s freesurface with mean water level Y ¼ 0. The equation of mass conservation

UX þ VY ¼ 0 ð1Þ

is a consequence of assuming constant density, a physically reasonable assumption for gravity water waves [46]. Under theassumption of inviscid flow (which is realistic since experimental evidence confirms that the length scales associated withan adjustment of the velocity distribution due to laminar viscosity or turbulent mixing are long compared to typical wave-lengths [4]) the equation of motion is Euler’s equation

UT þ UUX þ VUY ¼ � 1q PX ;

VT þ UVX þ VVY ¼ � 1q PY � g;

(ð2Þ

where P is the pressure, g is the constant acceleration of gravity and q is the constant density of water. We also have theboundary conditions

P ¼ Patm on Y ¼ HðX; TÞ; ð3Þ

where Patm is the (constant) atmospheric pressure at the water’s free surface,

V ¼ HT þ UHX on Y ¼ HðX; TÞ; ð4Þ

and

V ¼ 0 on Y ¼ �h0: ð5Þ

The conditions (4) and (5) express the fact that water particles can not cross the free surface, respectively, the imperme-able rigid bed, while (3) decouples the motion of the water from that of the air above it in the absence of surface tension; forwavelengths larger than a few mm (and in our case we deal with hundreds of km) the effects of surface tension are known tobe negligible [4]. We will consider irrotational flows with zero vorticity

UY � VX ¼ 0; ð6Þ

a hypothesis that allows for uniform currents but neglects the effects of non-uniform currents in the fluid [28,30]. We sup-pose that initially (at time T ¼ 0) a disturbance of the flat surface of still water was created and we analyze the subsequentmotion of the water. The balance between the restoring gravity force and the inertia of the system (1)–(6) governs the evo-lution of the mass of water and our primary objective is to understand the behaviour in time of the free surface Y ¼ HðX; TÞ.

2.1. Non-dimensionalisation

Finding exact solutions to the nonlinear governing equations for water waves is not possible even with the aid of the mostadvanced computers. Of special interest in the context of tsunamis are solutions in the form of localized traveling waves—solitary waves. Considerable progress in understanding solitary waves based on the governing equations for water waves hasbeen made: localized disturbances of a flat water surface in the form of bump-like gravity waves of elevation propagating

422 A. Constantin / Wave Motion 46 (2009) 420–426

without change of form have to be two-dimensional [32], the existence of two-dimensional solitary wave solutions of smalland large amplitude was proved in [3], these waves have to be waves of elevation, their profile is always symmetric aboutthe crest [33], and a qualitative description of the particle motion in such a wave motion is available [17]. However, anin-depth study of solitary wave interactions using the governing equations is not yet within reach. To make progress onehas therefore to make approximations that lead to simplified model equations. Since the linear theory of waves of smallamplitude fails to yield any approximation to solitary waves (see [60]), weakly nonlinear approximations to the governingequations for water waves have to be made. To this end we need to use the length scales, time scales, etc. that appear nat-urally by defining a set of nondimensional variables. The significance of this process lies in the fact that some parametersarise which will enable us to define approximate linear or weakly nonlinear versions of the governing equations and bound-ary conditions—to derive approximations to the governing equations it is useful to write them in non-dimensional form sothat the terms involved can be compared and one can give a meaning to ‘‘small with respect to”. We assume that the two-dimensional waves under investigation have acquired a certain pattern. Without necessarily restricting our attention towave trains (that is, periodic traveling waves [14]), we assume that the wave pattern under investigation represents aweakly irregular perturbation of a wave train in the sense that averages over suitable times/distances resemble a wave train.Since h0 is the average depth of the water, the non-dimensionalisation Y0 of Y should be

Y ¼ h0y; ð7Þ

which is to be understood as replacing the dimensional, physical variable Y by h0y, where y is now a non-dimensional versionof the original Y. The non-dimensionalisation of the horizontal spatial variable is also obvious: if k is some average or typicalwavelength of the wave, we set

X ¼ kx: ð8Þ

The corresponding non-dimensionalisation of time is [43]

T ¼ kffiffiffiffiffiffiffiffigh0

p t: ð9Þ

The governing equation for irrotational water waves equations read, in nondimensionalized form [2] (see also [15,43]),

d2Uxx þUyy ¼ 0 in XðtÞ;Uy ¼ 0; on y ¼ �1;ft þ efxUx þ e

d2 Uy ¼ 0 on y ¼ ef;

Ut þ e2 U2

x þ e2d2 U2

y þ f ¼ 0 on y ¼ ef;

8>>>><>>>>:

ð10Þ

where x # efðx; tÞ is a parametrization of the free surface at time t,

XðtÞ ¼ fðx; yÞ; �1 < y < efðx; tÞg

is the fluid domain delimited above by the free surface and below by the flat bed fy ¼ �1g, and where Uð�; �; tÞ : XðtÞ ! R isthe velocity potential associated to the flow, so that the two-dimensional velocity field is given by ðUx;UyÞ. Finally, e and dare two dimensionless parameters (the amplitude parameter, respectively, the shallowness parameter), defined as

e ¼ ah0; d ¼ h0

k: ð11Þ

Making assumptions on the respective size of e and d, one derives (simpler) asymptotic models from (10) by using theDirichlet–Neumann operator to transform the boundary-value problem into a system of evolutions equations—one equationfor the free surface parametrization f and the other for the trace of the velocity potential U at the free surface (see the rig-orous detailed discussion in [2]).

2.2. Long waves of small amplitude: linear and weakly nonlinear theory

We are interested in small-amplitude long waves, that is, in the limits e! 0 and d! 0. Noticing the way in which e and doccur in Eqs. (10), the regime

e ¼ Oðd2Þ ð12Þ

emerges naturally since one obtains a problem involving only one small parameter, e. It is within this regime [56,57] that atfirst order the evolution of the waves is governed by the linear wave equation

ftt � fxx ¼ 0 on y ¼ 0; ð13Þ

with the general solution

fðx; tÞ ¼ fþðx� tÞ þ f�ðxþ tÞ;

A. Constantin / Wave Motion 46 (2009) 420–426 423

where the sign � refers to a wave of profile f� moving with unchanged shape to the right/left at constant unit speed by virtueof the non-dimensionalisation process, the corresponding dimensional speed being

ffiffiffiffiffiffiffiffigh0

p. Choosing the wave moving to the

right, using the method of multiple scales one can obtain in the region of ðx; tÞ-space where

s ¼ et ¼ Oð1Þ; n ¼ x� t ¼ Oð1Þ; ð14Þ

more precise information about the evolution of the water’s free surface by taking into account weakly nonlinear interac-tions. This is achieved by showing that to the next order of approximation, the evolution of the leading order of the free sur-face is described by the Korteweg–de Vries (KdV) equation instead of the linear Eq. (13). Allowing for waves traveling in bothdirections one obtains the Boussinesq system [2]. Since the formal derivation of KdV is standard (see for example thedetailed discussions in [1,15,21,34]), we refrain from reproducing it here. For our purposes it is more important to specifyin what sense solutions to KdV or Boussinesq approximate the free surface from (10). The sharpest rigorous result in thisdirection is recent [2] and ensures that, given e0 > 0, there exists T0 > 0 such that if (12) holds, then if one defines

feðx; tÞ ¼ fþðs; x� tÞ;

where s ¼ te and fþðs; nÞ solves the KdV equation

fþs þ32

fþnnn þ16

fþfþn ¼ 0;

then for some C > 0 independent of e 2 ð0; e0Þ one has

jfðx; tÞ � feðx; tÞj 6 Ce2t; t 2 0;T0

e

� �

for the solution to (10) with the same initial data (and a similar estimate holds for the velocity fields). A similar approxima-tion of order Oðe2tÞ to the water-wave Eqs. (10) is given by the solution of the Boussinseq system in the case when the wavepropagation in not unidirectional [2]. Moreover, in the case of a non-flat bed with small variations of the order of the size ofthe surface waves, meaning that if b measures the amplitude of the variations of the bottom topography, then b

h0¼ OðeÞ, the

constant-coefficient KdV equation may be replaced by a variable-coefficient KdV equation [38]; and similarly for the Bous-sinesq system, with the same scaling and approximation properties [42]. From the point of view of identifying the physicalregime where a balance of nonlinear and dispersive effects occurs (leading to mathematically interesting model equationslike KdV whose solutions approximate well the free surface) a systematic approach keeping track of the two fundamentaldimensionless numbers e and d is required. The parameters e and d are useful in identifying several important regimesfor the propagation of two-dimensional water waves:

1. Shallow-water, large amplitude regime ðd� 1; e � 1Þ leading at first order to the shallow-water equations [43] and atsecond order to the Green–Naghdi model [37] which takes into account the dispersive effects neglected by the shallowwater equations. Notice that with increasing wavelength d! 0 the stability properties of traveling water waves improveconsiderably cf. [29], and the orbital stability of these waves (meaning that their shape is stable under small perturba-tions) explains why these patterns are physically recognizable.

2. Shallow-water, medium amplitude regime ðd� 1; e � dÞ leading to the Serre equations [59] and to the Camassa–Holm(CH) equation [12] (see [25]). CH captures more nonlinear effects than the classical model equations for water waves (ofKdV-type) but, while the equation has a lot of structural properties (it satisfies the Least Action Principle as a by-productof its geometric interpretation as geodesic flow [23,24], it is integrable [18,26], and its solitary waves are stable solitons[27,36]), the soliton interaction process is more intricate in comparison with that for KdV so that it is mostly in the con-text of breaking waves that this equation gained prominence [9,10,16]. Thus CH is not appropriate for gaining insight intothe propagation of tsunamis.

3. Shallow-water, small amplitude or long-wave regime ðd� 1; e � d2Þ leading to a variety of Boussinesq systems which, asdiscussed above, at first order reduce to the linear wave equation with the motion of the free surface described as the sumof two uncoupled counter-propagating waves modulated by a KdV equation. KdV belongs to a wider class of equations(the BBM equations [7], first used by Peregrine [55] and sometimes also called the regularized long-wave equations)which provide an approximation of the governing equations for water waves of the same accuracy as KdV [25]. Whilethe solitary wave solutions for BBM and KdV have similar expressions and both are orbitally stable [6,48], the KdV solitarywaves are solitons [34] while those of BBM are not [50]. Furthermore, BBM is not integrable [53] while the integrability ofKdV via inverse scattering allows for a clear description in terms of a finite number of solitons (followed by an oscillatorytail that gradually fades out) of the evolution of an arbitrary initial profile that is smooth and sufficiently localized inspace. For this reason the master nonlinear dispersive equation for unidirectional wave propagation is KdV.

4. Deep-water, small steepness regime ðd� 1; ed� 1Þ leading to the full-dispersion Matsuno equations [51].

Justifying at once all the physical regimes defined by the various ranges of the parameters e and d is possible by proving(see [2]) a well-posedness theorem for (10) which gives an estimate of the existence time of the solution which is uniformwith respect to e and d, obtaining thus the exact validity ranges for the aforementioned models. Whether soliton theory

424 A. Constantin / Wave Motion 46 (2009) 420–426

applies in a given context is therefore decided by the geophysical scales involved. The propagation of tsunami waves has tobe understood within the long-wave regime but the role of dispersion has to be clarified.

2.3. KdV is not relevant to the December 2004 or to the May 1960 tsunami

We now examine the geophysical scales involved in the 2004 and 1960 tsunami events to decide whether KdV (or a Bous-sinesq-type system) can be used in any of these two cases as a model for the propagation of the tsunami waves across thesea. While dispersion is not relevant in the generation phase, this effect may modify the wave propagation slightly when themotion takes place over a long time and may become important in shallow water [39,49,54]; the issue is whether a balancebetween dispersion and nonlinearity can occur over the given geophysical scales. Taking into account the performednon-dimensionalisation (8) and (9), the region where we expect a KdV-type balance to occur is given by

1 A fa

X � Tffiffiffiffiffiffiffiffigh0

pk

¼ Oð1Þ; eTffiffiffiffiffiffiffiffigh0

pk

¼ Oð1Þ;

in the original physical variables. The second relation yields Tffiffiffiffiffiffigh0

pk ¼ O 1

e

� �, which implemented in the first leads us to X

k ¼ O 1e

� �so that the length scale on which one sees KdV-type dynamics is

X ¼ Oðe�1 kÞ: ð15Þ

To put the conclusion (15) in the right perspective, notice that it corrects1 the classical scale for the evolution of KdV, be-lieved to be

X ¼ Oðe�1 h0Þ ð16Þ

cf. the recent survey [58] (see the discussion on pages 12–13), which also presents the classical theory put forward in thepapers [40,41]. This correction has far-reaching consequences for tsunamis since the change of scale is considerably largeras h0

k � 1 for tsunami waves propagating at sea. Indeed, tsunamis have wavelengths of several hundreds km while the deep-est part of the world’s seas is the Mariana Trench (located in the western North Pacific Ocean, to the east and south of theMariana Islands, near Guam), with maximum depth of about 10;911 m.

In the shallow water small amplitude regime (12) required for the appearance of KdV and Boussinesq as valid approxi-mations to the governing equations for water waves (see the discussion in [2]), the length scale (15) can be expressed ase�1k ¼ a�3=2h5=2

0 . This means that in the regime (12), if we compute the typical wave amplitude a on the basis of some initialprofile, the KdV balance will occur roughly at distances of order

d ¼ h5=20 a�3=2 ð17Þ

from the initial disturbance, where h0 is the typical depth which remains roughly constant throughout this propagationdistance.

On 26 December 2004 tsunami waves were generated by an earthquake occurring off the west coast of northern Sumatra.The tsunami severely damaged coastal communities in Indonesia, Thailand, Sri Lanka, and India, being one of the deadliestnatural disasters in history [63,61]. The tsunami waves moving towards Sumatra and Thailand hit the Aceh Province insouthern Thailand after crossing the Andaman Basin in less than an hour, with maximum wave height of the tsunami, asit came ashore at this location, about 10 m [22]. Other tsunami waves moved across the Bay of Bengal: 3 h after initiation,the first tsunami wave hit the southern tip of Sri Lanka as a leading elevation wave with a wave height less than 1 m,followed 10 min later by a second large elevation wave with wave height of 10 m [47]. For the December 2004 tsunami accu-rate measurements about 2 h after the main earthquake took place were provided by a radar altimeter on board a satellitealong a track traversing the Indian Ocean/Bay of Bengal [31]; on this basis we choose a ¼ 1 m, k ¼ 180 km as in [31]. Sincethe central Bay of Bengal has a relatively flat bed with average depth h0 ¼ 4 km, we compute e ¼ a

h0� 25 10�5 and

d ¼ h0k � 22 10�3. Since e � d2, the tsunami fits well into the regime (12), which opens up the possibility of the appearance

of KdV as an approximation to the governing equations. However, (17) shows that a propagation distance of the order of tensof thousand km is needed for the KdV balance, so that the distances of less than 1600 km from the epicentre of the earth-quake to the coasts of India and Sri Lanka were much too short for KdV dynamics to develop on this occasion. As for the tsu-nami waves that propagated eastward towards the western coast of the Malay Peninsula, where the principal Thai resortswere hit, they propagated over a distance of about 700 km through the Andaman Basin, which is a shallower sea of quitevariable depth. For purposes of our illustrative calculations let us follow [31] and take the approximate depth to be 1 km.The length scales required for the appearance of KdV, given by (17), are again of the order of tens of thousands km so thatKdV is not relevant to the modeling. Our conclusions are in agreement with some other recent findings [19–22,58]. At thispoint it is worth pointing out that due to the small propagation distances involved, the conclusion about the inapplicabilityof KdV could be reached in [58] using the wrong length scale (16). It is also of interest that since the validity of (12) wasestablished, the rigorous length scale (15) is equivalent to (17). On the other hand, the conclusions of [19–22] were reachedby noticing that the parameter d can be scaled out in favor of e in (10), as follows:

ct confirmed by H. Segur (private communication).

A. Constantin / Wave Motion 46 (2009) 420–426 425

x # xdffiffiffiep ; y! y; t # t

dffiffiffiep ; f # f; U # U

dffiffiffiep :

The advantage of this scaling lies in that it produces the system (10) with d2 replaced by e, for arbitrary d. This opens upthe possibility to prove that, provided the required length- and time-scales are available, for arbitrary wavelengths d, and notjust for those bound to e via (12), KdV will arise as a valid approximation for the evolution of the free surface waves. How-ever, these observations are not yet backed up by rigorous proofs, as is the case within the regime (12) cf. [2]. In this senseour present considerations improve upon those made in [19–22].

The tsunami of May 22, 1960 was caused by the last (and, at magnitude 9.5 on the moment-magnitude scale the largestearthquake ever recorded [35,5]) of several earthquakes occurring along 1000 km of fault parallel to the Chilean coastline.The epicentre was within 200 km off the coast, the submarine lift of 1 m and subsidence of 1:6 m ensued over a coastalstretch of 300 km [11]. A succession of large waves thus created propagated east/southeast, hitting the Chilean coast withwaves of several m height (and at 4 out of 27 locations where these waves were recorded by tide gauge stations [8] thewaves were even in excess of 10 m) and obliterating several coastal towns with a total loss of lifes of more than 5000[11]. Another series of waves propagated in the northwest direction across the Pacific Ocean, the tsunami reaching Hawaiiwith wave heights of 10 m after 14.8 h, at a distance of about 10;000 km from the epicentre of the earthquake [13]. Sevenhours later and about 7000 km further away, tsunami waves of 6 m height hit the Japanese islands of Honsu and Hokkaido[13]. Since extremely long distances of propagation are involved in this tsunami, the possibility of applicability of KdVdynamics might seem likely cf. [57,58,45]. To decide whether this is the case we have to take into consideration the geophys-ical scales. The ocean floor of the Pacific Ocean between Chile and Hawaii being relatively flat, with a mean depth of about4300 m, we take h0 ¼ 4:3 km in (17) and we conclude that for the occurence of the KdV balance within the propagation dis-tance d ¼ 10;000 km, an initial wave amplitude

a ¼ h5=30 d�2=3

>45=3 105

1014=3 m ¼ 4 � ð160Þ1=3 � 22 m ð18Þ

is required. However, the wave measurements performed on the waves propagating southeast show very few waves in ex-cess of 10 m height, and one should take into account that the Chilean coast presents gradual bottom slopes extending for150 km from coast to a deep sea trench about 3 km deep along the coast [52], a feature which greatly amplifies the magni-tudes of waves approaching the shore. That an initial wave amplitude in excess of 22 m is out of question is also supportedby the fact that on smaller islands in the Pacific Ocean fronted by steep offshore slopes the recorded waves were less than2 m in height [11]. We deduce that the 1960 tsunami can not be regarded as a manifestation of soliton theory. Concerningthe tsunami waves propagating on this occasion from Hawaii to Japan, taking h0 ¼ 6 km and d ¼ 7000 km, the estimate (17)would require waves with amplitudes

a ¼ h5=30 d�2=3 ¼ 60 � 6

7

� �2=3

m > 48 m

near Hawaii. This second estimate is, however, less reliable than (18) since after encountering the Hawaiian Islands, the two-dimensional character of the tsunami waves was most likely lost due to diffraction, reflection and wave breaking.

Let us conclude by pointing out that to predict the tsunami’s arrival time at Hawaii to within a minute the simple wavespeed formula

ffiffiffiffiffiffiffiffigh0

pfor (13) was used [11,63]. This further substantiates our conclusion that a balance between nonlinearity

and dispersion could not have developed over the geophysical scales of this event, which excludes the applicability of KdVand suggests that at sea linear wave propagation captures the main features while close to the shore where the depth dimin-ishes nonlinear effects lead to height amplification and wave breaking.

3. Discussion

The fundamental question of whether tsunami waves enter the regime of validity of KdV as an approximation to the gov-erning equations for water waves is not settled just because KdV is a shallow water model arising within the regime (12)relating the two fundamental parameters e and d of shallow water theory, even if this argumentation is made in several clas-sical as well as more recent research papers [31,45,56,62]. We showed that the geophysical scales involved in both the 2004and the 1960 tsunami lead to time- and space-scales that are orders of magnitude smaller than those required for KdV the-ory. Therefore in these two instants KdV is not an appropriate model. While one can take the view that simple evolutionequations like KdV and other models derived for unidirectional propagation of water waves above a flat bed are not appro-priate for the modeling of tsunamis in view of the variable bathymetry and perhaps variable-coefficient Boussinesq-likemodels should be implemented, our main contention is that the geophysical scales involved do not permit dispersion to playan important role in the propagation of the tsunamis. As pointed out in Section 2, the rigorous results in [42] ensure thateven in the case of a non-flat bed with small variations, scaling and approximation properties similar to those presentedabove for a flat bad hold and we can conclude similarly.

Acknowledgement

The author is grateful to both referees for constructive criticism.

426 A. Constantin / Wave Motion 46 (2009) 420–426

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