On the solitary wave paradigm for tsunamis

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2008jc004932, 2008 On the solitary wave paradigm for tsunamis Per A. Madsen, 1 David R. Fuhrman, 1 and Hemming A. Schäffer 1 Received 28 May 2008; revised 25 August 2008; accepted 13 October 2008; published 13 December [1] Since the 1970s, solitary waves have commonly been used to model tsunamis especially in experimental and mathematical studies. Unfortunately, the link to geophysical scales is not well established, and in this work, we question the geophysical relevance of this paradigm. In part 1, we simulate the evolution of initial rectangular-shaped humps of water propagating large distances over a constant depth. The objective is to clarify under which circumstances the front of the wave can develop into an undular bore with a leading soliton. In this connection, we discuss and test various measures for the threshold distance necessary for nonlinear and dispersive effects to manifest in a transient wave train. In part 2, we simulate the shoaling of long smooth transient and periodic waves on a mild slope and conclude that these waves are effectively non-dispersive. In this connection, we discuss the relevance of finite amplitude solitary wave theory in laboratory studies of tsunamis. We conclude that order-of-magnitude errors in effective temporal and spatial duration occur when this theory is used as an approximation for long waves on a sloping bottom. In part 3, we investigate the phenomenon of disintegration of long waves into shorter waves, which has been observed, for example, in connection with the Indian Ocean tsunami in This happens if the front of the tsunami becomes sufficiently steep, and as a result, the front turns into an undular bore. We discuss the importance of these very short waves in connection with breaking and runup and conclude that they do not justify a solitary wave model for the bulk tsunami. Citation: Madsen, P. A., D. R. Fuhrman, and H. A. Schäffer (2008), On the solitary wave paradigm for tsunamis, J. Geophys. Res., 113,, doi: /2008jc Introduction [2] Tsunamis caused by earthquakes typically evolve from the deep ocean as extremely long waves with small steepness. By nature the tsunami consists of a number of transient and non-periodic waves, and during the propagation from the ocean to the nearshore area, these waves gradually modify with respect to amplitudes, wave lengths and wave periods. A major amplification of amplitude and flow velocity occurs during the last stages of shoaling and runup. [3] In this work, we focus on the popular methods used for experimental and analytical studies of tsunami runup and the associated processes. Since the early 1970s, it has been frequently assumed that solitary (or cnoidal) waves can be used to model some of the important features of tsunamis approaching the beach and shoreline, and that these theories, originating from the KdV equation, can define the proper input waves for physical or mathematical models of tsunamis. Examples from the literature are numerous see, e.g., Goring [1978], Synolakis [1986, 1987], Synolakis and Deb [1988], Synolakis et al. [1988], 1 Department of Mechanical Engineering, Technical University of Denmark, Kgs Lyngby, Denmark. Copyright 2008 by the American Geophysical Union /08/2008JC Tadepalli and Synolakis [1994], Yeh et al. [1994], Briggs et al. [1995], Liu et al. [1995], Li [2000], Li and Raichlen [2001, 2002, 2003], Tonkin et al. [2003], Jensen et al. [2003], Kobayashi and Lawrence [2004], Craig [2006], Synolakis and Bernard [2006] and Lakshmanan [2007]. Another example of the popularity of this concept is the recent NOAA Technical Memorandum [Synolakis et al., 2007] discussing necessary analytical and experimental benchmarking for numerical tsunami models. In this memorandum 16 out of 45 references cover solitary waves, and although we do acknowledge that these waves can be helpful for the verification of a numerical model, the memorandum leaves the reader with the impression that the solitary wave (or N-waves composed by solitary waves) should be the preferred model of a tsunami. [4] A potential problem with this solitary wave paradigm is however that the link to geophysical tsunamis has never really been established and that its justification seems to be rather weak. Typical justifications from the literature are, e.g., Goring [1978]: Solitary waves were chosen as a model of tsunamis because (1) it can be shown theoretically that waves which have net positive volume, eventually, if the propagation distance is sufficient, will break up into a series of solitary waves, (2) for analysis, solitary waves have the advantage that, although nonlinear, they can be described with just two parameters, (3) solitary waves propagate with constant form in constant depth ; Synolakis 1of22

2 [1987]: Solitary waves are believed to model some important aspects of the coastal effects of tsunamis well ; Yeh et al. [1994]: Because of its stable form and the fact that a leading wave of tsunamis often emerges as a solitary wave after a long period of propagation, it has become customary to use a solitary wave as a model of tsunami formation offshore ; Li and Raichlen [2001]: Solitary waves or combinations of negative and positive solitary-like waves are often used to simulate the runup and shoreward inundation of these catastrophic waves. Some of these arguments are weak, while others take for granted that geophysical scales provide sufficient propagation time and space needed for solitary wave formation - a point we shall question in this work. [5] However how did the solitary wave paradigm for tsunamis arise in the first place? To answer this question, we start with a brief discussion of milestones in the promotion of KdV theory. In the 1960s and 1970s the KdV equation (first derived by Korteweg and de Vries [1895]), was exposed to extensive research by, e.g., Zabusky and Kruskal [1965], Gardner et al. [1967], Zabusky [1968], Zabusky and Galvin [1971] and Segur [1973]. In this connection, the theory of solitons was invented by Zabusky and Kruskal [1965], who coined this name to represent a special class of waves that propagate with permanent form, interact with other strongly nonlinear waves without loosing their identity, and are related to solitary waves and localized so that they decay at infinity. According to Drazin and Johnson [1989], the word soliton covers the case when several of these waves are present simultaneously, and a soliton becomes a solitary wave when it separates completely from the other solitons. [6] An important milestone with respect to tsunamis was the PhD dissertation by Hammack [1972], and the follow-up papers by Hammack [1973], Segur [1973] and Hammack and Segur [1974, 1978a, 1978b]. They addressed the generation and propagation of tsunamis in an ocean of uniform depth, and studied experimentally the impact of an impulsively raised or lowered portion of the seafloor. They found that a positive initial surface disturbance of arbitrary shape, will indeed eventually lead to the formation of leading solitons or solitary waves. Since then this conclusion has been frequently used as one of the justifications for the solitary wave paradigm, as is apparent in the quotations above. [7] A few years later, Goring [1978] presented experimental and theoretical investigations with the intent of studying the transformation of tsunamis or tsunami-like waves as they propagate from an ocean of constant depth on to the continental shelf at a much smaller water depth. Both abrupt and gradual changes in depth were investigated and the study concentrated on phenomena such as reflection, transmission and propagation on the shelf. The input waves used by Goring were primarily solitary waves given by hðx; tþ ¼ H sech 2 ðk s ðx ctþþ; K s ¼ 1 rffiffiffiffiffiffi 3H ; ð1þ h 4h i.e., exact solutions to the KdV equation for constant depth. Imbedded in this solution is the assumption of a balance between nonlinearity and dispersion, which leads to a horizontal scale defined by K s and being proportional to nonlinearity H/h. As we shall demonstrate in the following sections, this scale is rarely in good agreement with geophysical tsunamis. [8] Some years later, a third tsunami-related dissertation was made by Synolakis [1986], who investigated the characteristics of breaking and non-breaking solitary waves and their run-up on plane beaches. Synolakis provided an elegant solution to the initial value problem of solitary waves propagating from a constant depth offshore region onto a beach with a constant slope. The analytical solutions to the linear as well as nonlinear shallow water equations were established and explicit expressions for the maximum runup of the solitary wave were provided. This was an important achievement in its own right, and the work was published by Synolakis [1987] and extended by Synolakis and Deb [1988] and Synolakis et al. [1988] to also account for the runup of cnoidal waves on a plane beach. Unfortunately, the relevance to geophysical tsunamis was not established in any of these works. [9] In 1994, Tadepalli and Synolakis proposed the idea to replace the solitary wave input with so-called N-shaped solitary-like waves, which could be designed to be either leading depression N-waves (LDN) or leading elevation N-waves (LEN). This approach was inspired by a number of tsunami observations which revealed that the shoreline often first recedes before advancing up the beach. Evidence of this phenomenon was found, for example, at Flores Island in Indonesia 1992; at the Pacific coast of Nicaragua 1992; at Okushiri, Japan in 1993; East Java, Indonesia in 1994; Mindoro, Philippines in 1994; Manzanillo, Mexico in 1995; Chimboto, Peru in 1996 and now also recently in Thailand The input waves considered by Tadepalli and Synolakis were, for example, the so-called isosceles N-wave given by hðx; 0Þ ¼ 3 p ffiffi 3 2 H sech2 ðkx ð x 1 ÞÞtanhðKx ð x 1 ÞÞ; sffiffiffiffiffiffiffiffiffiffiffiffi K ¼ 3 rffiffi H 3 ; ð2þ 2h h 4 and the generalized N-wave given by hðx; 0Þ ¼ ahðx x 2 Þsech 2 ðk s ðx x 1 ÞÞ; K s ¼ 1 rffiffiffiffiffiffi 3H ; ð3þ h 4h where a < 0 was a scaling parameter introduced to ensure that the wave height of the N-wave was H. They showed that the maximum runup for a leading depression wave (LDN) is higher than for the equivalent leading elevation wave (LEN) as well as the solitary wave. [10] In a recent review Synolakis and Bernard [2006] conclude that the work by Tadepalli and Synolakis [1994] at the time was controversial, for it was not reconcilable with the solitary wave paradigm. The leading depression waves were believed to be hydrodynamically unstable in contrast to the classical KdV-type solitary wave. Synolakis and Bernard also claim that Tadepalli and Synolakis were the first to challenge the paradigm of a solitary wave as an appropriate initial condition for tsunami modeling. However, in our view, the possibility of tsunamis having leading depression waves was conceptually demonstrated much 2of22

3 earlier, for example, in the impulsive bottom down-thrust experiments by Hammack and Segur [1974] and it was also mentioned by Mei [1989, p. 47]. We do agree that Tadepalli and Synolakis [1994, 1996] were among the first to work out the analytical run-up of leading depression waves [see also Pelinovsky and Mazova, 1992], but our main concern about their work is that unfortunately they maintained the idea of a horizontal scale defined in terms of the nonlinearity H/h. This makes the scale of their N-waves similar if not identical to the scale of the solitary wave, and in this respect they continued to support the paradigm that the nonlinear KdV equation (valid on a horizontal bottom) governs the spatial and temporal scales of tsunamis. [11] With the accumulated evidence from field observations it is nowadays generally acknowledged that incident tsunamis approaching beaches behave either as leading depression waves or leading elevation waves depending on the source of sea-bottom rupture. Nevertheless, the literature from the last years is full of examples using solitary waves for experimental and analytical studies of tsunamis [see, e.g., Yeh et al., 1994; Briggs et al., 1995; Liu et al., 1995; Li, 2000; Li and Raichlen, 2001, 2002, 2003; Tonkin et al., 2003; Jensen et al., 2003; Kobayashi and Lawrence, 2004; Craig, 2006; Lakshmanan, 2007; Synolakis et al., 2007]. So in our view the solitary wave paradigm still prevails in many circles, and in this work we intend to discuss its lack of justification and demonstrate how it fails to represent important geophysical time- and length-scales typical of tsunamis. [12] The paper is organized as follows: section 2 provides a brief summary of solitary and cnoidal theory and we propose simple estimates of the effective time- and lengthscales of these waves. Our critical review of the solitary wave paradigm is divided into three parts: Part 1 (section 3) concentrates on the development of initial humps of water (of rectangular shape) propagating long distances over a constant water depth. A range of different nonlinearities is considered, and various threshold estimates from the literature are tested with a view to investigate the circumstances under which solitons can develop. Part 2 (section 4) concentrates on the shoaling process for smooth long waves coming from the deep ocean. We investigate the role of dispersion in this process and monitor the evolution of skewness and asymmetry. We also discuss the physical relevance of the frequent use of finite-amplitude solitary wave theory for the generation of input waves to laboratory studies of tsunamis. Part 3 (section 5) concentrates on soliton fission close to the shore. We investigate and simulate the process of disintegration which happens when the front of the tsunami becomes sufficiently steep. Finally, in section 6, we summarize our conclusions and discuss the prevailing solitary wave paradigm with reference to the recent literature. A brief discussion of typical geophysical scales including fault dimensions, average depth and width of oceans and typical periods and wave heights measured in connection with the Indian Ocean tsunami 2004 is given in the Appendix. 2. Cnoidal and Solitary Wave Theories [13] In this section we make a brief summary of low order solutions for solitary waves and cnoidal waves travelling in a single direction on a constant depth. Both solutions satisfy the constant depth KdV equation under the assumption that nonlinearity and dispersion balance each other to produce at lowest order a dispersive nonlinear wave of constant form. For later use, we suggest simple estimates of the effective time-and length-scales of these waves, which will be compared to typical geophysical scales in the following sections Solitary Wave Solution [14] The non-periodic solitary wave is characterized by having a single crest whose amplitude diminishes to zero as c! ±1. Boussinesq [1872] and Rayleigh [1876] were the first to derive this solution, with the objective of explaining the observations of Russell [1844]: where hðx; tþ ¼ H sech 2 ðk s cþ; c x c s t; ð4þ K s ¼ 1 rffiffiffiffiffiffiffiffi 3 H p ; c s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ghþ ð HÞ: ð5þ h 4 h Since this solution has only a single crest in c, i.e., in x- and t-space, in principle it corresponds to a wave period T or a wave length L of infinity. In practice, however, it makes more sense to talk about the effective duration in space and time, which we choose to define simply by L s 2p K s ; and T s 2p W s ; where W s K s c s : ð6þ This estimate is a straight forward generalization of the conventional expressions for periodic waves, and it is similar to (but simpler than) the one suggested by Kobayashi and Karjadi [1994]. We note that the surface profile at c =±L s /2 drops to less than 0.7% of the crest value. Alternatively, we may express the significance of equation (6) by the volume integral V s Z Ls=2 L s=2 hðx; t 0 Þdx ¼ 2H tanh p; ð7þ K s in which the deviation of tanh p from unity defines the difference from integrating the surface profile to ± infinity. It is clear that outside the range of L s /2 c L s /2 very little happens Cnoidal Wave Solution [15] Korteweg and de Vries [1895] extended the work by Boussinesq and Rayleigh by deriving analytical solutions for nonlinear periodic waves on constant depth and they coined their solution cnoidal wave theory: where h ¼ H Bþ cn 2 ½K c c; mš ; c x cc t; ð8þ K c 2K L ¼ 1 rffiffiffiffiffiffiffiffiffiffi 3 H ; h 4 mh p c c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ghþ ð AHÞ; L ¼ c c T; ð9þ 3of22

4 and A 1 m 2 m 3 E ; B 1 K m 1 m E : ð10þ K Here cn [z, m] is the cosine-elliptic function which is defined in terms of the incomplete elliptic integral of the first kind; K and E are the complete elliptic integrals of the first and second kind, respectively; while m is the fundamental elliptic parameter which satisfies the implicit equation U r HL2 h 3 ¼ 16 3 mk2 : ð11þ Solutions for a given wave period T, wave height H and water depth h are easily found by iterating inp equations (9) (11) starting from the approximation L T ffiffiffiffiffi gh. [16] Note that increasing values of the Ursell number U r quickly leads to m! 1.0. In this case, the cnoidal solution converges toward the solitary solution as K c! 1 rffiffiffiffiffiffiffiffi 3 H ; h 4 h p c c! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ghþ ð HÞ; h! H sech 2 ðk c cþ: ð12þ [17] In contrast to the solitary wave, the cnoidal wave is periodic and has a well defined wave length and wave period. Nevertheless, for larger Ursell numbers it also makes sense to introduce the concept of an effective duration in space and time, because the profile becomes very peaky and eventually converges toward the solitary profile. Analogous to equation (6), it is natural to introduce the definitions L c 2p K c ; and T c 2p W c ; where W c K c c c ; ð13þ with K c and c c defined by equation (9). By the use of equation (9), we notice that L c L ¼ T c T ¼ p K ; ð14þ and obviously this only makes sense if K p. This limit corresponds to m and U r Part 1: Evolution of Rectangular Displacements Over Constant Depth [18] In this section, we consider transient waves caused by initial rectangular displacements of the free surface. We study the development of these waves as they propagate for long distances over a constant depth, which may represent the deep ocean or the flat part of the continental shelf. This part of our work has been inspired by the experiments made by Hammack and Segur [1974], who concluded that if the initial surface displacement integrates to a positive value, then the wave train will eventually disintegrate into N leading solitons, organized with the tallest in front and all travelling faster than the linear shallow water celerity c 0 = 4of22 p ffiffiffiffiffiffiffi g. The solitons will be followed by an oscillatory tail travelling slower than c 0. These conclusions of Hammack and Segur [1974] are frequently quoted in the literature in discussions of the solitary wave paradigm, both in terms of its justification [see, e.g., Goring, 1978; Synolakis, 1987], and in terms of the general persistence of the paradigm (see the recent reviews by Synolakis and Bernard [2006] and Segur [2007]). [19] The question is however when can this phenomenon be expected to happen, and will it actually happen on a geophysical scale? It is obvious that the magnitudes of nonlinearity and dispersion play an important role in this process, and we therefore start with a discussion of the threshold distance (or evolution time) necessary for the effects of nonlinearity and dispersion to manifest for a given initial state. A number of estimates from the literature are derived and discussed, and in addition we propose some alternative measures. Next, we simulate several test cases with positive initial displacements and with different combinations of nonlinearity and dispersion, with a view to determine when solitons can be expected and how the various theoretical thresholds perform. [20] The vehicle for this investigation is a numerical model, which solves the high-order Boussinesq formulation by Madsen et al. [2006]. The equations incorporate fifth-derivative operators and are expanded from middepth. Linear and nonlinear properties are accurately represented up to wave number times water depth kh 25, while the interior velocity field is accurate up to kh 12 even for highly nonlinear waves. The numerical solution procedure is based on finite difference discretizations on an equidistant grid, and an explicit four-stage fourth-order Runge-Kutta scheme is used for the time integration. A detailed description of the scheme can be found in Madsen et al. [2002] for one horizontal dimension, and in Fuhrman and Bingham [2004] for two horizontal dimensions. All simulations in the following are restricted to 1D propagation Thresholds for Manifestation of Nonlinearity and Dispersion [21] Solitary and cnoidal waves are dispersive nonlinear waves of constant form and they satisfy the KdV equation (constant depth) under the assumption that nonlinearity (e H/ ) and dispersion (m 2 h 2 0 /L 2 ) balance each other, with H being the wave height, L the wave length and the water depth. [22] Craig [2006] recently concluded that the 2004 Sumatra tsunami fits quite well into the regime of dispersive nonlinear waves because of the fact that the nonlinearity e and the dispersion m were of the same order of magnitude (with H 1m, L 180 km, 4 km). The problem with his conclusion is that such a balance is a necessary but not a sufficient requirement, and in reality it takes a certain threshold time (et) and travelling distance (ex) before the influence of nonlinearity and dispersion can manifest in a transient disturbance. [23] Such thresholds have been estimated by, e.g., Hammack and Segur [1978b], Mei et al. [2005], Segur [2007], Constantin [2007] and Constantin and Johnson

5 [2006, 2008] with different conclusions, and in the following we summarize and discuss these results. To this end, we first express the KdV equation in terms of a non-dimensional moving coordinate system defined by which yields c 2 x c 0t ; t 2 c 0 L L t; c p 0 ffiffiffiffiffiffiffi g; þ e 2 2 þ m ¼ 0; 2 ð16þ where z is the surface elevation normalized by the wave amplitude Case of m 2 < e < 1, i.e., Nonlinearity Exceeds Dispersion [24] At first, we assume that the nonlinearity is small but dominates over dispersion, i.e., m 2 = ge, where g <1.We then follow the method by Mei et al. [2005], their chapter 12.10, and look for a perturbation solution z = z 0 + ez which satisfies equation (16) for small values of e. This leads to a hierarchy of equations in powers of e, and we obtain the solutions 3 z 0 ¼ Fðc 2 Þ; and z 1 ¼ t 2 2 F df þ g dc 2 6 d ; ð17þ where F = z(c 2, 0), i.e., the initial surface displacement. Note that for g 1, we can simply drop the third-derivative term in equation (17). As long as z 0 ez 1, we have a linear nondispersive solution. However, this solution will eventually break down, since z 1 grows linearly with time t 2. The break down occurs when ez 1! z 0, i.e., when t 2! e 1, and according to equation (15) this leads to the threshold time and distance et em rffiffiffiffi g ðemþ 1 ; ex em ¼ c 0 e t em ðemþ 1 : ð18þ When the travelling time exceeds this threshold, the linear nondispersive equations should be replaced by the nonlinear nondispersive equations. This is the result obtained (in a slightly different way) by Mei et al. [2005], in their equation ( ) Case of e < m 2 < 1, i.e., Dispersion Exceeds Nonlinearity [25] Next, we consider the case where dispersion is small but dominates over nonlinearity from the beginning. In this case, it is necessary to make the perturbation expansion in powers of m 2 instead of e. Now the nonlinear term drops out of equation (17) and the criterion for break down of the linear nondispersive solution is modified to m 2 z 1! z 0, i.e., t 2! m 2. According to equation (15), this leads to the threshold time and distance rffiffiffiffi ex m g ¼ et m m 3 : ð19þ This is the result obtained by Mei et al. [2005], in their equation ( ) Case of e = O(m 2 ), i.e., Nonlinearity and Dispersion in Balance [26] The interesting case in relation to the solitary wave paradigm is when nonlinearity and dispersion are of the same order of magnitude, i.e., g = O(1). In this case, we can again make the perturbation in terms of e and this leads directly to the previous result equation (18). Now we can however utilize that e m 2 and replace m by e 1/2 in equation (18), which leads to rffiffiffiffi ex e g ¼ et e e 3=2 : ð20þ This is the result recently obtained by Constantin [2007] and Constantin and Johnson [2006, 2008] for estimating the relevance of nonlinear dispersive wave theory, but in fact this scaling was mentioned already by Meyer [1967]. We emphasize that when e m 2, the threshold estimates given by equations (18), (19) and (20) all become equivalent. However, for most cases e and m 2 will be different from the beginning, and consequently the three threshold estimates will generally provide very different results Preliminary Concern About Existing Thresholds [27] In order to use the thresholds just described, it is necessary to determine representative values of e and m for a given geophysical tsunami. In this connection it is common practice to estimate e by the initial height of the disturbance and m by the initial width of the disturbance, see, e.g., Hammack and Segur [1978b], Mei et al. [2005], Craig [2006] and Segur [2007]. Hence typical large-scale values in the ocean would be, for example, H 1m,L 200 km and h 4 km leading to e and m There is however a fundamental problem in estimating the relevant dispersion and it very much depends on the actual wave profile. If the profile is smooth, dispersion can indeed be estimated by the overall wave length of the phenomenon (see Part 3), but if the profile is sharp as in the rectangular waves considered in this section, the steep front will immediately provoke a series of much shorter waves with corresponding higher values of m. These values are not at all related to the overall value of L. For this reason we expect a large degree of uncertainty in the thresholds involving m, i.e., equations (18) and (19) Linear Dispersive Theory for the Leading Waves [28] So far we have assumed that the linear nondispersive equation is the relevant leading order approximation. This is however not necessarily true for rectangular disturbances, where dispersion plays an important role from the very beginning even for the leading waves. In this case the leading order approximation is the linear KdV equation, þ m ¼ OðÞ; e 2 ð21þ and the question to be answered is when equation (21) needs to be replaced by the full KdV equation (16). 5of22

6 Table 1. Threshold Values for Nonlinearity and Dispersion Covering Sections Section m e (20): ex e / ,000 1,540 (31): ex AI / , (34): ex*/ ,000 (18): ex em / , ,000 (19): ex m / 14, (28): ex d / 232, (29): ex HS / , [29] The exact solution to equation (21), which represents a wave of transition because of an initial infinitely long step-disturbance hðx; 0Þ ¼ 0; x > 0 a; x < 0 can be expressed in terms of the integral of the Airy function [see, e.g., Meyer, 1967; Whitham, 1974]: where hc; ð tþ ¼ a 2 Z 1 Zðc;tÞ AiðÞds; s c x c rffiffiffiffi 0t g ; t t ; Zðc; tþ c 2 1=3 : t ð22þ ð23þ [30] For a disturbance initially covering the finite width region from 2b < x < 0, we can generalize equation (22) to where h d ðc; tþ ¼ a 2 Z Zðc;tÞþdt ð Þ Zðc;tÞ dt ð Þ 2b 2 1=3 : t AiðÞds; s ð24þ ð25þ Note that equations (22) and (24) are similarity solutions expressed in terms of Z, which combines the spatial and temporal coordinates. For a fixed time, Z is proportional to c, which measures the distance from the wave front in the moving coordinate system. For a fixed value of the moving coordinate, Z is proportional to t 1/3. For a finite but wide disturbance there is hardly any difference between equations (22) and (24), but since the effective width is defined by d, which is also proportional to t 1/3, this width will actually shrink over time. Hence even for relatively wide disturbances, differences between equations (22) and (24) will eventually show up. [31] When d becomes sufficiently small we may approximate equation (24) by its Taylor expansion about Z, which leads to h d ðc; tþ a 2 daiðzþþd2 ðzþþ d3 2 Ai0 6 Ai00 ðzþþo d 4 : ð26þ This is the asymptotic solution given by Hammack and Segur [1978b], see also Mei et al. [2005], their equation ( ). Retaining just the first term in equation (26) yields h d ðc; tþ ab 2 1=3 Ai c 2! 1=3 ; ð27þ t t as given by Mei et al. [2005], chapter Note that in equations (26) and (27), the amplitude of the leading wave is proportional to the total volume of the initial displacement (V = ab), a somewhat surprising conclusion previously made by Hammack and Segur [1978b]. This is however an asymptotic result and not generally applicable as seen from equation (24). [32] For equations (26) and (27) to be valid, we should formally require that d 1, but in comparison with the target equation (24) it turns out that d 0.5 is sufficient to obtain acceptable accuracy at least for the leading wave. In combination with equation (25), this leads to the requirement rffiffiffiffi g et d 16 2b 3 ; ð28þ which scales like et m from equation (19) but with an additional factor 16. As we shall demonstrate in the following subsections, et d turns out to be so large, that the asymptotic solutions equations (26) and (27) rarely have any relevance for geophysical tsunamis (see Table 1). Nevertheless, Hammack and Segur [1978b] used equation (27) as the leading order solution in a perturbation expansion to the nonlinear KdV equation, and they found that the linear KdV equation should be replaced by the full KdV equation when exceeding the threshold ex HS rffiffiffiffi g ¼ et HS m 3; e ð29þ see also Mei et al. [2005], their equation ( ). This threshold has two drawbacks: First, it involves m, which is difficult to estimate as discussed earlier. Second, it is based on an asymptotic linear solution which may never be reached. Consequently, you would typically find that et d is far greater than et HS, which leaves equation (29) as invalid. [33] As an alternative we propose a new threshold, which considers the time-varying dispersion for the leading waves in a rectangular disturbance. First, we assume that the disturbance is wide enough for equation (22) to be valid at least for the leading wave and later we determine a condition for the validity of this assumption. The front of this solution will move with approximately c 0 and the first and second crests of the leading waves will maintain crest elevations of 1.27 and 1.16 times a/2, and they will be 6of22

7 located at Z 1 = and Z 2 = Hence according to the definition of Z(c, t), the distance between these crests in c -space will gradually increase over time. This corresponds to a temporal decrease of local dispersion, and we estimate m for the leading waves by m ¼ ðc 1 c 2 Þ 1 ¼ ðz 1 Z 2 Þ 1 2 1=3 : t ð30þ At some instant the decreasing m 2 will match the nonlinearity e at hand, and we make the assumption that this instant et AI heralds the relevance of the full KdV equation. The threshold for this event becomes et AI ¼ 2ðZ 1 Z 2 Þ 3 e 3=2 0:0621e 3=2 ; rffiffiffiffi ex AI g ¼ et AI ¼ et AI ; ð31þ which scales like et e from equation (20), but with an additional factor [34] Finally, we check exactly how wide the initial disturbance needs to be for equation (31) to be valid. Basically this requires that the distance between the first two crests in equations (22) and (24) are approximately the same, and this turns out to be the case if d(t) ^ 10. Since d(t) will decrease over time, this criterion needs to be satisfied at least up to the point of et AI, and this leads to the criterion 2b h > 10ðZ 1 Z 2 Þ 1 e 1=2 3:14e 1=2 : ð32þ 0 We may express this criterion in terms of an Ursell number U r = e/m 0 2 > 10, where m 0 = /(2b) Threshold for the Separation of Solitons [35] Once the KdV dynamics start to become active, it still takes significant time and distance before the first soliton actually develops and separates from the rest of the wave train to become a solitary wave. In the case of only a single soliton, the separation time can be estimated by t*= L s /(c s c 0 ), where L s is the effective length of the solitary wave and c s its celerity including amplitude dispersion. The distance covered by the front through this time is x* ¼ c s c s c 0 Ls : ð33þ In the determination of c s and L s we apply equations (5) (6), and estimate the relevant nonlinearity as be (with b 1.5), because the amplitude of the leading soliton will eventually be twice the level of the original right-going disturbance. For small nonlinearities equation (33) can now be approximated by rffiffiffiffi x* g ¼ t* p 8p ffiffi ðbeþ 3=2 7:9e 3=2 : ð34þ 3 Notice that this threshold also scales like et e from equation (20), but it exceeds it by a factor 8. If many solitons develop, it will take longer time for all of them to separate, because the height difference between each of them can be rather small Simulation of Positive Rectangular Displacements [36] To elucidate the effect of dispersion in a non-ambiguous manner, we simulate the propagation of rectangular waves over large horizontal distances with constant depth. The initial surface displacement (with zero initial velocity) is located next to an upstream wall, which is a plane of symmetry for right and left going wave motions. We specify an initial displacement which is a rectangular wave of height a and length b, hence the corresponding fault width is actually L = 2b occupying the region b < x < b. Numerical results are generally presented with and without including nonlinear terms in the Boussinesq model, and they are shown from a moving framep offfiffiffiffiffiffiffiffiffi reference defined by (x c 0 t)/ at various fixed times t g=. In this frame the linear nondispersive solution will appear at all time as a rectangle with the height of a/2 and extending from b to b (except for t < b/c 0 where the height is a for 0 < x < b c 0 t). The nonlinearity will generally be estimated as e = a/(2 ), while dispersion will be estimated by m 0 = /(2b). Table 1 summarizes the four test cases including the resulting threshold values A Narrow Displacement With Nonlinearity e 0.05 [37] We start by considering the case of a/ = 0.1, which is unrealistic in terms of geophysical tsunamis in the ocean, but has the advantage that solitons will develop rather quickly so the basic phenomena can be easily observed. We use a narrow displacement b/ = 12.2, which is similar to Figure 3 in Hammack and Segur [1974]. Initial measures of dispersion and nonlinearity in the right-going wave read m and e = 0.05, i.e., nonlinearity is apparently dominating from the beginning. This naturally leads to the questions: How long will the linear nondispersive model be applicable and when should it be replaced by the nonlinear nondispersive model? The relevant threshold for pthis ffiffiffiffiffiffiffiffiffi question appears to be equation (18), which leads toet em g= 485. pinffiffiffiffiffiffiffiffiffi contrast, the thresholds pffiffiffiffiffiffiffiffiffiequations (20) and (19) lead to et e g= 90 and et m g= 14,500. [38] On the other hand, we have previously discussed that it might be problematic to estimate the relevant dispersion by m 0 = /(2b), as the rectangular shape of the disturbance will create much shorter waves at the start. The threshold equation (31), which is based on a time-varying m for the leading waves in an infinitely long step-disturbance is applicable since the requirement equation pffiffiffiffiffiffiffiffiffi (32) yields 2b/ > 14. This threshold yields et AI g= 6, which implies that the nonlinear dispersive model should be used from the very beginning. [39] Figures pffiffiffiffiffiffiffiffiffi 1a 1b show the numerical results obtained at times t g= = 35 and 90, and in both plots we notice obvious differences between the linear dispersive solution and the nonlinear dispersive solution. This clearly advocates for the nonlinear dispersive model from the start. The separation time for the firstp soliton ffiffiffiffiffiffiffiffiffi is estimated by equation (34), which yields t* g= 700. This is the time for Figure 1c, which reveals that the first soliton has indeed separated from the tail but not yet from the second soliton. We notice that the initial displacement has disintegrated into two leading peaks with amplitudes H 1 / = of22

8 Figure 1. Computed surface elevations evolving from a positive initial pffiffiffiffiffiffiffiffiffi rectangular displacement pffiffiffiffiffiffiffiffiffi of water p(height: ffiffiffiffiffiffiffiffiffi a, width: 2b). pffiffiffiffiffiffiffiffiffi Scales: a/ = 0.10, b/ = (a) t g= = 35; (b) t g= = 90; (c) t g= = 700; (d) t g= = Full line: nonlinear dispersive model; dashed line: linear dispersive model; cross: theoretical linear finite width solution according to equation (24); dots on d: solitary wave solutions. and H 2 / = 0.05, which are separated by a distance of approximately D/ 15. On the basis of these amplitudes, we can make a more accurate estimate of the time it takes for the first two solitons to separate: In equation (33), let c s and L s be determined by the higher wave and preplace ffiffiffiffiffiffiffiffiffi c 0 by c s determined by the lower wave. This leads to t g= pffiffiffiffiffiffiffiffiffifigure 1d shows the wave train at time t g= At this stage the linear solution shows a dominating leading wave with its crest slightly delayed compared to the moving coordinate system. In contrast, the nonlinear solution reveals that two solitons have separated completely from each other and also from the dispersive oscillatory package which falls behind. The overlay with solitary wave theory shows a nearly perfect match. The volume integrals V s defined by equation (7) represent 57% and 42% (i.e., a combined 99%) of the initial volume of right-going water V = ab. [40] Finally, we emphasize that the theoretical linear finite width solution obtained from equation (24) agrees very well with the linear dispersive simulations at all four instances. In contrast, the asymptotic linear solution equation (27) is not relevant pffiffiffiffiffiffiffiffiffi as equation (28) predicts that it will only be valid for t g= > 232, A Wide Displacement With Nonlinearity e 0.05 [41] Next, we increase the width of the initial displacement to b/ = 100. This changes the initial dispersion to m , while nonlinearity is unchanged pffiffiffiffiffiffiffiffiffi e Now the threshold estimates become et e g= 90 pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi (unchanged), et em g= 4000 and et m g= Note that et em predicts that a wide initial surface displacement requires much longer travelling time than a narrow displacement for nonlinear effects to manifest. This is true for smooth disturbances (see Part 3), but it is not true for rectangular wave shapes. Once again the steep front will generate a series of shorter waves rather independently of the total width of the disturbance, so that the classical estimate of m 0 becomes irrelevant. In contrast the threshold equation (31) is independent of the width pffiffiffiffiffiffiffiffiffi of the initial displacement, and again it predicts that et AI g= 6, i.e., the full KdV equation should be applied from the start. This is also confirmed by thep numerical ffiffiffiffiffiffiffiffiffi simulations shown in Figures 2a 2c at times t g= = 90, 700, and of22

9 Figure 2. Computed surface elevations evolving from pa ffiffiffiffiffiffiffiffiffi positive initial prectangular ffiffiffiffiffiffiffiffiffi hump pofffiffiffiffiffiffiffiffiffi water (height: a, pwidth: ffiffiffiffiffiffiffiffiffi 2b). Scales: a/ = 0.10, b/ = 100. (a) t g= = 90; (b) t g= = 700; (c) t g= = 2000; (d) t g= = Full line: nonlinear dispersive model; dashed line: linear dispersive model; cross: theoretical linear infinite width solution according to equation (22); dots on b: solitary wave pffiffiffiffiffiffiffiffiffi solution; dots on d: the theoretical solution by Gurevich and Pitaevskii [1974] determined at time t g= = [42] In Figure 2a the differences between linear and nonlinear simulations pffiffiffiffiffiffiffiffiffi are noticeable but moderate. Figure 2b (plotted at t g= = 700 corresponding to t*) shows that the first soliton has almost separated from the tail, and the overlay of the solitary wave theory is in pretty good agreement with the computation. The volume integral of the leading solitary wave yields 6.5% of V = ab, hence it is only representing a very pffiffiffiffiffiffiffiffiffi small part of the transient wave. In Figure 2c (plotted at t g= = 2000), the nonlinear KdV solution shows five leading solitons, so clearly et em strongly overestimates the necessary travelling time for the linear nondispersive theory to break down. At this instant, there is a very distinct difference between the KdV solution and the linear dispersive solution. The theoretical linear infinite width solution obtained from equation (22) agrees very well with the linear dispersive simulation. Again, the asymptotic linear solution equation (27) pffiffiffiffiffiffiffiffiffi is not relevant as equation (28) predicts a threshold p of ffiffiffiffiffiffiffiffiffi et d g= [43] Figure 2d (also plotted at t g= = 2000) compares the nonlinear KdV solution with the analytical solution by Gurevich and Pitaevskii [1974] for an undular bore. They made their theoretical solution to the oscillatory transition between two different water levels (of infinite width) and used a modulated cnoidal wave train in which the amplitude, the mean level, the speed and the wave number were all slowly varying functions of x and t [see also Grimshaw et al., 2007]. They showed that on any individual crest in the cnoidal wavetrain, the elliptic parameter m will eventually approach unity, i.e., the undular bore will eventually become a train of solitary waves. Their solution is asymptotic and the leading wave always has an amplitude which is twice the jump height with an elliptic parameter of unity. Therefore we cannot expect the solution top beffiffiffiffiffiffiffiffiffi time-true, and it has been necessary to shift the time to t g= = 1700 in Figure 2d. Qualitatively however there is a good agreement between the numerical and analytical solution. The asymptotic solution has the peculiarity of starting with a crest An Ocean Displacement With Nonlinearity e [44] We now turn to a case, which is geophysically realistic for tsunamis in the deep ocean (see the Appendix). Typical scales would be = 4 km, a = 2 m and b = 200 km 9of22

10 Figure 3. Computed surface elevations evolving from pa ffiffiffiffiffiffiffiffiffi positive initial rectangular hump of water (height: a, width: 2b). Scales: a/ = , b/ = 50. t g= = 4900; Full line: nonlinear dispersive model; dashed line: linear dispersive model; cross: theoretical linear finite width solution according to equation (24). leading to e = and m This is 200 times less nonlinear than the previous case. The different classical thresholds now lead to ex e / 253,000, ex em / 400,000, ex m / 1,000,000. The threshold equation (31) yields ex AI / 16,000, but it is strictly not applicable, as the requirement equation (32) leads to 2b/ > 198, which is not fulfilled. Finally, the soliton separation distance reads x*/ 2,000,000. All of these are significantly larger than the width of any ocean or the circumference of the earth at the equator (see the Appendix), so we can clearly rule out any manifestation of KdV dynamics or appearances of leading solitons in the ocean. pffiffiffiffiffiffiffiffiffifigure 3 shows the computed wave profiles at time t g= = This is as far as the fetch of any ocean (see Table A1 in the Appendix), and at this stage we notice 2 3 peaks, which travel more or less with the linear shallow water celerity. Differences between the nonlinear and linear simulation can hardly be detected. The agreement with the theoretical linear finite width solution obtained from equation (24) is again very good. [45] It should be mentioned that Tadepalli and Synolakis [1996] simulated long distance propagation of leading depression waves (LDN) with a view to determine the effective propagation distance for leading solitary waves to emerge. They concluded that no distinct solitary waves emerged when LDN waves were propagated depths with nonlinearities of A Shelf Displacement With Nonlinearity e [46] When the tsunami reaches the continental shelf, the depth is reduced to approximately h 1 = 150 m (see Table A1 in the Appendix). The continental slope which connects the ocean floor with the shelf break is typically quite steep, i.e., 1/5 1/20, and in comparison with typical tsunami wave lengths this acts like a step. Now the reflection (b R ), transmission (b T ) and shoaling (b S ) coefficients can be estimated by [see, e.g., Mei et al., 2005] pffiffiffiffi pffiffiffiffi h 1 b R ¼ pffiffiffiffi p ffiffiffiffi ; b T ¼ 2 p ffiffiffiffi pffiffiffiffi p ffiffiffiffi ; b S ¼ h 1=4 1 ; þ h 1 þ h 1 which leads to b R = 0.68, b T = 1.68, and b S = We notice that the reflection is quite significant, but that the transmission of the tsunami is still quite large although reduced compared to Green s shoaling law. Hence on the basis of H 0 =1min = 4000 m, we can expect the wave height to be H m in h 1 = 150 m. At the same time the wave length will be reduced from approximately L km to L 1 80 km. This would indicate nonlinearity and dispersion levels of approximately e and m [47] We have not modeled the actual transition from the ocean to the shelf, but have generated an initial positive rectangular displacement directly on the shelf using a = 2.25m and b = 100 km. The shelf is assumed to be flat with a relative width of w/h 1 = 2000 (this exceeds the width of most continental shelves, see the Appendix). This leads to e and m 2 0 = , i.e., by far dominated by nonlinearity. In this case the different thresholds lead to ex AI / 96, ex e / 1,540, ex em / 180,000, while the soliton separation distance becomes x*/ 12,000. p[48] ffiffiffiffiffiffiffiffiffi Figures 4a 4d show the computed solutions at times t g=h 1 = 100, 500, 1500 and 2000 corresponding to similar distances in x/h 1. Note that only a small fraction of the solution is shown covering approximately 100h 1 = 15 km < b. The linear dispersive numerical solution (dashed) compares very well with the theoretical infinite width solution equation (22) (red crosses). In Figure 4a (corresponding to t = et AI ) the distance between the two first crests in the linear solution is L/h 11.7, which yields m , i.e., in balance with e. Later, in Figures 4b 4d this distance increases to L/h 20.0, 28.9, and It is also clear that the linear and nonlinear dispersive solutions gradually deviate more and more from each other: Typically, the nonlinear solution moves faster than the linear solution because of amplitude dispersion, and while the leading crest in the linear solution falls more and more behind in the moving coordinate system, the nonlinear solution keeps pace. Crest-to-crest distances grow faster in the linear solution than in the nonlinear solution. It is evident that nonlinear pffiffiffiffiffiffiffiffiffi dispersive theory is relevant from early on, i.e., from t g=h On the other hand, the solution is far from a separated soliton predicted to occur at x*/h 1 12,000, which exceeds the width of a typical shelf by a factor 4. Just as in Figure 2, the transient wave is an undular bore rather than a train of solitary waves, but the development is still too immature to provide a reasonable match with the solution of Gurevich and Pitaevskii [1974], for which the amplitude of the leading wave is twice the disturbance. 10 of 22

11 Figure 4. Computed surface elevations evolving frompa ffiffiffiffiffiffiffiffiffi positive initial prectangular ffiffiffiffiffiffiffiffiffi hump pofffiffiffiffiffiffiffiffiffi water (height: a, pwidth: ffiffiffiffiffiffiffiffiffi2b). Scales: a/ = 0.015, b/ = 667. (a) t g= = 100; (b) t g= = 500; (c) t g= = 1500; (d)t g= = Full line: nonlinear dispersive model; dashed line: linear dispersive model; cross: theoretical linear infinite width solution according to equation (22). [49] Finally, it should be mentioned that Mei et al. [2005], on their page 694, used ex em to conclude that linear nondispersive theory is valid everywhere on the shelf. This seems to be contradicted by the results in Figures 4a 4d, which all show importance of both dispersion and nonlinearity because of the initial rectangular shape of the disturbance. Just like in Figures 1 and 2, the reliability of ex em is hampered by the fact that the initial wave form immediately starts to generate much shorter undulations and ex AI is a much better threshold estimate than ex em. 4. Part 2: Shoaling From the Ocean to the Nearshore Area [50] In part 1, we concluded that solitary waves would not evolve in connection with typical tsunamis propagating in the deep ocean or over the flat part of the continental shelf. This however does not automatically rule out their presence in connection with uneven bathymetry, for example, during shoaling toward smaller water depths where nonlinearity rapidly increases. To study this issue we first simulate the shoaling of a transient wave from the deep ocean to the nearshore over a mild and constant slope. Second, we simulate shoaling of a periodic wave with a typical tsunami period of 13 minutes. In both cases, the objective is to monitor the evolution of skewness and asymmetry, and to check if disintegration into solitons occurs. Finally, we discuss the applicability of cnoidal and solitary wave profiles for long waves on a sloping bottom. Throughout this section, we utilize the high-order Boussinesq model applied in the previous section Shoaling of a Transient Wave on a Constant Slope [51] We first simulate the evolution of a transient wave train as it shoals up a mildly sloping beach. For this purpose we continue our simulation from Figure 3, and let a rectangular hump of water with initial nonlinearity e = , first propagate 5000 on a constant depth ( ) and then climb a mildly sloping beach with slope s = 1/200 to a terminating depth of h/ = It turns out that 11 of 22

12 Figure 5. Timeseries from shoaling of a transient wave on a mild slope s = 1/200. Continuation of Figure 3 with initial conditions a/ = and b/ = 50. 1: h/ = 1.0; 2: h/ =0.1;3:h/ = 0.01; 4: h/ = there is practically no reflection from the foot of the slope because of the mild inclination. Figure 5 shows the time series computed at the toe of the slope, i.e., h/ = 1, and at three locations on the slope with h/ = 0.1, 0.01 and During the shoaling process, the leading crest grows slightly faster than the second crest and because of amplitude dispersion, pthis ffiffiffiffiffiffiffiffiffi makes the crest to crest periods grow slightly from t g= 42.2, 42.5, 42.7 to If we convert numbers to geophysical scales with = 4000m,the periods correspond to approximately 850s, which just happens to be similar to what was measured on the boat Mercator near Phuket, Thailand in 2004 (see Appendix). As long as nonlinearity stays low or moderate, the crest amplitudes follow Green s law very well, but eventually the growth of crest amplitudes will be smaller than predicted by this law. Note that Synolakis [1991] actually proved the validity of Green s law also for the runup of solitary waves. The most important observation is that the shoaling proceeds without any signs of disintegration into new and shorter waves, i.e., solitons or even small undulations do not evolve. This issue will be covered in more detail in Part Shoaling of a Long Periodic Wave on a Constant Slope [52] Next, we make a similar shoaling test on a slope of 1/200 but this time starting from h = 2000 m with a sinusoidal wave with height H 0 = 1.45 m and period T = 780s. This period has been chosen to match the observation from Mercator 2004 (see Appendix). Figure 6 shows the resulting time series computed during the shoaling process at three depths (1: h = 100 m; 2: h =40m;3:h = 20 m). We notice that the initial sinusoidal wave profile develops with a gradual increase of asymmetry associated with a steepening of the wave front. Skewness, on the other hand, stays very small at all three locations. If the simulation is continued to h =14m (not shown), the wave height reaches H 5.5 m in qualitative agreement with observations from Mercator 2004 (confirming the relevance of the choice of initial wave height). Again there is no sign of disintegration into new and shorter waves and no soliton formation. Instead nonlinearity dominates completely over dispersion and a simulation based on the nonlinear shallow water equations (not shown) leads to exactly the same solution. This confirms earlier conjectures made by, e.g., Constantin and Johnson [2008] and Didenkulova et al. [2007] Applicability of Cnoidal and Solitary Wave Profiles [53] An important issue, which we would like to address, is if cnoidal and solitary wave theories can provide reasonable approximations to long waves on a sloping bottom during the shoaling process. This question is relevant because so many experimental studies have been made with a solitary wave input of relatively strong nonlinearity [see, e.g., Synolakis, 1987; Yeh et al., 1994; Briggs et al., 1995; Li and Raichlen, 2001; Tonkin et al., 2003; Jensen et al., 2003; and Kobayashi and Lawrence, 2004], with the intent of studying dynamics relevant to tsunamis. [54] Table 2 summarizes the results of using these theories at nine different water depths ranging from 4000 m to 10 m. The variation of the wave height with water depth is taken from Green s law for long waves, and the reference wave height H = 5.0 m has been chosen to occur in a depth of h = 14 m to match the measurement at Mercator (see Appendix). The solitary wave solution (columns 4 5) is given by equations (4) (5) and is completely defined by the two columns of h and H. Hence we can work out the local values of effective duration in space and time, i.e., L s and T s as defined by equation (6). In particular we notice that the 12 of 22

13 Figure 6. Timeseries from shoaling of a periodic linear wave on a mild slope s = 1/200. Period T = 780 s; initial depth = 2000 m; initial height H 0 = 1.45 m. (a) Timeseries at h = 100 m; (b) Timeseries at h = 40 m; (c) Timeseries at h =20m. duration varies from T s = 140 minutes at h = 4000m, tot s = 8sath = 10 m. This represents an enormous change of time scale (about a factor 1000). [55] The similar list of cnoidal solutions (columns 6 7) is based on the wave period T = 13 minutes. Now, on the basis of T, h and H we have worked out the local values of the Ursell number and of T c and L c (determined from equation (13) provided that U r > 51). From column 8 it is clear that the local Ursell number skyrockets from U r = 0.45 at h = 4000 m, to U r = 500,000 at h =10m. [56] Figures 7a 7d compare the theoretical cnoidal and solitary wave profiles at four different water depths from Table 2. At h = 2000m (Figure 7a) where H/h = and U r = 2.2 there is little difference between the cnoidal and the sinusoidal solutions, while the solitary solution leads to a duration T s = 64 minutes, which is more than 4 times the periodic ones. At h = 500m (Figure 7b) where H/h = and U r = 49, the cnoidal profile deviates clearly from the sinusoidal one by having a distinct skewness. The solitary solution is quite different from the cnoidal solution, but in fact their effective durations are almost the same with T s = 808 s. At h = 100 m (Figure 7c) where H/h = 0.03 and U r = 1875 the cnoidal and solitary profiles are similar and they deviate mainly due to differences in the trough levels. The effective durations T s and T c are clearly the same and they are much smaller than the actual period of the cnoidal wave. We notice a very strong build up of skewness in both profiles. Finally, at h = 40 m (Figure 7d) where H/h = 0.10 and U r = 16,000, the cnoidal and solitary profiles are practically identical, and their effective durations have shrunk to 45 s. [57] This is in complete contrast to the profiles computed during shoaling, see again Figures 6a 6c. Hence obviously it is not a good idea to approximate a long wave on a mildly sloping bottom by locally constant depth theory such as, for example, cnoidal theory (or streamfunction theory for that matter). This conclusion sounds obvious but nevertheless one can often get away with such an approximation for wind waves in shoaling water. However, for tsunamis, which are extremely long, the mistake becomes much more important, which can be illustrated by the fact that it is not the slope s but the relative slope sl/h which really matters. Only if this quantity is small, the constant depth solution provide a reasonable approximation. [58] It is evident from Figures 7a 7d and Table 2, that solitary and cnoidal wave profiles are extremely sensitive to Table 2. Solitary Theory and Cnoidal Theory (With T = 780 s) at Different Water Depths h (m) H (m) H/h T s L s T c (s) L c U r s 100 m m 500, s 170 m m 206, s 303 m m 83, s 935 m m 15, s 4.1 km km 1, s 18.4 km km s 56.7 km km min 175 km km min 540 km km min 1660 km km of 22

14 the value of nonlinearity, i.e., H/h. This is also illustrated by Figure 8, which shows the relative shallowness L s /h as a function of H/h. Only for rather small values of nonlinearity do we get really long waves, while nonlinearities in the range of leads to waves with wind wave characteristics. For comparison the measurements from Mercator near Phuket in 2004 indicated typical nearshore values of order H/h 0.36 and L/h 700. Obviously these data do not fit into the framework of KdV theory. 5. Part 3: Soliton Fission Close to the Shore [59] Soliton fission is known to occur in connection with the transition of solitary waves from one horizontal section with depth h 1 to another horizontal section with depth h 2 < h 1. Madsen and Mei [1969] and Goring [1978] studied this phenomenon and found that while being on the slope, the solitary wave maintained its shape except for an increase in asymmetry and the crest followed more or less Green s law. However, reaching the crest of the shelf, the solitary wave quickly started to disintegrate and gradually developed into several new but shorter solitary waves. As only moderate depth ratios of order 2 3 were considered by Madsen and Mei and Goring, relatively few waves developed and nonlinearity and dispersion played important roles everywhere in the model domain. [60] The question is however what happens to a very long asymmetric nondispersive wave (such as the one shown in Figure 6), when it hits a shallow and flat region, where nonlinearity is significant? To answer this question, we have continued the simulation from Figure 6 on to a horizontal shelf at the depth of 20m. In Figure 6c, which was shown in the same depth but on a sloping bottom, the wave profile showed significant asymmetry but no sign of disintegration. In Figure 9 the spatial variation of the profile is shown at two instances after the wave front has traveled 5 km and 9 km on the flat shelf. At the first location, the wave front becomes very steep and a few short undulations show up near the wave crest. At the second location, a series of short and steep undulations have developed in the wave front. Hence it turns out that in contrast to wind waves and solitary waves, the asymmetry of the long nondispersive wave will continue to grow also on a flat bottom until the wave front becomes almost vertical. Up to this point dispersion will play almost no role. When the wave front becomes sufficiently steep, an undular bore will develop and only at that time will dispersion start to play a role. This conclusion is in agreement with Peregrine [1966], Didenkulova et al. [2007] and Constantin and Johnson [2006, 2008]. [61] To investigate this phenomenon in more detail, we consider a symmetric smooth wave input (thus disregarding the asymmetry build up on the slope) and propagate this wave over a constant depth with nonlinearity of e = 0.1. Two different smooth input profiles are investigated: Figure 7. Surface elevation profiles. Solitary wave: thick full line; cnoidal wave: dashed line; sinusoidal wave: thin full line. (a) = 2000 m, H = 1.45 m, T s = 64 min, T = 780 s; (b) = 500 m, H = 2.05 m, T s = 808 s, T = 780s; (c) = 100 m, H = 3.06 m, T s = 130s, T = 780 s; (d) =40m,H = 3.85 m, T s =45s,T = 780 s; Further information: see Table 2. hð0; tþ ¼ a 0 sin wt; and hð0; tþ ¼ a 0 sech 2 wðt t 0 Þ; with wave period T = 780 s, wave amplitude a 0 =2.0m, water depth h = 20 m and phase shift t 0 = 500s. [62] At first we simulate this problem by the method of characteristics solving the nonlinear shallow water equa- 14 of 22

15 Figure 8. (H/h). Solitary waves. The effective wave length (L s ) over depth (h) as a function of the nonlinearity tions, i.e., disregarding dispersion completely. Following Stoker [1958], his equations ( ) ( ), the slope of any characteristic through the point (x, t) = (0, t) reads where cðtþ ¼ dx dt ¼ 3cðtÞ 2c 0 þ u 0 ; ð35þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ghþ ð hð0; tþþ; c 0 ¼ ffiffiffiffiffi gh ; u0 ¼ 0: ð36þ Along each characteristic the surface elevation will stay constant and equal to h(0, t) and to follow this particular elevation in (x, t) -space, we integrate equation (35) to obtain xt; ð t Þ ¼ ðt tþð3cðtþ 2c 0 þ u 0 Þ: ð37þ Now the spatial variation of the surface profile for a given time t 1 can be determined from a parametric plot of x(t 1, t) and h(0, t) wit t t. This profile will steepen over time, and Stoker provided the following criterion for crossing characteristics, which corresponds to a vertical wave front: ð x b ðtþ ¼ 3cðtÞ 2c 0 þ u 0 Þ 2 ; t b ðtþ ¼ t þ 3cðtÞ 2c 0 þ u 0 ; 3c t ðtþ 3c t ðtþ ð38þ where c t is the derivative of c with respect to t. To find the first occurrence of breaking, it is necessary to determine the minima of x b (t) and t b (t) for a span of t -values leading to t b min, x b min and t b min. Figure 9. Disintegration of a periodic non-dispersive long wave during propagation in shallow water. Continuation of the case from Figure 6, with the wave now propagating on a shelf with constant depth h = 20 m. The horizontal axis shows the distance from the start of the shelf. 15 of 22

16 Figure 10. Disintegration of a periodic non-dispersive long wave during propagation in shallow water. Input: sinusoidal wave, period T = 780 s; amplitude a 0 = 2.0 m; depth h = 20 m. Simulated by a Boussinesq model (dx =10m;dt = 0.7 s). 1: t 0 = 827 s; 2: t 0 = 1163 s; 3: t 0 = 1500 s. Full line: computed dispersive solution; dashed line: analytical non-dispersive solution based on equations (36) (37). [63] For the case of the sine-wave, breaking occurs for t b min = 0, i.e., the wave profile becomes vertical at the foot of the wave front, and the location can be expressed analytically as rffiffiffi bh ¼ g tmin b ¼ 2 pffiffiffiffiffi gh h 3 a 0 w ¼ 2 ð 3 ka 0Þ 1 : ð39þ x min For the given data equation (39) yields x b min 11,592m and t b min 828 s. For the case of the single wave defined by sech 2, breaking occurs much closer to the wave crest (t b min 410 s), and the solution to equation (38) yields x b min 18,623m and t b min 1629 s. Beyond this point the characteristic method is formally invalid. [64] Next, we make simulations using the Boussinesq model including dispersive terms. Figures show a comparison between the computed dispersive solution and the analytical nondispersive solution. Up to the point where the first characteristics start to cross, the two solutions are almost identical. Beyond this point however an undular bore starts to develop in the numerical simulation, while the analytical solution becomes double-valued. The observed disintegration into short waves riding at the front of the long wave is clearly related to the undular bore formation found in Figure 2 in connection with rectangular shaped disturbances. The analytical solution by Gurevich and Pitaevskii [1974] is no longer directly applicable, but it is still true that the leading undulation will develop into a separate solitary wave given sufficient time and propagation distance [see also Peregrine, 1966]. [65] We notice that the time and distance for the birth of the undular bore is very well predicted by (38) for both wave profiles. The reason is that the slope of the free surface grows very rapidly near the intersection of characteristics, so that it hardly takes any time or distance to increase it from say 0.1 to infinity. In this connection it should be emphasized that (39) is actually proportional to ex em /h given by equation (18). Hence, while we found ex em to be more or less useless in connection with the rectangular shaped waves in Part 1, this expression actually makes sense in connection with smooth waves. On the other hand, we prefer (38) and (39), which are quantitative measures leaving no uncertainty on the definition of m. [66] The threshold ex AI given by equation (31) is by definition irrelevant for the present case, but more surpris- 16 of 22

17 Figure 11. Disintegration of a periodic non-dispersive long wave during propagation in shallow water. Input: single wave, period T = 780 s; amplitude a 0 = 2.0 m; depth h = 20 m. Simulated by a Boussinesq model (dx =10m;dt = 0.7 s). (1) t 0 = 1050 s; (2) t 0 = 1629 s; (3) t 0 = 2100 s; (4) t 0 = 2450 s. Full line: computed dispersive solution; dashed line: analytical non-dispersive solution based on equations (36) (37). ingly, ex e given by equation (20) also appears to be off the point: It yields ex e 632m, which is seen to be much too short according to Figures Next, we consider the estimate of the separation time t* given by equation (34). This worked quite well in Part 1, where undulations were present from the beginning, but in case of a smooth disturbance it obviously has to be combined with a prediction of when the undulations start to appear. A simple estimate would be t** = t b min + t*, but it will underestimate the necessary time because the undulations are quite small at time t b min. In the present case this yields t** 1080s (sinusoidal wave) and t** 1880 s (single wave) and we notice from Figures 10 and 11 that at this time the leading wave has not yet become a perfect solitary wave, but it is indeed in the process of becoming one. [67] What is the influence of dispersion on the short waves riding at the front of the long wave? Well first of all, the undulations do not appear at all if dispersion is absent (as seen from the analytical model by Stoker). Secondly, it turns out that the higher the dispersion, the longer the crest-to-crest distance in time as well as in space. Actually, it turns out that if we switch off the dispersive terms in the numerical model, the undulations will become a lot shorter but they will still show up because of numerical dispersion, and consequently a coarser grid will lead to longer wave lengths and periods. With the correct Boussinesq dispersion combined with a fine grid resolution, the order of magnitude of wave lengths and periods in the undular bore turn out to be similar to the solitary wave solution with the relevant nonlinearity, which in our case yields L s 375 m and T s 25 s for a nonlinearity of This also implies that the stronger the nonlinearity, the shorter the wave lengths and periods. [68] Clearly the front of the tsunami can develop into an undular bore when the steepness of the front exceeds a certain threshold. This does not require a constant depth region but can happen during shoaling if the wave becomes sufficiently nonlinear. Typically, this happens in very shallow water, i.e., close to the beach and consequently the undular bore does not have sufficient time or travel distance to develop leading solitons. 17 of 22

18 Figure 12. The Sumatra 2004 tsunami reaching the island Koh Jum off the coast of Thailand (copyright Anders Grawin, 2006). [69] Recently, Grue et al. [2008] presented 1D results for the formation of undular bores and solitary waves in the Strait of Malacca for the 2004 Indian Ocean tsunami. For several reasons, we believe the nonlinearity of their computation is exaggerated. First of all the initial tsunami height was taken as 5.2 m on a 160m depth with reference to model results by Zaitsev et al. [2005]. This height appears to be very large considering the 5.0m height measured by Mercator in only 14 m depth. In fact, Grue et al. did mention that their nonlinearity was about twice that of Glimsdal et al. [2006], probably even comparing with the most severe of Grimsdal et al s four fault scenarios. Secondly, the tsunami would have to turn about 90 degrees to run south-east into this strait diverging from its main cause toward the coastline of Thailand and Malaysia. In this diffraction process the height would likely be reduced. Thirdly, the narrowness of the strait would lead to tsunami height decay because of the refraction to the adjacent coastlines (Grue et al. already describe how the tsunami front at their input boundary was bending slightly toward the Malaysian Peninsula.). This effect is obviously not included in the 1DH model although it would counteract the nonlinear evolution over the unusually long relative fetch. With at length of about 400 km and an average depth of around 80 m, the modeled part of the Strait has a fetch of about 5000 times the average water depth. This is more than twice the typical continental shelf fetch listed in the Appendix. However a relative fetch of this magnitude would typically (as for the Strait of Malacca) require propagation along the shelf (or shore) rather than perpendicular to the shelf and thus be subject to refractional effects reducing nonlinearity and counteracting the ability to shed solitary waves. Because of the strong nonlinearity and the large fetch, Grue et al. found that a series of solitary waves evolved and separated with a typical wave height of 12 m and arriving at about one-minute intervals. If this really happened in 2004, it would have been a spectacular effect, but we very much doubt it and to our knowledge there are no observations supporting it. [70] The formation of undular bores was actually observed in connection with the 1983 Nihonkai Chubu tsunami [Shuto, 1985], who reported that short waves of the order 10s 15s were riding on top of the main tsunami, which was of the order 5 30 minutes. They also appear on photographs taken in connection with the 2004 Sumatra tsunami (see Figure 12). In fact Li and Raichlen [2003] took these short waves as evidence supporting the use of a solitary wave representation of a tsunami. We strongly disagree with this conclusion for the following reasons: Although these short waves can have a local effect on wave forces on coastal structures [see, e.g., Asakura et al., 2002; Ikeno et al., 2006], they will play hardly any role in the runup and inundation caused by the main and much longer trailing tsunami. It is misleading to focus on a s event riding on a minute flood wave which carries so much more water on to the beach. So, if Figure 12 shows bore undulations, why does it not show the underlying tsunami? To answer this, Figure 13 repeats the result of Figure 10 by extending it to long-crested waves and taking a birds view from the beach. We noticed that only the undulations show up on the picture while there is no sign of the underlying long wave due to its very mild gradient. There is clearly a resemblance between Figure 13 and the photograph in Figure 12. [71] Finally, it should be emphasized that the role of wave breaking for tsunami runup is very often exaggerated. Breaking could be observed on several videos recorded by tourists in Thailand 2004, and it can also clearly be seen from Figures 12 and 14, which reveals two breaking wave fronts separated by a relatively short distance. The fact is, however, that it is the short waves riding on top of the tsunami which are breaking rather than the main tsunami itself. Therefore we doubt that the runup will be much influenced by the breaking process. This questions the 18 of 22

19 Figure 13. A perspective view of the undular bore from Figure 10 (bottom panel). Notice that only the short waves are visible while the underlying long wave does not show. tsunami relevance of experimental studies of breaking short solitary waves such as presented by Li [2000] and Li and Raichlen [2001, 2002, 2003]. 6. Final Conclusions on the Solitary Wave Paradigm [72] In the literature it has become common practice to study the runup of tsunamis and the associated processes such as breaking, scour and sediment transport by investigating the canonical problem of solitary waves generated over a constant depth entrance region, and then climbing a sloping beach to the shore. As examples we mention Synolakis [1986, 1987], Synolakis and Deb [1988], Synolakis et al. [1988], Yeh et al. [1994], Briggs et al. [1995], Liu et al. [1995], Li [2000], Li and Raichlen [2001, 2002, 2003], Tonkin et al. [2003], Jensen et al. [2003] and Kobayashi and Lawrence [2004]. [73] Yeh et al. [1994], Briggs et al. [1995], and Liu et al. [1995] made experimental studies of the runup of tsunamis on circular islands and used as input solitary waves with nonlinearity H/ = Although these waves were Figure 14. The Sumatra 2004 tsunami reaching Hat Ray Leah beach near Krabi, Thailand (copyright Scanpix, 2006). 19 of 22