## 1. Introduction

[2] The natural disaster caused by the earthquake and subsequent tsunami in the Indian Ocean on 26 December 2004 has led to a large number of recent publications on tectonics, earthquakes, tectonically generated water waves, and wave runup along the coastline. There is, moreover, an enhanced attention to potential generation of disastrous flood waves in the ocean, coastal waters, fjords/inlets, and lakes by submarine slides and landslides. Publications by, for example, *Liu* [2005], *Dalrymple et al.* [2006], *Glimsdal et al.* [2006], *Pelinovsky* [2006], *Liu et al.* [2008], and the references cited in the publications, describe recent progress of the analysis and prediction of tsunamis. This includes derivation of the basic equations and how they are simplified and solved numerically and implemented in larger computer codes for practical computations of the waves on ocean scale and in coastal water. The wave analysis codes are combined with seismic models which predict the motion of the seafloor that generates the waves. A recent book edited by *Yalciner et al.* [2003] is particularly devoted to slide-generated waves and runup on shore. Analysis and prediction of runup has recently been studied by *Didenkulova et al.* [2006] and *Kanoglu and Synolakis* [2006].

[3] Nonlinear shallow water theory (NLSW) has been implemented in the TUNAMI code for practical analysis of tsunamis [see *Intergovernmental Oceanographic Commission.*, 1997; *Zaitsev et al.*, 2005; *Pelinovsky*, 2006]. Other codes to predict tsunami motion on basin scale as well as in shallow seas, including the motion on shore, are described by *Dalrymple et al.* [2006]. They employ models like linear long-wave theory and NLSW and discuss the effect of frequency dispersion in terms of the Boussinesq equations. Predictions of runup using the FUNWAVE program and NLSW code agreed well with the ranges measured in field surveys after the 26 December 2004 tsunami in the Indian Ocean at a few locations along the coastline. They concluded, among other things, that the effect of dispersion should be included in the modeling in order to properly model the dispersive wave train following the leading tsunami. Formation of undular bores in shallow water was not discussed.

[4] Numerical results for the waves in the Bay of Bengal and wave runup at Banda Aceh and Yala/Sri Lanka were described by *Horrillo et al.* [2006] and *Ioualalen et al.* [2007]. Using the shallow water equations, a variant of the nonlinear Boussinesq equations, and the full Navier-Stokes equations, the latter implemented with a volume of fluid method, *Horillo et al.* [2006] were particularly looking for the effect of dispersion on the generated waves. Predictions of the leading waves by the nonlinear Boussinesq and full Navier-Stokes equations agreed quite well but generally differed from the shallow water equations in that study. Formation of short undular bores was not discussed. Using the nonlinear long-wave equations and a variant of the modified momentum conservation equations, where nonlinearity, dispersion, the Coriolis force, and even baroclinicity and compressibility were accounted for, *Rivera* [2006] modeled the Indian Ocean tsunami. He primarily presented results regarding the initial drying and flooding of the coasts and compared the simulation to observations of sea elevation by satellite along its track. No attention was given to formation of undular bores in regions with shallow water.

[5] *Glimsdal et al.* [2006] performed simulations using several models, including the linear shallow water equations, a variant of the low-order Boussinesq equations, including weakly nonlinear and dispersive effects, and ray theory for linear hydrostatic waves. They investigated the waves due to four different source characteristics of the fault, concluding that the largest uncertainty in the wave computations of the Indian Ocean tsunami relates to the generation phase of the waves. While dispersion was not seen in the generation phase, this effect may modify the wave propagation slightly when the motion takes place over a long time and may become important in shallow water. Idealized simulations of one-dimensional wave propagation along a section in the shallow Strait of Malacca exhibited that an undular bore was generated in a region between 2 and 7 km from the shoreline, depending on the source characteristics of the fault. We shall compare our calculations to the results obtained by *Glimsdal et al.* [2006]; see section 3.8. Undular bores were also observed in calculations of the Nicaragua earthquake tsunami [see *Shuto*, 1985]. The generation process and potential formation of solitons in the shallow sea were not discussed.

[6] The present investigation focuses on undular bores and solitons that are generated by a long tsunami propagating into shallow water. Motivation comes from observations of short waves in the form of undular bores in the Strait of Malacca generated by the Indian Ocean tsunami; see video of possible breaking undular bore, Malaysia, 26 December 2004 (available at http://www.dagbladet.no/download/malaysia_wave.wmv) The video indicates a short-wave period of 15–20 s at the beach. Recent computations by *Pelinovsky et al.* [2005] using Korteweg-deVries (KdV) theory indicate that very short wave phenomena may take place during the motion in the shallow sea and that a fuller model than KdV is required to correctly model the formation of the short waves. The essential point here is to study the process where a long depression wave with a subsequent wave of elevation steepens in its back face during the propagation into shallow water and how a train of short waves is generated in the back face of the leading depression.

[7] We model the wave motion along a 420 km long section, given as A–B which is midway between Sumatra Island and the Malaysian Peninsula; see Figure 1a. The depth profile of the sea is indicated in Figure 1b, ranging from 156 m at position A (the starting point) and reducing to 70 m at the first of three bottom ridges after 200 km. The two other ridges have depths of 65 and 36 m and are located after 300 and 375 km, respectively.

[8] The wave motion at the entrance of the strait results from a realistic simulation of the Indian Ocean tsunami and is used as initial condition in the computations of the motion in the strait. The input wave is characterized by a leading depression and subsequent elevation and has a height that corresponds to observations at a yacht located outside Phuket, somewhat north of the starting point of the simulation; see Figure 2c and the video of tsunami observations by the echo sounder on yacht *Mercator*, Phuket (available at http://www.knmi.nl/VinkCMS/news_detail.jsp?id = 19222).

[9] Two models are used to predict the wave motion: a weakly nonlinear dispersive Korteweg-de Vries model and a fully nonlinear and fully dispersive model (the full Euler equations) derived recently by *Fructus and Grue* [2007]. Descriptions of the models are given in section 2. Results of the simulations are described in section 3. Summary and conclusions are given in section 4.