On the open sea propagation of water waves generated by a moving bed

Abstract

Within the framework of linear theory, applicable far from the shore, we investigate the two-dimensional propagation of waves generated in the ocean by a sudden seabed deformation.

On the open sea propagation of two-dimensional rotational water waves generated by a moving bed

“…We consider an analogous setup as in [11]: we consider two-dimensional surface waves with the unbounded X-direction corresponding to the direction of wave propagation. The water's free surface is Y = d + F (X, T ), where d is the average depth of the sea, the seabed being Y = H(X, T ).…”

“…Since it appears [3] that up to 75% of all recorded tsunamis were generated by undersea earthquakes, in this paper we focus on this type. Three events stand out in the last 100 years: the 22 May 1960 Chile tsunami (caused by the largest earthquake ever recorded and with waves propagating across the Pacific Ocean, causing havoc in Hawaii and in Japan), the 26 December 2004 tsunami (that killed more than 200.000 people around the shores of the Indian Ocean), and the tsunami off the coast of Japan on 11 March 2011 (that killed thousands of people and triggered a nuclear accident). In all three cases the earthquake was undersea, along a fault line, in which case the generated waves are typically two-dimensional and with long wavelengths, of the order of 100 km (see the discussions in [1,3,4]).…”

“…Some authors (see [5][6][7]) advocated that KdV-like models apply, but even in the case of propagation over the maximal distance that is possible (across the Pacific Ocean, which was the case for the 1960 Chile tsunami), the scale analysis reveals that weakly nonlinear theory is not adequate (see the discussion in [1,3,[8][9][10]): the effects remain negligible in the open sea. Advances in the modelling of the tsunami wave propagation in the open sea, triggered by an undersea earthquake, were obtained in [4,11] for the setting of a still sea prior to the perturbation. In this paper we extend these considerations to accommodate the presence of underlying currents.…”

Nonlinear water waves

“…The first part starts from the assumption that the variations of the bed are on a scale comparable to those of the water's free surface, so that linear theory becomes relevant; cf. the considerations made in Constantin & Germain [5]. The second part requires an analysis of the wave dynamics in each particular region, and the specific features of the bottom topography are essential.…”

Shallow water equations for equatorial tsunami waves

“…Let us finally remark that, in contrast with the 2004 tsunami, which caused a devastating flood on the coasts of Sri Lanka, the 1883 tsunami generated by the volcanoes at Krakatoa was hardly notable in Sri Lanka, although both tsunamis were triggered in a similar region by comparably strong seismic events. The huge difference in the effects on the coast of Sri Lanka could be explained by the fact that the earlier event happened in the month of August as opposed to the 2004 tsunami which occurred in the month of December when under the northeast monsoon the Equatorial Indian Ocean current propagates along the equator towards Sri Lanka, thus enhancing the effect; see the discussion in [34,35,[37][38][39]. It is, therefore, of interest to investigate the effects of tropical currents on the propagation of tsunami waves in the spirit of [14] taking into account also the influence of the Coriolis effect, see [7,8] for recent discussions of equatorial current fields.…”