On the solitary wave paradigm for tsunamis

Abstract

[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 2004. 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.

Transformation of a shoaling undular bore

“…Such a configuration is relevant to the modelling of the near-shore tsunami propagation (see e.g. Madsen et al (2008), Tissier et al (2011) and references therein) but is also of a broader significance in the context of the general description of the dispersive shock wave propagation in weakly inhomogeneous media.…”

Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation

“…In particular, this will emphasise the important role of the undular bore solutions of the KdV equation, as seen in some tsunami observations and numerical simulations, see Arcas and Segur (2012) and Grue et al (2008) for instance. It is pertinent to note here that the critique of the validity of KdV models by Madsen et al (2008), Madsen and Schaffer (2010) and Arcas and Segur (2012), amongst others, is based on solitary wave dynamics, and we suggest that this is an overly restrictive view of the value of KdV models.…”

“…This indicates inter alia that the nonlinear and dispersive effects contained in the KdV model can indeed develop over plausible distances when applied as here to the near shore region. In much deeper water, where say h 0 ¼ 1:6 km; ¼ 0:02 for an initial amplitude of again 1 m, the corresponding length scale x s h 0 À1 ¼ 2760 km; 80 times larger, indicating inapplicability of a KdV model, as argued in a different manner by Madsen et al (2008), Madsen and Schaffer (2010) and Arcas and Segur (2012) for instance.…”

Characteristics of tsunami motion and energy budget during runup and rundown processes over a plane beach

“…Recently, new insights provided by Constantin and Johnson, 17 Constantin, 18 and Madsen et al, 19 had seriously questioned the paradigm of describing tsunamis as the solitary waves or N-waves. They demonstrated that initial surface displacement induced by submarine earthquakes is highly relative to the seismic activities, and the time-and space-scales of geophysical tsunamis are typically orders of magnitude different from the scales embedded in the classical Korteweg-deVries (KdV) tie between the dispersion and the nonlinearity.…”

“…They demonstrated that initial surface displacement induced by submarine earthquakes is highly relative to the seismic activities, and the time-and space-scales of geophysical tsunamis are typically orders of magnitude different from the scales embedded in the classical Korteweg-deVries (KdV) tie between the dispersion and the nonlinearity. Madsen et al 19 indicated that the bulk of the tsunami is very poorly represented by the solitary waves, and the front of the tsunami hardly ever develops into the solitary waves. Constantin 18 discussed the relevance of soliton theory to the modeling of tsunamis in the context of the real tsunamis records.…”