Simulation of the Transformation of Internal Solitary Waves on Oceanic Shelves

Abstract

Internal solitary waves transform as they propagate shoreward over the continental shelf into the coastal zone, from a combination of the horizontal variability of the oceanic hydrology (density and current stratification) and the variable depth. If this background environment varies sufficiently slowly in comparison with an individual solitary wave, then that wave possesses a soliton-like form with varying amplitude and phase. This stage is studied in detail in the framework of the variable-coefficient extended Korteweg-de Vries equation where the variation of the solitary wave parameters can be described analytically through an asymptotic description as a slowly varying solitary wave. Direct numerical simulation of the variable-coefficient extended Korteweg-de Vries equation is performed for several oceanic shelves (North West shelf of Australia, Malin shelf edge, and Arctic shelf ) to demonstrate the applicability of the asymptotic theory. It is shown that the solitary wave may maintain its soliton-like form for large distances (up to 100 km), and this fact helps to explain why internal solitons are widely observed in the world's oceans. In some cases the background stratification contains critical points (where the coefficients of the nonlinear terms in the extended Korteweg-de Vries equation change sign), or does not vary sufficiently slowly; in such cases the solitary wave deforms into a group of secondary waves. This stage is studied numerically.

Transformation of a shoaling undular bore

“…The relevant model here is an extended KdV (Gardner) equation which is often used to model oceanic internal solitary waves over bottom shelves (see e.g. Grimshaw et al (2004)). We also stress that the availability of the full modulation solution (an analog of the Gurevich-Pitaevskii solution (3.4)) for the initial "flat-bottom" undular bore is not a pre-requisite in our analysis, and a similar study can be undertaken for the systems where the initial evolution of the undular bore is described by a non-integrable dispersive equation (see El (2005) for the relevant generalisation of the Gurevich-Pitaevskii problem).…”

Modelling internal solitary waves on the Australian North West Shelf

“…In this present paper we complement and extend that study by focusing solely on the NWS, and including the effects of the Earth's rotation, which was not considered by Grimshaw et al (2004). As in Grimshaw et al (2004) the main goal is to determine the breakdown of an initial soliton into a complex waveform, taking into account the spatial variability of hydrology and the depth variation in the coastal zone. The basic generalized KdV equation is briefly described in section 2.…”

“…For instance, the Princeton Ocean Model, based on the nonlinear primitive equations, shows a high degree of spatial variability in the amplitude and phase of internal wave currents and vertical displacements [Craig, 1988;Holloway, 1996;Holloway and Barnes, 1998]. On the other hand, weakly nonlinear and weakly dispersive models based on generalizations of the Korteweg-de Vries (KdV) equation also demonstrate the appearance of intense short-scale solitons from the internal tide on the NWS [Smyth & Holloway, 1988;Holloway et al, 1997Holloway et al, , 1999Holloway et al, , 2001Grimshaw et al, 2004]. Together these numerical and theoretical modeling studies demonstrate the important role of each of nonlinearity, dispersion, the Earth's rotation, the bottom slope and the horizontal variability of density stratification and the background current.…”

“…These combined effects lead to the existence of a wide variety of nonlinear waves on the NWS, including forward and backward shocks, solitary waves of positive and negative polarities, table-like (or thick) solitons. 4 In a recent paper Grimshaw et al (2004) used a generalized Korteweg-de Vries equation to simulate the propagation of an internal solitary wave in a real oceanic environment; they studied the present NWS, and also the Malin Shelf off the west coast of Scotland, and the Arctic Shelf in the Laptev Sea. In this present paper we complement and extend that study by focusing solely on the NWS, and including the effects of the Earth's rotation, which was not considered by Grimshaw et al (2004).…”

“…4 In a recent paper Grimshaw et al (2004) used a generalized Korteweg-de Vries equation to simulate the propagation of an internal solitary wave in a real oceanic environment; they studied the present NWS, and also the Malin Shelf off the west coast of Scotland, and the Arctic Shelf in the Laptev Sea. In this present paper we complement and extend that study by focusing solely on the NWS, and including the effects of the Earth's rotation, which was not considered by Grimshaw et al (2004). As in Grimshaw et al (2004) the main goal is to determine the breakdown of an initial soliton into a complex waveform, taking into account the spatial variability of hydrology and the depth variation in the coastal zone.…”