Shallow water asymptotic models for the propagation of internal waves

TL;DR

This paper develops and justifies a comprehensive Green-Naghdi model for internal wave propagation in two shallow fluid layers, accommodating non-flat topography and two-dimensional horizontal domains, unifying various existing models.

Contribution

It introduces a generalized Green-Naghdi model that includes non-flat topography and 2D horizontal dimensions, unifying and extending previous internal wave models.

Findings

Green-Naghdi model derived and justified for complex topography

Models such as shallow water, Boussinesq, and Camassa-Holm are shown to descend from this model

The approach can be adapted to different regimes beyond shallow water

Abstract

We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topography, and horizontal dimension d = 2. Its derivation follows directly from classical results concerning the one-layer case, and we believe such strategy may be used to construct interesting models in different regimes than the shallow-water/shallow-water studied in the present work.