Decoupled and unidirectional asymptotic models for the propagation of internal waves

TL;DR

This paper evaluates various scalar asymptotic models, including new coupled models, for internal wave propagation in a two-fluid system, providing rigorous justification and comparing decoupled and unidirectional approaches.

Contribution

It introduces and rigorously justifies new coupled asymptotic models for internal waves, alongside analysis of decoupled and unidirectional scalar equations.

Findings

Decoupled scalar models approximate internal wave propagation effectively.

New coupled models, including Green-Naghdi type, are rigorously justified.

Unidirectional models offer greater accuracy for specific initial data.

Abstract

We study the relevance of various scalar equations, such as inviscid Burgers', Korteweg-de Vries (KdV), extended KdV, and higher order equations (of Camassa-Holm type), as asymptotic models for the propagation of internal waves in a two-fluid system. These scalar evolution equations may be justified with two approaches. The first method consists in approximating the flow with two decoupled, counterpropagating waves, each one satisfying such an equation. One also recovers homologous equations when focusing on a given direction of propagation, and seeking unidirectional approximate solutions. This second justification is more restrictive as for the admissible initial data, but yields greater accuracy. Additionally, we present several new coupled asymptotic models: a Green-Naghdi type model, its simplified version in the so-called Camassa-Holm regime, and a weakly decoupled model. All of the models are rigorously justified in the sense of consistency.