The multilayer shallow water system in the limit of small density contrast

TL;DR

This paper rigorously analyzes the multilayer shallow water system with small density contrast, justifying common approximations and describing the asymptotic behavior, including initial layers and wave interactions, under hyperbolicity and smallness conditions.

Contribution

It provides a rigorous mathematical justification for the rigid-lid and Boussinesq approximations in multilayer shallow water flows in the small density contrast limit.

Findings

Well-posedness on large time intervals under certain conditions

Justification of rigid-lid and Boussinesq approximations

Existence of initial layers for ill-prepared data

Abstract

We study the inviscid multilayer Saint-Venant (or shallow-water) system in the limit of small density contrast. We show that, under reasonable hyperbolicity conditions on the flow and a smallness assumption on the initial surface deformation, the system is well-posed on a large time interval, despite the singular limit. By studying the asymptotic limit, we provide a rigorous justification of the widely used rigid-lid and Boussinesq approximations for multilayered shallow water flows. The asymptotic behaviour is similar to that of the incompressible limit for Euler equations, in the sense that there exists a small initial layer in time for ill-prepared initial data, accounting for rapidly propagating "acoustic" waves (here, the so-called barotropic mode) which interact only weakly with the "incompressible" component (here, baroclinic).