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

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).

Minor modifications with respect to v1. To appear in Asymptotic Analysis

References in corpus (3)

• Fully nonlinear long-waves models in presence of vorticity
• On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast
• Local well-posedness of the multi-layer shallow-water model with free surface

Cited by in corpus (2)

• An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification
• On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity