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

Article type: Research Article

Authors: Duchêne, Vincent

Affiliations: IRMAR – UMR6625, CNRS and Université de Rennes 1, Rennes, France. E-mail: vincent.duchene@univ-rennes1.fr

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).

Keywords: internal waves, multilayer shallow water, small density contrast, singular limit, mode decomposition, rigid-lid approximation, Boussinesq approximation

DOI: 10.3233/ASY-161366

Journal: Asymptotic Analysis, vol. 98, no. 3, pp. 189-235, 2016