On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast

Vincent Duchêne (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

The rigid-lid approximation is a commonly used simplification in the study of density-stratified fluids in oceanography. Roughly speaking, one assumes that the displacements of the surface are negligible compared with interface displacements. In this paper, we offer a rigorous justification of this approximation in the case of two shallow layers of immiscible fluids with constant and quasi-equal mass density. More precisely, we control the difference between the solutions of the Cauchy problem predicted by the shallow-water (Saint-Venant) system in the rigid-lid and free-surface configuration. We show that in the limit of small density contrast, the flow may be accurately described as the superposition of a baroclinic (or slow) mode, which is well predicted by the rigid-lid approximation; and a barotropic (or fast) mode, whose initial smallness persists for large time. We also describe explicitly the first-order behavior of the deformation of the surface, and discuss the case of non-small initial barotropic mode.

Mots clés: Internal waves; Rigid-lid approximation; Boussinesq approximation; Asymptotic analysis; Hyperbolic system

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Vincent Duchêne. On the rigid-lid approximation for two shallow layers of immiscible fluids with small density contrast. Journal of Nonlinear Science, 2014, 24 (4), pp.579-632. ⟨10.1007/s00332-014-9200-2⟩. ⟨hal-00861093v3⟩