Journal of Nonlinear Science

, Volume 24, Issue 4, pp 579–632 | Cite as

On the Rigid-Lid Approximation for Two Shallow Layers of Immiscible Fluids with Small Density Contrast

  • Vincent Duchêne


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 a 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 a nonsmall initial barotropic mode.


Internal waves Rigid-lid approximation Boussinesq approximation Asymptotic analysis Hyperbolic system 



The author is grateful to Christophe Cheverry, Jean-François Coulombel, and Frédéric Rousset for helpful advice and stimulating discussions. This work was partially supported by Project ANR-13-BS01-0003-01 DYFICOLTI.


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Université de Rennes, Institut de Recherche Mathématique deRennesFrance

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