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Boussinesq/Boussinesq systems for internal waveswith a free surface, and the KdV approximation

Published online by Cambridge University Press:  03 October 2011

Vincent Duchêne
Vincent Duchêne*
Affiliation:
Département de Mathématiques et Applications, UMR 8553, École Normale Supérieure, 45 rue d'Ulm, 75230 Paris Cedex 05, France. [email protected]
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Abstract

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes,Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83–103]. We study the well-posedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538–566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C. Besse,C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1427–1432;C. Besse and C.H. Bruneau,Math. Mod. Methods Appl. Sci. 8 (1998) 1363–1386].

Keywords

Internal wavesfree surfacerigid lid configurationlong wavesKorteweg-de Vries approximationBoussinesq models
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012 , pp. 145 - 185
Copyright
© EDP Sciences, SMAI, 2011

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