LANNES, David; MARCHE, Fabien

doi:10.1111/sapm.12110

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LANNES, David
Institut de Mathématiques de Bordeaux [IMB]

MARCHE, Fabien
Institut Montpelliérain Alexander Grothendieck [IMAG]
Littoral, Environment: MOdels and Numerics [LEMON]

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Article de revue

Ce document a été publié dans

Studies in Applied Mathematics. 2015-11-22, vol. 136, n° 4, p. 382–423

Wiley-Blackwell

Résumé en anglais

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [9] that the motion of the waves could be described using an extended Green-Naghdi system. In this paper we propose an analysis of these equations, and show that they can be used to get some new insight into wave-current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also propose a robust and simple numerical scheme validated on several examples. Finally, we give some examples of wave-current interactions with a non trivial vorticity field and topography effects.< Réduire

Mots clés en anglais

solitary waves

vorticity

Boussinesq

nonlinear dispersive equations

Green-Naghdi

shallow water

Water waves

Finite-Volume discretization

URI

https://oskar-bordeaux.fr/handle/20.500.12278/194261

DOI

http://dx.doi.org/10.1111/sapm.12110

Project ANR

Frontières, numérique, dispersion. - ANR-13-BS01-0009
DYnamique des Fluides, Couches Limites, Tourbillons et Interfaces - ANR-13-BS01-0003

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251