Imperfect bifurcation for the quasi-geostrophic shallow-water equations

Abstract : We study analytical and numerical aspects of the bifurcation diagram of simply-connected rotating vortex patch equilibria for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations are a generalisation of the Euler equations and contain an additional parameter, the Rossby deformation length ε −1 , which enters in the relation between streamfunction and (potential) vorticity. The Euler equations are recovered in the limit ε → 0. We prove, close to circular (Rankine) vortices, the persistence of the bifurcation diagram for arbitrary Rossby deformation length. However we show that the twofold branch, corresponding to Kirchhoff ellipses for the Euler equations, is never connected even for small values ε, and indeed is split into a countable set of disjoint connected branches. Accurate numerical calculations of the global structure of the bifurcation diagram and of the limiting equilibrium states are also presented to complement the mathematical analysis.
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Contributeur : Taoufik Hmidi <>
Soumis le : mardi 9 janvier 2018 - 19:59:12
Dernière modification le : lundi 25 juin 2018 - 14:19:12
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  • HAL Id : hal-01679400, version 1

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Taoufik Hmidi, David Dritschel, Coralie Renault. Imperfect bifurcation for the quasi-geostrophic shallow-water equations. 2018. 〈hal-01679400〉

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