Mathematical study of the betaplane model: Equatorial waves and convergence results
Mémoires de la Société Mathématique de France, no. 107 (2006) , 122 p.

We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the model, we describe the various waves involved and study in detail the resonances associated to those waves. We then exhibit the formal limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and prove its wellposedness. Finally we prove three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a strong convergence result of the filtered solutions towards the unique strong solution to the limit system, and finally a “hybrid” strong convergence result of the filtered solutions towards a weak solution to the limit system. In particular we obtain that there are no confined equatorial waves in the mean motion as the rotation becomes large.

On s’intéresse à un modèle de fluides en rotation rapide, décrivant le mouvement de l’océan dans la zone équatoriale. Ce modèle est connu sous le nom de Saint-Venant, ou système « shallow water », auquel on ajoute un terme de rotation dont l’amplitude est linéaire en la latitude ; en particulier il s’annule à l’équateur. Après une introduction physique au modèle, on décrit les différentes ondes en jeu et l’on étudie en détail les résonances associées à ces ondes. On exhibe ensuite un système limite formel (dans la limite d’une forte rotation), obtenu comme d’habitude en filtrant les ondes, et l’on démontre qu’il est bien posé. Enfin on démontre trois types de résultats de convergence : un théorème de convergence faible vers un système géostrophique linéaire, un théorème de convergence forte des solutions filtrées vers la solution unique du système limite, et enfin un résultat « hybride » de convergence forte des solutions filtrées vers une solution faible du système limite. En particulier on démontre l’absence d’ondes équatoriales confinées dans le mouvement moyen, quand la rotation augmente.

DOI: 10.24033/msmf.419
Classification: 35Q30, 35P99, 76U05, 86A10
Keywords: Rotating fluid, betaplane, Kelvin wave, Poincaré wave, Rossby wave, equatorial trapping, weak compactness, filtering method, harmonic analysis
Mot clés : Fluides tournants, Bétaplan, ordre de Kelvin, ordre de Poincaré, ordre de Rossby, confinement équatorial, compacité faible, filtrage, analyse harmonique
     author = {Gallagher, Isabelle and Saint-Raymond, Laure},
     title = {Mathematical study of the betaplane model: {Equatorial} waves and convergence results},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {107},
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     doi = {10.24033/msmf.419},
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     zbl = {1151.35070},
     language = {en},
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Gallagher, Isabelle; Saint-Raymond, Laure. Mathematical study of the betaplane model: Equatorial waves and convergence results. Mémoires de la Société Mathématique de France, Serie 2, no. 107 (2006), 122 p. doi : 10.24033/msmf.419.

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