# American Institute of Mathematical Sciences

September  2008, 1(3): 461-480. doi: 10.3934/dcdss.2008.1.461

## A mathematical review of the analysis of the betaplane model and equatorial waves

 1 Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France

Received  January 2008 Revised  March 2008 Published  June 2008

This review paper is devoted to the presentation of recent progress in the mathematical analysis of equatorial waves. After a short presentation of the physical background, we present some of the main mathematical results related to the problem.
More precisely we are interested in the study of the shallow water equations set in the vicinity of the equator: in that situation the Coriolis force vanishes and its linearization near zero leads to the so-called betaplane model. Our aim is to study the asymptotics of this model in the limit of small Rossby and Froude numbers. We show in a first part the existence and uniqueness of bounded (strong) solutions on a uniform time, and we study their weak limit. In a second part we give a more precise account of the asymptotics by characterizing the possible defects of compactness to that limit, in the framework of weak solutions only.
These results are based on the studies [6]-[8] on the one hand, and [11] on the other.
Citation: Isabelle Gallagher. A mathematical review of the analysis of the betaplane model and equatorial waves. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 461-480. doi: 10.3934/dcdss.2008.1.461
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