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
[1]

Elena Kartashova. Nonlinear resonances of water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 607-621. doi: 10.3934/dcdsb.2009.12.607

[2]

David Henry, Hung-Chu Hsu. Instability of equatorial water waves in the $f-$plane. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 909-916. doi: 10.3934/dcds.2015.35.909

[3]

Anatoly Abrashkin. Wind generated equatorial Gerstner-type waves. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4443-4453. doi: 10.3934/dcds.2019181

[4]

Tony Lyons. Geophysical internal equatorial waves of extreme form. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4471-4486. doi: 10.3934/dcds.2019183

[5]

Jifeng Chu, Joachim Escher. Steady periodic equatorial water waves with vorticity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4713-4729. doi: 10.3934/dcds.2019191

[6]

Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1

[7]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41

[8]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75

[9]

Delia Ionescu-Kruse, Anca-Voichita Matioc. Small-amplitude equatorial water waves with constant vorticity: Dispersion relations and particle trajectories. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3045-3060. doi: 10.3934/dcds.2014.34.3045

[10]

S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271

[11]

Sergey A. Suslov. Two-equation model of mean flow resonances in subcritical flow systems. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 165-176. doi: 10.3934/dcdss.2008.1.165

[12]

Biswajit Basu. On an exact solution of a nonlinear three-dimensional model in ocean flows with equatorial undercurrent and linear variation in density. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4783-4796. doi: 10.3934/dcds.2019195

[13]

Xiao-Qiang Zhao, Wendi Wang. Fisher waves in an epidemic model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1117-1128. doi: 10.3934/dcdsb.2004.4.1117

[14]

Darryl D. Holm, Cornelia Vizman. Dual pairs in resonances. Journal of Geometric Mechanics, 2012, 4 (3) : 297-311. doi: 10.3934/jgm.2012.4.297

[15]

Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems & Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225

[16]

Kazuyuki Yagasaki. Degenerate resonances in forced oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 423-438. doi: 10.3934/dcdsb.2003.3.423

[17]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051

[18]

Todd Kapitula, Björn Sandstede. Eigenvalues and resonances using the Evans function. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 857-869. doi: 10.3934/dcds.2004.10.857

[19]

Anna Geyer, Ronald Quirchmayr. Shallow water models for stratified equatorial flows. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4533-4545. doi: 10.3934/dcds.2019186

[20]

John R. King, Judith Pérez-Velázquez, H.M. Byrne. Singular travelling waves in a model for tumour encapsulation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 195-230. doi: 10.3934/dcds.2009.25.195

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]