Fast Wave Averaging for the Equatorial Shallow Water Equations

Dutrifoy, Alexandre; Majda, Andrew J.

doi:10.1080/03605300601188730

Cited by 20 publications

(25 citation statements)

References 27 publications

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“…This is the result that we will describe in the following as it is both easier and more general than the previous works [6] and [7]. We will omit however in this presentation the possible O(ε) variations in the direction x 1 to simplify the analysis; the interested reader can consult [8] for the more general case.…”

Section: 1mentioning

confidence: 84%

“…From a mathematical point of view, very few results have been obtained on this question, and this survey will mainly concentrate on four recent results, three (by A. Dutrifoy and A. Majda [6]- [7] and by A. Dutrifoy, A. Majda and S. Schochet [8]) concerning the non viscous case and the behaviour of the mean flow (with in particular the proof of the existence of uniformly bounded, unique solutions), and one (by the author and L. Saint-Raymond [11]) concerning the viscous case with the consideration of the mean flow as well as the description of oscillations around the mean flow, and their resonances (this time in the case of weak, possibly non unique solutions).…”

Section: 3mentioning

confidence: 99%

“…Once a bounded family of solutions is constructed, it is natural to consider its asymptotic behaviour as the Rossby number goes to zero. That is the second main result of [7], where a linear limit system is obtained for the mean flow.…”

Section: 1mentioning

confidence: 99%

“…The first paper, [6], concerns a special situation where the x 1 coordinate is assumed to vary slowly with ε: the solutions only depend on εx 1 instead of x 1 , which simplifies somewhat the analysis. In the second paper, [7], this assumption is dropped and the full non viscous shallow water system presented above is studied.…”

Section: 1mentioning

confidence: 99%

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

Gallagher¹

2008

Discrete &Amp; Continuous Dynamical Systems - S

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Abstract. 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.1. Physical background and goal of the study.1.1. Physical background. The aim of this review paper is to present a mathematical description of oceanic flows in the equatorial zone of the Earth. We are interested in domains extending over many thousands of kilometers, and on such scales the forces with dominating influence are the gravity and the Coriolis force. Our goal is to try to understand how those forces counterbalance each other to impose the so-called geostrophic constraint on the mean motion, and to describe the oscillations which are generated around this geostrophic equilibrium.In this survey we will be concentrating on the equatorial zone. Let us however briefly discuss the situation at midlatitudes, which has been more extensively studied from a mathematical point of view, in the past years. At midlatitudes, on "small" geographical zones (namely for a small enough interval around a given latitude far enough from the equator) one can neglect the variations of the Coriolis force due to the curvature of the Earth, and this leads to a singular perturbation problem with constant coefficients. The corresponding asymptotics (in the limit of a large rotation) have been studied by a number of authors, depending on the boundary conditions, the generality of the initial data... Roughly speaking one may summarize the situation in that case by stating that a limiting behaviour (in the limit of a small Rossby number, meaning that the rotation of the Earth is predominant over the motion under study) can be exhibited, and some sort of convergence (weak or strong depending on the boundaries or on the initial data) can be proved. We refer for instance to the review by R. Temam and M. Ziane [21] or by the author and L. Saint-Raymond [10], or to the work by J.-Y. Chemin, B. Desjardins, the author and E. Grenier [3] for more details.Here we are interested in a geographical zone where the variations of the Coriolis force do play a role (indeed the Coriolis force is identically zero at the equator and has a different sign on each hemisphere); we also wish to...

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Section: 1mentioning

confidence: 84%

Section: 3mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

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

Gallagher¹

2008

Discrete &Amp; Continuous Dynamical Systems - S

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“…Under the usual assumption that f (y) ≡ y, the existence of solutions to (1) and (2) on a time interval independent of ε and a characterization of their behavior as ε tends to zero have been proved in [7,8] for general unbalanced initial data and even, in the case of (2), in the presence of forcing terms of order ε −1 , introduced to mimic the effects of convective heating. For well-prepared initial data, a simpler proof of similar results on a multiscale system combining (1) and (2) has been given in [9].…”

Section: Introductionmentioning

confidence: 99%

Fast Averaging for Long- and Short-wave Scaled Equatorial Shallow Water Equations with Coriolis Parameter Deviating from Linearity

Dutrifoy

2014

Arch Rational Mech Anal

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The equatorial shallow water equations at low Froude number form a symmetric hyperbolic system with large terms containing a variable coefficient, the Coriolis parameter f , which depends on the latitude. The limiting behavior of the solutions as the Froude number tends to zero was investigated rigorously a few years ago, using the common approximation that the variations of f with latitude are linear. In that case, the large terms have a peculiar structure, due to special properties of the harmonic oscillator Hamiltonian, which can be exploited to prove strong uniform a priori estimates in adapted functional spaces. It is shown here that these estimates still hold when f deviates from linearity, even though the special properties on which the proofs were based have no obvious generalization. As in the linear case, existence, uniqueness and convergence properties of the solutions corresponding to general unbalanced data are deduced from the estimates.

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A simple justification of the singular limit for equatorial shallow‐water dynamics

Dutrifoy

Majda²,

Schochet

2008

Comm Pure Appl Math

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The equatorial shallow-water equations at low Froude number form a symmetric hyperbolic system with large variable-coefficient terms. Although such systems are not covered by the classical Klainerman-Majda theory of singular limits, the first two authors recently proved that solutions exist uniformly and converge to the solutions of the long-wave equations as the height and Froude number tend to 0. Their proof exploits the special structure of the equations by expanding solutions in series of parabolic cylinder functions. A simpler proof of a slight generalization is presented here in the spirit of the classical theory.

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Fast Wave Averaging for the Equatorial Shallow Water Equations

Cited by 20 publications

References 27 publications

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

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

Fast Averaging for Long- and Short-wave Scaled Equatorial Shallow Water Equations with Coriolis Parameter Deviating from Linearity

A simple justification of the singular limit for equatorial shallow‐water dynamics

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