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Mathematical study of the betaplane model: equatorial waves and convergence results. (English) Zbl 1151.35070
The authors study the oceanic flows in the equatorial zone. Their study follows other previous studies in the field created by other authors and by themselves, alone or in collaboration, where the oceanic flow is considered from different points of view. In the present one, they focus on quasigeostrophic, oceanic flows in the shallow-water approximation. The analysis is restricted to equatorial flows, where the Coriolis force cancels. The mathematical model for the ocean in the equatorial zone is presented in the first chapter of the paper, Introduction, where the ocean is considered as an incompressible viscous fluid with free surface submitted to gravitation in the following assumptions:
1. The density of the fluid is homogeneous; 2. The pressure law is given by the hydrostatic approximation; 3. the motion is essentially horizontal and does not depend on the vertical coordinate, leading to the so-called shallow water approximation. According to these assumptions, the authors write the movement system of equations in the form: \[ \partial_t h+\nabla(hu)= 0, \]
\[ \partial_t(hu)+ \nabla(hu\otimes u)+ f(hu)^\perp+ (1/\text{Fr}^2) h\nabla h- h\nabla K(h)- A(h, u)= 0, \]
where \(u\) is the horizontal mean velocity field, \(h\) is the depth due to the free surface, \(f\) is the vertical component of the Earth rotation, Fr is the Froude number, \(K\), \(A\) are the capillarity and viscosity operators, respectively, and \(u^\perp\) is the vector \((u_2,- u_1)\).
The study is made only on a thin strip around the equator, so that the longitude and the lattitude can be considered as Cartesian coordinates and supplement the above relations with boundary conditions. For this reason \(\sin\varphi\sim\varphi\), \(\cos\varphi\sim 1\), betaplane approximation, where \(\varphi\) is the lattitude. So, the authors have to solve a boundary value problem describing the motion of the ocean in the equatorial zone; the various waves and the resonances associated to them are involved. There are, also, presented the formal limit system, when the rotation becomes large, and its well-posedness. Finally, there are proved three types of convergence results: weak convergence, strong convergence and hybrid convergence.

MSC:
35Q30 Navier-Stokes equations
76U05 General theory of rotating fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
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