Modeling and mathematical analysis of models in oceanography

Abstract

This thesis is dedicated to the modeling and the mathematical analysis of asymptotic models used in oceanography describing the propagation of internal waves at the interface between two layers of fluids of different densities, under the only influence of gravity.We aim here at constructing and justifying new asymptotic models taking into account variable topography. To this end, we assume several smallness assumptions on the depth of the water and on the deformations at the interface and at the bottom. We are interested in two topographic regimes, one for variations of medium amplitude and one for slow variations with large amplitude.In the first part of this thesis we rigorously justify and mathematically study (existence, uniqueness, stability and convergence of the solution) two classes of asymptotic models. A class of coupled models and a class of scalar models. The latter class is characterized by the description of the propagation of unidirectional internal waves. In the second part we propose a numerical resolution for the coupled asymptotic model derived in the first part restricted to the flat bottom case. This existing model in the literature has been rewritten under a new formulation more suitable for numericalresolution with the same order of precision as the standard one but with improved frequency dispersion. Finally, we present several numerical simulations to validate our scheme.