Abstract : In a first part, we present some Shallow Water equations. About the actual model, we firstly remark that, depending on the link between the viscosity and the aspect ratio, keeping the complete Coriolis force expression is essential: this gives a new model, with a so-called "cosine effect". We then show that the proofs of existence of weak solutions can be adapted to this new system. Numerical simulations of some waves underline the fact that this term is of importance. Next we study the influence of the limit conditions (surface, bottom) on Shallow-Water type models. We also present some models obtained using multiple scales in space and time. Finally we analyze a new model of sedimentation from a theoretical and numerical point of view and then we give some results for visco-plastic fluids.
In a second part, we are interested in the limit equation, namely the Quasi-Geostrophic (QG) equations and the lake equations. The numerical study of the 2d QG equations enables us to emphasize the role of the cosine effect from the Coriolis force. Depending on the topography we consider, we show that this effect can turn out to be not negligible. Still about the QG equations, we give a numerical scheme, based on asymptotic developments, which capture the boundary layer well and also give the opportunity to add a topography term to the solution for a flat bottom, without re-computing everything. Lastly we explain how to get the lake equations with cosine effect and we prove that the properties of existence of solutions to such equations are still valid.