Analysis of the hydrostatic approximation in oceanography with compressionterm

The hydrostatic approximation of the incompressible 3D stationary Navier-St okes equations is widely used in oceanography and other applied sciences. I t appears through a limit process due to the anisotropy of the domain in us e, an ocean, and it is usually studied as such. We consider in this paper a n equivalent formulation to this hydrostatic approximation that includes Co riolis force and an additional pressure term that comes from taking into ac count the pressure in the state equation for the density. It therefore mode ls a slight dependence of the density upon compression terms. We study this model as an independent mathematical object and prove an existence theorem by means of a mixed variational formulation. The proof uses a family of fi nite element spaces to discretize the problem coupled with a limit process that yields the solution. We finish this paper with an existence and unique ness result for the evolutionary linear problem associated to this model. T his problem includes the same additional pressure term and Coriolis force. Mathematics Subject Classification. 35Q30, 76D05.