The hydrostatic approximation of the incompressible 3D stationary
Navier-Stokes equations is widely used in oceanography and other
applied sciences. It appears through a limit process due to
the anisotropy of the domain in use, an ocean, and it is usually studied as
such.
We consider in this paper an equivalent formulation to this
hydrostatic approximation that includes Coriolis force and an additional
pressure term that comes from taking into account the
pressure in the state equation for the density. It therefore models 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 finite
element spaces to discretize the problem coupled with a limit
process that yields the solution.
We finish this paper with an existence and uniqueness result for
the evolutionary linear problem associated to
this model. This problem includes the same additional pressure term and
Coriolis force.