Rotating fluids in a cylinder

We study various singularly perturbed models related to rotating flows in a cylinder. At first we consider the three dimensional incompressible Navier--Stokes equations with turbulent viscosity, in the low Rossby limit. We prove a strong convergence result for ill prepared data, under a geometrical assumption on the cylinder section and a genericity condition on the singular operator.
In a second section, we discuss the compressible Navier--Stokes equations with anisotropic viscosity tensor in the combined low Mach and low Rossby number limit. In the case of well prepared initial data, we prove that global weak solutions with Dirichlet boundary conditions converge to the solution of a two--dimensional quasi-geostrophic model taking into account the compressibility. In the case of ill prepared data, we only show that we can hope a strong convergence result under the same kind of assumptions as in the incompressible case.