Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics

Geophysical fluids all exhibit a common feature: their aspect ratio ( depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier-Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium.