Full Navier-Stokes equations in shallow water: the hydrostatic approximation

Geophysical fluids all exhibit a common feature: the ratio depth vs horizon tal width is very small. This leads to an asymptotic model widely used in m eteorology, oceanography and limnology, namely the hydrostatic approximatio n of the time-dependent incompressible Navier-Stokes equations. It relies o n the hypothesis that pressure increases linearly in the vertical direction . We prove a convergence and existence theorem for this model by means of a nisotropic estimates and a new time-compactness criterium. (C) 1999 Academi e des sciences/Editions scientifiques et medicales Elsevier SAS.