On the influence of gravity in the dynamics of geophysical flows

In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($Ma$, $Ro$ and $Fr$, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime ${Ma}/{Fr} rightarrow 0$, we consider scaling for the Froude number which go beyond the "critical" value $Fr=sqrt{Ma}$. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.