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 ($ {\rm{Ma}} $, $ {\rm{Ro}} $ and $ {\rm{Fr}} $, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $ {\rm{Ma}}/{\rm{Fr}}\, \rightarrow\, 0 $, we consider scaling for the Froude number which go beyond the "critical" value $ {\rm{Fr\, = \, \sqrt{\rm{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.