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 ( $\mathrm{M}\mathrm{a}$, $\mathrm{R}\mathrm{o}$ and $\mathrm{F}\mathrm{r}$, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime $\mathrm{M}\mathrm{a}/\mathrm{F}\mathrm{r}\to 0$, we consider scaling for the Froude number which go beyond the "critical" value $\mathrm{F}\mathrm{r}=\sqrt{\mathrm{M}\mathrm{a}}$. 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.