On the influence of gravity in the dynamics of geophysical flows

TL;DR

This paper investigates the effects of gravity on the multiscale dynamics of geophysical flows modeled by the Navier-Stokes system, deriving limiting behaviors under various scalings of key dimensionless parameters.

Contribution

It provides a rigorous derivation of limiting systems for geophysical flows with gravity under non-critical Froude number scalings using compensated compactness.

Findings

Derived limiting systems for different gravity scalings.

Showed the importance of gravitational force structure for convergence.

Extended understanding of low stratification regimes in geophysical flows.

Abstract

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.