Fast rotating non-homogeneous fluids in thin domains and the Ekman pumping effect

TL;DR

This paper rigorously analyzes the Ekman pumping effect in fast rotating, non-homogeneous incompressible fluids within thin domains, using a novel approach that avoids complex boundary layer analysis.

Contribution

It provides the first asymptotic analysis of fast rotating incompressible fluids with variable density in a 3D thin domain, extending previous work on constant density and compressible flows.

Findings

Justifies the Ekman pumping phenomenon for non-homogeneous fluids.

Circumvents boundary layer complexity in the analysis.

First study of its kind in 3D for variable density fluids.

Abstract

In this paper, we perform the fast rotation limit $\varepsilon\rightarrow0^+$ of the density-dependent incompressible Navier-Stokes-Coriolis system in a thin strip $\Omega_\varepsilon\,:=\,\mathbb{R}^2\times\,]-\ell_\varepsilon,\ell_\varepsilon[\,$, where $\varepsilon\in\,]0,1]$ is the size of the Rossby number and $\ell_\varepsilon>0$ for any $\varepsilon>0$. By letting $\ell_\varepsilon\longrightarrow0^+$ for $\varepsilon\rightarrow0^+$ and considering Navier-slip boundary conditions at the boundary of $\Omega_\varepsilon$, we give a rigorous justification of the phenomenon of the Ekman pumping in the context of non-homogeneous fluids. With respect to previous studies (performed for flows of contant density and for compressible fluids), our approach has the advantage of circumventing the complicated analysis of boundary layers. To the best of our knowledge, this is the first study dealing with the asymptotic analysis of fast rotating incompressible fluids with variable density in a $3$-D setting. In this respect, we remark that the case $\ell_\varepsilon\geq\ell>0$ for all $\varepsilon>0$ remains largely open at present.