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

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.