Fast Rotating Non-homogeneous Fluids in Thin Domains and the Ekman Pumping Effect

Abstract

In this paper, we perform the fast rotation limit varepsilon rightarrow 0^+ of the density-dependent incompressible Navier–Stokes–Coriolis system in a thin strip Omega _varepsilon :=,{mathbb {R}}^2times , left. right] -ell _varepsilon ,ell _varepsilon left[ right. ,, where varepsilon in ,left. right] 0,1left. right] is the size of the Rossby number and ell _varepsilon >0 for any varepsilon >0. By letting ell _varepsilon longrightarrow 0^+ for varepsilon rightarrow 0^+ 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 geqslant ell >0 for all varepsilon >0 remains largely open at present.