  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.
