  In the present paper we study the fast rotation limit for viscous
incompressible fluids with variable density, whose motion is influenced by the
Coriolis force. We restrict our analysis to two dimensional flows. In the case
when the initial density is a small perturbation of a constant state, we
recover in the limit the convergence to the homogeneous incompressible
Navier-Stokes equations (up to an additional term, due to density
fluctuations). For general non-homogeneous fluids, the limit equations are
instead linear, and the limit dynamics is described in terms of the vorticity
and the density oscillation function: we lack enough regularity on the latter
to prove convergence on the momentum equation itself. The proof of both results
relies on a compensated compactness argument, which enables one to treat also
the possible presence of vacuum.
