  In the present paper, we study the fast rotation limit for the
density-dependent incompressible Euler equations in two space dimensions with
the presence of the Coriolis force. In the case when the initial densities are
small perturbation of a constant profile, we show the convergence of solutions
towards the solutions of a quasi-homogeneous incompressible Euler system. The
proof relies on a combination of uniform estimates in high regularity norms
with a compensated compactness argument for passing to the limit. This
technique allows us to treat the case of ill-prepared initial data.
