Fast rotation and inviscid limits for the SQG equation with general ill-prepared initial data

In the present paper, we study the fast rotation and inviscid limits for the 2-D dissipative surface quasi-geostrophic equation with a dispersive forcing term, in the domain $\Omega =\T \times \R$. In the case when we perform the fast rotation limit (keeping the viscosity fixed), in the context of general ill-prepared initial data, we prove that the limit dynamics is described by a linear equation with parabolic structure. Conversely, performing the combined fast rotation and inviscid limits, we show that the means of the target initial datum $\oline \vartheta_0$ are conserved along the motion. The proof of the convergence is based on a compensated compactness argument which allows, on the one hand, to get compactness properties for suitable quantities hidden in the wave system and, on the other hand, to exclude the oscillatory part of waves at the limit.