  In the present paper, we study the combined incompressible and fast rotation
limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and
gravitational forces, in the regime of small Mach, Froude and Rossby numbers
and for general ill-prepared initial data. We consider both the isotropic
scaling (where all the numbers have the same order of magnitude) and the
multi-scale case (where some effect is predominant with respect to the others).
In the case when the Mach number is of higher order than the Rossby number, we
prove that the limit dynamics is described by an incompressible
Oberbeck-Boussinesq system, where the velocity field is horizontal (according
to the Taylor-Proudman theorem), but vertical effects on the temperature
equation are not negligible. Instead, when the Mach and Rossby numbers have the
same order of magnitude, and in absence of the centrifugal force, we show
convergence to a quasi-geostrophic equation for a stream function of the limit
velocity field, coupled with a transport-diffusion equation for a new unknown,
which links the target density and temperature profiles.
  The proof of the convergence is based on a compensated compactness argument.
The key point is to identify some compactness properties hidden in the system
of acoustic-Poincar\'e waves. Compared to previous results, our method enables
first of all to treat the whole range of parameters in the multi-scale problem,
and also to consider a low Froude number regime with the somehow critical
choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture
some (low) stratification effects in the limit.
