  The primary goal of this paper is to develop robust methods to handle two
ubiquitous features appearing in the modeling of geophysical flows: (i) the
anisotropy of the viscous stress tensor, (ii) stratification effects. We focus
on the barotropic Navier-Stokes equations with Coriolis and gravitational
forces. Two results are the main contributions of the paper. Firstly, we
establish a local well-posedness result for finite-energy solutions, via a
maximal regularity approach. This method allows us to circumvent the use of the
effective viscous flux, which plays a key role in the weak solutions theories
of Lions-Feireisl and Hoff, but seems to be restricted to isotropic viscous
stress tensors. Moreover, our approach is sturdy enough to take into account
non constant reference density states; this is crucial when dealing with
stratification effects. Secondly, we study the structure of the solutions to
the previous model in the regime when the Rossby, Mach and Froude numbers are
of the same order of magnitude. We prove an error estimate on the relative
entropy between actual solutions and their approximation by a large-scale
quasi-geostrophic flow supplemented with Ekman boundary layers. Our analysis
holds for a large class of barotropic pressure laws.
