  We consider a system describing the long-time dynamics of an hydrodynamical,
density-dependent flow under the effects of gravitational forces. We prove that
if the Froude number is sufficiently small such system is globally well posed
with respect to a $ H^s, \ s>1/2 $ Sobolev regularity. Moreover if the Froude
number converges to zero we prove that the solutions of the aforementioned
system converge (strongly) to a stratified three-dimensional Navier-Stokes
system. No smallness assumption is assumed on the initial data.
