Derivation of limit equation for a singular perturbation of a 3D periodic Boussinesq system

We consider a system describing the 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 two-dimensional Navier-Stokes system. No smallness assumption is assumed on the initial data