Global existence and convergence for non-dimensionalized incompressible Navier-Stokes equations in low Froude number regime

We prove that the incompressible, density dependent, Navier-Stokes equations are globally well posed in a low Froude number regime. The density is supposed to be increasing in depth and linearized around a stable state. Moreover if the Froude number tends to zero we prove that such system converges (strongly) to a two-dimensional, stratified Navier-Stokes system with full diffusivity. No smallness assumption is made on the initial data.