Global well-posedness and asymptotics for a penalized Boussinesq-type system without dispersion

J.-Y. Chemin proved the convergence (as the Rossby number $ϵ$ goes to zero) of the solutions of the Primitive Equations to the solution of the 3D quasi-geostrophic system when the Froude number F = 1 that is when no dispersive property is available. The result was proved in the particular case where the kinematic viscosity $\nu$ and the thermal diffusivity $\nu$ ’ are close. In this article we generalize this result for any choice of the viscosities, the key idea is to rely on a special feature of the quasi-geostrophic structure.