Asymptotic analysis of the primitive equations under the small depth assumption

In this article we study the asymptotic behavior of solutions of the primitive equations (PEs) as the depth of the domain goes to zero. We prove that the solutions of the PEs can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded (in time and space) initial time layer. The barotropic flow is solution of the 2D Navier-Stokes equations with Coriolis force coupled with density. By employing a comparison theorem, the baroclinic flow can be approximated by a quasi-stationary nonlinear GFD-Stokes problem. This article presents a mathematically rigorous justification that the barotropic flow dominates the baroclinic flow in the motion of the atmosphere and ocean.