Asymptotic derivation of a Navier condition for the primitive equations

This paper is devoted to the establishment of a friction boundary condition associated with the hydrostatic Navier-Stokes equations (also called the primitive equations). Usually the Navier boundary condition is used, as a wall law, for a fluid governed by the Navier-Stokes equations when we want to modelize roughness. We consider here an anisotropic Navier boundary condition corresponding to the anisotropic Navier-Stokes equations. By an asymptotic analysis with respect to the aspect ratio of the domain, we obtain a friction boundary condition for the limit system, that is the primitive equations. This asymptotic boundary condition only concerns the trace of the horizontal components of the velocity field and the trace of its vertical derivative.