Asymptotic derivation of a Navier condition for the primitive equations

Abstract

This paper is devoted to the establishment of a friction boundary condition associated with hydrostatic Navier-Stokes equations (also called primitive equations). Usually the Navier boundary condition is used, as a wall law, for a fluid governed by Navier-Stokes equations when we want to model 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 for 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.