Mathematical analysis and optimal control problems for the perturbation of the primitive equations of the ocean with vertical viscosity

In this paper we consider an oceanic domain in R3, in which there exists, at initial time, a current Uo, a pressure po and a density po. The perturbation U, p and p of the velocity, the pressure and the density are induced by a perturbation of the mean windstress. The equations are of Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the density. They are linearized around a given mean circulation and modified by the physical assumptions including the Boussinesq approximation and the Hydrostatic approximation with vertical viscosity. The existence and uniqueness of the solution for the variational problem are studied for the three-dimensional problem, and for the two-dimensional cyclic problem derived by assuming a sinusoidal .x-dependence for the perturbation of mean flow. The latter corresponds to a modelization of tropical instability waves which are illustrated by El Nino phenomenon.