Regularity results and optimal control problems for the perturbation of boussinesq equations of the ocean
Résumé
We study in this article a method which computes the variability of current, density and pressure in an oceanic domain. 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 Boussinesq approximation: density variations are neglected except in the terms of gravity acceleration. The existence and uniqueness of a solution are proved for two sets of equations: first the three-dimensional problem and then 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 the El Nino phenomenon. The value of the pressure p on the surface of ocean is of great interest for physical interpretation. To define that quantity, it is necessary to have the regularity $p\in H^1$. We have proved that the perturbation (u,ρ,p) of mean circulation is such that: $u\in L^2(0,T,H^2)$, $ρ\in L^2(0,T, H^2)$ and $p\in L^2(0,T, H^ 1), provided the perturbation of the windstress is sufficiently regular and satisfies compatibility relations. It is proved by means of an extension method, with even-odd reflection. We then develop a problem of control. The observation is the Variability of pressure on the surface of ocean. The control is the variability of windstress f, which acts as to forcing of the perturbation. We prove the existence and uniqueness of an optimal control, which is characterized by a set of equations including the direct problem and the adjoint problem. These results are valid for the three-dimensional problem and the two-dimensional cyclic problem.