In this paper we consider an oceanic domain included in R^3 in which there exist, at initial time, a current U0, a pressure p0 and a density ρ0. The perturbations U, p and ρ of the velocity, the pressure and the density are induced by a perturbation of the mean wind-stress. The equations are of Navier-Stokes type for the velocity and pressure, of transport-diffusion type for the density. They are modified by the physical assumptions including the Boussinesq approximation and the hydrostatic approximation with vertical viscosity. The existence of the solution for the variational problem is studied. We prove some results about the regularity and then deduce the uniqueness. This result is valid in a domain Ω with corners. It is proved by means of an extension method, with even-odd reflection. In order to study the asymptotic behavior of the perturbation, we introduce some operators and give some properties for these operators. Under some assumptions, we prove that the perturbation tends to 0 as t tends to +∞ and that the solution of the variational problem is a strong solution for every time t in [0,+∞ ].