To model the title problem, the authors consider linearized Navier-Stokes equations in hydrostatic and Boussinesq approximations. The existence and uniqueness of the solution are proved for the three-dimensional problem and for the two-dimensional cyclic problem which describes the “El-Nino” phenomenon. The cyclic equations are also solved numerically in a realistic situation corresponding to the tropical region of the Pacific Ocean. The numerical algorithm uses the method of characteristics for time discretization and \(Q_1\) finite elements for space discretization. The results are in good agreement with physical observations.