Nonlinear Asymptotic Stability of Munk's Solution of the General Circulation Problem

Abstract

The Lyapunov direct method is applied to the quasigeostrophic dynamics to deduce a nonlinear, asymptotic stability criterion, in the energy norm, for the wind-driven, one-layered oceanic circulation. The lateral diffusion of relative vorticity as a frictional mechanism and the related additional boundary conditions for a closed-basin ocean are specifically taken into account. Then, the criterion is used to prove the stability of the classical Munk's solution, viewed as the basic state in the limit of vanishing nonlinearity. The time derivative of the total perturbation energy is less than the product of the perturbation enstrophy and a nonlinear functional of the basic state: this is the key point in the derivation of the general stability criterion. Moreover, using the same kind of argument yields a similar stability analysis when Munk's no-stress boundary conditions are substituted with the more physical no-slip conditions. Finally, as a consequence of the arbitrariness of the considered perturbations, it follows that an asymptotically stable steady state, if it exists, is unique and acts as an attractor for any (analytical or numerical) solution, whatever the initial conditions may be.