THE FINITE-DIFFERENCE BAROTROPIC VORTICITY EQUATION IN OCEAN CIRCULATION MODELING - BASIC PROPERTIES OF THE SOLUTIONS

We study a discrete form of the barotropic vorticity equation used in the numerical studies of the ocean general circulation. Our goal is to contribute to the understanding of the numerical solutions, introducing some new tools of investigation. We begin by examining the basic properties of the discrete solutions for time-independent forcing. All the solutions are shown to be ultimately bounded in a finite region in the phase space and an estimate of the energy and enstrophy bounds is made. Steady-state solutions are shown always to exist for abitrary forcing and non-zero dissipation. A sufficient condition of asymptotic stability and uniqueness for the steady-state solutions is derived. The limit solutions for zero dissipation and infinite forcing are considered. For the infinite forcing case, a class of asymptotic steady-state solutions φk is recovered, proportional to the eigenvectors of the Laplacian operator. In this limit, the discrete problem is shown to be different from the continuous problem that allows for a wider class of asymptotic solutions. The analytical results have been used to study a number of numerical solutions computed for the single-gyre and double-gyre wind stresses and for various values of the forcing and dissipation parameters. We focus on solutions at high non-linearity, obtained either for decreasing dissipation or for increasing forcing. When the dissipation is decreased, the energy of the solutions increases at a slower rate than the upper bound. For sufficiently small values of the dissipation parameters, the steady-state solutions become unstable and the computed solutions are time dependent. The transition to instability is characterized by the rapid increase of the non-linear contribution to the Jacobian matrix. When the forcing is increased, the energy of the solutions increases and tends to the theoretical bound. These solutions are always stationary and asymptotically stable. The non-linear contribution to the Jacobian matrix increases linearly with the forcing. The steady-state solutions in this case converge to the asymptotic solutions φ1 and φ2 for the single-gyre and the double-gyre respectively. © 1990.