On the evolution of Rossby waves, generated by wind stress in a closed basin, incorporating total mass conservation

Abstract

The evolution of Rossby waves generated in a closed basin by applied stationary wind stress is considered, taking into account the total mass conservation constraint. The behavior of the wave field for large time t and the formation of a Sverdrup regime in the open ocean are analyzed by neglecting the frictionally induced wave decay. The consistent quasigeostrophic formulation of the problem is developed and the conditions are found for the applicability of relations derived. Using the vertical mode expansion reduces the 3D problem to a set of 2D problems. A method of solving the forced 2D problem with the mass constraint is suggested. The method consists of solving several particular problems, the most important of which are the forced problem for the auxiliary stream function Ψτ with zero boundary values and the Volterra integral equation of the second kind for time-dependent boundary values of the stream function Ψ. To simplify the analysis, the 1D model of the forced problem considered is offered. The analytical solution of the problem with zero boundary values of the stream function Ψ is found. It is known that, in general, such a solution does not stabilize with time but for large values of m (the ratio of the basin dimension to the Rossby radius of deformation) and t it tends to the Sverdrup solution in the open ocean. The time-averaged stream function Ψ is introduced, which tends to the Sverdrup solution in the open ocean for all m. A solution of the consistent problem satisfying the total mass conservation constraint is obtained and analyzed. It is pointed out that the stream function Ψ does not tend to the Sverdrup solution outside the western boundary layer for large values of m and t, which differs drastically from the corresponding behavior of Ψ in the problem with zero boundary values. This demonstrates the failure of the solution of the inconsistent problem with zero boundary values to describe the wave motion for large t. It is shown that the establishment of the Sverdrup solution for large t can be seen only if the time-averaged stream function Ψ is considered. It is proved for all m that outside the western boundary layer Ψ tends to the Sverdrup solution (minus the integral of Ψ in x over the whole basin) for large t. The evolution of the total energy is also discussed. © 1993.