Analytical-Numerical Method for Analyzing Small Perturbations of Geostrophic Ocean Currents with a General Parabolic Vertical Profile of Velocity.

An analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral non-self-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small values of the wave number . It is shown that, for small , there exist two bounded eigenvalues and a countable set of unboundedly growing eigenvalues. For a varying wave number , the trajectories of eigenvalues are calculated for various dimensionless parameters of the problem. As a result, it is shown that the growth rate of unstable perturbations depends significantly on the physical parameters of the model. [ABSTRACT FROM AUTHOR]