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 $k$. It is shown that, for small $k$, there exist two bounded eigenvalues and a countable set of unboundedly growing eigenvalues. For a varying wave number $k$, 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.