Spectral analysis of model Couette flows in application to the ocean

A method for analysis of the evolution equation of potential vorticity in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum for analyzing the stability of small perturbations of ocean currents with a linear vertical profile of the main flow is developed. The problem depends on several dimensionless parameters and reduces to solving a spectral non-self-adjoint problem containing a small parameter multiplying the highest derivative. A specific feature of this problem is that the spectral parameter enters into both the equation and the boundary conditions. Depending on the types of the boundary conditions, problems I and II, differing in specifying either a perturbations of pressure or its second derivative, are studied. Asymptotic expansions of the eigenfunctions and eigenvalues for small wavenumbers $k$ are found. It is found that, in problem I, as $k\to+0$, there are two finite eigenvalues and a countable set of unlimitedly increasing eigenvalues lying on the line $\operatorname{Re} (c)=\tfrac{1}{2}$. In problem II, as $k\to+0$, there are only unlimitedly increasing eigenvalues. A high-precision analytical-numerical method for calculating the eigenfunctions and eigenvalues of both problems for a wide range of physical parameters and wavenumbers $k$ is developed. It is shown that, with variation in the wavenumber $k$, some pairs of eigenvalues form double eigenvalues, which, with increasing $k$, split into simple eigenvalues, symmetric with respect to the line $\operatorname{Re} (c)=\tfrac{1}{2}$. A large number of simple and double eigenvalues are calculated with high accuracy, and the trajectories of eigenvalues with variation in $k$, as well as the dependence of the flow instability on the problem parameters, are analyzed.