Analytical-numerical method for solving an Orr–Sommerfeld-type problem for analysis of instability of ocean currents

Stable and unstable disturbances of ocean currents are studied by analyzing a spectral problem based on the evolution potential vorticity equation in the quasi-geostrophic approximation. The problem is reduced to a fourth-order nonself-adjoint differential equation with a small parameter multiplying the highest derivative and with several dimensionless physical parameters. A feature of the problem is that the spectral parameter is involved in both the equation and boundary conditions for the third derivative. The problem is considered in two versions, namely, with a boundary condition setting the function or its second derivative to zero. An efficient analytical-numerical method is constructed for solving the problem. According to this method, even and odd functions are computed using power series expansions of the solution at boundary and middle points of the layer. An equation for the desired spectrum of the problem is derived by matching the expansions at an interior point. The asymptotic expansions of solutions and eigenvalues for small values of the wave number $k\to 0$ are studied. It is found that the problem for even and odd solutions with the boundary condition for the second derivative has a single finite eigenvalue and a countable set of indefinitely increasing eigenvalues as $k\to 0$. The problem with the boundary condition for the function has only a countable set of indefinitely increasing eigenvalues as $k\to 0$. The eigenvalues are computed for various parameters of the problem. The numerical results show that a current can be unstable in a wide range of $k$.