Accurate method of solving modified Orr–Sommerfeld problem for analysis of the oceanic currents instability in the Arctic basin.

The efficient numerical method for solving the modified Orr - Sommerfeld problem has been elaborated. The govering equation defines the operator pencil of polynomial type with spectral parameter "$c$ " entering the equation and boundary condition. This equation describes long-wave stable and unstable perturbations of geostrophic currents with linear vertical shear. The model includes vertical density diffusion; it is used to study generation of large scale intrusions in the Arctic Ocean.
The method for evaluation of eigenfunctions and eigenvalues is based on using power expansions at boundary and internal points of the layer and smooth matching of that expansions. The equation for Wronskian of 4 basic solutions, $W\left ( c \right )=0$, enables us to compute the discrete spectrum of the problem. The numerical analysis of the spectrum for odd eigenfunctions shows, that the first eigenvalue $c_{1}$ corresponds to unstable currents, and all the rest eigenvalues correspond to stable currents.
The asymptotics of eigenvalue $c_{1}$ for small value of parameter of the problem was studied and the main term of the asymptotics was found. Numerical data for eigenvalues $c_{n}$ have been obtained for a wide range of the Peclet number (modified Reinolds number), $R\in \left ( 0,10^{6} \right )$, with high accuracy $10^{-20}$.
The results obtained confirm and supplement early published analytical considerations justifying the conclusion that geostrophic current can be unstable due to vertical diffusion of density.

Keywords:spectral problem, Orr - Sommerfeld equation, eigenvalue, eigenfunction, unstableflow