On the eigenvalue spectra for a model problem describing formation of the large-scale intrusions in the Arctic basin

Abstract

To study the geostrophic flow instability with a linear vertical velocity shear in a vertically bounded layer the eigenvalue problem solution is considered. The vertical buoyancy diffusion effect on stable and unstable flow perturbations dynamics is taken into account in the model equation. The problem is reduced to a numerical solution of a non-self-adjoint third order differential equation with a small parameter at the highest derivative under the boundary conditions typical for the ocean. The solutions are sought as the power expansions at zero. The eigenvalues calculation leads to the search for the roots of a high-degree polynomial. The eigenvalue spectra are presented for various values of the problem dimensionless parameter. The results of the eigenvalues calculation are compared with the results obtained by an alternative method for solving an equivalent problem. It is noteworthy that the flow instability examined is an oscillatory instability, which is fundamentally different from the typical monotonous instability of fronts on the scales of the intrusion formation everywhere except for the equatorial zone. The results obtained are significant for analyzing the mechanisms of intrusive layers formation, which occur in the Arctic basin under conditions of absolutely stable stratification (decrease of the mean temperature with depth is accompanied by increase of the mean salinity).