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A spectral problem of the Orr–Sommerfeld type is considered to describe unstable disturbances of oceanic geostrophic currents with a linear vertical velocity profile by taking into account the vertical diffusion of buoyancy and friction. Numerical solutions are obtained for different values of the dimensionless parameters of the problem. Calculations of the spectra of eigenvalues and growth rates are compared with those of a similar problem for an ideal fluid. It is shown that (a) dissipation expands the range of wave numbers of unstable disturbances, (b) dissipation can increase the growth rates of baroclinic disturbances, (c) disturbances driven by the instability of the critical layer can grow faster than baroclinic disturbances, and (d) currents with a width equal to or less than the Rossby radius can be unstable; nearly circular (axisymmetric) unstable disturbances (submesoscale eddies) can develop in narrow currents or frontal zones. [ABSTRACT FROM AUTHOR]

Abstract—A spectral problem of the Orr–Sommerfeld type for describing stable and unstable perturbations of oceanic geostrophic flows with linear vertical velocity shear is considered (parabolic vertical velocity profile). Calculations of eigenvalues, increments of growth rate of unstable modes, and eigenfunctions of the fastest growing perturbations are presented. It is found that the instability of the geostrophic flow is observed over a wide range of horizontal scales: in addition to longwave perturbations with a phase velocity exceeding the maximum velocity of the geostrophic current and perturbations with the scales of the Rossby radius, shortwave modes exist with scales much smaller than the Rossby radius (submesoscale structures). The results of the model are used to describe intrusions in the Arctic basin, which are observed under conditions of absolutely stable stratification. [ABSTRACT FROM AUTHOR]

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability. [ABSTRACT FROM AUTHOR]