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An analysis is made of unstable perturbations of a geostrophic current of finite transverse scale with a vertical parabolic velocity profile of a general form (with linear and constant velocity shear) in a vertically bounded layer. The model is based on the potential vortex equation in the quasi-geostrophic approximation taking into account the vertical diffusion of momentum and mass. The equation and boundary conditions are reduced to a spectral eigenvalue problem of the Orr–Sommerfeld type. To search for eigenfunctions and eigenvalues, a high-precision analytic–numerical method was used. Particular attention is paid to the study of unstable perturbations with a phase velocity exceeding the maximum flow velocity. Such instability should be distinguished from baroclinic instability and critical layer instability. It is found that the indicated instability can develop in ocean currents when the problem parameters vary in a wide range of values. It is also obtained that, at an increase in the Prandtl number, the phase velocity of such perturbations increases and can significantly exceed the maximum flow velocity. However, the occurrence of such unstable perturbations is possible only when the maximum flow velocity is located in the inner region of the layer (but not necessarily in its center). It has also been found that narrow currents (the transverse scale is equal to or smaller than the Rossby radius) with a parabolic vertical profile can be unstable. The most unstable perturbations have approximately equal scales along and across the flow; that is, they are circular perturbations. A discussion of various types of geostrophic current instability with a vertical parabolic velocity profile as applied to the ocean is presented. [ABSTRACT FROM AUTHOR]

An analytical-numerical method is developed for solving a problem for the potential vorticity equation in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum. The method is used to analyze small perturbations of ocean currents of finite transverse scale with a general parabolic vertical profile of velocity. For the arising spectral non-self-adjoint problem, asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small values of the wave number . It is shown that, for small , there exist two bounded eigenvalues and a countable set of unboundedly growing eigenvalues. For a varying wave number , the trajectories of eigenvalues are calculated for various dimensionless parameters of the problem. As a result, it is shown that the growth rate of unstable perturbations depends significantly on the physical parameters of the model. [ABSTRACT FROM AUTHOR]

The paper presents an analysis of stable and unstable perturbations of ocean currents of a finite transverse scale with a parabolic vertical profiles of velocity (Poiseuille–Couette-type flow), based on the potential vorticity equation in the quasi-geostrophic approximation and taking into account both linear and constant flow velocity shear. The model takes into account the effect of vertical diffusion of buoyancy and vertical friction and assumes that the maximum mean current velocity takes place at the boundary of the layer. The analysis is based on the small perturbation method. The problem depends on several physical parameters and reduces to solving a spectral non-self-adjoint problem for a fourth-order equation with a small parameter at the highest derivative. Asymptotic expansions of the eigenfunctions and eigenvalues are constructed for small values of the wavenumber k. Using the method of continuation in the parameter k, the trajectories of the eigenvalues are calculated for different values of the problem's physical parameters. A detailed analysis of how the features of the vertical flow structure influence the characteristics of stable and unstable perturbations is presented. It is shown that the phase velocities of unstable perturbations can vary significantly depending on the linear vertical shear of the flow velocity. [ABSTRACT FROM AUTHOR]

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]