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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]

Based on the equation for potential vorticity in the quasi-geostrophic approximation, an analysis of stable and unstable perturbations of ocean currents of a finite transverse scale with a vertical linear velocity profile (Couette-type flows) is presented. The model takes into account the influence of vertical diffusion of buoyancy, friction, and the beta effect (the change in the Coriolis parameter with latitude). 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 eigenfunctions and eigenvalues are constructed for small values of the wavenumber . Using the continuation in the parameter , trajectories of the eigenvalues are calculated, which made it possible to compare the influence of the beta effect on unstable perturbations of the first and higher order modes. It is shown that the flow instability depends in a complex way on the physical parameters of the flow. [ABSTRACT FROM AUTHOR]

A method for analysis of the evolution equation of potential vorticity in the quasi-geostrophic approximation with allowance for vertical diffusion of mass and momentum for analyzing the stability of small perturbations of ocean currents with a linear vertical profile of the main flow is developed. The problem depends on several dimensionless parameters and reduces to solving a spectral non-self-adjoint problem containing a small parameter multiplying the highest derivative. A specific feature of this problem is that the spectral parameter enters into both the equation and the boundary conditions. Depending on the types of the boundary conditions, problems I and II, differing in specifying either a perturbations of pressure or its second derivative, are studied. Asymptotic expansions of the eigenfunctions and eigenvalues for small wavenumbers are found. It is found that, in problem I, as , there are two finite eigenvalues and a countable set of unlimitedly increasing eigenvalues lying on the line . In problem II, as , there are only unlimitedly increasing eigenvalues. A high-precision analytical-numerical method for calculating the eigenfunctions and eigenvalues of both problems for a wide range of physical parameters and wavenumbers k is developed. It is shown that, with variation in the wavenumber , some pairs of eigenvalues form double eigenvalues, which, with increasing , split into simple eigenvalues, symmetric with respect to the line . A large number of simple and double eigenvalues are calculated with high accuracy, and the trajectories of eigenvalues with variation in k, as well as the dependence of the flow instability on the problem parameters, are analyzed. [ABSTRACT FROM AUTHOR]