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A new efficient 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 wave numbers and the existence of a countable set of complex eigenvalues with an unboundedly decreasing imaginary part is shown. On the integration interval , a system of three neighborhoods is introduced and a solution in each of them is constructed in the form of power series expansions, which are matched smoothly, so that the eigenfunctions and eigenvalues are efficiently calculated with high accuracy. For a varying wave number , the trajectories of complex eigenvalues are computed for various parameters of the problem and the existence of double eigenvalues is shown. The complex picture of instability developing in the simulated flow depending on physical parameters of the problem is briefly described. [ABSTRACT FROM AUTHOR]

For the problem of conformal mapping of a half-plane onto a -shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions . The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton's iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices. [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 problem of finding parameters of the Schwarz–Christoffel integral for a conformal mapping of a canonical domain onto an L-shaped one is solved analytically for arbitrary geometric parameters of the domain. The unknown preimage is represented in the form of a series in powers of a small parameter with coefficients written in closed form, and an estimate for the moduli of the coefficients is obtained. We find asymptotics for the crowding effect (crowding of preimages), which is especially pronounced for elongated domains. are computing The mapping and its inverse are computed using series with closed-form coefficients, whose domains of convergence collectively cover the entire (closed) mapped domain. Combining with linear fractional mappings and the elliptic sine function yields mappings of the half-plane, disk, and rectangle onto an L-shaped domain. Numerical implementations of the constructed mappings demonstrate the high efficiency of the applied methods. [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]

This work continues previous studies of the authors and provides an analytical solution to the plane problem of symmetric cavitating flow of an ideal fluid over a wedge for Tulin's cavity closure model with a double-spiral vortex. The solution is expressed in terms of the Lauricella hypergeometric function. A numerical implementation of the solution is described in detail, and an asymptotic analysis of the solution is given. The spiral structure of vortices closing the cavity is studied, and the vortex size is estimated. An asymptotic representation of the wake width as is found. Additionally, the asymptotics of the drag coefficient and the relative sizes of the cavity as the cavitation number tends to zero are established. [ABSTRACT FROM AUTHOR]

The solution to the problem of cavitating flow of an ideal fluid over a wedge is represented in explicit form in terms of the Gauss and Appell hypergeometric functions for a number of classical cavity closure models. A numerical implementation of the solution is described in detail. Asymptotic representations of the drag coefficient and the cavity sizes as the cavitation number tends to zero are obtained. [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]

Abstract: Stable and unstable disturbances of ocean currents are studied by analyzing a spectral problem based on the evolution potential vorticity equation in the quasi-geostrophic approximation. The problem is reduced to a fourth-order nonself-adjoint differential equation with a small parameter multiplying the highest derivative and with several dimensionless physical parameters. A feature of the problem is that the spectral parameter is involved in both the equation and boundary conditions for the third derivative. The problem is considered in two versions, namely, with a boundary condition setting the function or its second derivative to zero. An efficient analytical-numerical method is constructed for solving the problem. According to this method, even and odd functions are computed using power series expansions of the solution at boundary and middle points of the layer. An equation for the desired spectrum of the problem is derived by matching the expansions at an interior point. The asymptotic expansions of solutions and eigenvalues for small values of the wave number are studied. It is found that the problem for even and odd solutions with the boundary condition for the second derivative has a single finite eigenvalue and a countable set of indefinitely increasing eigenvalues as . The problem with the boundary condition for the function has only a countable set of indefinitely increasing eigenvalues as . The eigenvalues are computed for various parameters of the problem. The numerical results show that a current can be unstable in a wide range of . [ABSTRACT FROM AUTHOR]