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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]

A model of stationary axisymmetric air motion in the lower layer of a typhoon is expounded and substantiated. The model takes into account air viscosity and compressibility, as well as the Coriolis force and friction against the Earth surface. The model is reduced to a system of three nonlinear differential equations for the horizontal velocity and density of air averaged in the vertical direction. Introducing the function ββ describing the vertical air flux at the top of the layer, the problem is decomposed into the two following ones solved successively: a boundary value problem for the tangential velocity component and a Cauchy problem for air density. After the solution of these problems, the remaining unknowns (the radial velocity component and pressure averaged in the vertical direction) are obtained explicitly. An efficient numerical implementation of the model is performed. The calculation results are in good accordance with observation data. The numerical experiments performed in this work reveal the existence of two maxima in the dependence of air velocity on the distance from the typhoon center in its lower layer. [ABSTRACT FROM AUTHOR]

By means of an analytic method the exact value of torsional rigidity for a beam with frame-type profile and asymptotics for stress intensity factor for a beam with crack have been obtained. [ABSTRACT FROM AUTHOR]

Nonlinear evolution of a trans-Alfvénic shock wave (TASW), i.e., such at which the flow velocity passes through the Alfvén velocity, is computed in a magnetohydrodynamic approximation. The analytical theory suggests that an infinitesimal perturbation of a TASW results in its disintegration, and, thus, in finite variation of the flow, or in transformation into some other unsteady configuration. In the present paper, this result is confirmed by numerical simulations. It is shown that the disintegration time is close to its minimum value equal to the shock thickness divided by a velocity of separation of the emerging secondary structures. The secondary TASW that appears after the disintegration is again unstable with respect to disintegration. When the perturbation has a cyclic nature, the TASW undergoes oscillatory disintegration, during which it repeatedly transforms into another TASW. This process manifests itself as a train of shock and rarefaction waves, which consecutively emerge at one edge of the train and merge at the other edge. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]