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Analytical-numerical method for solving an Orr-Sommerfeld-type problem for analysis of instability of ocean currents. (English. Russian original) Zbl 1403.34065

Comput. Math. Math. Phys. 58, No. 6, 976-992 (2018); translation from Zh. Vychisl. Mat. Mat. Fiz. 58, No. 6, 1022-1039 (2018).
Summary: 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 \(k\) 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 \(k \rightarrow 0\). The problem with the boundary condition for the function has only a countable set of indefinitely increasing eigenvalues as \(k \rightarrow 0\). 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 \(k\).

MSC:

34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34B09 Boundary eigenvalue problems for ordinary differential equations
86A05 Hydrology, hydrography, oceanography
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
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[6] A. V. Boiko, A. V. Dovgal, G. R. Grek, and V. V. Kozlov, Physics of Transitional Shear Flows (Inst. Komp’yut. Issled., Moscow, 2006; Springer, Berlin, 2012).
[9] S. L. Skorokhodov and N. P. Kuzmina, “Efficient method for solving a modified Orr-Sommerfeld problem for stability analysis of currents in the Arctic Ocean,” Tavrich. Vestn. Inf. Mat., No. 3, 88-97 (2016).
[10] A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973). · Zbl 0265.35002
[11] A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981). · Zbl 0449.34001
[12] M. A. Lavren’tev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable (Nauka, Moscow, 1973) [in Russian].
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