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On the influence of the beta effect on the spectral characteristics of unstable perturbations of ocean currents. (English. Russian original) Zbl 1456.76056

Comput. Math. Math. Phys. 60, No. 11, 1900-1912 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 11, 1962-1974 (2020).
Summary: 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 \(k\). Using the continuation in the parameter \(k\), 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.

MSC:

76E20 Stability and instability of geophysical and astrophysical flows
76U60 Geophysical flows
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
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[1] Kuzmina, N. P., About one hypothesis on the generation of large-scale intrusions in the Arctic Ocean, Fundam. Prikl. Gidrofiz., 9, 15-26 (2016)
[2] Kuzmina, N. P., Generation of large-scale intrusions at baroclinic fronts: An analytical consideration with a reference to the Arctic Ocean, Ocean Sci., 12, 1269-1277 (2016)
[3] Kuzmina, N. P.; Skorokhodov, S. L.; Zhurbas, N. V.; Lyzhkov, D. A., On instability of geostrophic current with linear vertical shear at length scales of interleaving, Izv. Atmos. Ocean. Phys., 54, 47-55 (2018)
[4] Kuzmina, N. P.; Skorokhodov, S. L.; Zhurbas, N. V.; Lyzhkov, D. A., Description of the perturbations of oceanic geostrophic currents with linear vertical velocity shear taking into account friction and diffusion of density, Izv. Atmos. Ocean. Phys., 55, 207-217 (2019)
[5] Skorokhodov, S. L.; Kuzmina, N. P., Analytical-numerical method for solving an Orr-Sommerfeld-type problem for analysis of instability of ocean currents, Comput. Math. Math. Phys., 58, 976-992 (2018) · Zbl 1403.34065
[6] Skorokhodov, S. L.; Kuzmina, N. P., Spectral analysis of model Couette flows in application to the ocean, Comput. Math. Math. Phys., 59, 815-835 (2019) · Zbl 1458.76048
[7] 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).
[8] N. P. Kuzmina, S. L. Skorokhodov, N. V. Zhurbas, and D. A. Lyzhkov, “Spectral problem of Orr-Sommerfeld type for analysis of instability of currents in Arctic basin,” Abstracts of the International Scientific-Engineering Conference on Modern Problems in Ocean Thermohydromechanics (SPTO-2017), November 28-30,2017 (Inst. Okeanol. Ross. Akad. Nauk, Moscow, 2017), pp. 87-90.
[9] Charney, J. G., The dynamics of long waves in a baroclinic westerly current, J. Meteorol., 4, 135-162 (1947)
[10] Green, J. S. A., A problem in baroclinic stability, Quart. J. R. Meteorol. Soc., 86, 237-251 (1960)
[11] Demuth, M.; Hansmann, M.; Katriel, G., Eigenvalues of non-self-adjoint operators: A comparison of two approaches, Operator Theory: Adv. Appl., 232, 107-163 (2013) · Zbl 1280.47005
[12] Reddy, S. C.; Schmid, P. J.; Henningson, D. S., Pseudospectra of the Orr-Sommerfeld operator, SIAM J. Appl. Math., 53, 15-47 (1993) · Zbl 0778.34060
[13] Trefethen, L. N., Pseudospectra of linear operators, SIAM Rev., 39, 383-406 (1997) · Zbl 0896.15006
[14] A. V. Boiko, G. R. Grek, A. V. Dovgal, and V. V. Kozlov, Physics of Transitional Shear Flows (Inst. Komp’yut. Issled., Moscow, 2006; Springer, Berlin, 2012).
[15] Skorokhodov, S. L., Numerical analysis of the spectrum of the Orr-Sommerfeld problem, Comput. Math. Math. Phys., 47, 1603-1621 (2007)
[16] Skorokhodov, S. L., Branch points of eigenvalues of the Orr-Sommerfeld operator, Dokl. Math., 76, 744-749 (2007) · Zbl 1149.76019
[17] Nayfeh, A. H., Perturbation Methods (1973), New York: Wiley, New York · Zbl 0265.35002
[18] Nayfeh, A. H., Introduction to Perturbation Techniques (1981), New York: Wiley, New York · Zbl 0449.34001
[19] Lavren’tev, M. A.; Shabat, B. V., Methods of the Theory of Functions of Complex Variable (1973), Moscow: Nauka, Moscow
[20] Kuzmina, N. P., On the parameterization of interleaving and turbulent mixing using CTD data from the Azores frontal zone, J. Mar. Syst., 23, 285-302 (2000)
[21] N. P. Kuzmina, S. L. Skorokhodov, N. V. Zhurbas, and D. A. Lyzhkov, “On instability of geostrophic current with a constant vertical shear taking into account mass and momentum diffusion,” Proceedings of International Symposium on Mesoscale and Submesoscale Processes in the Hydrosphere and Atmosphere (MSP-2018), October 30-November 2,2018 (Inst. Okeanol. Ross. Akad. Nauk, Moscow, 2018), pp. 205-208. doi:10.29006/978-5-990149-4-1-2018-57
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