Spectral analysis of model Couette flows in application to the ocean. (English. Russian original) Zbl 1458.76048

Comput. Math. Math. Phys. 59, No. 5, 815-835 (2019); translation from Zh. Vychisl. Mat. Mat. Fiz. 59, No. 5, 867-888 (2019).
Summary: 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 \(k\) are found. It is found that, in problem I, as \(k \to + 0\), there are two finite eigenvalues and a countable set of unlimitedly increasing eigenvalues lying on the line \(\operatorname{Re} (c) = \tfrac{1}{2}\). In problem II, as \(k \to + 0\), 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 \(k\), some pairs of eigenvalues form double eigenvalues, which, with increasing \(k\), split into simple eigenvalues, symmetric with respect to the line \(\operatorname{Re} (c) = \tfrac{1}{2}\). 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.


76E20 Stability and instability of geophysical and astrophysical flows
76M22 Spectral methods applied to problems in fluid mechanics
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
86A05 Hydrology, hydrography, oceanography
Full Text: DOI


[1] N. P. Kuzmina, “About one hypothesis on the generation of large-scale intrusions in the Arctic Ocean,” Fundam. Prikl. Gidrofiz. 9 (2), 15-26 (2016).
[2] N. P. Kuzmina, “Generation of large-scale intrusions at baroclinic fronts: An analytical consideration with a reference to the Arctic Ocean,” Ocean Sci. 12, 1269-1277 (2016). https://doi.org/10.5194/os-12-1269-2016
[3] N. P. Kuzmina, S. L. Skorokhodov, N. V. Zhurbas, and D. A. Lyzhkov, “On instability of geostrophic current with linear vertical shear at length scales of interleaving,” Izv. Atmos. Ocean. Phys. 54 (1), 47-55 (2018).
[4] S. L. Skorokhodov and N. P. Kuzmina, “Analytical-numerical method for solving an Orr-Sommerfeld-type problem for analysis of instability of ocean currents,” Comput. Math. Math. Phys. 58 (6), 976-992 (2018). · Zbl 1403.34065
[5] 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).
[6] 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.
[7] M. Demuth, M. Hansmann, and G. Katriel, “Eigenvalues of non-self-adjoint operators: A comparison of two approaches,” Operator Theory: Adv. Appl. 232, 107-163 (2013). · Zbl 1280.47005
[8] S. C. Reddy, P. J. Schmid, and D. S. Henningson, “Pseudospectra of the Orr-Sommerfeld operator,” SIAM J. Appl. Math. 53 (1), 15-47 (1993). · Zbl 0778.34060
[9] L. N. Trefethen, “Pseudospectra of linear operators,” SIAM Rev. 39 (3), 383-406 (1997). · Zbl 0896.15006
[10] 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).
[11] S. L. Skorokhodov, “Numerical analysis of the spectrum of the Orr-Sommerfeld problem,” Comput. Math. Math. Phys. 47 (10), 1603-1621 (2007).
[12] S. L. Skorokhodov, “Branch points of eigenvalues of the Orr-Sommerfeld operator,” Dokl. Math. 76 (2), 744-749 (2007). · Zbl 1149.76019
[13] A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973). · Zbl 0265.35002
[14] A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981). · Zbl 0449.34001
[15] M. A. Lavren’tev and B. V. Shabat, Methods of the Theory of Functions of Complex Variable (Nauka, Moscow, 1973) [in Russian].
[16] S. A. Orszag, “Accurate solution of the Orr-Sommerfeld equation,” J. Fluid Mech. 50 (4), 689-703 (1971). · Zbl 0237.76027
[17] A. Shkalikov, “Stokes lines and the ‘spectral tie’ in the Orr-Sommerfeld problem,” Usp. Mat. Nauk 53 (4), 140 (1998).
[18] A. A. Shkalikov, “Spectral portraits of the Orr-Sommerfeld operator with large Reynolds numbers,” J. Math. Sci. 124 (6), 5417-5441 (2004). · Zbl 1074.76018
[19] N. Kuzmina, B. Rudels, V. Zhurbas, and T. Stipa, “On the structure and dynamical features of intrusive layering in the Eurasian basin in the Arctic Ocean,” J. Geophys. Res. 116, C00D11 (2011). https://doi.org/10.1029/2010JC006920
[20] N. P. Kuzmina, S. L. Skorokhodov, N. V. Zhurbas, and D. A. Lyzhkov, “On the instability of geostrophic current with a constant vertical shear taking into account mass and momentum diffusion,” Proceedings of the International Symposium on Mesoscale and Submesoscale Processes in Hydrosphere and Atmosphere (MSP-2018), October 30-November 2,2018 (Inst. Okeanol. Ross. Akad. Nauk, Moscow, 2018), pp. 205-208. https://doi.org/10.29006/978-5-990149-4-1-2018-57
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.