Advertisement

Journal of Nonlinear Science

, Volume 29, Issue 4, pp 1659–1700 | Cite as

On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

  • Leonid G. KurakinEmail author
  • Irina A. Lysenko
  • Irina V. Ostrovskaya
  • Mikhail A. Sokolovskiy
Article
  • 97 Downloads

Abstract

A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle of radius R in one of the layers is presented. The vortices have identical intensity and length scale is \(\gamma ^{-1}>0\). The problem has three parameters: N, \(\gamma R\) and \(\beta \), where \(\beta \) is the ratio of the fluid layer thicknesses. The stability of the stationary rotation is interpreted as orbital stability. The instability of the stationary rotation is instability of system reduced equilibrium. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space \((N,\gamma R,\beta )\) is divided on three parts: \(\varvec{A}\) is the domain of stability in an exact nonlinear setting, \(\varvec{B}\) is the linear stability domain, where the stability problem requires the nonlinear analysis, and \(\varvec{C}\) is the instability domain. The case \(\varvec{A}\) takes place for \(N=2,3,4\) for all possible values of parameters \(\gamma R\) and \(\beta \). In the case of \(N=5\), we have two domains: \(\varvec{A}\) and \(\varvec{B}\). In the case \(N=6\), part \(\varvec{B}\) is curve, which divides the space of parameters \((\gamma R, \beta )\) into the domains: \(\varvec{A}\) and \(\varvec{C}\). In the case of \(N=7\), there are all three domains: \(\varvec{A}\), \(\varvec{B}\) and \(\varvec{C}\). The instability domain \(\varvec{C}\) takes place always if \(N=2n\geqslant 8\). In the case of \(N=2\ell +1\geqslant 9\), there are two domains: \(\varvec{B}\) and \(\varvec{C}\). The results of research are presented in two versions: for parameter \(\beta \) and parameter \(\alpha \), where \(\alpha \) is the difference between layer thicknesses. A number of statements about the stability of the Thomson N-gon is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles. The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.

Keywords

N-vortex problem Point vortices Two-layer fluid Hamiltonian dynamics Stability 

Mathematics Subject Classification

76B47 76E20 34D20 

Notes

Acknowledgements

The work of the first three authors was supported by the Ministry of Education and Science of the Russian Federation, Southern Federal University (Project No. 1.5169.2017/8.9). MAS was supported by the Ministry of Education and Science of the Russian Federation (Project No. 14.W.03.31.0006, numerical simulation), Russian Science Foundation (Project No. 14-50-00095, application to ocean) and Russian Foundation for Basic Research (Projects Nos. 16-55-150001 and 16-05-00121, vortex dynamics). The authors are grateful to M. Yu. Zhukov for valuable discussions. We express our gratitude to the anonymous reviewer #2 for useful comments and recommendations.

References

  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Applied Mathematics Series, vol. 55. National Bureau of standards, Gaithersburg (1964)zbMATHGoogle Scholar
  2. Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2003)CrossRefGoogle Scholar
  3. Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv. 18, 85–191 (1963)MathSciNetCrossRefGoogle Scholar
  4. Borisov, A.V., Mamaev, I.S.: Mathematical Methods in the Dynamics of Vortex Structures. Institute of Computer Sciences, Moscow (2005)zbMATHGoogle Scholar
  5. Gantmacher, F.R.: Lectures on Analytical Mechanics. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury, Nauka, Moscow (1966)Google Scholar
  6. Gryanik, V.M.: Dynamics of singular geostrophic vortices in a two-level model of atmosphere (ocean). Izv. Atmos. Ocean. Phys. 19(3), 227–240 (1983)MathSciNetGoogle Scholar
  7. Havelock, T.H.: The stability of motion of rectilinear vortices in ring formation. Philos. Mag. 11(70), 617–633 (1931)CrossRefzbMATHGoogle Scholar
  8. Karapetyan, A.V.: The Routh Theorem and Its Extensions. Colloquia Mathematica Societatis Janos Bolyai: Qualitative Theory of Differential Equations, vol. 53, pp. 271–290. North Holland, Amsterdam (1990)Google Scholar
  9. Kelvin, W.T.: Floating Magnets (Illustrating Vortex Systems). Mathematical and Physical Papers, vol. 4, pp. 162–164. Cambridge University press, Cambridge (1910)Google Scholar
  10. Kolmogorov, A.N.: On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR 98, 527–530 (1954)MathSciNetGoogle Scholar
  11. Kurakin, L.G.: Stability, resonances, and instability of the regular vortex polygons in the circular domain. Dokl. Phys. 49(11), 658–661 (2004)CrossRefGoogle Scholar
  12. Kurakin, L.G., Ostrovskaya, I.V.: Nonlinear stability analysis of a regular vortex pentagon outside a circle. Regul. Chaotic Dyn. 17(5), 385–396 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kurakin, L.G., Ostrovskaya, I.V.: On stability of the Thomson’s vortex \(N\)-gon in the geostrophic model of the point Bessel vortices. Regul. Chaotic Dyn. 22(7), 865–879 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kurakin, L.G., Yudovich, V.I.: The stability of stationary rotation of a regular vortex polygon. Chaos 12(3), 574–595 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kurakin, L.G., Yudovich, V.I.: On nonlinear stability of steady rotation of a regular vortex polygon. Dokl. Phys. 47(6), 465–470 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kurakin, L.G., Ostrovskaya, I.V., Sokolovskiy, M.A.: On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid. Regul. Chaotic Dyn. 21(3), 291–334 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kurakin, L.G., Melekhov, A.P., Ostrovskaya, I.V.: A survey of the stability criteria of Thomson’s vortex polygons outside a circular domain. Bol. Soc. Mat. Mex. 22, 733–744 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Markeev, A.P.: Libration Points in Celestial Mechanics and Space Dynamics. Nauka, Moscow (1978)Google Scholar
  19. Morikawa, G.K., Swenson, E.V.: Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14(6), 1058–1073 (1971)CrossRefGoogle Scholar
  20. Moser, J.: Lectures on Hamiltonian Systems, vol. 81. Memoirs of the American Mathematical Society, Providence (1968)zbMATHGoogle Scholar
  21. Newton, P.K.: The N-Vortex Problem: Analytical Techniques. Applied Mathematical Sciences, vol. 145. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  22. Proskuryakov, I.V.: A Collection of Problems in Linear Algebra. Textbook (Sbornik zadach po linejnoj algebre. Uchebnoe posobie), 7th ed. Moscow, Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury, Nauka (1984)Google Scholar
  23. Routh, E.J.: A Treatise on the Stability of a Given State Motion. Macmillan, London (1877)Google Scholar
  24. Saffman, P.G.: Vortex Dynamics. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  25. Sokolovskiy, M.A., Verron, J.: Dynamics of Vortex Structures in a Stratified Rotating Fluid. Atmospheric and Oceanographic Sciences Library, vol. 47. Springer, Cham (2014)CrossRefzbMATHGoogle Scholar
  26. Stewart, H.J.: Periodic properties of the semi-permanent atmospheric pressure systems. Q. Appl. Math. 1, 262 (1943)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Stewart, H.J.: Hydrodynamic problems arising from the investigation of the transverse circulation in the atmosphere. Bull. Am. Math. Soc. 51, 781–799 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  28. Thomson, W.: Floating magnets (illustrating vortex systems). Nature 18, 13–14 (1878)CrossRefGoogle Scholar
  29. Thomson, J.J.: A Treatise on the Motion of Vortex Rings, pp. 94–108. Macmillan, London (1883)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Leonid G. Kurakin
    • 1
    • 2
    Email author
  • Irina A. Lysenko
    • 1
  • Irina V. Ostrovskaya
    • 1
  • Mikhail A. Sokolovskiy
    • 3
    • 4
  1. 1.Institute for Mathematics, Mechanics and Computer SciencesSouthern Federal UniversityRostov-on-DonRussia
  2. 2.Southern Mathematical InstituteVladikavkaz Scientific Center of RASVladikavkazRussia
  3. 3.Water Problems InstituteRASMoscowRussia
  4. 4.Shirshov Institute of OceanologyRASMoscowRussia

Personalised recommendations