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


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.


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

Mathematics Subject Classification

76B47 76E20 34D20 



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.


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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

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