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Regular and Chaotic Dynamics

, Volume 21, Issue 3, pp 291–334 | Cite as

On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

  • Leonid G. Kurakin
  • Irina V. Ostrovskaya
  • Mikhail A. Sokolovskiy
Article

Abstract

A two-layer quasigeostrophic model is considered in the f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.

A limiting case of a homogeneous fluid is also considered.

The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O G , formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.

The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.

Keywords

discrete multipole vortex structure two-layer rotating fluid stability 

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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Leonid G. Kurakin
    • 1
    • 2
  • 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 Scienific Center of RASVladikavkazRussia
  3. 3.Water Problems Institute, RASMoscowRussia
  4. 4.P. P. Shirshov Institute of Oceanology, RASMoscowRussia

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