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Regular and Chaotic Dynamics, 2016, Volume 21, Issue 3, Pages 291–334
DOI: https://doi.org/10.1134/S1560354716030059
(Mi rcd80)
 

This article is cited in 12 scientific papers (total in 12 papers)

On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-layer/Homogeneous Rotating Fluid

Leonid G. Kurakinab, Irina V. Ostrovskayab, Mikhail A. Sokolovskiycd

a Southern Mathematical Institute, Vladikavkaz Scienific Center of RAS, ul. Markusa 22, Vladikavkaz, 362027, Russia
b Institute for Mathematics, Mechanics and Computer Sciences, Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, 344090, Russia
c Water Problems Institute, RAS, ul. Gubkina 3, Moscow, 119333, Russia
d P. P. Shirshov Institute of Oceanology, RAS, pr. Nakhimovski 36, Moscow, 117997, Russia
Citations (12)
References:
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 $\Gamma$ 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 $(R, \Gamma, \alpha)$, where $\alpha$ is the difference between layer thicknesses. 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 $\mathcal{G}$ is applied. The two definitions of stability used in the study are Routh stability and $\mathcal{G}$-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the $\mathcal{G}$-stability is the stability of a three-parameter invariant set $O_{\mathcal{G}}$, 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.
Funding agency Grant number
Ministry of Education and Science of the Russian Federation 1.1398.2014/K
Russian Science Foundation 14-50-00095
The studies of the first two authors were supported within the framework of the design part of the State Assignment to SFU in the Sphere of Scientific Activity (Assignment No. 1.1398.2014/K), and the study of the third author was supported by the Russian Scientific Foundation, project No. 14-50-00095.
Received: 18.01.2016
Accepted: 03.04.2016
Bibliographic databases:
Document Type: Article
MSC: 76U05, 76B47, 76E20
Language: English
Citation: Leonid G. Kurakin, Irina V. Ostrovskaya, Mikhail A. Sokolovskiy, “On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-layer/Homogeneous Rotating Fluid”, Regul. Chaotic Dyn., 21:3 (2016), 291–334
Citation in format AMSBIB
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\by Leonid G. Kurakin, Irina V. Ostrovskaya, Mikhail A. Sokolovskiy
\paper On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Two-layer/Homogeneous Rotating Fluid
\jour Regul. Chaotic Dyn.
\yr 2016
\vol 21
\issue 3
\pages 291--334
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