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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
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.
Received: 18.01.2016 Accepted: 03.04.2016
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
Linking options:
https://www.mathnet.ru/eng/rcd80 https://www.mathnet.ru/eng/rcd/v21/i3/p291
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