Irina Ostrovskaya

Milchakova st. 8a, Rostov-on-Don, 344090, Russia
Southern federal University

Publications:

Kurakin L. G., Ostrovskaya I. V.
Abstract
A stability analysis of the stationary rotation of a system of $N$ identical point Bessel vortices lying uniformly on a circle of radius $R$ is presented. The vortices have identical intensity $\Gamma$ and length scale $\gamma^{-1}>0$. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for $N=2,\ldots,6$ are studied sequentially. The case of odd $N=2\ell+1\geqslant 7$ vortices and the case of even $N=2n\geqslant 8$ vortices are considered separately. It is shown that the $(2\ell+1)$-gon is exponentially unstable for $0 < \gamma R < R_*(N)$. However, this $(2\ell+1)$-gon is stable for $\gamma R\geqslant R_*(N)$ in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even $N=2n\geqslant 8$ vortex $2n$-gon is exponentially unstable for $R>0$.
Keywords: $N$-vortex problem, point Bessel vortices, Hamiltonian dynamics, stability
Citation: Kurakin L. G., Ostrovskaya I. V.,  On Stability of Thomson’s Vortex $N$-gon in the Geostrophic Model of the Point Bessel Vortices, Regular and Chaotic Dynamics, 2017, vol. 22, no. 7, pp. 865-879
DOI:10.1134/S1560354717070085
Kurakin L. G., Ostrovskaya I. V., Sokolovskiy M. A.
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 where the structure consists of a central vortex of arbitrary effective intensity $\Gamma$ and $N$ ($N = 4, 5$ and $6$) identical peripheral vortices. The identical vortices, each having a unit effective intensity, are uniformly distributed over a circle of radius $R$ in the lower 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 nondimensional thicknesses. The cases $N=2, 3$ were investigated by us earlier.
The theory of stability of steady-state motions of dynamical 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 structure, 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 two-layer vortex structure. 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 results of theoretical analysis are sustained by numerical calculations of vortex trajectories.
Keywords: discrete vortex structure, two-layer rotating fluid, stability
DOI:10.1134/S1560354724580019

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