Irina Ostrovskaya
Publications:
Kurakin L. G., Ostrovskaya I. V.
On Stability of Thomson’s Vortex $N$gon in the Geostrophic Model of the Point Bessel Vortices
2017, vol. 22, no. 7, pp. 865879
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 $R>0$.
