Аннотация:
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 Γ and length scale γ−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,…,6 are studied sequentially. The case of odd N=2ℓ+1⩾7 vortices and the case
of even N=2n⩾8 vortices are considered separately. It is shown that the (2ℓ+1)-gon is exponentially unstable
for 0<γR<R∗(N). However, this (2ℓ+1)-gon is stable for γR⩾R∗(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N=2n⩾8 vortex 2n-gon is exponentially unstable for R>0.
Ключевые слова:N-vortex problem, point Bessel vortices, Hamiltonian dynamics, stability.
This research was supported by the Ministry of Education and Science of the Russian Federation, Southern Federal University (Project № 1.5169.2017/8.9).
Поступила в редакцию: 31.08.2017 Принята в печать: 30.10.2017
Образец цитирования:
Leonid G. Kurakin, Irina V. Ostrovskaya, “On Stability of Thomson’s Vortex N-gon in the Geostrophic Model of the Point Bessel Vortices”, Regul. Chaotic Dyn., 22:7 (2017), 865–879
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\by Leonid G. Kurakin, Irina V. Ostrovskaya
\paper On Stability of Thomson’s Vortex $N$-gon in the Geostrophic Model of the Point Bessel Vortices
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
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\pages 865--879
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Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd296
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Эта публикация цитируется в следующих 12 статьяx:
Leonid G. Kurakin, Irina V. Ostrovskaya, Mikhail A. Sokolovskiy, “On the Stability of Discrete N+1 Vortices in a Two-Layer Rotating Fluid: The Cases N=4,5,6”, Regul. Chaot. Dyn., 2024
Leonid Kurakin, Irina Ostrovskaya, “On the influence of circulation on the linear stability of a system of a moving cylinder and two identical parallel vortex filaments”, Bol. Soc. Mat. Mex., 29:3 (2023)
Jean N. Reinaud, “Circular Vortex Arrays in Generalised Euler’s
and Quasi-geostrophic Dynamics”, Regul. Chaotic Dyn., 27:3 (2022), 352–368
Habin Yim, Sun-Chul Kim, Sung-Ik Sohn, “Motion of three geostrophic Bessel vortices”, Physica D: Nonlinear Phenomena, 441 (2022), 133509
Jean N. Reinaud, “Finite-core quasi-geostrophic circular vortex arrays with a central vortex”, AIP Advances, 12:2 (2022), 025302
D. G. Dritschel, “Ring Configurations of Point Vortices in Polar Atmospheres”, Regul. Chaotic Dyn., 26:5 (2021), 467–481
Jean N. Reinaud, “Three-dimensional Quasi-geostrophic Staggered Vortex Arrays”, Regul. Chaotic Dyn., 26:5 (2021), 505–525
Leonid G. Kurakin, Irina V. Ostrovskaya, “Resonances in the Stability Problem of a Point Vortex
Quadrupole on a Plane”, Regul. Chaotic Dyn., 26:5 (2021), 526–542
L. G. Kurakin, I. A. Lysenko, “On the Stability of the Orbit and the Invariant Set of Thomson’s Vortex Polygon in a Two-Fluid Plasma”, Rus. J. Nonlin. Dyn., 16:1 (2020), 3–11
А. А. Килин, Е. М. Артемова, “Устойчивость правильных вихревых многоугольников в конденсате Бозе–Эйнштейна”, Изв. ИМИ УдГУ, 56 (2020), 20–29
L. G. Kurakin, I. V. Ostrovskaya, “On the Stability of Thomson's Vortex N-gon and a Vortex Tripole/Quadrupole in Geostrophic Models of Bessel Vortices and in a Two-Layer Rotating Fluid: a Review”, Rus. J. Nonlin. Dyn., 15:4 (2019), 533–542
L. G. Kurakin, I. A. Lysenko, I. V. Ostrovskaya, M. A. Sokolovskiy, “On stability of the Thomson's vortex n-gon in the geostrophic model of the point vortices in two-layer fluid”, J. Nonlinear Sci., 29:4 (2019), 1659–1700