On the stability of the orbit and the invariant set of Thomson’s vortex polygon in a two-fluid plasma. (English) Zbl 1453.82085

Summary: The motion of the system of \(N\) point vortices with identical intensity \(\Gamma\) in the Alfven model of a two-fluid plasma is considered. The stability of the stationary rotation of \(N\) identical vortices disposed uniformly on a circle with radius \(R\) is studied for \(N = 2,\ldots,5\). This problem has three parameters: the discrete parameter \(N\) and two continuous parameters \(R\) and \(c\), where \(c>0\) is the value characterizing the plasma. Two different definitions of the stability are used - the orbital stability and the stability of a three-dimensional invariant set founded by the orbits of a continuous family of stationary rotations. Instability is interpreted as instability of equilibrium of the reduced system. An analytical analysis of eigenvalues of the linearization matrix and the quadratic part of the Hamiltonian is given. As a result, the parameter space \((N,R,c)\) of this problem for two stability definitions considered is divided into three parts: the domain \(\boldsymbol{A}\) of stability in an exact nonlinear problem setting, the linear stability domain \(\boldsymbol{B} \), where the nonlinear analysis is needed, and the domain of exponential instability \(\boldsymbol{C} \). The application of the stability theory of invariant sets for the systems with a few integrals for \(N=2,3,4\) allows one to obtain new statements about the stability in the domains, where nonlinear analysis is needed in investigating the orbital stability.


82D10 Statistical mechanics of plasmas
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76W05 Magnetohydrodynamics and electrohydrodynamics
76E25 Stability and instability of magnetohydrodynamic and electrohydrodynamic flows
76U05 General theory of rotating fluids
Full Text: DOI MNR


[1] Alfvén, H., “On the Existence of Electromagnetic-Hydromagnetic Waves”, Arc. f. Mat. Ast. Fys., 29B:2 (1942), 7 pp. · Zbl 0027.19003
[2] Batchelor, G. K., “On the Spontaneous Magnetic Field in a Conducting Liquid in Turbulent Motion”, Proc. Roy. Soc. London Ser. A, 201:1066 (1950), 405-416 · Zbl 0040.14203
[3] Bergmans, J., Kuvshinov, B. N., Lakhin, V. P., and Schep, T. J., “Spectral Stability of Alfvén Filament Configurations”, Phys. Plasmas, 7:6 (2000), 2388-2403
[4] Borisov, A. V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, R&C Dynamics, Institute of Computer Science, Izhevsk, 2005, 368 pp. (Russian) · Zbl 1119.76001
[5] Havelock, T. H., “The Stability of Motion of Rectilinear Vortices in Ring Formation”, Philos. Mag., 11:70 (1931), 617-633 · JFM 57.1109.03
[6] Karapetyan, A. V., “Invariant Sets of Mechanical Systems: Lyapunov”s Methods in Stability and Control”, Math. Comput. Modelling, 36:6 (2002), 643-661 · Zbl 1129.70323
[7] Kelvin, W. T., Mathematical and Physical Papers, v. 4, Cambridge Univ. Press, Cambridge, 1910 · JFM 41.0019.02
[8] Krall, N. A. and Trivelpiece, A. W., Principles of Plasma Physics, McGraw-Hill, New York, 1973, 674 pp.
[9] Kurakin, L. G. and Yudovich, V. I., “The Stability of Stationary Rotation of a Regular Vortex Polygon”, Chaos, 12:3 (2002), 574-595 · Zbl 1080.76520
[10] Dokl. Akad. Nauk, 384:4 (2002), 476-482 (Russian)
[11] Dokl. Akad. Nauk, 399:1 (2004), 52-55 (Russian)
[12] Kurakin, L. G. and Ostrovskaya, I. V., “Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle”, Regul. Chaotic Dyn., 17:5 (2012), 385-396 · Zbl 1252.76017
[13] Kurakin, L. G., Ostrovskaya, I. V., and Sokolovskiy, M. A., “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 · Zbl 1346.76204
[14] Kurakin, L. G. and Ostrovskaya, I. V., “On Stability of the Thomson”s Vortex \(N\)-Gon in the Geostrophic Model of the Point Bessel Vortices”, Regul. Chaotic Dyn., 22:7 (2017), 865-879 · Zbl 1401.76037
[15] Kurakin, L. G., Lysenko, I. A., Ostrovskaya, I. V., and Sokolovskiy, M. A., “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 · Zbl 1423.76076
[16] Lysenko, I. A., “On Stability of a Vortex Triangle, Square and Pentagon in the Two-Fluid Plasma”, Izv. Vyssh. Uchebn. Zaved. Severo-Kavkazskii Region. Natural Science, 2019, no. 1, 17-23 (Russian)
[17] Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Nauka, Moscow, 1978, 312 pp. (Russian)
[18] Morikawa, G. K. and Swenson, E. V., “Interacting Motion of Rectilinear Geostrophic Vortices”, Phys. Fluids, 14:6 (1971), 1058-1073
[19] Routh, E. J., A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion, Macmillan, London, 1877, 108 pp. · JFM 17.0315.02
[20] Sokolovskiy, M. A. and Verron, J., Dynamics of Vortex Structures in a Stratified Rotating Fluid, Atmos. Oceanogr. Sci. Libr., 47, Springer, Cham, 2014, XII, 382 pp. · Zbl 1384.86001
[21] Stewart, H. J., “Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems”, Quart. Appl. Math., 1 (1943), 262-267 · Zbl 0063.07196
[22] Stewart, H. J., “Hydrodynamic Problems Arising from the Investigation of the Transverse Circulation in the Atmosphere”, Bull. Amer. Math. Soc., 51 (1945), 781-799 · Zbl 0063.07197
[23] Thomson, W., “Floating Magnets (Illustrating Vortex Systems)”, Nature, 18 (1878), 13-14
[24] Thomson, J. J., Treatise on the Motion of Vortex Rings, Macmillan, London, 1883, 156 pp. · JFM 15.0854.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.