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 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.
Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D. L., Vortex Crystals, Adv. Appl. Mech., 2013, vol. 39, pp. 1–79.
Saffman, P.G., Vortex Dynamics, Cambridge Monogr. Mech. Appl. Math., New York: Cambridge Univ. Press, 1992.
Sokolovskiy, M.A. and Verron, J., Dynamics of Vortex Structures in a Stratified Rotating Fluid, Atmos. Oceanogr. Sci. Libr., vol. 47, Cham: Springer, 2014.
Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
Kurakin, L. G., Melekhov, A. P., and Ostrovskaya, I. V., A Survey of the Stability Criteria of Thomson’s Vortex Polygons outside a Circular Domain, Bol. Soc. Mat. Mex., 2016, vol. 22, no. 2, pp. 733–744.
Thomson, W., Floating Magnets, Nature, 1878, vol. 18, pp. 13–14.
Kelvin, W.T., Mathematical and Physical Papers: Vol. 4, Cambridge: Cambridge Univ. Press, 1910, pp. 162–164.
Thomson, J. J., On the Motion of Vortex Rings, London: Macmillan, 1883, pp. 94–108.
Havelock, T.H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag., 1931, vol. 11, no. 70, pp. 617–633.
Kurakin, L. G. and Yudovich, V. I., The Stability of Stationary Rotation of a Regular Vortex Polygon, Chaos, 2002, vol. 12, no. 3, pp. 574–595.
Kurakin, L. G. and Yudovich, V. I., On the Nonlinear Stability of the Steady Rotation of a Regular Vortex Polygon, Dokl. Phys., 2002, vol. 47, no. 6, pp. 465–470; see also: Dokl. Akad. Nauk, 2002, vol. 384, no. 4, pp. 476–482.
Stewart, H. J., Periodic Properties of the Semi-Permanent Atmospheric Pressure Systems, Quart. Appl. Math., 1943, vol. 1, pp. 262–267.
Stewart, H. J., Hydrodynamic Problems Arising from the Investigation of the Transverse Circulation in the Atmosphere, Bull. Amer. Math. Soc., 1945, vol. 51, pp. 781–799.
Morikawa, G.K. and Swenson, E. V., Interacting Motion of Rectilinear Geostrophic Vortices, Phys. Fluids, 1971, vol. 14, no. 6, pp. 1058–1073.
Routh, E. J., A Treatise on Stability of a Given State of Motion, London: McMillan, 1877.
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/HomogeneousRotating Fluid, Regul. Chaotic Dyn., 2016, vol. 21, no. 3, pp. 291–334.
Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., The Bifurcation Analysis and the Conley Index in Mechanics, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 457–478.
Proskuryakov, I.V., A Collection of Problems in Linear Algebra, 7th ed.,Moscow: Nauka, 1984 (Russian).
Kurakin, L. G., Stability, Resonances, and Instability of the Regular Vortex Polygons in the Circular Domain, Dokl. Phys., 2004, vol. 49, no. 11, pp. 658–661; see also: Dokl. Akad. Nauk, 2004, vol. 399, no. 1, pp. 52–55.
Kurakin, L. G. and Ostrovskaya, I.V., Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle, Regul. Chaotic Dyn., 2012, vol. 17, no. 5, pp. 385–396.
Markeev, A.P., bration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).
Kolmogorov, A.N., Preservation of Conditionally Periodic Movements with Small Change in the Hamilton Function, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Volta Memorial Conference, Como, 1977), G. Casati, J. Ford (Eds.), Lect. Notes Phys.Monogr., vol. 93, Berlin: Springer, 1979, pp. 51–56; see also: Dokl. Akad. Nauk SSSR (N. S.), 1954, vol. 98, pp. 527–530 (Russian).
Arnol’d, V. I., Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics, Russian Math. Surveys, 1963, vol. 18, no. 6, pp. 85–191; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 6(114), pp. 91–192.
Moser, J. K., Lectures on Hamiltonian Systems, Mem. Amer. Math. Soc., vol. 81, Providence, R.I.: AMS, 1968.
Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz, I.A. Stegun (Eds.), New York: Dover, 1992.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kurakin, L.G., Ostrovskaya, I.V. On Stability of Thomson’s Vortex N-gon in the Geostrophic Model of the Point Bessel Vortices. Regul. Chaot. Dyn. 22, 865–879 (2017). https://doi.org/10.1134/S1560354717070085
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354717070085