Regular and Chaotic Dynamics

, Volume 21, Issue 3, pp 291–334 | Cite as

On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

  • Leonid G. Kurakin
  • Irina V. Ostrovskaya
  • Mikhail A. Sokolovskiy


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 when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.

A limiting case of a homogeneous fluid is also considered.

The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O G , formed by the orbits of a continuous family of steady-state rotations of a multipole. 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 cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.


discrete multipole vortex structure two-layer rotating fluid stability 


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  1. 1.
    Agee, E. M., Snow, J. T., and Clare, P. R., Multiple Vortex in the Tornado Cyclone and the Occurence of Tornado Families, Mon. Wea. Rev., 1976, vol. 104, no. 5, pp. 552–563.CrossRefGoogle Scholar
  2. 2.
    Aguiar, A.C.B., Read, P.L., Wordsworth, R.D., Salter, T., and Yamazaki, Y.H., A Laboratory Model of Saturn’s North Polar Hexagon, Icarus, 2009, vol. 206, no. 2, pp. 755–763.CrossRefGoogle Scholar
  3. 3.
    Allison, M., Godfrey, D. A., and Beebe, R. F., A Wave Dynamical Interpretation of Saturn’s Polar Hexagon, Science, 1990, vol. 247, no. 4946, pp. 1061–1063.CrossRefGoogle Scholar
  4. 4.
    Aref, H., Motion of Three Vortices, Phys. Fluids, 1979, vol. 22, no. 3, pp. 393–400.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aref, H., Stability of Reactive Equilibria of Three Vortices, Phys. Fluids, 2009, vol. 21, no. 9, 094101, 22 pp.MATHCrossRefGoogle Scholar
  6. 6.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D. L., Vortex Crystals, Adv. Appl. Mech., 2003, vol. 39, pp. 1–79.CrossRefGoogle Scholar
  7. 7.
    Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.Google Scholar
  8. 8.
    Ashwin, J. and Ganesh, R., Coherent Vortices in Strongly Coupled Liquids, Phys. Rev. Lett., 2011, vol. 106, no. 13, 135001, 4 pp.CrossRefGoogle Scholar
  9. 9.
    Baey, J.-M. and Carton, X., Vortex Multipoles in Two-Layer Rotating Shallow-Water Flows, J. Fluid Mech., 2002, vol. 460, pp. 151–175.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Barba, L.A., Nonshielded Multipolar Vortices at High Reynolds Number, Phys. Rev. E, 2006, vol. 73, no. 6, 065303, 4 pp.CrossRefGoogle Scholar
  11. 11.
    Barba, L.A. and Leonard, A., Emergence and Evolution of Tripole Vortices from Net-Circulation Initial Conditions, Phys. Fluids, 2007, vol. 19, no. 1, 017101, 16 pp.MATHCrossRefGoogle Scholar
  12. 12.
    Beckers, M. and van Heijst, G. J. F., The Observation of a Triangular Vortex in a Rotating Fluid, Fluid Dynam. Res., 1998, vol. 22, no. 5, pp. 265–279.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Blackmore, D., Ting, L., and Knio, O., Studies of Perturbed Three-Vortex Dynamics, J. Math. Phys., 2007, vol. 48, no. 6, 065402, 32 pp.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    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.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    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).Google Scholar
  16. 16.
    Burbea, J., Motions of Vortex Patches, Lett. Math. Phys., 1982, vol. 6, no. 1, pp. 1–16.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Burbea, J. and Landau, M., The Kelvin Waves in Vortex Dynamics and Their Stability, J. Comput. Phys., 1982, vol. 45, no. 1, pp. 127–156.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Cabral, H.E. and Schmidt, D. S., Stability of Relative Equilibria in the Problem of N + 1 Vortices, SIAM J. Math. Anal., 2000, vol. 31, no. 2, pp. 231–250.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Caillol, P. and Grimshaw, R., Steady Multipolar Planar Vortices with Nonlinear Critical Layers, Geophys. Astrophys. Fluid Dyn., 2004, vol. 98, no. 6, pp. 473–506.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Campbell, L. J., Transverse Normal Modes of Finite Vortex Arrays, Phys. Rev. A, 1981, vol. 24, no. 1, pp. 514–534.CrossRefGoogle Scholar
  21. 21.
    Carnevale, G. F. and Kloosterziel, R.C., Emergence and Evolution of Triangular Vortices, J. Fluid Mech., 1994, vol. 259, pp. 305–331.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Carton, X. J., Hydrodynamical Modelling of Oceanic Vortices, Surv. Geophys., 2001, vol. 22, no. 3, pp. 179–263.CrossRefGoogle Scholar
  23. 23.
    Carton, X. J., Instability of Surface Quasigeostrophic Vortices, J. Atmos. Sci., 2009, vol. 66, no. 4, pp. 1051–1062.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Carton, X., Ciani, D., Verron, J., Reinaud, J., and Sokolovskiy, M., Vortex Merger in Surface Quasi-Geostrophy, Geophys. Astrophys. Fluid Dyn., 2016, vol. 110, no. 1, pp. 1–22.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Carton, X. J., Flierl, G.R., and Polvani, L.M., The Generation of Tripoles from Unstable Axisymmetric Isolated Vortex Structures, Europhys. Lett., 1989, vol. 9, no. 4, pp. 339–344.CrossRefGoogle Scholar
  26. 26.
    Carton, X. and Legras, B., The Life-Cycle of Tripoles in Two-Dimensional Incompressible Flows, J. Fluid Mech., 1994, vol. 267, pp. 53–82.MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Castro, A., Córdoba, D.C., and Gómez-Serrano, J., Uniformly Rotating Analytic Global Patch Solutions for Active Scalars, Preprint, arXiv:1508.10655v1 (2015), 30 pp.Google Scholar
  28. 28.
    Chaplygin, S.A., On a Pulsating Cylindrical Vortex, Regul. Chaotic Dyn., 1899, vol. 10, no. 1, pp. 13–22.MathSciNetGoogle Scholar
  29. 29.
    Corréard, S.M. and Carton, X. J., Formation and Stability of Tripolar Vortices in Stratified Geostrophic Flows, Il Nuovo Cimento C, 1999, vol. 22, no. 6, pp. 767–777.Google Scholar
  30. 30.
    Crowdy, D.G., A Class of Exact Multipolar Vortices, Phys. Fluids, 1999, vol. 11, no. 9, pp. 2556–2564.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Crowdy, D.G., The Construction of Exact Multipolar Equilibria of the Two-Dimensional Euler Equations, Phys. Fluids, 2002, vol. 14, no. 1, pp. 257–267.MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Deem, G. S. and Zabusky, N. J., Vortex Waves: Stationary “V States”, Interactions, Recurrence, and Breaking, Phys. Rev. Lett., 1978, vol. 40, no. 13, pp. 859–862.CrossRefGoogle Scholar
  33. 33.
    Dhanak, M.R. and Marshall, M.P., Motion of an Elliptical Vortex under Applied Periodic Strain, Phys. Fluids A, 1993, vol. 5, no. 5, pp. 1224–1230.MATHCrossRefGoogle Scholar
  34. 34.
    Donnelly, R. J., Quantized Vortices in Helium II, Cambridge Studies in Low Temperature Physics, vol. 3, Cambridge: Cambridge Univ. Press, 1991.Google Scholar
  35. 35.
    Dritschel, D. G., The Stability and Energetics of Corotating Uniform Vortices, J. Fluid Mech., 1985, vol. 157, pp. 95–134.MATHCrossRefGoogle Scholar
  36. 36.
    Dritschel, D.G., The Nonlinear Evolution of Rotating Configurations of Uniform Vorticity, J. Fluid Mech., 1986, vol. 172, pp. 157–182.MATHCrossRefGoogle Scholar
  37. 37.
    Dritschel, D. G., The Stability of Elliptical Vortices in an External Straining Flow, J. Fluid Mech., 1990, vol. 210, pp. 223–261.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Dubosq, S. and Víudez, Á., Three-DimensionalMesoscale Dipole Frontal Collisions, J. Phys. Oceanogr., 2007, vol. 37, no. 9, pp. 2331–2344.CrossRefGoogle Scholar
  39. 39.
    Durkin, D. and Fajans, J., Experiments on Two-Dimensional Vortex Patterns, Phys. Fluids, 2000, vol. 12, no. 2, pp. 289–293.MATHCrossRefGoogle Scholar
  40. 40.
    Filyushkin, B.N. and Sokolovskiy, M. A., Modeling the Evolution of Intrathermocline Lenses in the Atlantic Ocean, J. Marine Res., 2011, vol. 69, nos. 2–3, pp. 191–220.CrossRefGoogle Scholar
  41. 41.
    Filyushkin, B.N., Sokolovskiy, M. A., Kozhelupova, N.G., and Vagina, I.M., Dynamics of Intrathermocline Lenses, Dokl. Earth Sci., 2010, vol. 434, no. 5, pp. 688–681.MATHGoogle Scholar
  42. 42.
    Fine, K. S., Cass, A. C., Flynn, W. G., and Dryscoll, C. F., Relaxation of 2D Turbulence to Vortex Crystal, Phys. Rev. Lett., 1995, vol. 75, no. 18, pp. 3277–3282.CrossRefGoogle Scholar
  43. 43.
    Flierl, G.R., On the Instability of Geostrophic Vortices, J. Fluid Mech., 1988, vol. 197, pp. 349–388.MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Flór, J.-B., Govers, W. S. S., van Heijst, G. J.F., and van Sluis, R., Formation of a Tripolar Vortex in a Stratified Fluid, Appl. Sci. Res., 1993, vol. 51, nos. 1–2, pp. 405–409.CrossRefGoogle Scholar
  45. 45.
    Flór, J.-B. and van Heijst, G. J. F., An Experimental Study of Dipolar Vortex Structures in a Stratified Fluid, J. Fluid Mech., 1994, vol. 279, pp. 101–133.CrossRefGoogle Scholar
  46. 46.
    Flór, J.-B. and van Heijst, G. J. F., Stable and Unstable Monopolar Vortices in a Stratified Fluid, J. Fluid Mech., 1996, vol. 311, pp. 257–287.MathSciNetCrossRefGoogle Scholar
  47. 47.
    Fujita, T.T. and Wakimoto, R.M., Five Scales of Airflow Associated with a Series of Downbursts on 16 July 1980, Mon. Wea. Rev., 1981, vol. 109, no. 6, pp. 1438–1456.CrossRefGoogle Scholar
  48. 48.
    Godfrey, D.A., A Hexagonal Feature around Saturn’s North Pole, Icarus, 1988, vol. 76, no. 2, pp. 335–356.CrossRefGoogle Scholar
  49. 49.
    Goryachev, D. N., On Some Cases of Motion of Rectilinear Parallel Vortex Filaments, Magister Dissertation, Moscow: Imp. Moscow Univ., 1898, 106 pp. (Russian).Google Scholar
  50. 50.
    Gröbli, W., Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden, Vierteljschr. Naturf. Ges. Zürich, 1877, vol. 22, pp. 37–81, 129–165.MATHGoogle Scholar
  51. 51.
    Gryanik, V. M., Dynamics of Singular Geostrophical Vortices in a 2-Level Model of the Atmosphere (Ocean), Izv. Atmos. Ocean Phys., 1983, vol. 19, no. 3, pp. 227–240.MathSciNetGoogle Scholar
  52. 52.
    Hamad, N., Millot, C., and Taupier-Letaget, I., The Surface Circulation in the Eastern Basin of the Mediterranean Sea, Sci. Mar., 2006, vol. 70, no. 3, pp. 457–503.Google Scholar
  53. 53.
    Hassainia, Z. and Hmidi, T., On the V-States for the Generalized Quasi-Geostrophic Equations, Comm. Math. Phys., 2015, vol. 337, no. 1, pp. 321–371.MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Havelock, T.H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag., 1931, vol. 11, no. 70, pp. 617–633.MATHCrossRefGoogle Scholar
  55. 55.
    Hmidi, T. and Mateu, J., Bifurcation of Rotating Patches from Kirchhoff Vortices, Preprint, arXiv:15008.04589v1 (2015), 21 pp.Google Scholar
  56. 56.
    Hmidi, T. and Mateu, J., Degenerate Bifurcation of the Rotating Patches, Preprint, arXiv:1510.04657v1 (2015), 39 pp.Google Scholar
  57. 57.
    Hmidi, T. and Mateu, J., Existence of Corotating and Counter-Rotating Vortex Pairs for Active Scalar Equations, Preprint, arXiv:1601.02242v1 (2016), 46 pp.Google Scholar
  58. 58.
    Hmidi, T., Mateu, J., and Verdera, J., Boundary Regularity of Rotating Vortex Patches, Arch. Ration. Mech. Anal., 2013, vol. 209, no. 1, pp. 171–208.MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Hmidi, T., Mateu, J., and Verdera, J., On Rotating Doubly Connected Vortices, J. Differential Equations, 2015, vol. 258, no. 4, pp. 1395–1429.MathSciNetMATHCrossRefGoogle Scholar
  60. 60.
    Hogg, N.G. and Stommel, H.M., The Heton, an Elementary Interaction between Discrete Baroclinic Geostrophic Vortices, and Its Implications Concerning Eddy Heat-Flow, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci., 1985, vol. 397, no. 1812, pp. 1–20.MATHCrossRefGoogle Scholar
  61. 61.
    Hooker, S.B., Brown, J. W., Kirwan, A. D., Lindermann, G. L., and Mied, R.P., Kinematics of a Warm-Core Dipole Ring, J. Geophys. Res. Oceans, 1995, vol. 100, no.C12, pp. 24797–24809.CrossRefGoogle Scholar
  62. 62.
    de la Hoz, F., Hassainia, Z., and Hmidi, T., Doubly Connected V-States for the Generalized Surface Quasi-Geostrophic Equations, Preprint, arXiv:1412.4587v2 (2015), 56 pp.MATHGoogle Scholar
  63. 63.
    Jischke, M. and Parang, M., On Laboratory Simulation of Tornado-Like Vortices, J. Atmos. Sci., 1974, vol. 31, no. 2, pp. 506–512.CrossRefGoogle Scholar
  64. 64.
    Joukowskii, N.E., Vortex Theory of a Rowing Screw, Tr. Otdel. Fiz. Nauk Ob-va Lubit. Estestvozn., 1912, vol. 16, no. 1, pp. 1–31.Google Scholar
  65. 65.
    Karapetyan, A.V., The Routh Theorem and Its Extensions, in Qualitative Theory of Differential Equations (Szeged, 1988), Colloq. Math. Soc. János Bolyai, vol. 53, Amsterdam: North Holland, 1990, pp. 271–290.Google Scholar
  66. 66.
    Karapetyan, A.V., The Stability of Steady Motions, Moscow: Editorial URSS, 1998 (Russian).Google Scholar
  67. 67.
    Karapetyan, A.V., Invariant Sets of Mechanical Systems: Lyapunov’s Methods in Stability and Control, Math. Comput. Modelling, 2002, vol. 36, no. 6, pp. 643–661.MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Kennelly, M.A., Evans, R. H., and Joyce, T. M., Small-Scale Cyclones on the Periphery of Gulf Stream Warm-Core Rings, J. Geophys. Res. Oceans, 1985, vol. 90, no.C5, pp. 8845–8857.CrossRefGoogle Scholar
  69. 69.
    Khazin, L.G., Regular Polygons of Point Vortices and Resonance Instability of Steady States, Sov. Phys. Dokl., 1976, vol. 230, no. 4, pp. 799–802.Google Scholar
  70. 70.
    Kida, S., Motion of an Elliptical Vortex in an Uniform Shear Flow, J. Phys. Soc. Japan, 1981, vol. 50, no. 10, pp. 3517–3520.CrossRefGoogle Scholar
  71. 71.
    Kirchhoff, G., Vorlesungen über mathematische Physik: Vol. 1. Mechanik, Leipzig: Teubner, 1876.MATHGoogle Scholar
  72. 72.
    Kizner, Z., Stability of Point-Vortex Multipoles Revisited, Phys. Fluids, 2001, vol. 23, no. 6, 064104, 11 pp.MATHGoogle Scholar
  73. 73.
    Kizner, Z., On the Stability of Two-Layer Geostrophic Point-Vortex Multipoles, Phys. Fluids, 2014, vol. 26, no. 4, 046602, 18 pp.MATHCrossRefGoogle Scholar
  74. 74.
    Kizner, Z., Berson, D., and Khvoles, R., Non-Circular Baroclinic Modons: Constructing Stationary Solutions, J. Fluid Mech., 2003, vol. 489, pp. 199–228.MathSciNetMATHCrossRefGoogle Scholar
  75. 75.
    Kizner, Z. and Khvoles, R., The Tripole Vortex: Experimental Evidence and Explicit Solutions, Phys. Rev. E, 2004, vol. 70, no. 1, 016307, 4 pp.MathSciNetCrossRefGoogle Scholar
  76. 76.
    Kizner, Z., Khvoles, R., and McWilliams, J.C., Rotating Multipoles on the f-and planes, Phys. Fluids, 2007, vol. 19, no. 1, 016603, 13 pp.MATHCrossRefGoogle Scholar
  77. 77.
    Kloosterziel, R.C. and van Heijst, G. J. F., On Tripolar Vortices, in Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, J.C. J. Nihoul and B.M. Jamart (Eds.), Amsterdam: Elsevier, 1989, pp. 609–625.CrossRefGoogle Scholar
  78. 78.
    Kloosterziel, R.C. and van Heijst, G. J. F., An Experimental Study of Unstable Barotropic Vortices in a Rotating Fluid, J. Fluid Mech., 1991, vol. 223, pp. 1–24.CrossRefGoogle Scholar
  79. 79.
    Kossin, J.P. and Schubert, W.H., Mesovortices, Polygonal Flow Patterns, and Rapid Pressure Falls in Hurricane-Like Vortices, J. Atmos. Sci., 2001, vol. 58, no. 15, pp. 2196–2209.CrossRefGoogle Scholar
  80. 80.
    Kossin, J.P. and Schubert, W.H., and Montgomery, M. T., Unstable Interactions between a Hurricane’s Primary Eyewall and a Secondary Ring of Enhanced Vorticity, J. Atmos. Sci., 2000, vol. 57, no. 24, pp. 3893–3917.MathSciNetCrossRefGoogle Scholar
  81. 81.
    Kozlov, V. F., Construction of the Stationary States of Vortex Patches by the Method of Perturbations, Izv. Atmos. Ocean Phys., 1991, vol. 27, no. 1, pp. 115–130.MathSciNetGoogle Scholar
  82. 82.
    Kozlov, V.V., On the Degree of Instability, J. Appl. Math. Mech., 1993, vol. 57, no. 5, pp. 14–19.MathSciNetCrossRefGoogle Scholar
  83. 83.
    Kulikov, D.V., Mikkelsen, R., Naumov, I. V., and Okulov, V. L., Diagnostics of Bubble-Mode Vortex Breakdown in Swirling Flow in a Large-Aspect-Ratio Cylinder, Tech. Phys. Lett, 2014, vol. 40, no. 4, pp. 87–94.Google Scholar
  84. 84.
    Kunze, E. and Lueck, R., Velocity Profiles in a Warm-Core Ring, J. Phys. Oceanogr., 1986, vol. 16, no. 5, pp. 991–995.CrossRefGoogle Scholar
  85. 85.
    Kurakin, L. G., On the Stability of the Regular N-Sided Polygon of Vortices, Dokl. Phys., 1994, vol. 335, no. 6, pp. 729–731.MATHGoogle Scholar
  86. 86.
    Kurakin, L. G., Stability, Resonances, and Instability of the Regular Vortex Polygons in the Circular Domain, Dokl. Phys., 2004, vol. 399, no. 1, pp. 52–55.MathSciNetGoogle Scholar
  87. 87.
    Kurakin, L. G., On Stability of a Regular Vortex Polygon in the Circular Domain, J. Math. Fluid Mech., 2005, vol. 7, suppl. 3, S376–S386.MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    Kurakin, L. G., On the Stability of Thomson’s Vortex Configurations inside a Circular Domain, Regul. Chaotic Dyn., 2010, vol. 15, no. 1, pp. 40–58.MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Kurakin, L. G., The Stability of the Steady Rotation of a System of Three Equidistant Vortices outside a Circle, J. Appl. Math. Mech., 2011, vol. 75, no. 2, pp. 327–337.MathSciNetMATHCrossRefGoogle Scholar
  90. 90.
    Kurakin, L. G., Influence of Annular Boundaries on Thomson’s Vortex Polygon Stability, Chaos, 2014, vol. 14, no. 2, 023105, 12 pp.MathSciNetGoogle Scholar
  91. 91.
    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. Mexicana, 2016, DOI:10.1007/s40590-016-0121-y.Google Scholar
  92. 92.
    Kurakin, L. G. and Ostrovskaya, I.V., Stability of the Thomson Vortex Polygon with Evenly Many Vortices outside a Circular Domain, Siberian Math. J., 2010, vol. 51, no. 3, pp. 584–598.MathSciNetMATHCrossRefGoogle Scholar
  93. 93.
    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.MathSciNetMATHCrossRefGoogle Scholar
  94. 94.
    Kurakin, L. G., Ostrovskaya, I.V., and Sokolovskiy, M.A., Stability of Discrete Vortex Multipoles in Homogeneous and Two-Layer Rotating Fluid, Dokl. Phys., 2015, vol. 462, no. 2, pp. 161–167.MathSciNetGoogle Scholar
  95. 95.
    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.MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Kurakin, L. G. and Yudovich, V. I., On Nonlinear Stability of Steady Rotation of a Regular Vortex Polygon, Dokl. Phys., 2002, vol. 384, no. 4, pp. 476–482.MathSciNetMATHGoogle Scholar
  97. 97.
    Lappa, M., Rotating Thermal Flows in Natural and Industrial Processes, Chichester: Wiley, 2012.MATHCrossRefGoogle Scholar
  98. 98.
    Larichev, V.D. and Reznik, G.M., On Collisions between Two-Dimensional Solitary Rossby Waves, Okeanologiya, 1983, vol. 23, no. 5, pp. 725–734.Google Scholar
  99. 99.
    Legras, B. and Dritschel, D. G., The Elliptical Model of Two-Dimensional Vortex Dynamics: 1. The Basic State, Phys. Fluids A, 1991, vol. 3, no. 5, pp. 845–854.MathSciNetMATHCrossRefGoogle Scholar
  100. 100.
    Legras, B., Santangelo, P., and Benzi, R., High-Resolution Numerical Experiments for Forced Two-Dimensional Turbulence, Europhys. Lett., 1988, vol. 5, no. 1, pp. 37–42.CrossRefGoogle Scholar
  101. 101.
    Levi-Civita, T., Sur la recherche des solutions particulières des systèmes différenties et sur les mouvments stationaires, Prace Mat.-Fiz., 1906, vol. 17, no. 1, pp. 1–40.MATHGoogle Scholar
  102. 102.
    Luz, D., Berry, D. L., Piccioni, G., Drossart, P., Politi, R., Wilson, C.F., Erard, S., and Nuccilli, F., Venus’s Southern Polar Vortex Reveals Precessing Circulation, Science, 2011, vol. 332, no. 6029, pp. 577–580.CrossRefGoogle Scholar
  103. 103.
    Luzzatto-Fegiz, P. and Williamson, C.H.K., Stability of Elliptical Vortices from “Imperfect-Velocity-Impulse” Diagrams, Theor. Comput. Fluid Dyn., 2010, vol. 24, nos. 1–4, pp. 181–188.MATHCrossRefGoogle Scholar
  104. 104.
    Luzzatto-Fegiz, P. and Williamson, C.H.K., An Efficient and General Numerical Method to Compute Steady Uniform Vortices, J. Comput. Phys., 2011, vol. 230, no. 17, pp. 6495–6511.MathSciNetMATHCrossRefGoogle Scholar
  105. 105.
    Lyapunov, A. M., On Constant Screw Motions of a Rigid Body in a Liquid, Soobshch. Kharkov. Mat. Obshch., Ser. 2, 1888, vol. 1, nos. 1–2, pp. 7–60 (Russian).Google Scholar
  106. 106.
    Makarov, V. G. and Kizner, Z., Stability and Evolution of Uniform-Vorticity Dipoles, J. Fluid Mech., 2011, vol. 672, pp. 307–325.MathSciNetMATHCrossRefGoogle Scholar
  107. 107.
    Malkin, I.G., On Stability of Periodic Solutions of Dynamic Systems, Prikl. Mat. Mekh., 1944, vol. 8, no. 4, pp. 327–335 (Russian).MathSciNetMATHGoogle Scholar
  108. 108.
    Malkin, I.G., Stability Theory of Motion, Moscow: Nauka, 1966.MATHGoogle Scholar
  109. 109.
    Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).Google Scholar
  110. 110.
    Marsden, J. and Weinstein, A., Reduction of Symplectic Manifolds with Symmetry, Rep. Mathematical Phys., 1974, vol. 5, no. 1, pp. 121–130.MathSciNetMATHCrossRefGoogle Scholar
  111. 111.
    Mayer, A.M., Floating Magnets, Nature, 1878, vol. 18, pp. 258–260.CrossRefGoogle Scholar
  112. 112.
    Mayer, A.M., A Note on Experiments with Floating Magnets, Am. J. Sci. Arts, Ser.3, 1878, vol. 15, no. 88, pp. 276–277.Google Scholar
  113. 113.
    Meleshko, V.V. and Aref, H., A Bibliography of Vortex Dynamics 1858–1956, Adv. Appl. Mech., 2007, vol. 41, pp. 197–292.CrossRefGoogle Scholar
  114. 114.
    Meleshko, V.V. and Konstantinov, M.Yu., Dynamics of Vortex Structures, Kiev: Naukova Dumka, 1993 (Russian).Google Scholar
  115. 115.
    Mertz, G. T., Stability of Body-Centered Polygonal Configurations of Ideal Vortices, Phys. Fluids, 1978, vol. 21, no. 7, pp. 1092–1095.MATHCrossRefGoogle Scholar
  116. 116.
    Mitchell, T. B. and Rossi, L. F., The Evolution of Kirchhoff Elliptic Vortices, Phys. Fluids, 2008, vol. 20, no. 5, 054103, 12 pp.MATHCrossRefGoogle Scholar
  117. 117.
    Morikawa, G.K. and Swenson, E.V., Interacting Motion of Rectilinear Geostrophic Vortices, Phys. Fluids, 1971, vol. 14, no. 6, pp. 1058–1073.CrossRefGoogle Scholar
  118. 118.
    Morton, W.V., Vortex Polygons, Proc. R. Irish Acad., Sect. A, 1935, vol. 42, pp. 21–29.MATHGoogle Scholar
  119. 119.
    Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.Google Scholar
  120. 120.
    Newton, P.K. and Ostrovskyi, V., Energy-Momentum Stability of Icosahedral Configurations of Point Vortices on a Sphere, J. Nonlinear Sci., 2012, vol. 22, no. 4, pp. 499–515.MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Nikolenko, N.V., Stability of the Family of Cycles of Dynamical Systems, Izv. Vyssh. Uchebn. Zaved. Matem., 1973, no. 7, pp. 63–69 (Russian).MathSciNetGoogle Scholar
  122. 122.
    Oey, L.-Y., Loop Current and Deep Eddies, J. Phys. Oceanogr., 2008, vol. 38, no. 7, pp. 1426–1449.CrossRefGoogle Scholar
  123. 123.
    Okamoto, A., Hara, K., Nagaoka, K., Yoshimura, S., Vranje?s, J., Kono, M., and Tanaka, M.Y., Experimental Observation of a Tripolar Vortex in a Plasma, Phys. Plasmas, 2003, vol. 10, no. 6, pp. 2211–2216.CrossRefGoogle Scholar
  124. 124.
    Okulov, V. L., On the Stability of Multiple Helical Vortices, J. Fluid Mech., 2004, vol. 521, pp. 319–342.MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Okulov, V. L., Naumov, I.V., and Sørensen, J.N., Self-Organized Vortex Multiplets in Swirling Flow, Tech. Phys. Lett., 2008, vol. 34, no. 15, pp. 89–95.Google Scholar
  126. 126.
    Okulov, V. L. and Sørensen, J.N., Instability of a Vortex Wake behind Wind Turbines, Dokl. Phys., 2004, vol. 399, no. 6, pp. 775–779.Google Scholar
  127. 127.
    Orlandi, P. and van Heijst, G. J. F., Numerical Simulations of Tripolar Vortices in 2D Flows, Fluid Dyn. Res., 1992, vol. 9, no. 3, pp. 179–206.CrossRefGoogle Scholar
  128. 128.
    Overman, E.A., Steady-State Solutions of the Euler Equations in Two Dimensions: 2. Local Analysis of Limiting V-States, SIAM J.Appl. Math., 1986, vol. 46, no. 5, pp. 765–800.MathSciNetMATHCrossRefGoogle Scholar
  129. 129.
    Patrick, G.W., Relative Equilibria in Hamiltonian System: The Dynamic Interpretation of Nonlinear Stability on a Reduced Phase Space, J. Geom. Phys., 1992, vol. 9, no. 2, pp. 111–119.MathSciNetMATHCrossRefGoogle Scholar
  130. 130.
    Peng, S., Robinson, W.A., and Li, S., Mechanisms for the NAO Responses to the North Atlantic SST Tripole, J. Climate, 2003, vol. 16, no. 12, pp. 1987–2004.CrossRefGoogle Scholar
  131. 131.
    Pingree, R.D. and Le Cann, B., Three Anticyclonic Slope Water Oceanic eDDIES (SWODDIES) in the Southern Bay of Biscay in 1990, Deep Sea Res. Part 1 Oceanogr. Res. Pap., 1992, vol. 39, nos. 7/8A, pp. 1147–1175.CrossRefGoogle Scholar
  132. 132.
    Polvani, L.M. and Flierl, G.R., Generalized Kirchhoff Vortices, Phys. Fluids, 1986, vol. 29, no. 8, pp. 2376–2379.MATHCrossRefGoogle Scholar
  133. 133.
    Polvani, L.M., Two-Layer Geostrophic Vortex Dynamics: 2.Alignment and Two-Layer V-States, J. Fluid Mech., 1991, vol. 225, pp. 241–270.MathSciNetMATHCrossRefGoogle Scholar
  134. 134.
    Polvani, L.M., Zabusky, N. J., and Flier, G.R., Two-Layer Geostrophic Vortex Dynamics: 1. Upper-Layer V-States and Merger, J. Fluid Mech., 1989, vol. 205, pp. 215–242.MathSciNetMATHCrossRefGoogle Scholar
  135. 135.
    Pozharitsky, G. K., On the Construction of Lyapunov Functions from the Integrals of the Equations of the Perturbed Motion, J. Appl. Math. Mech., 1958, vol. 22, no. 2, pp. 145–154.Google Scholar
  136. 136.
    Robert, R. and Rosier, C., Long Range Predictability of Atmospheric Flows, Nonlinear Proc. Geophys., 2001, vol. 8, nos. 1/2, pp. 55–67.CrossRefGoogle Scholar
  137. 137.
    Rodríguez-Marroyo, R., Víudez, A., and Ruiz, S., Vortex Merger in Oceanic Tripoles, J. Phys. Oceanogr., 2011, vol. 41, no. 6, pp. 1239–1251.CrossRefGoogle Scholar
  138. 138.
    Routh, E. J., A Treatise on Stability of a Given State of Motion, London: McMillan, 1877.Google Scholar
  139. 139.
    Rubanovskii, V.N., The Bifurcation and Stability of Steady Motions, Theor. Appl. Mech., 1974, vol. 5, no. 1, pp. 67–79 (Russian).MathSciNetMATHGoogle Scholar
  140. 140.
    Rumyantsev, V.V. and Oziraner, A. S., Motion Stability and Stabilization with Respect to Part of Variables, Moscow: Nauka, 1987 (Russian).MATHGoogle Scholar
  141. 141.
    Saffman, P.G., Vortex Dynamics, Cambridge: Cambridge Univ. Press, 1992.MATHGoogle Scholar
  142. 142.
    Sahai, R., Multi-Polar Structures in Young Planetary and Protoplanetary Nebulae, in Planetary Nebulae: Their Evolution and Role in the Universe: Proc. of the 209th Symposium of the International Astronomical Union (Canberra, Australia, 19–23 November, 2001), pp. 471–479.Google Scholar
  143. 143.
    Salvadori, L., Un osservazione su di un criteria di stabilitá del Routh, Rend. Accad. Sci. Fis. Math. Napoli (IV), 1953, vol. 20, pp. 267–272.Google Scholar
  144. 144.
    Schubert, W.H., Montgomery, M.T., Taft, R. K., Guinn, T.A., Fulton, S.R., Kossin, J.P., and Edwards, J.P., Polygonal Eyewalls, Asymmetric Eye Contraction, and Potential Vorticity Mixing in Hurricanes, J. Atmos. Sci., 1999, vol. 56, no. 9, pp. 1197–1223.CrossRefGoogle Scholar
  145. 145.
    Shteinbuch-Fridman, B., Makarov, V., Carton, X., and Kizner, Z., Two-Layer Geostrophic Tripoles Comprised by Patches Uniform Potential Vorticity, Phys. Fluids, 2015, vol. 27, no. 3, 036602, 11 pp.MATHCrossRefGoogle Scholar
  146. 146.
    Simo, J.C., Lewis, D., and Marsden, J. E., Stability of Relative Equilibria: P. 1. The educed Energy-Momentum Method, Arch. Ration. Mech Anal., 1991, vol. 115, no. 1, pp. 15–59.MathSciNetMATHCrossRefGoogle Scholar
  147. 147.
    Sokolovskiy, M.A., Head-On Collisions of Distributed Hetons, Trans. Dokl. USSR Acad. Sci. Earth Sci. Sect., 1989, vol. 306, no. 1, pp. 198–202.Google Scholar
  148. 148.
    Sokolovskiy, M.A. and Carton, X., Baroclinic Multipole Formation from Heton Interaction, Fluid Dyn. Res., 2010, vol. 42, no. 4, 045501, 31 pp.MathSciNetMATHCrossRefGoogle Scholar
  149. 149.
    Sokolovskiy, M.A., Koshel, K.V., and Verron, J., Three-Vortex Quasi-Geostrophic Dynamics in a Two-Layer Fluid: P. 1. Analysis of Relative and Absolute Motions, J. Fluid Mech., 2013, vol. 717, pp. 232–254.MathSciNetMATHCrossRefGoogle Scholar
  150. 150.
    Sokolovskiy, M.A. and Verron, J., Finite-Core Hetons: Stability and Interactions, J. Fluid Mech., 2000, vol. 423, pp. 127–154.MathSciNetMATHCrossRefGoogle Scholar
  151. 151.
    Sokolovskiy, M.A. and Verron, J., Some Properties of motion of A+1 vortices in a Two-Layer Rotating Fluid, Nelin. Dinam., 2006, vol. 2, no. 1, pp. 27–54 (Russian).CrossRefGoogle Scholar
  152. 152.
    Sokolovskiy, M.A. and Verron, J., On the motion of A + 1 Vortices in a Two-Layer Rotating Fluid, in Proc. of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (Moscow, 25–30 August, 2006), A.V. Borisov, V.V. Kozlov, I. S. Mamaev, and M.A. Sokolovisky (Eds.), Dordrecht: Springer, 2008, pp. 481–490.CrossRefGoogle Scholar
  153. 153.
    Sokolovskiy, M.A. and Verron, J., Dynamics of Vortex Structures in a Stratified Rotating Fluid, Atmos. Oceanogr. Sci. Lib., vol. 47, Cham: Springer, 2014.Google Scholar
  154. 154.
    Sutyrin, G.G., McWilliams, J.C., and Saravanan, R., Co-Rotating Stationary States and Vertical Alignment of Geostrophic Vortices with Thin Cores, J. Fluid Mech., 1998, vol. 357, pp. 321–349.MathSciNetMATHCrossRefGoogle Scholar
  155. 155.
    Synge, J. L., On the Motion of Three Vortices, Canad. J. Math., 1949, vol. 1, pp. 257–270.MathSciNetMATHCrossRefGoogle Scholar
  156. 156.
    Tang, Y., Nonlinear Stability of Vortex Patches, Trans. Amer. Math. Soc., 1987, vol. 304, no. 2, pp. 617–638.MathSciNetMATHCrossRefGoogle Scholar
  157. 157.
    Tavantzis, J. and Ting, L., The Dynamics of Three Vortices Revisited, Phys. Fluids, 1988, vol. 31, no. 6, pp. 1392–1409.MathSciNetMATHCrossRefGoogle Scholar
  158. 158.
    Thomson, W., Floating Magnets, Nature, 1878, vol. 18, pp. 13–14.CrossRefGoogle Scholar
  159. 159.
    Thomson, J. J., Treatise on the Motion of Vortex Rings, London: Macmillan, 1883, pp. 94–108.Google Scholar
  160. 160.
    Trieling, R.R., van Heijst, G. J. F., and Kizner, Z., Laboratory Experiments on Multipolar Vortices in a Rotating Fluid, Phys. Fluids, 2010, vol. 22, no. 9, 094104, 12 pp.CrossRefGoogle Scholar
  161. 161.
    van Heijst, G. J.F., Topography Effects on Vortices in a Rotating Fluid, Meccanica, 1994, vol. 29, no. 4, pp. 431–451.MathSciNetMATHCrossRefGoogle Scholar
  162. 162.
    van Heijst, G. J. F. and Kloosterziel, R.C., Tripolar Vortices in a Rotating Fluid, Nature, 1989, vol. 338, pp. 569–571.CrossRefGoogle Scholar
  163. 163.
    van Heijst, G. J. F., Kloosterziel, R. C., and Williams, C.W.M., Laboratory Experiments on the Tripolar Vortex in a Rotating Fluid, J. Fluid Mech., 1991, vol. 225, pp. 301–331.CrossRefGoogle Scholar
  164. 164.
    Vranjes, J., Tripolar Vortex in Plasma Flow, Planet. Space Sci., 1999, vol. 47, no. 12, pp. 1531–1535.CrossRefGoogle Scholar
  165. 165.
    Vranjes, J., Maríc, G., and Shukla, P.K., Tripolar Vortices and Vortex Chains in Dusty Plasma, Phys. Lett. A, 1999, vol. 258, nos. 4–6, pp. 317–322.CrossRefGoogle Scholar
  166. 166.
    Vranjes, J., Okamoto, A., Yoshimura, S., Poedts, S., Kono, M., and Tanaka, M.Y., Analytical Description of a Neutral-Induced Tripole Vortex in a Plasma, Phys. Rev. Lett., 2002, vol. 89, no. 26, 265002, 4 pp.CrossRefGoogle Scholar
  167. 167.
    Wan, Y.H., The Stability of Rotating Vortex Patches, Comm. Math. Phys., 1986, vol. 107, no. 1, pp. 1–20.MathSciNetMATHCrossRefGoogle Scholar
  168. 168.
    Ward, N.B., The Exploration of Certain Features of Tornado Dynamics Using a Laboratory Model, J. Atmos. Sci., 1972, vol. 29, no. 6, pp. 1194–1204.CrossRefGoogle Scholar
  169. 169.
    Wu, H.M., Overman, E.A. II, and Zabusky, N. J., Steady-State Solutions of the Euler Equations: Rotating and Translating V-States with Limiting Cases: 1. Numerical Algorithms and Results, J. Comput. Phys., 1984, vol. 53, no. 1, pp. 42–71.MathSciNetMATHCrossRefGoogle Scholar
  170. 170.
    Wurman, J., Kosiba, K., Robinson, P., and Marshall, T., The Role ofMultiple-Vortex Tornado Structure in Causing Storm Researcher Ratalities, Bull. Amer. Meteor. Soc., 2014, vol. 95, no. 1, pp. 31–45.CrossRefGoogle Scholar
  171. 171.
    Xu, F.-H., Chang, Y.-L., Oey, L.-Y., and Hamilton, P., Loop Current Growth and Eddy Shedding Using Models and Observations: Analyses of the July 2011 Eddy-Shedding Event, J. Phys. Oceanogr., 2013, vol. 43, no. 5, pp. 1015–1027.CrossRefGoogle Scholar
  172. 172.
    Yarmchuck, E., Gordon, M., and Packard, R., Observation of Stationary Vortex Array in Rotating Superfluid Helium, Phys. Rev. Lett., 1979, vol. 43, no. 3, pp. 214–218.CrossRefGoogle Scholar
  173. 173.
    Yarmchuck, E. and Packard, R., Photographic Studies of Quantized Vortex Lines, J. Low Temp. Phys., 1982, vol. 46, nos. 5/6, pp. 479–515.CrossRefGoogle Scholar
  174. 174.
    Yudovich, V. I., On the Stability of Forced Oscillations of the Liquid, Dokl. Akad. Nauk SSSR, 1970, vol. 195, no. 2, pp. 292–295 (Russian).MathSciNetGoogle Scholar
  175. 175.
    Yudovich, V. I., On the Stability of Self-Excited Oscillations in a Liquid, Dokl. Akad. Nauk SSSR, 1970, vol. 195, no. 3, pp. 574–576 (Russian).MathSciNetGoogle Scholar
  176. 176.
    Yudovich, V. I., Mathematical Problems of Hydrodynamic Stability Theory, Dissertation, Moscow: Institute for Problems of Mechanics, 1972 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  • Leonid G. Kurakin
    • 1
    • 2
  • Irina V. Ostrovskaya
    • 1
  • Mikhail A. Sokolovskiy
    • 3
    • 4
  1. 1.Institute for Mathematics, Mechanics and Computer SciencesSouthern Federal UniversityRostov-on-DonRussia
  2. 2.Southern Mathematical InstituteVladikavkaz Scienific Center of RASVladikavkazRussia
  3. 3.Water Problems Institute, RASMoscowRussia
  4. 4.P. P. Shirshov Institute of Oceanology, RASMoscowRussia

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