Skip to main content
SpringerLink
Log in

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

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  4. Aref, H., Motion of Three Vortices, Phys. Fluids, 1979, vol. 22, no. 3, pp. 393–400.

    Article  MathSciNet  MATH  Google Scholar 

  5. Aref, H., Stability of Reactive Equilibria of Three Vortices, Phys. Fluids, 2009, vol. 21, no. 9, 094101, 22 pp.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  7. Arnol’d, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1997.

  8. Ashwin, J. and Ganesh, R., Coherent Vortices in Strongly Coupled Liquids, Phys. Rev. Lett., 2011, vol. 106, no. 13, 135001, 4 pp.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  10. Barba, L.A., Nonshielded Multipolar Vortices at High Reynolds Number, Phys. Rev. E, 2006, vol. 73, no. 6, 065303, 4 pp.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Burbea, J., Motions of Vortex Patches, Lett. Math. Phys., 1982, vol. 6, no. 1, pp. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  20. Campbell, L. J., Transverse Normal Modes of Finite Vortex Arrays, Phys. Rev. A, 1981, vol. 24, no. 1, pp. 514–534.

    Article  Google Scholar 

  21. Carnevale, G. F. and Kloosterziel, R.C., Emergence and Evolution of Triangular Vortices, J. Fluid Mech., 1994, vol. 259, pp. 305–331.

    Article  MathSciNet  Google Scholar 

  22. Carton, X. J., Hydrodynamical Modelling of Oceanic Vortices, Surv. Geophys., 2001, vol. 22, no. 3, pp. 179–263.

    Article  Google Scholar 

  23. Carton, X. J., Instability of Surface Quasigeostrophic Vortices, J. Atmos. Sci., 2009, vol. 66, no. 4, pp. 1051–1062.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Chaplygin, S.A., On a Pulsating Cylindrical Vortex, Regul. Chaotic Dyn., 1899, vol. 10, no. 1, pp. 13–22.

    MathSciNet  Google Scholar 

  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. Crowdy, D.G., A Class of Exact Multipolar Vortices, Phys. Fluids, 1999, vol. 11, no. 9, pp. 2556–2564.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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. Dritschel, D. G., The Stability and Energetics of Corotating Uniform Vortices, J. Fluid Mech., 1985, vol. 157, pp. 95–134.

    Article  MATH  Google Scholar 

  36. Dritschel, D.G., The Nonlinear Evolution of Rotating Configurations of Uniform Vorticity, J. Fluid Mech., 1986, vol. 172, pp. 157–182.

    Article  MATH  Google Scholar 

  37. Dritschel, D. G., The Stability of Elliptical Vortices in an External Straining Flow, J. Fluid Mech., 1990, vol. 210, pp. 223–261.

    Article  MathSciNet  MATH  Google Scholar 

  38. Dubosq, S. and Víudez, Á., Three-DimensionalMesoscale Dipole Frontal Collisions, J. Phys. Oceanogr., 2007, vol. 37, no. 9, pp. 2331–2344.

    Article  Google Scholar 

  39. Durkin, D. and Fajans, J., Experiments on Two-Dimensional Vortex Patterns, Phys. Fluids, 2000, vol. 12, no. 2, pp. 289–293.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    Article  Google Scholar 

  43. Flierl, G.R., On the Instability of Geostrophic Vortices, J. Fluid Mech., 1988, vol. 197, pp. 349–388.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  48. Godfrey, D.A., A Hexagonal Feature around Saturn’s North Pole, Icarus, 1988, vol. 76, no. 2, pp. 335–356.

    Article  Google Scholar 

  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. Gröbli, W., Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden, Vierteljschr. Naturf. Ges. Zürich, 1877, vol. 22, pp. 37–81, 129–165.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  54. Havelock, T.H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag., 1931, vol. 11, no. 70, pp. 617–633.

    Article  MATH  Google Scholar 

  55. Hmidi, T. and Mateu, J., Bifurcation of Rotating Patches from Kirchhoff Vortices, Preprint, arXiv:15008.04589v1 (2015), 21 pp.

    Google Scholar 

  56. Hmidi, T. and Mateu, J., Degenerate Bifurcation of the Rotating Patches, Preprint, arXiv:1510.04657v1 (2015), 39 pp.

    Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  59. Hmidi, T., Mateu, J., and Verdera, J., On Rotating Doubly Connected Vortices, J. Differential Equations, 2015, vol. 258, no. 4, pp. 1395–1429.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    MATH  Google Scholar 

  63. Jischke, M. and Parang, M., On Laboratory Simulation of Tornado-Like Vortices, J. Atmos. Sci., 1974, vol. 31, no. 2, pp. 506–512.

    Article  Google Scholar 

  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. 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. Karapetyan, A.V., The Stability of Steady Motions, Moscow: Editorial URSS, 1998 (Russian).

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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. Kida, S., Motion of an Elliptical Vortex in an Uniform Shear Flow, J. Phys. Soc. Japan, 1981, vol. 50, no. 10, pp. 3517–3520.

    Article  Google Scholar 

  71. Kirchhoff, G., Vorlesungen über mathematische Physik: Vol. 1. Mechanik, Leipzig: Teubner, 1876.

    MATH  Google Scholar 

  72. Kizner, Z., Stability of Point-Vortex Multipoles Revisited, Phys. Fluids, 2001, vol. 23, no. 6, 064104, 11 pp.

    MATH  Google Scholar 

  73. Kizner, Z., On the Stability of Two-Layer Geostrophic Point-Vortex Multipoles, Phys. Fluids, 2014, vol. 26, no. 4, 046602, 18 pp.

    Article  MATH  Google Scholar 

  74. Kizner, Z., Berson, D., and Khvoles, R., Non-Circular Baroclinic Modons: Constructing Stationary Solutions, J. Fluid Mech., 2003, vol. 489, pp. 199–228.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  82. Kozlov, V.V., On the Degree of Instability, J. Appl. Math. Mech., 1993, vol. 57, no. 5, pp. 14–19.

    Article  MathSciNet  Google Scholar 

  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. Kunze, E. and Lueck, R., Velocity Profiles in a Warm-Core Ring, J. Phys. Oceanogr., 1986, vol. 16, no. 5, pp. 991–995.

    Article  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  90. Kurakin, L. G., Influence of Annular Boundaries on Thomson’s Vortex Polygon Stability, Chaos, 2014, vol. 14, no. 2, 023105, 12 pp.

    MathSciNet  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    MathSciNet  MATH  Google Scholar 

  97. Lappa, M., Rotating Thermal Flows in Natural and Industrial Processes, Chichester: Wiley, 2012.

    Book  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Makarov, V. G. and Kizner, Z., Stability and Evolution of Uniform-Vorticity Dipoles, J. Fluid Mech., 2011, vol. 672, pp. 307–325.

    Article  MathSciNet  MATH  Google Scholar 

  107. Malkin, I.G., On Stability of Periodic Solutions of Dynamic Systems, Prikl. Mat. Mekh., 1944, vol. 8, no. 4, pp. 327–335 (Russian).

    MathSciNet  MATH  Google Scholar 

  108. Malkin, I.G., Stability Theory of Motion, Moscow: Nauka, 1966.

    MATH  Google Scholar 

  109. Markeev, A.P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).

    Google Scholar 

  110. Marsden, J. and Weinstein, A., Reduction of Symplectic Manifolds with Symmetry, Rep. Mathematical Phys., 1974, vol. 5, no. 1, pp. 121–130.

    Article  MathSciNet  MATH  Google Scholar 

  111. Mayer, A.M., Floating Magnets, Nature, 1878, vol. 18, pp. 258–260.

    Article  Google Scholar 

  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. Meleshko, V.V. and Aref, H., A Bibliography of Vortex Dynamics 1858–1956, Adv. Appl. Mech., 2007, vol. 41, pp. 197–292.

    Article  Google Scholar 

  114. Meleshko, V.V. and Konstantinov, M.Yu., Dynamics of Vortex Structures, Kiev: Naukova Dumka, 1993 (Russian).

    Google Scholar 

  115. Mertz, G. T., Stability of Body-Centered Polygonal Configurations of Ideal Vortices, Phys. Fluids, 1978, vol. 21, no. 7, pp. 1092–1095.

    Article  MATH  Google Scholar 

  116. Mitchell, T. B. and Rossi, L. F., The Evolution of Kirchhoff Elliptic Vortices, Phys. Fluids, 2008, vol. 20, no. 5, 054103, 12 pp.

    Article  MATH  Google Scholar 

  117. Morikawa, G.K. and Swenson, E.V., Interacting Motion of Rectilinear Geostrophic Vortices, Phys. Fluids, 1971, vol. 14, no. 6, pp. 1058–1073.

    Article  Google Scholar 

  118. Morton, W.V., Vortex Polygons, Proc. R. Irish Acad., Sect. A, 1935, vol. 42, pp. 21–29.

    MATH  Google Scholar 

  119. Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.

    Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  122. Oey, L.-Y., Loop Current and Deep Eddies, J. Phys. Oceanogr., 2008, vol. 38, no. 7, pp. 1426–1449.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  124. Okulov, V. L., On the Stability of Multiple Helical Vortices, J. Fluid Mech., 2004, vol. 521, pp. 319–342.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  132. Polvani, L.M. and Flierl, G.R., Generalized Kirchhoff Vortices, Phys. Fluids, 1986, vol. 29, no. 8, pp. 2376–2379.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Robert, R. and Rosier, C., Long Range Predictability of Atmospheric Flows, Nonlinear Proc. Geophys., 2001, vol. 8, nos. 1/2, pp. 55–67.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  138. Routh, E. J., A Treatise on Stability of a Given State of Motion, London: McMillan, 1877.

    Google Scholar 

  139. Rubanovskii, V.N., The Bifurcation and Stability of Steady Motions, Theor. Appl. Mech., 1974, vol. 5, no. 1, pp. 67–79 (Russian).

    MathSciNet  MATH  Google Scholar 

  140. Rumyantsev, V.V. and Oziraner, A. S., Motion Stability and Stabilization with Respect to Part of Variables, Moscow: Nauka, 1987 (Russian).

    MATH  Google Scholar 

  141. Saffman, P.G., Vortex Dynamics, Cambridge: Cambridge Univ. Press, 1992.

    MATH  Google Scholar 

  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. 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. 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.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Sokolovskiy, M.A. and Carton, X., Baroclinic Multipole Formation from Heton Interaction, Fluid Dyn. Res., 2010, vol. 42, no. 4, 045501, 31 pp.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  150. Sokolovskiy, M.A. and Verron, J., Finite-Core Hetons: Stability and Interactions, J. Fluid Mech., 2000, vol. 423, pp. 127–154.

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  155. Synge, J. L., On the Motion of Three Vortices, Canad. J. Math., 1949, vol. 1, pp. 257–270.

    Article  MathSciNet  MATH  Google Scholar 

  156. Tang, Y., Nonlinear Stability of Vortex Patches, Trans. Amer. Math. Soc., 1987, vol. 304, no. 2, pp. 617–638.

    Article  MathSciNet  MATH  Google Scholar 

  157. Tavantzis, J. and Ting, L., The Dynamics of Three Vortices Revisited, Phys. Fluids, 1988, vol. 31, no. 6, pp. 1392–1409.

    Article  MathSciNet  MATH  Google Scholar 

  158. Thomson, W., Floating Magnets, Nature, 1878, vol. 18, pp. 13–14.

    Article  Google Scholar 

  159. Thomson, J. J., Treatise on the Motion of Vortex Rings, London: Macmillan, 1883, pp. 94–108.

    Google Scholar 

  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.

    Article  Google Scholar 

  161. van Heijst, G. J.F., Topography Effects on Vortices in a Rotating Fluid, Meccanica, 1994, vol. 29, no. 4, pp. 431–451.

    Article  MathSciNet  MATH  Google Scholar 

  162. van Heijst, G. J. F. and Kloosterziel, R.C., Tripolar Vortices in a Rotating Fluid, Nature, 1989, vol. 338, pp. 569–571.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  164. Vranjes, J., Tripolar Vortex in Plasma Flow, Planet. Space Sci., 1999, vol. 47, no. 12, pp. 1531–1535.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  167. Wan, Y.H., The Stability of Rotating Vortex Patches, Comm. Math. Phys., 1986, vol. 107, no. 1, pp. 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  176. Yudovich, V. I., Mathematical Problems of Hydrodynamic Stability Theory, Dissertation, Moscow: Institute for Problems of Mechanics, 1972 (Russian).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Mathematics, Mechanics and Computer Sciences, Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, 344090, Russia

    Leonid G. Kurakin & Irina V. Ostrovskaya

  2. Southern Mathematical Institute, Vladikavkaz Scienific Center of RAS, ul. Markusa 22, Vladikavkaz, 362027, Russia

    Leonid G. Kurakin

  3. Water Problems Institute, RAS, ul. Gubkina 3, Moscow, 119333, Russia

    Mikhail A. Sokolovskiy

  4. P. P. Shirshov Institute of Oceanology, RAS, pr. Nakhimovski 36, Moscow, 117997, Russia

    Mikhail A. Sokolovskiy

Authors
  1. Leonid G. Kurakin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Irina V. Ostrovskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mikhail A. Sokolovskiy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leonid G. Kurakin.

Additional information

To the blessed memory of Hassan Aref and Slava Meleshko

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kurakin, L.G., Ostrovskaya, I.V. & Sokolovskiy, M.A. On the stability of discrete tripole, quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid. Regul. Chaot. Dyn. 21, 291–334 (2016). https://doi.org/10.1134/S1560354716030059

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354716030059

Keywords

Search

Navigation