Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-18T19:46:54.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part 1. Analysis of relative and absolute motions

Published online by Cambridge University Press:  01 February 2013

M. A. Sokolovskiy ,
K. V. Koshel  and
J. Verron
M. A. Sokolovskiy
Affiliation:
Water Problems Institute, RAS, 3, Gubkina Str., Moscow, 119991, Russia Southern Federal University, 8a Mil’chakova Str., Rostov-on-Don, 344090, Russia
K. V. Koshel*
Affiliation:
V. I. Il’ichev Pacific Oceanological Institute, FEB RAS, 43 Baltiyskaya Str., Vladivostok, 690041, Russia Far Eastern Federal University, 8 Sukhanova Str., Vladivostok, 690950, Russia
J. Verron
Affiliation:
Laboratoire des Ecoulements Géophysiques et Industriels, CNRS, BP 52, 38041, Grenoble, CEDEX 9, France
*
†Email address for correspondence: kvkoshel@poi.dvo.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

The results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Geophysical and Geological Flows: Quasi-geostrophic flows Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 232 - 254
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aref, H. 1979 Motion of three vortices. Phys. Fluids 22, 393–400.CrossRefGoogle Scholar
Aref, H. 1983 Integrable, chaos and turbulent vortex motion in two-dimensional flows. Annu. Rev. Fluid Mech. 15, 345–389.Google Scholar
Aref, H. 1986 The numerical experiment in fluid mechanics. J. Fluid Mech. 173, 15–41.CrossRefGoogle Scholar
Aref, H. 1989 Three-vortex motion with zero total circulation: addendum. Z. Angew. Math. Phys. 40, 495–500.Google Scholar
Aref, H. 2009 Stability of relative equilibria of three vortices. Phys. Fluids 21, 094101.CrossRefGoogle Scholar
Aref, H. 2010 Self-similar motion of three point vortices. Phys. Fluids 22, 057104.Google Scholar
Aref, H., Jones, S. W., Mofina, S. & Zawadski, I. 1989 Vortices, kinematics and chaos. Physica D 37, 423–440.Google Scholar
Aref, H. & Pomphrey, N. 1980 Integrable and chaotic motions of four vortices. Phys. Lett. A 78, 297–300.CrossRefGoogle Scholar
Aref, H. & Pomphrey, N. 1982 Integrable and chaotic motions of four vortices. I. The case of identical vortices. Proc. R. Soc. Lond. A 380, 359–387.Google Scholar
Aref, H. & Stremler, M. A. 1999 Four-vortex motion with zero total circulation and impulse. Phys. Fluids 11, 3704–3715.Google Scholar
Blackmore, D., Ting, L. & Knio, O. 2007 Studies of perturbed three vortex dynamics. J. Math. Phys. 48, 065402.Google Scholar
Bogomolov, V. A. 1977 Vorticity dynamics on a sphere. Fluid Dyn. 6, 57–65.Google Scholar
Bogomolov, V. A. 1985 On motion on a rotating sphere. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 21, 391–396.Google Scholar
Borisov, A. V. & Mamaev, I. S. 2005 Mathematical Methods of Dynamics of Vortex Structures. Institute of Computer Sciences.Google Scholar
Boyland, P., Stremler, M. & Aref, H. 2003 Topological fluid mechanics of point vortex motions. Physica D 175, 69–95.Google Scholar
Eckhardt, B. 1988 Integrable four vortex motion. Phys. Fluids 31, 2796–2801.CrossRefGoogle Scholar
Eckhardt, B. & Aref, H. 1988 Integrable and chaotic motions of four vortices. II. Collision dynamics of vortex pairs. Phil. Trans. R. Soc. Lond. A 326, 655–696.Google Scholar
Gröbli, W. 1877 Specialle Probleme ünber die Bewegung Geradliniger Paralleler Wirbelfäden. Züricher und Furrer.Google Scholar
Gryanik, V. M. 1983 Dynamics of singular geostrophic vortices in a two-layer model of the atmosphere (ocean). Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 19, 227–240.Google Scholar
Gryanik, V. M. 1988 Localized vortical disturbances – ‘vortex charges’ and ‘vortex threads’ in a baroclinic differentially rotating fluid. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 24, 1251–1261.Google Scholar
Gryanik, V. M., Sokolovskiy, M. A. & Verron, J. 2006 Dynamics of heton-like vortices. Regul. Chaot. Dyn. 11, 383–434.Google Scholar
Gudimenko, A. I. 2008 Dynamics of perturbed equilateral and collinear configurations of three point vortices. Regul. Chaot. Dyn. 13, 85–95.CrossRefGoogle Scholar
Gudimenko, A. I. & Zakharenko, A. D. 2010 Qualitative analysis of relative motion of three vortices. Russ. J. Non-linear Dyn. 6, 307–326 (in Russian).Google Scholar
Hogg, N. G. & Stommel, H. M. 1985 The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implications concerning eddy heat-flow. Proc. R. Soc. Lond. A 397, 1–20.Google Scholar
Izrailsky, Yu. G., Koshel, K. V. & Stepanov, D. V. 2008 Determination of optimal excitation frequency range in background flows. Chaos 18, 013107.CrossRefGoogle ScholarPubMed
Jamaloodeen, M. I. & Newton, P. K. 2007 Two-layer quasigeostrophic potential vorticity model. J. Math. Phys. 48, 065601.CrossRefGoogle Scholar
Kimura, Y. 1988 Chaos and collapse of a system of point vortices. Fluid Dyn. Res. 3, 98–104.Google Scholar
Kizner, Z. 2006 Stability and transitions of hetonic quartets and baroclinic modons. Phys. Fluids 18, 056601.Google Scholar
Koshel, K. V. & Prants, S. V. 2006 Chaotic advection in the ocean. Physics-Uspekhi (Advances in Physical Sciences) 49, 1151–1178.Google Scholar
Koshel, K. V., Sokolovskiy, M. A. & Davies, P. A. 2008 Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle. Fluid Dyn. Res. 40, 695–736.Google Scholar
Koshel, K. V., Sokolovskiy, M. A. & Verron, J. 2013 Three-vortex quasi-geostrophic dynamics in a two layer fluid. Part 2. Regular and chaotic advection around the perturbed steady states. J. Fluid Mech. 717, 255–280.Google Scholar
Kozlov, V. V. 1998 General Theory of Vortices. Izd. dom Udmurtskii Universitet.Google Scholar
Kozlov, V. V. 1996 Symmetries, Topology and Resonances in Hamiltonian Mechanics. Springer.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.Google Scholar
Meleshko, V. V. & van Heijst, G. J. F. 1994 Interacting two-dimensional vortex structures: point vortices, contour kinematics and stirring properties. Chaos, Solitons Fractals 4, 977–1010.Google Scholar
Meleshko, V. V. & Konstantinov, M. Yu. 1993 Dynamics of Vortex Structures, p. 280. Naukova Dumka.Google Scholar
Meleshko, V. V., Konstantinov, M. Yu., Gurzhi, A. A. & Konovaljuk, T. P. 1992 Advection of a vortex pair atmosphere in a velocity field of point vortices. Phys. Fluids A 4, 2779–2797.CrossRefGoogle Scholar
Newton, P. K. 2001 The N-vortex Problem. Analytical Techniques. Applied Mathematical Sciences Series, vol. 145 , Springer.Google Scholar
Novikov, E. A. 1976 Dynamics and statistics of a system of vortices. Sov. Phys. JETP 41, 937–943.Google Scholar
Novikov, E. A. & Sedov, Yu. B. 1979a Stochastic properties of a four-vortex system. Sov. Phys. JETP 48, 440–444.Google Scholar
Novikov, E. A. & Sedov, Yu. B. 1979b Stochastization of vortices. J. Expl Theor. Phys. Lett. 29, 677–679.Google Scholar
Poincaré, H. 1893 Theorie des Tourbillons. G. Carré.Google Scholar
Rott, N. 1989 Three-vortex motion with zero total circulation. Z. Angew. Math. Phys. 40, 473–494.CrossRefGoogle Scholar
Rott, N. 1990 Constrained three- and four-vortex problems. Phys. Fluids A 2, 1477–1480.Google Scholar
Ryzhov, E. A. & Koshel, K. V. 2010 Chaotic transport and mixing of a passive admixture by vortex flows behind obstacles. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 46, 184–191.Google Scholar
Simó, K. 2001 New families of solutions to the N-body problem. In Proc. ECM 2000, Barcelona. Progress in Mathematics, vol. 201. pp. 101–115. Birkhäuser.Google Scholar
Sokolovskiy, M. A., Koshel, K. V. & Carton, X. 2010 Baroclinic multipole evolution in shear and strain. Geophys. Astrophys. Fluid Dyn. 105, 506–535.Google Scholar
Sokolovskiy, M. A. & Verron, J. 2000a Finite-core hetons: stability and interactions. J. Fluid Mech. 423, 127–154.CrossRefGoogle Scholar
Sokolovskiy, M. A. & Verron, J. 2000b Four-vortex motion in the two layer approximation: integrable case. Regul. Chaot. Dyn. 5, 413–436.CrossRefGoogle Scholar
Sokolovskiy, M. A. & Verron, J. 2002a New stationary solutions of the three-vortex problem in a two-layer fluid. Dokl. Phys. 47, 233–237.CrossRefGoogle Scholar
Sokolovskiy, M. A. & Verron, J. 2002b Dynamics of the triangular two-layer vortex structures with zero total intensity. Regul. Chaot. Dyn. 7, 435–472.Google Scholar
Sokolovskiy, M. A. & Verron, J. 2004 Dynamics of the three vortices in two-layer rotating fluid. Regul. Chaot. Dyn. 9, 417–438.CrossRefGoogle Scholar
Sokolovskiy, M. A. & Verron, J. 2006 Some properties of motion of $A+ 1$ vortices in a two-layer rotating fluid. Russ. J. Non-linear Dyn. 2, 27–54 (in Russian).Google Scholar
Stremler, M. A. & Aref, H. 1999 Motion of three point vortices in a periodic parallelogram. J. Fluid Mech. 392, 101–128.Google Scholar
Synge, J. L. 1949 On the motion of three vortices. Can. J. Maths 1, 257–270.Google Scholar
Tavantzis, J. & Ting, L. 1988 The dynamics of three vortices revisited. Phys. Fluids 31, 1392–1409.CrossRefGoogle Scholar
Zaslavsky, G. M. 2007 The Physics of Chaos in Hamiltonian Systems, 2nd edn. p. 316. Imperial College Press.Google Scholar
Ziglin, S. L. 1980 Nonintegrability of a problem on the motion of four point vortices. Sov. Math. Dokl. 21, 296–299.Google Scholar