Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-08T02:02:54.224Z Has data issue: false hasContentIssue false
  • English
  • Français

Transitions and oscillatory regimes in two-layer geostrophic hetons and tripoles

Published online by Cambridge University Press:  01 December 2016

Biana Shteinbuch-Fridman ,
Viacheslav Makarov  and
Ziv Kizner
Biana Shteinbuch-Fridman
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
Viacheslav Makarov
Affiliation:
Instituto Politécnico Nacional, Centro Interdisciplinario de Ciencias Marinas, La Paz, Baja California Sur 23096, Mexico
Ziv Kizner*
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel Department of Physics, Bar-Ilan University, Ramat-Gan 5290002, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate numerically the transitions and oscillatory regimes in two-layer quasigeostrophic hetons and tripoles composed of patches of uniform potential vorticity (PV). The contour-surgery algorithms are employed, in which either some symmetries are preserved, or asymmetric evolution of the vortex structures is allowed, induced by generally asymmetric numerical noise. The fluid layers are assumed equally thick. First, the evolution of hetons is considered. A heton, a steadily translating pair of vortices residing in different layers, is antisymmetric in the sense that the two PV patches are opposite in sign and symmetric in shape about the axis of translation. A feebly stable heton, when exposed to weak antisymmetric perturbations, responds by developing an oscillation, which culminates in a transition to a new, substantially robust oscillating heton. The results obtained reinforce our earlier findings regarding the modon-to-modon transition (Kizner et al., J. Fluid Mech., vol. 468, 2002, pp. 239–270; Kizner, Phys. Fluids, vol. 18 (5), 2006, 056601; Kizner, UTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (ed. Borisov et al.), IUTAM Bookseries, vol. 6, 2008, pp. 125–133. Springer) and clarify the transition mechanism. Asymmetric perturbations might cause a heton-to-tripole transition. Next we consider the transitions and oscillations in carousel tripoles exposed to weak, generally asymmetric perturbations. A carousel tripole is a steadily rotating centrally symmetric ensemble of three PV patches, with the central vortex being located in one layer and the two remaining, satellite vortices in the other layer. Depending on the tripoles’ size, hence also on the shape of the satellite vortices, three different types of transition are revealed, the transition to a ringed (shielded) monopole being one of them. Whereas the transition of a ringed monopole into a tripole is a known phenomenon, the reverse transition in baroclinic flows is detected for the first time.

JFM classification

Vortex Flows: Contour dynamics Geophysical and Geological Flows: Rotating flows Vortex Flows: Vortex breakdown
Type
Papers
Information
Journal of Fluid Mechanics , Volume 810 , 10 January 2017 , pp. 535 - 553
Check for updates
Copyright
© 2016 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

Baey, J. & Carton, X. 2002 Vortex multipoles in two-layer rotating shallow-water flows. J. Fluid Mech. 460, 151–175.Google Scholar
Carnevale, G. & Kloosterziel, R. C. 1994 Emergence and evolution of triangular vortices. J. Fluid Mech. 259, 305–331.Google Scholar
Carton, X. & Correard, S. 1999 Baroclinic tripolar geostrophic vortices. In IUTAM/SIMFLOW Symposium on Simulation and Identification of Organized Structures Inflows (ed. Sorensen, J. N. et al. ), pp. 181–190. Kluwer.CrossRefGoogle Scholar
Carton, X., Flierl, G. R. & Polvani, L. M. 1989 The generation of tripoles from unstable axisymmetric isolated vortex structures. Europhys. Lett. 9 (4), 339–344.Google Scholar
Carton, X. & Legras, B. 1994 The life-cycle of the barotropic tripolar vortex. J. Fluid Mech. 267, 53–82.CrossRefGoogle Scholar
Dritschel, D. G. 1986 The nonlinear evolution of rotating configurations of uniform vorticity. J. Fluid Mech. 172, 157–182.Google Scholar
Dritschel, D. G. 1988 Contour surgery – a topological reconnection scheme for extended integrations using contour dynamics. J. Comput. Phys. 77, 240–266.Google Scholar
Dritschel, D. G. 1989 Contour dynamics and contour surgery: numerical algorithms for extended, high-resolution modelling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comput. Phys. Rep. 10, 77–146.Google Scholar
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269–303.Google Scholar
Flierl, G. R. 1988 On the stability of geostrophic vortices. J. Fluid Mech. 197, 349–388.Google Scholar
Flór, J. B., Govers, W. S. S., van Heijst, G. J. F. & van Sluis, R. 1993 Formation of a tripolar vortex in a stratified fluid. In Advances in Turbulence IV (ed. Nieuwstadt, F. T. M.), Applied Scientific Research, vol. 51, pp. 405–409. Kluwer.Google Scholar
Griffiths, R. W. & Hopfinger, E. J. 1986 Experiments with baroclinic vortex pairs in a rotating fluid. J. Fluid Mech. 173, 501–518.Google Scholar
Griffiths, R. W. & Hopfinger, E. J. 1987 Coalescing of geostrophic vortices. J. Fluid Mech. 178, 73–97.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 (3), 227–240.Google Scholar
Gryanik, V. M. 1988 Localized vortices – ‘vortex charges’ and ‘vortex filaments’ in a baroclinic differentially rotating fluid. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 24 (12), 919–926.Google Scholar
Gryanik, V. M., Borth, H. & Olbers, D. 2004 The theory of quasigeostrophic von Kármán vortex streets in two-layer fluids on beta-plane. J. Fluid Mech. 505, 23–57.Google Scholar
Gryanik, V. M., Sokolovskiy, M. A. & Verron, J. 2006 Dynamics of heton-like vortices. Regular Chaotic Dyn. 11 (3), 417–438.Google Scholar
van Heijst, G. J. F. & Kloosterziel, R. C. 1989 Tripolar vortices in a rotating fluid. Nature 338, 569–571.Google Scholar
van Heijst, G. J. F., Kloosterziel, R. C. & Williams, C. W. M. 1991 Laboratory experiments on the tripolar vortices in a rotating fluid. J. Fluid Mech. 225, 301–331.Google Scholar
Helfrich, K. R. & Send, U. 1988 Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech. 197, 331–348.Google Scholar
Hogg, N. G. & Stommel, H. M. 1985a 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
Hogg, N. G. & Stommel, H. M. 1985b Hetonic explosions: the breakup and spread of warm pools as explained by baroclinic point vortices. J. Atmos. Sci. 42 (14), 1465–1476.Google Scholar
Khvoles, R., McWilliams, J. C. & Kizner, Z. 2007 Non-coincidence of separatrices in two-layer modons. Phys. Fluids 19 (5), 056602.Google Scholar
Kizner, Z. 1984 Rossby solitons with axisymmetric baroclinic modes. Dokl. USSR Acad. Sci. 275, 1495–1498.Google Scholar
Kizner, Z. 1997 Solitary Rossby waves with baroclinic modes. J. Mar. Res. 55, 671–685.Google Scholar
Kizner, Z. 2006 Stability and transitions of hetonic quartets and baroclinic modons. Phys. Fluids 18 (5), 056601.Google Scholar
Kizner, Z. 2008 Exploring the transitions in baroclinic modons. In IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (ed. Borisov et al. ), IUTAM Bookseries, vol. 6, pp. 125–133. Springer.Google Scholar
Kizner, Z. 2011 Stability of point-vortex multipoles revisited. Phys. Fluids 23 (6), 064104.Google Scholar
Kizner, Z. 2014 On the stability of two-layer geostrophic point-vortex multipoles. Phys. Fluids 26 (4), 046602.Google Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2002 Baroclinic modon equilibria on the beta-plane: stability and transitions. J. Fluid Mech. 468, 239–270.Google Scholar
Kizner, Z., Berson, D., Reznik, G. & Sutyrin, G. 2003a The theory of beta-plane baroclinic topographic modons. Geophys. Astrophys. Fluid Dyn. 97 (3), 175–211.Google Scholar
Kizner, Z., Berson, D. & Khvoles, R. 2003b Noncircular baroclinic beta-plane modons: Constructing stationary solutions. J. Fluid Mech. 489, 199–228.Google Scholar
Kizner, Z. & Khvoles, R. 2004a The tripole vortex: experimental evidence and explicit solutions. Phys. Rev. E 70 (1), 016307.Google Scholar
Kizner, Z. & Khvoles, R. 2004b Two variations on the theme of Lamb–Chaplygin. Regular Chaotic Dyn. 9 (4), 509–518.Google Scholar
Kizner, Z., Khvoles, R. & McWilliams, J. C. 2007 Rotating multipoles on the f- and 𝛾-planes. Phys. Fluids 19 (1), 016603.Google Scholar
Kloosterziel, R. C. & Carnevale, G. F. 1999 On the evolution and saturation of instabilities of two-dimensional isolated circular vortices. J. Fluid Mech. 388, 217–251.Google Scholar
Kloosterziel, R. C. & van Heijst, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.Google Scholar
Kozlov, V. F., Makarov, V. G. & Sokolovskiy, M. A. 1986 Numerical model of the baroclinic instability of axially symmetric eddies in two-layer ocean. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 22 (8), 674–678.Google Scholar
Legras, B., Santangelo, P. & Benzi, R. 1988 High-resolution numerical experiments for forced two-dimensional turbulence. Europhys. Lett. 5 (1), 37–42.Google Scholar
Makarov, V. G & Kizner, Z. 2011 Stability and evolution of uniform-vorticity hetons. J. Fluid Mech. 672, 307–325.CrossRefGoogle Scholar
Makarov, V. G., Sokolovskiy, M. A. & Kizner, Z. 2012 Doubly symmetric finite-core heton equilibria. J. Fluid Mech. 708, 397–417.Google Scholar
Morel, Y. G. & Carton, X. J. 1994 Multipolar vortices in two-dimensional incompressible flows. J. Fluid Mech. 267, 23–51.Google Scholar
Polvani, L. M. 1991 Two-layer geostrophic vortex dynamics. Part 2. Alignment and two-layer V-states. J. Fluid Mech. 225, 241–270.Google Scholar
Polvani, L. M. & Carton, X. J. 1990 The tripole: a new coherent vortex structure of incompressible two-dimensional flows. Geophys. Astrophys. Fluid Dyn. 51, 87–102.Google Scholar
Polvani, L. M., Zabusky, N. J. & Flierl, G. R. 1988 Applications of contour dynamics to two-layer quasigeostrophic flows. Fluid Dyn. Res. 3, 422–424.Google Scholar
Reinaud, J. N. & Carton, X. 2016 The interaction between two oppositely travelling, horizontally offset, antisymmetric quasigeostrophic hetons. J. Fluid Mech. 794, 409–443.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23 (12), 2339–2342.Google Scholar
Shteinbuch-Fridman, B., Makarov, V., Carton, X. & Kizner, Z. 2015 Two-layer geostrophic tripoles comprised by patches of uniform potential vorticity. Phys. Fluids 27 (3), 036602.Google Scholar
Sokolovskiy, M. A. & Carton, X. 2010 Baroclinic multipole formation from heton interaction. Fluid Dyn. Res. 42 (4), 045501.Google Scholar
Sokolovskiy, M. A. & Verron, J. 2000 Finite-core hetons: stability and interactions. J. Fluid Mech. 423, 127–154.Google Scholar
Sokolovskiy, M. A. & Verron, J. 2008 Motion of A + 1 vortices in a two-layer rotating fluid. In IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence (ed. Borisov et al. ), IUTAM Bookseries, vol. 6, pp. 481–490. Springer.Google Scholar
Trieling, R. R., van Heijst, G. J. F. & Kizner, Z. 2010 Laboratory experiments on multipolar vortices in a rotating fluid. Phys. Fluids 22 (9), 094104.Google Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.Google Scholar