Regular and Chaotic Dynamics

, Volume 15, Issue 1, pp 66–83 | Cite as

Homostrophic vortex interaction under external strain, in a coupled QG-SQG model

  • X. Perrot
  • J. N. Reinaud
  • X. Carton
  • D. G. Dritschel
Research Articles


The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.

Key words

coupled QG-SQG model point-vortex CASL 

MSC2000 numbers

86A05 76U05 76B70 76B47 76M25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    McWilliams, J.C., The Emergence of Isolated Coherent Vortices in Turbulent Flow, J. Fluid Mech., 1984, vol. 146, pp. 21–43.MATHCrossRefGoogle Scholar
  2. 2.
    McWilliams, J.C., Statistical Properties of Decaying Geostrophic Turbulence, J. Fluid Mech., 1989, vol. 198, pp. 199–230.CrossRefGoogle Scholar
  3. 3.
    Dritschel, D.G., Scott, R.K., Macaskill, C., Gottwald, G., and Tran, C.V., Late Time Evolution of Unforced Inviscid Two-dimensional Turbulence, J. Fluid Mech., 2009, vol. 640, pp. 217–235.CrossRefMathSciNetGoogle Scholar
  4. 4.
    McWilliams, J.C., The Vortices of Two-dimensional Turbulence, J. Fluid Mech., 1990a, vol. 219, pp. 361–385.CrossRefGoogle Scholar
  5. 5.
    McWilliams, J.C., The Vortices of Geostrophic Turbulence, J. Fluid Mech., 1990b, vol. 219, pp. 387–404.CrossRefGoogle Scholar
  6. 6.
    Dritschel, D.G., Vortex Properties of Two-dimensional Turbulence, Phys. Fluids A, 1993a, vol. 5, pp. 984–997.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dritschel, D.G. and Zabusky, N.J., On the Nature of Vortex Interactions and Models in Unforced Nearlyinviscid Two-dimensional Turbulence, Phys. Fluids, 1996, vol. 8, pp. 1252–1256.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dritschel, D.G., Scott, R.K., Macaskill, C., Gottwald, G., and Tran, C.V., Unifying Theory for Vortex Dynamics in Two-dimensional Turbulence, Phys. Rev. Lett., 2008, vol. 101, p. 094501.CrossRefGoogle Scholar
  9. 9.
    Carton, X., Hydrodynamical Modeling of Oceanic Vortices, Surveys in Geophysics, 2001, vol. 22, pp. 179–263.CrossRefGoogle Scholar
  10. 10.
    Charney, J.G., The Dynamics of Long Waves in a Baroclinic Westerly Current, J. Meteor., 1947, vol. 4, pp. 135–162.MathSciNetGoogle Scholar
  11. 11.
    Eady, E.T., Long Waves and Cyclone Waves, Tellus, 1949, vol. 1, pp. 33–52.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Juckes, M., Quasi-geostrophic Dynamics of the Tropopause, J. Atmos. Sci., 1994, vol. 51, pp. 2756–2768.CrossRefGoogle Scholar
  13. 13.
    Held, I.M., Pierrehumbert, R.T., Garner, S.T. and Swanson, K.L., Surface Quasi-geostrophic Dynamics, J. Fluid Mech., 1995, vol. 282, pp. 1–20.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lim, C. and Majda, A., Point Vortex Dynamics for Coupled Surface/Interior QG and propagating heton clusters in models for ocean convection, Geophys. and Astrophys. Fluid Dyn., 2001, vol. 94, pp. 177–220.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Sukhatme, J. and Pierrehumbert, R.T., Surface Quasi-geostrophic Turbulence: The Study of an Active Scalar, Chaos, 2002, vol. 12, pp. 439–450.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hakim, G.J., Snyder, C., and Muraki, D.J., A New Model for Cyclone-anticyclone Asymmetry, J. Atmos. Sci., 2002, vol. 59, pp. 2405–2420.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Tran, C.V. and Bowman, J.C., Energy Budgets in Charney-hasegawa-mima and Surface Quasigeostrophic Turbulence, Phys. Rev. E, 2003, vol. 68, 036304, 4 pp.Google Scholar
  18. 18.
    Scott, R.K., Local and Nonlocal Advection of a Passive Scalar, Phys. Fluids, 2006, vol. 18, p. 116601.CrossRefGoogle Scholar
  19. 19.
    Lapeyre, G. and Klein, P., Dynamics of the Upper Oceanic Layers in Terms of Surface Quasigeostrophy Theory, J. Phys. Oceanogr., 2006, vol. 36, pp. 165–176.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Wu, H.M., Overman, E.A., and Zabusky, N.J., Steady States of the Euler Equations in Two dimensions. Rotating and Translating V-states with Limiting cases. I. Numerical Algorithms and Results, J. Comp. Phys., 1984, vol. 53, pp. 42–71.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dritschel, D.G., The Stability and Energetics of Corotating Uniform Vortices, J. Fluid Mech., 1985, vol. 157, pp. 95–134.MATHCrossRefGoogle Scholar
  22. 22.
    Melander, M.V., Zabusky, N.J., and McWilliams, J.C., Asymmetric Vortex Merger in Two dimensions: Which Vortex is “victorious”? Phys. Fluids, 1987, vol. 30, pp. 2604–2610.CrossRefGoogle Scholar
  23. 23.
    Melander, M.V., Zabusky, N.J., and McWilliams, J.C., Symmetric Vortex Merger in Two Dimensions, J. Fluid Mech., 1988, vol. 195, pp. 303–340.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Waugh, D., The Efficiency of Symmetric Vortex Merger, Phys. Fluids, 1992, vol. A4, pp. 1745–1758.Google Scholar
  25. 25.
    Dritschel, D.G. and Waugh, D., Quantification of the Inelastic Interaction of Unequal Vortices in Two-dimensional Vortex Dynamics, Phys. Fluids, 1992, vol. A4, pp. 1737–1744.Google Scholar
  26. 26.
    Yasuda, I. and Flierl, G.R., Two-dimensional Asymmetric Vortex Merger: Merger Dynamics and Critical Merger Distance, Dyn. Atmos. Oceans, 1997, vol. 26, pp. 159–181.CrossRefGoogle Scholar
  27. 27.
    Trieling, R.R., Velasco-Fuentes, O.U. and van Heijst, G.J.F., Interaction of Two Unequal Corotating Vortices, Phys. Fluids, 2005, vol. 17, 087103, 17 pp.Google Scholar
  28. 28.
    Brandt, L.K. and Nomura, K.K., The Physics of Vortex Merger: Further Insight, Phys. Fluids, 2006, vol. 18, 051701, 4 pp.Google Scholar
  29. 29.
    Carton, X., Legras, B., and Maze, G., Two-dimensional Vortex Merger in an External Strain Field, Journal of Turbulence, 2002, vol. 3, Paper 45, 7 pp. (electronic).Google Scholar
  30. 30.
    Maze, G., Lapeyre, G., and Carton, X., Dynamics of a 2d Vortex Doublet under External Deformation, Regul. Chaotic Dyn., 2004, vol. 9, pp. 179–263.CrossRefMathSciNetGoogle Scholar
  31. 31.
    Liu, Z. and Roebber, P.J., Vortex-driven Sensitivity in Deformation Flow, J. Atmos. Sci., 2008, vol. 65, pp. 3819–3839.CrossRefGoogle Scholar
  32. 32.
    Perrot, X. and Carton, X., Vortex Interaction in an Unsteady Large-scale Shear-strain Flow, Proceedings of the IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence, Borisov, A.V. et al. (Eds), Dordrecht: Springer, 2008, pp. 373–382.CrossRefGoogle Scholar
  33. 33.
    Perrot, X. and Carton, X., Point-vortex Interaction in an Oscillatory Deformation Field: Hamiltonian Dynamics, Harmonic Resonance and Transition to Chaos, Discr. Cont. Dyn. Syst. B, 2009, vol. 11, pp. 971–995.MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Muraki, D.J. and Snyder, C., Vortex Dipoles for Surface Quasi-geostrophic Models, J. Atmos. Sci., 2007, vol. 64, pp. 2961–2967.CrossRefMathSciNetGoogle Scholar
  35. 35.
    Carton, X., Instability of Surface Quasigeostrophic Vortices, J. Atmos. Sci., 2009, vol. 66, pp. 1051–1062.CrossRefGoogle Scholar
  36. 36.
    Pedlosky, J., Geophysical Fluid Dynamics 2nd edition, New York: Springer-Verlag, 1987.MATHGoogle Scholar
  37. 37.
    Scott, R.K. and Dritschel, D.G., Quasi-geostrophic Vortices in Compressible Atmospheres, J. Fluid Mech., 2005, vol. 530, pp. 305–325.MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Dritschel, D.G. and Ambaum, M.H.P., A Contour-advective Semi-Lagrangian Algorithm for the Simulation of Fine-scale Conservative Fields, Q. J. R. Met. Soc., 1997, vol. 123, pp. 1097–1130.CrossRefGoogle Scholar
  39. 39.
    Reinaud, J.N. and Dritschel, D.G., The Merger of Vertically Offset Quasi-geostrophic Vortices, J. Fluid Mech., 2002, vol. 469, pp. 287–315.MATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Esfahanian, V., Ghader, S. and Mohebalhojeh, A.R., On the Use of Super Compact Scheme for the Spatial Differencing in Numerical Models of the Atmosphere, Q. J. R. Meteorol. Soc., 2005, vol. 131, pp. 2109–2129.CrossRefGoogle Scholar
  41. 41.
    Melander, M.V., Zabusky, N.J., and Styczek, A.S., A Moment Model for Vortex Interactions of the Two-dimensional Euler Equations. Part 1. Computational Validation of a Hamiltonian Elliptical Representation, J. Fluid Mech., 1986, vol. 167, pp. 95–115.MATHCrossRefGoogle Scholar
  42. 42.
    Dritschel, D.G., A Fast Contour Dynamics Method for Many-vortex Calculations in Two-dimensional Flows, Phys. Fluids A, 1993b, vol. 25, pp. 173–186.CrossRefMathSciNetGoogle Scholar
  43. 43.
    Vandermeirsch, F., Carton, X.J., and Morel, Y.G., Interaction Between an Eddy and a Zonal Jet. Part I. One-and-a-half Layer Model, Dyn. Atmos. Oceans, 2003a, vol. 36, pp. 247–270.CrossRefGoogle Scholar
  44. 44.
    Vandermeirsch, F., Carton, X.J., and Morel, Y.G., Interaction Between an Eddy and a Zonal Jet. Part ii. Two-and-a-half Layer Model, Dyn. Atmos. Oceans, 2003b, vol. 36, pp. 271–296.CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  • X. Perrot
    • 1
  • J. N. Reinaud
    • 2
  • X. Carton
    • 1
  • D. G. Dritschel
    • 2
  1. 1.Laboratoire de Physique des OcéansUBO, UEBBrestFrance
  2. 2.Mathematical Institute North HaughUniversity of St AndrewsSt AndrewsScotland, UK

Personalised recommendations