Skip to main content
SpringerLink
Log in

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

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

Abstract

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.

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. McWilliams, J.C., The Emergence of Isolated Coherent Vortices in Turbulent Flow, J. Fluid Mech., 1984, vol. 146, pp. 21–43.

    Article  MATH  Google Scholar 

  2. McWilliams, J.C., Statistical Properties of Decaying Geostrophic Turbulence, J. Fluid Mech., 1989, vol. 198, pp. 199–230.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  4. McWilliams, J.C., The Vortices of Two-dimensional Turbulence, J. Fluid Mech., 1990a, vol. 219, pp. 361–385.

    Article  Google Scholar 

  5. McWilliams, J.C., The Vortices of Geostrophic Turbulence, J. Fluid Mech., 1990b, vol. 219, pp. 387–404.

    Article  Google Scholar 

  6. Dritschel, D.G., Vortex Properties of Two-dimensional Turbulence, Phys. Fluids A, 1993a, vol. 5, pp. 984–997.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  9. Carton, X., Hydrodynamical Modeling of Oceanic Vortices, Surveys in Geophysics, 2001, vol. 22, pp. 179–263.

    Article  Google Scholar 

  10. Charney, J.G., The Dynamics of Long Waves in a Baroclinic Westerly Current, J. Meteor., 1947, vol. 4, pp. 135–162.

    MathSciNet  Google Scholar 

  11. Eady, E.T., Long Waves and Cyclone Waves, Tellus, 1949, vol. 1, pp. 33–52.

    Article  MathSciNet  Google Scholar 

  12. Juckes, M., Quasi-geostrophic Dynamics of the Tropopause, J. Atmos. Sci., 1994, vol. 51, pp. 2756–2768.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  15. Sukhatme, J. and Pierrehumbert, R.T., Surface Quasi-geostrophic Turbulence: The Study of an Active Scalar, Chaos, 2002, vol. 12, pp. 439–450.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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. Scott, R.K., Local and Nonlocal Advection of a Passive Scalar, Phys. Fluids, 2006, vol. 18, p. 116601.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  24. Waugh, D., The Efficiency of Symmetric Vortex Merger, Phys. Fluids, 1992, vol. A4, pp. 1745–1758.

    Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. 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. 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).

  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.

    Article  MathSciNet  Google Scholar 

  31. Liu, Z. and Roebber, P.J., Vortex-driven Sensitivity in Deformation Flow, J. Atmos. Sci., 2008, vol. 65, pp. 3819–3839.

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  34. Muraki, D.J. and Snyder, C., Vortex Dipoles for Surface Quasi-geostrophic Models, J. Atmos. Sci., 2007, vol. 64, pp. 2961–2967.

    Article  MathSciNet  Google Scholar 

  35. Carton, X., Instability of Surface Quasigeostrophic Vortices, J. Atmos. Sci., 2009, vol. 66, pp. 1051–1062.

    Article  Google Scholar 

  36. Pedlosky, J., Geophysical Fluid Dynamics 2nd edition, New York: Springer-Verlag, 1987.

    MATH  Google Scholar 

  37. Scott, R.K. and Dritschel, D.G., Quasi-geostrophic Vortices in Compressible Atmospheres, J. Fluid Mech., 2005, vol. 530, pp. 305–325.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique des Océans, UBO, UEB, 6 av. Le Gorgeu, 29200, Brest, France

    X. Perrot & X. Carton

  2. Mathematical Institute North Haugh, University of St Andrews, St Andrews, KY16 9SS, Scotland, UK

    J. N. Reinaud & D. G. Dritschel

Authors
  1. X. Perrot
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. N. Reinaud
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. X. Carton
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. G. Dritschel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. Perrot.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Perrot, X., Reinaud, J.N., Carton, X. et al. Homostrophic vortex interaction under external strain, in a coupled QG-SQG model. Regul. Chaot. Dyn. 15, 66–83 (2010). https://doi.org/10.1134/S1560354710010041

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354710010041

MSC2000 numbers

Key words

Search

Navigation