A Numerical Investigation of the Stability of Isolated Shallow Water Vortices

A. Stegner Laboratoire de Météorologie Dynamique, École Normale-Supérieure, Paris, France

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D. G. Dritschel Mathematical Institute, University of St. Andrews, North Haugh, St. Andrews, United Kingdom

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Abstract

Motivated by observational data and recent numerical simulations showing that ageostrophic effects may play an important role in the dynamics and transport of large-scale vortices in the atmosphere and the oceans, the authors examine the stability of a family of isolated vortices, numerically, using the contour-advective semi-Lagrangian algorithm. The full shallow-water equations (1½-layer model) are integrated in order to investigate vortices over a wide range of parameters. In order to characterize the cyclone–anticyclone asymmetry, the stability of a couple of vortices having velocity profiles of opposite sign is compared. It is found that ageostrophic effects (finite Rossby number) tend to stabilize anticyclones but destabilize cyclones. On the other hand, large-scale effects (small Burger number) are shown to stabilize all vortices for this reduced-gravity model. Here again, the anticyclones tend to be favored in this restabilization process. These results are compared with a linear stability analysis performed in the framework of the standard quasigeostrophic model that predicts a symmetric evolution for cyclones and anticyclones. The authors have shown that a significant departure from QG dynamics, due to ageostrophic and large-scale effects, appears in a range of parameters relevant to large-scale coherent structures in nature.

Corresponding author address: Dr. Alexandre Stegner, Laboratoire de Météorologie Dynamique, École Normale Supérieure, 24, rue Lhomond, 75005 Paris Cedex, France.

Email: stegner@lmd.ens.fr

Abstract

Motivated by observational data and recent numerical simulations showing that ageostrophic effects may play an important role in the dynamics and transport of large-scale vortices in the atmosphere and the oceans, the authors examine the stability of a family of isolated vortices, numerically, using the contour-advective semi-Lagrangian algorithm. The full shallow-water equations (1½-layer model) are integrated in order to investigate vortices over a wide range of parameters. In order to characterize the cyclone–anticyclone asymmetry, the stability of a couple of vortices having velocity profiles of opposite sign is compared. It is found that ageostrophic effects (finite Rossby number) tend to stabilize anticyclones but destabilize cyclones. On the other hand, large-scale effects (small Burger number) are shown to stabilize all vortices for this reduced-gravity model. Here again, the anticyclones tend to be favored in this restabilization process. These results are compared with a linear stability analysis performed in the framework of the standard quasigeostrophic model that predicts a symmetric evolution for cyclones and anticyclones. The authors have shown that a significant departure from QG dynamics, due to ageostrophic and large-scale effects, appears in a range of parameters relevant to large-scale coherent structures in nature.

Corresponding author address: Dr. Alexandre Stegner, Laboratoire de Météorologie Dynamique, École Normale Supérieure, 24, rue Lhomond, 75005 Paris Cedex, France.

Email: stegner@lmd.ens.fr

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  • Arai, M., and T. Yamagata, 1994: Asymmetric evolution of eddies in rotating shallow water. Chaos,4 (2), 163–175.

  • Aristegui, J., P. Sangra, S. Hernandez-Leon, M. Canton, A. Hernandez-Guerra, and J. L. Kerling, 1994: Island-induced eddies in the Canary Islands. Deep-Sea Res.,41, 1509–1525.

  • Arnold, V. I., 1978: Mathematical Methods of Classical Mechanics. Springer-Verlag, 462 pp.

  • Bartello, P., O. Metais, and M. Lesieur, 1994: Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech.,273, 1–29.

  • Benilov, E. S., and Coauthors, 1998: On the stability of large-amplitude vortices in a continuously stratified fluid on the f-plane. J. Fluid Mech.,355, 139–162.

  • Ben Jelloul, M., and V. Zeitlin 1999: Remarks on stability of the rotating shallow-water vortices in the frontal dynamics regime. Nuovo Cimento Soc. Ital. Fis. C,22, 931–941.

  • Carton, X., and B. Legras, 1994: The life-cycle of tripoles in two-dimensional incompressible flows. J. Fluid Mech.,267, 53–82.

  • ——, C. R. Flierl, and L. Polvani, 1989: The generation of tripoles from unstable axisymmetric isolated vortex structure. Europhys. Lett.,9, 339–344.

  • Cushman-Roisin, B., 1986: Frontal geostrophic dynamics. J. Phys. Oceanogr.,16, 132–143.

  • Dewar, W. K., and P. D. Killworth, 1995: On the stability of oceanic rings. J. Phys. Oceanogr.,25, 1467–1487.

  • Dritschel, D. G., and M. H. P. Ambaum, 1997: A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields. Quart. J. Roy. Meteor. Soc.,123, 1097–1130.

  • ——, L. M. Polvani, and A. R. Mohebalhojeh, 1999: The contour-advective semi-Lagrangian method for the shallow water equations. Mon. Wea. Rev.,127, 1551–1565.

  • Flierl, G. R., 1988: On the instability of geostrophic vortices. J. Fluid Mech.,197, 349–388.

  • Helfrich, K. R., and U. Send, 1988: Finite-amplitude evolution of two-layer geostrophic vortices. J. Fluid Mech.,197, 331–348.

  • Holm, D. D., J. E. Mardsen, T. Ratiu, and A. Weinstein, 1985: Nonlinear stability of fluid and plasma equilibria. Phys. Rep.,123, 1–116.

  • Ikeda, M., 1981: Instability and splitting of mesoscale rings using a two-layer quasigeostrophic model on an f-plane. J. Phys. Oceanogr.,11, 987–998.

  • Killworth, P. D., J. R. Blundell, and W. K. Dewar, 1997: Primitive equation instability of wide oceanic rings. Part I: Linear theory. J. Phys. Oceanogr.,27, 941–962.

  • Kloosterziel, R. C., and C. J. F. van Heijst, 1991: An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech.,223, 1–124.

  • Mutabazi, I., C. Normand, and J. E. Wesfreid, 1992: Gap size effects on centrifugally and rotationally driven instability. Phys. Fluids A,4, 1199–1205.

  • Olson, D. B., 1985: Two-layer diagnostic model of the long-term physical evolution of warm-core ring 82B. J. Geophys. Res.,90, 8813–8822.

  • ——, 1991: Rings in the ocean. Annu. Rev. Planet. Sci.,19, 283–311.

  • ——, and R. J. Evans, 1986: Rings of the Agulhas. Deep-Sea Res.,33, 27–42.

  • Orlandi, P., and G. J. F. van Heijst, 1992: Numerical simulations of tripolar vortices in 2D flows. Fluid Dyn. Res.,9, 179–206.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer-Verlag, 710 pp.

  • Polvani, L. M., J. C., McWilliams, M. A. Spall, and R. Ford, 1994:The coherent structures of shallow water turbulence: Deformation-radius effects, cyclone–anticyclone asymmetry, and gravity-wave generation. Chaos,4 (2), 177–186.

  • Potylitsin, P. G., and W. R. Peltier, 1998: Stratification effects on the stability of columnar vortices on the f-plane. J. Fluid Mech.,355, 45–79.

  • Ripa, P., 1987: On the stability of elliptical vortex solutions of the shallow water equations. J. Fluid Mech.,183, 343–363.

  • Smith, B. A., and Coauthors, 1979: The Jupiter system through the eyes of Voyager 1. Science,204, 951–972.

  • Waugh, D. W., and D. G. Dritschel, 1991: The stability of filamentary vorticity in two-dimensional geophysical vortex-dynamics models. J. Fluid Mech.,231, 575–598.

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