Hostname: page-component-5c6d5d7d68-tdptf Total loading time: 0 Render date: 2024-08-18T07:23:48.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Wave and vortex dynamics on the surface of a sphere

Published online by Cambridge University Press:  26 April 2006

Lorenzo M. Polvani  and
David G. Dritschel
Lorenzo M. Polvani
Affiliation:
Department of Applied Physics, Columbia University, New York, NY 10027, USA
David G. Dritschel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by the observed potential vorticity structure of the stratospheric polar vortex, we study the dynamics of linear and nonlinear waves on a zonal vorticity interface in a two-dimensional barotropic flow on the surface of a sphere (interfacial Rossby waves). After reviewing the linear problem, we determine, with the help of an iterative scheme, the shapes of steadily propagating nonlinear waves; a stability analysis reveals that they are (nonlinearly) stable up to very large amplitude.

We also consider multi-vortex equilibria on a sphere: we extend the results of Thompson (1883) and show that a (latitudinal) ring of point vortices is more unstable on the sphere than in the plane; notably, no more than three point vortices on the equator can be stable. We also determine the shapes of finite-area multi-vortex equilibria, and reveal additional modes of instability feeding off shape deformations which ultimately result in the complex merger of some or all of the vortices.

We discuss two specific applications to geophysical flows: for conditions similar to those of the wintertime terrestrial stratosphere, we show that perturbations to a polar vortex with azimuthal wavenumber 3 are close to being stationary, and hence are likely to be resonant with the tropospheric wave forcing; this is often observed in high-resolution numerical simulations as well as in the ozone data. Secondly, we show that the linear dispersion relation for interfacial Rossby waves yields a good fit to the phase velocity of the waves observed on Saturn's ‘ribbon’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 255 , October 1993 , pp. 35 - 64
Copyright
© 1993 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

Bowman, K. P. 1992 Observations of deformation and mixing of the total ozone field in the antarctic polar vortex. J. Atmos. Sci. (in press.)Google Scholar
Busse, F. H. 1975 Patterns of convection in spherical shells. J. Fluid Mech. 72, 67–85.Google Scholar
Deem, G. S. & Zabusky, N. J. 1978 Stationary V-states: interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859.Google Scholar
Dhanak, M. R. 1992 Stability of a regular polygon of finite vortices. J. Fluid Mech. 234, 297–316.Google Scholar
Dritschel, D. G. 1985 The stability and energetics of co-rotating uniform vortices. J. Fluid Mech. 157, 95–134.Google Scholar
Dritschel, D. G. 1988a Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasi-geostrophic flows. J. Fluid Mech. 191, 575–582.Google Scholar
Dritschel, D. G. 1988b The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511–532.Google Scholar
Dritschel, D. G. 1988c Contour dynamics/surgery on the sphere. J. Comput. Phys. 79, 477–483.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. & Legras, B. 1993 Vortex stripping. J. Fluid Mech. (submitted.)Google Scholar
Dritschel, D. G. & McIntyre, M. E. 1990 Does contour dynamics go singular? Phys. Fluids A 2, 748–753.Google Scholar
Dritschel, D. G. & Polvani, L. M. 1992 The roll-up of vorticity strips on the surface of a sphere. J. Fluid Mech. 234, 47–69.Google Scholar
Ingersoll, A. P., Beebe, R. E., Conrath, B. J. & Hunt, G. E. 1984 Structure and dynamics of Saturn's atmosphere. In Saturn (ed. T. Gehrels & M. S. Matthews). University of Arizona Press.
Juckes, M. N. & McIntyre, M. E. 1987 A high-resolution one-layer model of braking planetary waves in the stratosphere. Nature 328, 590–596.Google Scholar
Kimura, Y. & Okamoto, H. 1987 Vortex motion on a sphere. J. Phys. Soc. Japan 56, 4203–4206.Google Scholar
Lamb, H. H. 1932 Hydrodynamics. Dover.
McIntyre, M. E. 1991 Atmospheric dynamics: some fundamentals with observational implications. In Proc. Intl School of Physics ‘Enrico Fermi’, CXV Course (ed. J. C. Gille & G. Visconti).
Pierrehumbert, R. T. 1980 A family of steady translating vortex pairs with distributed vorticity. J. Fluid Mech. 99, 129–144.Google Scholar
Polvani, L. M. 1991 Two-layer geostrophic vortex dynamics. Part 2. Alignment an 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. 52, 87–102.Google Scholar
Polvani, L. M., Flierl, G. R. & Zabusky, N. J. 1989a Filamentation of coherent vortex structures via separatrix crossing: a quantitative estimate of onset time. Phys. Fluids A 2, 181–184.Google Scholar
Polvani, L. M. & Plumb, R. A. 1993 Rossby wave breaking, microbreaking, filamentation and secondary vortex formation: the dynamics of a perturbed vortex. J. Atmos. Science (in press.)Google Scholar
Polvani, L. M., Zabusky, N. J. & Flierl, G. R. 1989b Two-layer geostrophic vortex dynamics. Part 1. Upper-layer V-states and merger. J. Fluid Mech. 205, 215–242.Google Scholar
Saffman, P. G. & Schatzman, J. C. 1982 Stability of vortex street of finite vortices. J. Fluid Mech. 117, 171–185.Google Scholar
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes for a pair of uniform vortices. Phys. Fluids 23, 2339–2342.Google Scholar
Smith, B. A. et al. 1982 A new look at the Saturn system: the Voyager 2 images. Science 215, 504–537.Google Scholar
Sromovsky, L. A., Revercomb, H. E., Krauss, R. J. & Soumi, V. E. 1983 Voyager 2 observations of Saturn's northern mid-latitude cloud features: morphology, motions and evolution. J. Geophys. Res. 88, 8650–8666.Google Scholar
Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings, pp. 94–108. Macmillan.
Waugh, D. W. 1992 Contour surgery simulations of a forced polar vortex. J. Atmos. Sci. 50, 714–730.Google Scholar
Zabusky, N. J. 1981 Recent developments in contour dynamics for the Euler equations in two dimensions. Ann. N.Y. Acad. Sci. 373, 160–170.Google Scholar