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Formal stability of circular vortices

Published online by Cambridge University Press:  26 April 2006

R. C. Kloosterziel  and
G. F. Carnevale
R. C. Kloosterziel
Affiliation:
Institute for Nonlinear Science, University of California. San Diego. La Jolla. CA 92093, USA Present address: School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI 96822, USA.
G. F. Carnevale
Affiliation:
Scripps Institute of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 249 - 278
Copyright
© 1992 Cambridge University Press

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References

Arnol'D, V. I. 1965 Variational principles for three-dimensional steady-state flows of an ideal fluid. Prikl. Math. Mech. 29, 846–851 (English transl. Sov. Maths 6, 773–777 1965).Google Scholar
Arnol'D, V. I. 1966 Over an a priori estimate in the theory of hydrodynamic stability. Izv. Uchebn. Zaved. Mat. 54, no. 5, 3–5 (English transl. Am. Math. Soc. Transl., Series 2, 79, 267–269 (1969).Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Benzi, R., Pierini, S. & Vulpiani, A. 1982 On nonlinear hydrodynamic stability of planetary vortices. Geophys. Astrophys. Fluid Dyn. 20, 293–306.Google Scholar
Carnevale, G. F. & Frederiksen, J. S. 1987 Nonlinear stability and statistical mechanics of flow over topography. J. Fluid Mech. 175, 157–181.Google Scholar
Carnevale, G. F. & Shepherd, T. G. 1990 On the interpretation of Andrew's theorem. Geophys. Astrophys. Fluid Dyn. 51, 1–17.Google Scholar
Carton, X. J. & McWilliams, J. C. 1989 Barotropic and baroclinic instabilities of axisymmetric vortices in a quasi-geostrophic model. In Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence (ed. J. C. J. Nihoul & B. M. Jamart), pp. 225–244. Elsevier.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Dritschel, D. G. 1988 Nonlinear stability bounds for inviscid, two-dimensional, parallel or circular flows with monotonic vorticity, and the analogous three-dimensional quasigeostrophic flows. J. Fluid Mech. 191, 575–581.Google Scholar
Ewing, G. M. 1985 Calculus of Variations with Applications. Dover.
Flierl, G. R. 1988 On the instability of geostrophic vortices. J. Fluid Mech. 197, 349–388.Google Scholar
Gent, P. R. & McWilliams, J. C. 1986 The instability of circular vortices. Geophys. Astrophys. Fluid Dyn. 35, 209–233.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–116.Google Scholar
Kloosterziel, R. C. & Heijst, G. J. F. van 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.Google Scholar
Mcintyre, M. E. & Shepherd, T. G. 1987 An exact local conservation theorem for finiteamplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and Arnol'd's stability theorems. J. Fluid Mech. 181, 527–565.Google Scholar
Michalke, A. & Timme, A. 1967 On the inviscid stability of certain two-dimensional vortex-type flows. J. Fluid Mech. 29, 647–666.Google Scholar
Shepherd, T. G. 1987 Non-ergodicity of inviscid two-dimensional flow on a beta-plane and on the surface of a rotating sphere. J. Fluid Mech. 184, 289–302.Google Scholar
Stern, M. E. 1987 Horizontal entrainment and detrainment in large-scale eddies. J. Phys. Oceanogr. 17, 1688–1695.Google Scholar