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Baroclinic instability of Kirchhoff's elliptic vortex

Published online by Cambridge University Press:  26 April 2006

Takeshi Miyazaki  and
Hideshi Hanazaki
Takeshi Miyazaki
Affiliation:
Department of Mechanical and Control Engineering, University of Electro-Communications, Chofu, Tokyo 182, Japan
Hideshi Hanazaki
Affiliation:
Division of Atmospheric Environment, National Institute for Environmental Studies, Tsukuba, Ibaraki 305, Japan
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Abstract

The linear instability of Kirchhoff's elliptic vortex in a vertically stratified rotating fluid is investigated using the quasi-geostrophic, f-plane approximation. Any elliptic vortex is shown to be unstable to baroclinic disturbances of azimuthal wavenumber m = 1 (bending mode) and m = 2 (elliptical deformation). The axial wavenumber of the unstable bending mode approaches Λc = 1.7046 in the limit of small ellipticity, indicating that it is a short-wave baroclinic instability. The instability occurs when the bending wave rotates around the vortex axis with angular velocity identical to the rotation rate of the undisturbed elliptic vortex. On the other hand, the wavenumber of the elliptical deformation mode approaches zero in the same limit, showing that it is a long-wave sideband instability.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 261 , 25 February 1994 , pp. 253 - 271
Copyright
© 1994 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Natl Bur. Stand./Dover.
Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27, 417–430.Google Scholar
Blanch, G. 1966 Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, 51–97.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.
Clemm, D. S. 1969 Algorithm 352: Characteristic values and associated solutions of Mathieu's differential equation. Commun. Assoc. Computing Machinery 12, 399–407.Google Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.
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, 575–581.Google Scholar
Dritschel, D. G. 1988b The repeated filamentation of two-dimensional vorticity interfaces. J. Fluid Mech. 194, 511–547.Google Scholar
Dritschel, D. G. 1990 The stability of elliptical vortices in an external straining flow. J. Fluid Mech. 210, 223–261.Google Scholar
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
Kida, S. 1981 Motion of an elliptic vortex in a uniform shear flow. J. Phys. Soc. Japan 50, 3517–3520.Google Scholar
Kloosterziel, R. C. & Carnevale, G. F. 1992 Formal stability of circular vortices. J. Fluid Mech. 242, 249–278.Google Scholar
Love, A. E. H. 1893 On the stability of certain vortex motions. Proc. Lond. Math. Soc. 25, 18–42.Google Scholar
McWilliams, J. C. 1989 Statistical properties of decaying geostrophic turbulenc. J. Fluid Mech. 198, 199–230.Google Scholar
McWilliams, J. C. 1990 The vortices of geostrophic turbulence. J. Fluid Mech. 219, 387–404.Google Scholar
Meacham, S. P. 1992 Quasigeostrophic, ellipsoidal vortices in a stratified fluid. Dyn. Atmos. Oceans 16, 189–223.Google Scholar
Moore, D. W. & Saffman, P. G. 1971 Structure of a line vortex in an imposed strain. In Aircraft Wake Turbulence (ed. J. Olsen, A. Goldburg & N. Rogers), pp. 339–354. Plenum.
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Pedlosky, J. 1979 Geophysical Fluid Dynamics. Springer.
Robinson, A. C. & Saffman, P. G. 1984 Three-dimensional stability of an elliptical vortex in a straining field. J. Fluid Mech. 142, 451–466.Google Scholar
Sale, A. H. J. 1970 Remark on algorithm 352 [S22]; Characteristic values and associated solutions of Mathieu's differential equation. Commun Assoc. Computing Machinery 13, 750.Google Scholar
Tsai, C.-Y. & Widnall, S. E. 1976 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73, 721–733.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar