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Inviscid waves on a Lamb–Oseen vortex in a rotating stratified fluid: consequences for the elliptic instability

Published online by Cambridge University Press:  01 February 2008

STÉPHANE LE DIZÈS
STÉPHANE LE DIZÈS*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
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Abstract

The inviscid waves propagating on a Lamb–Oseen vortex in a rotating medium for an unstratified fluid and for a strongly stratified fluid are analysed using numerical and asymptotic approaches. By a local Lagrangian description, we first provide the characteristics of the local plane waves (inertia–gravity waves) as well as the local growth rate associated with the centrifugal instability when the vortex is unstable. A global WKBJ approach is then used to determine the frequencies of neutral core modes and neutral ring modes. We show that these global Kelvin modes only exist in restricted domains of the parameters. The consequences of these domain limitations for the occurrence of the elliptic instability are discussed. We argue that in an unstratified fluid the elliptic instability should be active in a small range of the Coriolis parameter which could not have been predicted from a local approach. The wavenumbers of the sinuous modes of the elliptic instability are provided as a function of the Coriolis parameter for both an unstratified fluid and a strongly stratified fluid.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 597 , 25 February 2008 , pp. 283 - 303
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Afanasyev, Y. D. 2002 Experiments on instability of columnar vortex pairs in rotating fluid. Geophys. Astrophys. Fluid Dyn. 96, 31–48.CrossRefGoogle Scholar
Afanasyev, Y. D. & Peltier, W. R. 1998 Three-dimensional instability of anticyclonic swirling flow in rotating fluid: laboratory experiments and related theoretical predictions. Phys. Fluids 10, 3194–3202.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Billant, P. & Chomaz, J.-M. 2000 a Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167–188.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 b Theoretical analysis of the zigzag instability of a columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 29–63.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2000 c Three-dimensional stability of a vertical columnar vortex pair in a stratified fluid. J. Fluid Mech. 419, 65–91.CrossRefGoogle Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.CrossRefGoogle Scholar
Billant, P., Chomaz, J.-M. & Otheguy, P. 2005 A general theory for the zigzag instability in stratified fluids. In Intl Conf. on High Reynolds Number Vortex Interactions, Toulouse, France, pp. 39–40.Google Scholar
Boubnov, B., Gledzer, E. & Hopfinger, E. 1995 Stratified circular Couette flow: instability and flow regimes. J. Fluid Mech. 292, 333–358.CrossRefGoogle Scholar
Cairns, R. A. 1979 The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech. 92, 1–14.CrossRefGoogle Scholar
Craik, A. D. D. 1989 The stability of unbounded two- and three-dimensional flows subject to body forces: some exact solutions. J. Fluid Mech. 198, 275–292.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fabre, D., Sipp, D. & Jacquin, L. 2006 The Kelvin waves and the singular modes of the Lamb–Oseen vortex. J. Fluid Mech. 551, 235–274.CrossRefGoogle Scholar
Friedlander, S. & Lipton-Lifschitz, A. 2003 Localized instabilities in fluids. In Handbook of Mathematical Fluid Dynamics (ed. Fiendlander, S. & Serre, D.), vol. 2, pp. 289–354. North-Holland.CrossRefGoogle Scholar
Fukumoto, Y. 2003 The three-dimensional instability of a strained vortex tube revisited. J. Fluid Mech. 493, 287–318.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Kloosterziel, R. C. & van Heijst, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.CrossRefGoogle Scholar
Lacaze, L., Ryan, K. & Le Dizès, S. 2007 Elliptic instability in a strained Batchelor vortex. J. Fluid Mech. 577, 341–361.CrossRefGoogle Scholar
Landau, L. & Lifchitz, E. 1967 Mécanique Quantique, Théorie non relativiste. Éditions MIR, Moscow.Google Scholar
Le Bars, M., LeDizès, S. Dizès, S. & LeGal, P. Gal, P. 2007 Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers. J. Fluid Mech. 585, 323–342.CrossRefGoogle Scholar
Le Dizès, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12 (11), 2762–2774.CrossRefGoogle Scholar
Le Dizès, S. 2004 Viscous critical-layer analysis of vortex normal modes. Stud. Appl. Maths 112, 315–332.CrossRefGoogle Scholar
Le Dizès, S. & Billant, P. 2006 Instability of an axisymmetric vortex in a stably stratified fluid. In 6th European Fluid Mechanics Conference, Stockholm, Sweden, p. 143.Google Scholar
Le Dizès, S. & Billant, P. 2007 Radiative instability in stratified vortices. Phys. Rev. Lett. (submitted).Google Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.CrossRefGoogle Scholar
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptic instability in a two-vortex flow. J. Fluid Mech. 471, 169–201.CrossRefGoogle Scholar
Leblanc, S. 2003 Internal wave resonances in strain flows. J. Fluid Mech. 477, 259–283.CrossRefGoogle Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.CrossRefGoogle Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3 (11), 2644–2651.CrossRefGoogle Scholar
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159.CrossRefGoogle Scholar
Miyazaki, T. 1993 Elliptical instability in a stably stratified rotating fluid. Phys. Fluids A 5 (11), 2702–2709.CrossRefGoogle Scholar
Miyazaki, T. & Fukumoto, Y. 1991 Axisymmetric waves on a vertical vortex in a stratified fluid. Phys. Fluids A 3, 606–616.CrossRefGoogle Scholar
Miyazaki, T. & Fukumoto, Y. 1992 Three-dimensional instability of strained vortices in stably stratified fluid. Phys. Fluids A 4, 2515–2522.CrossRefGoogle Scholar
Otheguy, P., Billant, P. & Chomaz, J.-M. 2006 a The effect of planetary rotation on the zigzg instability of co-rotating vortices in a stratified fluid. J. Fluid Mech. 553, 273–281.CrossRefGoogle Scholar
Otheguy, P., Chomaz, J.-M. & Billant, P. 2006 b Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid. J. Fluid Mech. 553, 253–272.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Schecter, D. A. & Montgomery, M. T. 2004 Damping and pumping of a vortex Rossby wave in a monotonic cyclone: critical layer stirring versus inertia-buoyancy wave emission. Phys. Fluids 16, 1334–48.CrossRefGoogle Scholar
Sipp, D. & Jacquin, L. 2003 Widnall instabilities in vortex pairs. Phys. Fluids 15, 1861–1874.CrossRefGoogle Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.CrossRefGoogle Scholar
Stegner, A., Pichon, T. & Beunier, M. 2005 Elliptical–inertial instability of rotating Kármán streets. Phys. Fluids 17, 066602.CrossRefGoogle Scholar
Waleffe, F. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids A 2 (1), 76–80.CrossRefGoogle Scholar
Withjack, E. & Chen, C. 1974 An experimental study of Couette instability of stratified fluids. J. Fluid Mech. 66, 725–737.CrossRefGoogle Scholar