Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-02-11T18:24:24.677Z Has data issue: false hasContentIssue false
  • English
  • Français

The strato-rotational instability of Taylor–Couette and Keplerian flows

Published online by Cambridge University Press:  23 August 2010

S. LE DIZÈS  and
X. RIEDINGER
S. LE DIZÈS*
Affiliation:
IRPHE, UMR 6594 CNRS, 49 rue F. Joliot Curie, F-13013 Marseille, France
X. RIEDINGER
Affiliation:
IRPHE, UMR 6594 CNRS, 49 rue F. Joliot Curie, F-13013 Marseille, France
*
†Email address for correspondence: ledizes@irphe.univ-mrs.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The linear inviscid stability of two families of centrifugally stable rotating flows in a stably stratified fluid of constant Brunt–Väisälä frequency N is analysed by using numerical and asymptotic methods. Both Taylor–Couette and Keplerian angular velocity profiles ΩTC = (1 − μ)/r2 + μ and ΩK = (1 − λ)/r2 + λ/r3/2 are considered between r = 1 (inner boundary) and r = d > 1 (outer boundary, or without boundary if d = ∞). The stability properties are obtained for flow parameters λ and μ ranging from 0 to +∞, and different values of d and N. The effect of the gap size is analysed first. By considering the potential flow (λ = μ = 0), we show how the instability associated with a mechanism of resonance for finite-gap changes into a radiative instability when d → ∞. Numerical results are compared with large axial wavenumber results and a very good agreement is obtained. For infinite gap (d = ∞), we show that the most unstable modes are obtained for large values of the azimuthal wavenumber for all λ and μ. We demonstrate that their properties can be captured by performing a local analysis near the inner cylinder in the limit of both large azimuthal and axial wavenumbers. The effect of the stratification is also analysed. We show that decreasing N is stabilizing. An asymptotic analysis for small N is also performed and shown to capture the properties of the most unstable mode of the potential flow in this limit.

JFM classification

Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 147 - 161
Copyright
Copyright © Cambridge University Press 2010

References

REFERENCES

Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Rev. Mod. Phys. 70 (1), 1–53.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Billant, P & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.Google Scholar
Billant, P. & Le Dizès, S. 2009 Waves on a columnar vortex in a strongly stratified fluid. Phys. Fluids 21, 106602.Google Scholar
Dubrulle, B., Dauchot, O., Daviaud, F., Longaretti, P.-Y., Richard, D. & Zahn, J.-P. 2005 a Stability and turbulent transport in Taylor–Couette flow from analysis of experimental data. Phys. Fluids 17, 095103.CrossRefGoogle Scholar
Dubrulle, B., Marie, L., Normand, C., Richard, D., Hersant, F. & Zahn, J.-P. 2005 b A hydrodynamic shear instability in stratified disks. Astron. Astrophys. 429, 1–13.CrossRefGoogle Scholar
Goldreich, P. & Lynden-Bell, D. 1965 Spiral arms as sheared gravitational instabilities. Mon. Not. R. Astron. Soc. 130, 125–158.CrossRefGoogle Scholar
Gula, J., Plougonven, R. & Zeitlin, V. 2009 Ageostrophic instabilities of fronts in a channel in a stratified rotating fluid. J. Fluid Mech. 627, 485–507.Google Scholar
Hopfinger, E. J. & van Heijst, G. J. F. 1993 Vortices in rotating fluids. Annu. Rev. Fluid Mech. 25, 241–289.CrossRefGoogle Scholar
Le Bars, M. & Le Gal, P. 2007 Experimental analysis of the stratorotational instability in a cylindrical Couette flow. Phys. Rev. Lett. 99, 064502.CrossRefGoogle Scholar
Le Dizès, S. & Billant, P. 2009 Radiative instability in stratified vortices. Phys. Fluids 21, 096602.CrossRefGoogle Scholar
Le Dizès, S. & Lacaze, L. 2005 An asymptotic description of vortex Kelvin modes. J. Fluid Mech. 542, 69–96.Google Scholar
Molemaker, M. J., McWilliams, J. C. & Yavneh, I. 2001 Instability and equilibration of centrifugally stable stratified Taylor–Couette flow. Phys. Rev. Lett. 86, 5270–5273.CrossRefGoogle ScholarPubMed
Narayan, R., Goldreich, P. & Goodman, J. 1987 Physics of modes in a differentially rotating system – analysis of the shearing sheet. Mon. Not. R. Astron. Soc. 228, 1–41.CrossRefGoogle Scholar
Riedinger, X., Le Dizès, S. & Meunier, P. 2010 a Viscous stability properties of a Lamb–Oseen vortex in a stratified fluid. J. Fluid Mech. 645, 255–278.Google Scholar
Riedinger, X., Meunier, P. & Le Dizès, S. 2010 b Instability of a vertical columnar vortex in a stratified fluid. Exp. Fluids (in press, doi:10.1007/S00348-010-0833-0 online).Google Scholar
Satomura, T. 1981 An investigation of shear instability in a shallow water. J. Meteorol. Soc. Japan 59, 148–167.CrossRefGoogle 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, 1334–1348.Google Scholar
Shalybkov, D. & Rüdiger, G. 2005 Non-axisymmetric instability of density-stratified Taylor–Couette flow. J. Phys.: Conf. Ser. 14, 128–137.Google Scholar
Withjack, E. M. & Chen, C. F. 1974 An experimental study of Couette instability of stratified fluids. J. Fluid Mech. 66, 725–737.CrossRefGoogle Scholar
Yavneh, I., McWilliams, J. C. & Molemaker, M. J. 2001 Non-axisymmetric instability of centrifugally stable stratified Taylor–Couette flow. J. Fluid Mech. 448, 1–21.CrossRefGoogle Scholar