Physics of Fluids
Search:
   
 
 
 
Previous Article
Nonlinear development of inertial instability in a barotropic shear
Inertial instability is investigated numerically in a two-dimensional setting in order to understand its nonlinear stage and saturation. To focus on fundamental mechanisms, a simple barotropic shear U...
Next Article
A Lagrangian approach to droplet condensation in atmospheric clouds
The condensation of microdroplets in model systems, reminiscent of atmospheric clouds, is investigated numerically and analytically. Droplets have been followed through a synthetic turbulent flow fiel...

Waves on a columnar vortex in a strongly stratified fluid

Phys. Fluids 21, 106602 (2009); doi:10.1063/1.3248366

Published 12 October 2009

You are not logged in to this journal. Log in

Paul Billant1 and Stéphane Le Dizès2
1Laboratoire d'Hydrodynamique (LADHYX), CNRS, Ecole Polytechnique, Palaiseau Cedex F-91128, France
2Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS, 49, rue F. Joliot-Curie, Marseille F-13013, France

This paper investigates the discrete bounded waves sustained by a vertical columnar Rankine vortex in a strongly stratified fluid. We show that these waves are very different from their well-known counterpart in homogeneous fluid (Kelvin vortex waves); they exist only for nonzero azimuthal wavenumber m, their frequency lies in the interval [0,mOmega] (Omega is the angular velocity in the vortex core) and they are unstable because of an outward radiation from the vortex. The instability mechanism is explained in terms of an over-reflection phenomenon by means of a Wentzel–Kramers–Brillouin–Jeffreys analysis for large axial wavenumber. ©2009 American Institute of Physics
History: Received 27 March 2009; accepted 21 September 2009; published 12 October 2009
Permalink: http://link.aip.org/link/?PHFLE6/21/106602/1
BUY THIS ARTICLE   (US$24)
Download PDF (573 kB) View Cart

KEYWORDS and PACS

Keywords
PACS

RELATED DATABASES


To view database links for this article,
you need to log in.
To view database links for this article,
you need to log in.

PUBLICATION DATA

ISSN:
1070-6631 (print)   1089-7666 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (28)

For access to fully linked references, you need to log in. For access to fully linked references, you need to Log in.
  1. L. Kelvin, “Vibrations of a columnar vortex,” Philos. Mag. 10, 155 (1880).
  2. P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, 1992).
  3. M. Rossi, Lec. Notes Phys. 555, 40 (2000).
  4. The Kelvin vortex waves should not be confused with the other type of Kelvin waves in geophysical fluid dynamics, i.e., the gravitational waves along a boundary in a rotating fluid.
  5. D. Fabre, D. Sipp, and L. Jacquin, “The Kelvin waves and the singular modes of the Lamb–Oseen vortex,” J. Fluid Mech. 551, 235 (2006).
  6. S. C. Crow, “Stability theory for a pair of trailing vortices,” AIAA J. 8, 2172 (1970).
  7. C. -Y. Tsai and S. E. Widnall, “The stability of short waves on a straight vortex filament in a weak externally imposed strain field,” J. Fluid Mech. 73, 721 (1976).
  8. P. Billant and J. -M. Chomaz, “Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid,” J. Fluid Mech. 418, 167 (2000).
  9. P. Otheguy, P. Billant, and J. M. Chomaz, “Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid,” J. Fluid Mech. 584, 103 (2007).
  10. S. Le Dizès, “Inviscid waves on a Lamb-Oseen vortex in a rotating stratified fluid: Consequences on the elliptic instability,” J. Fluid Mech. 597, 283 (2008).
  11. T. Miyazaki and Y. Fukumoto, “Axisymmetric waves on a vertical vortex in a stratified fluid,” Phys. Fluids A 3, 606 (1991).
  12. D. A. Schecter and M. T. Montgomery, “Damping and pumping of a vortex Rossby wave in a monotonic cyclone: Critical layer stirring versus inertia-buoyancy wave emission,” Phys. Fluids 16, 1334 (2004).
  13. D. A. Schecter and M. T. Montgomery, “Conditions that inhibit the spontaneous radiation of spiral inertia-gravity waves from an intense mesoscale cyclone,” J. Atmos. Sci. 63, 435 (2006).
  14. E. Broadbent and D. W. Moore, “Acoustic destabilization of vortices,” Philos. Trans. R. Soc. London, Ser. A 290, 353 (1979).
  15. R. Ford, “The instability of an axisymmetric vortex with monotonic potential vorticity in rotating shallow water,” J. Fluid Mech. 280, 303 (1994).
  16. R. Plougonven and V. Zeitlin, “Internal gravity wave emission from a pancake vortex: An example of wave-vortex interaction in strongly stratified flows,” Phys. Fluids 14, 1259 (2002).
  17. D. A. Schecter and M. T. Montgomery, “Waves in a cloudy vortex,” J. Atmos. Sci. 64, 314 (2007).
  18. D. Hodyss and D. S. Nolan, “The Rossby-inertia-buoyancy instability in baroclinic vortices,” Phys. Fluids 20, 096602 (2008).
  19. D. A. Schecter, “The spontaneous imbalance of an atmospheric vortex at high Rossby number,” J. Atmos. Sci. 65, 2498 (2008).
  20. P. Billant and J. -M. Chomaz, “Self-similarity of strongly stratified inviscid flows,” Phys. Fluids 13, 1645 (2001).
  21. Note that Eq. (7) can be put into the form of a modified Mathieu equation with the change of variable u=ln(r2omega/(mOmegaR2))/2. However, this is not very helpful because the modified Mathieu functions are not tabulated for arbitrary parameters.
  22. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  23. P. H. Roberts, “On vortex waves in compressible fluids. I. The hollow-core vortex,” Philos. Trans. R. Soc. London, Ser. A 459, 331 (2003).
  24. S. Le Dizès and P. Billant, “Radiative instability in stratified vortices,” Phys. Fluids 21, 096602 (2009).
  25. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  26. R. Lindzen, B. Farell, and K. Tung, “The concept of wave overreflection and its application to baroclinic instability,” J. Atmos. Sci. 37, 44 (1980).
  27. R. S. Lindzen and J. W. Barker, “Instability and wave over-reflection in stably stratified shear flow,” J. Fluid Mech. 151, 189 (1985).
  28. M. L. Waite and P. K. Smolarkiewicz, “Instability and breakdown of a vertical vortex pair in a strongly stratified fluid,” J. Fluid Mech. 606, 239 (2008).

CITING ARTICLES

For access to citing articles, you need to log in.
For access to citing articles, you need to Log in.