Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-23T09:23:39.308Z Has data issue: false hasContentIssue false
  • English
  • Français

Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.

Published online by Cambridge University Press:  21 July 2010

P. BILLANT ,
A. DELONCLE ,
J.-M. CHOMAZ  and
P. OTHEGUY
P. BILLANT*
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau Cedex, France
A. DELONCLE
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau Cedex, France
J.-M. CHOMAZ
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau Cedex, France
P. OTHEGUY
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau Cedex, France
*
†Email address for correspondence: billant@ladhyx.polytechnique.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The three-dimensional stability of vertical vortex pairs in stratified and rotating fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-rotating and counter-rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh = Γ/(2πR2N) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Väisälä frequency) is lower than unity independently of the Rossby number Ro = Γ/(4πR2Ωb) (Ωb is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2πb2) (b is the separation distance between the vortices) and is almost independent of Fh and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects.

When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous fluids. However, its growth rate is lower than in homogeneous fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution.

Physically, the different stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Instability Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 660 , 10 October 2010 , pp. 396 - 429
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Billant, P. 2010 Zigzag instability of vortex pairs in stratified and rotating fluid. Part 1. General stability equations. J. Fluid Mech. doi:10.1017/S0022112010002818.Google Scholar
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 vertical columnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 29–63.Google 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. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.Google Scholar
Billant, P., Dritschel, D. G. & Chomaz, J.-M. 2006 Bending and twisting instabilities of columnar elliptical vortices in a rotating strongly stratified fluid. J. Fluid Mech. 561, 73–102.Google Scholar
Boulanger, N., Meunier, P. & Le Dizès, S. 2008 Tilt-induced instability of a stratified vortex. J. Fluid Mech. 596, 1–20.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343–368.Google Scholar
Bristol, R. L., Ortega, J. M., Marcus, P. S. & Savas, O. 2004 On cooperative instabilities of parallel vortex pairs. J. Fluid Mech. 517, 331–358.CrossRefGoogle Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Deloncle, A., Billant, P. & Chomaz, J.-M. 2008 Nonlinear evolution of the zigzag instability in stratified fluids: a shortcut on the route to dissipation. J. Fluid Mech. 599, 229–239.CrossRefGoogle Scholar
Dritschel, D. G. & de la Torre Juárez, M. 1996 The instability and breakdown of tall columnar vortices in a quasi-geostrophic fluid. J. Fluid Mech. 328, 129–160.Google Scholar
Fincham, A. M., Maxworthy, T. & Spedding, G. R. 1996 Energy dissipation and vortex structure in freely decaying, stratified grid turbulence. Dyn. Atmos. Oceans 23, 155–169.CrossRefGoogle Scholar
Godeferd, F. S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy. J. Fluid Mech. 486, 115–159.Google Scholar
Hebert, D. A. & deBruynKops, S. M. 2006 Relationship between vertical shear rate and kinetic energy dissipation rate in stably stratified flows. Geophys. Res. Lett. 33, L06602.Google Scholar
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.CrossRefGoogle Scholar
Jimenez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18, 1580–1582.Google Scholar
Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearly parallel vortex filaments. J. Fluid Mech. 288, 201–248.Google Scholar
Leblanc, S. 2003 Internal wave resonances in strain flows. J. Fluid Mech. 477, 259–283.Google Scholar
Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. Fluid Mech. 360, 85–119.Google Scholar
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.Google Scholar
Lilly, D. K., Bassett, G., Droegemeier, K. & Bartello, P. 1998 Stratified turbulence in the atmospheric mesoscales. Theor. Comput. Fluid Dyn. 11 (3–4), 139–153.Google Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.Google Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83–108.Google Scholar
Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533, 125–159.CrossRefGoogle Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. 346, 413–425.Google Scholar
Otheguy, P., Billant, P. & Chomaz, J. M. 2006 a The effect of planetary rotation on the zigzag instability of co-rotating vortices in a stratified fluid. J. Fluid Mech. 553, 273–281.Google Scholar
Otheguy, P., Billant, P. & Chomaz, J. M. 2007 Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid. J. Fluid Mech. 584, 103–123.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.Google Scholar
Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
Praud, O., Fincham, A. M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratified fluid. J. Fluid Mech. 522, 1–33.CrossRefGoogle Scholar
Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy. Phys. Fluids 15 (7), 2047–2059.Google Scholar
Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification. Annu. Rev. Fluid Mech. 32, 617–657.CrossRefGoogle Scholar
Riley, J. J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysical turbulence measurements. J. Atmos. Sci. 65 (7), 2416–2424.Google Scholar
Riley, J. J., Metcalfe, W. & Weissman, M. A. 1981 Direct numerical simulations of homogeneous turbulence in density-stratified fluids. In Proceedings of the AIP Conference on Nonlinear Properties of Internal Waves (ed. Bruce, B. J.), pp. 79–112. American Institute of Physics.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.Google Scholar
Waite, M. L. & Smolarkiewicz, P. K. 2008 Instability and breakdown of a vertical vortex pair in a strongly stratified fluid. J. Fluid Mech. 606, 239–273.Google Scholar
Widnall, S. E., Bliss, D. & Zalay, A. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence (ed. Goldburg, A., Olsen, J. H. & Rogers, M.), pp. 305–329. Plenum.CrossRefGoogle Scholar