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Theoretical analysis of the zigzag instability of a vertical co-rotating vortex pair in a strongly stratified fluid

Published online by Cambridge University Press:  25 July 2007

PANTXIKA OTHEGUY ,
PAUL BILLANT  and
JEAN-MARC CHOMAZ
PANTXIKA OTHEGUY
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
PAUL BILLANT
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
JEAN-MARC CHOMAZ
Affiliation:
LadHyX, CNRS, École Polytechnique, F–91128 Palaiseau Cedex, France
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Abstract

A long-wavelength stability analysis of two co-rotating Gaussian vertical vortices in an inviscid strongly stratified fluid is conducted for vortices separated by a large distance b compared to their radius a (b ≫ a). This analysis predicts and explains the zigzag instability found by a numerical stability analysis in a companion paper (Otheguy, Chomaz & Billant, J. Fluid. Mech. vol. 553, 2006, p. 253). The zigzag instability results from the coupling between the bending perturbations of each vortex and the external strain that one vortex induces on the other S = Γ/2 π b2, where Γ is the circulation of the vortices. The analysis predicts that the maximum growth rate of the instability is twice the strain S and that the most unstable vertical wavelength λ scales as the buoyancy length, defined by LB = Γ/πaN, multiplied by the ratio b/a, i.e. λ ∝ Fhb, where Fh = Γ/πa2N is the horizontal Froude number. The asymptotic results are in very good agreement with the numerical results.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 103 - 123
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.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.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. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Bonnier, M., Eiff, O. & Bonneton, P. 2000 On the density structure of far-wake vortices in a stratified fluid. Dyn. Atmos. Oceans 31, 117–137.CrossRefGoogle Scholar
Eloy, C. & Le Dizès, S. 1999 Three-dimensional instability of Burgers and Lamb-Oseen vortices in a strain field. J. Fluid Mech. 378, 145–166.CrossRefGoogle 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.CrossRefGoogle Scholar
Godoy-Diana, R., Chomaz, J. M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence. J. Fluid Mech. 202, 97–115.CrossRefGoogle Scholar
Holford, J. M. & Linden, P. F. 1999 Turbulent mixing in a stratified fluid. Dyn. Atmos. Oceans 30, 173–198.CrossRefGoogle Scholar
Jimenez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18, 1580–1582.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Le Dizès, S., & Laporte, F. 2002 Theoretical predictions for the elliptical instability in a two-vortex flow. J. Fluid Mech. 471, 169–201.CrossRefGoogle Scholar
Leibovich, S., Brown, S. N. & Patel, Y. 1986 Bending waves on inviscid columnar vortices. J. Fluid Mech. 173, 595–624.CrossRefGoogle Scholar
Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos. Sci. 40, 749–761.2.0.CO;2>CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Moffatt, H. K., Kida, S. & Ohkitani, K. 1994 Stretched vortices-the sinews of turbulence-large reynolds number asymptotics. J. Fluid Mech. 259, 241–264.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., Chomaz, J.-M. & Billant, P. 2006 Elliptic and zigzag instabilities on co-rotating vertical vortices in a stratified fluid. J. Fluid Mech. 553, 253–272.CrossRefGoogle 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. & 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., Metcalfe, W. & Weissman, M. A. 1981 Direct numerical simulations of homogeneous turbulence in density-stratified fluids. Proc. AIP Conf. Nonlinear Properties of Internal Waves (ed. West, B. J.), pp. 79–112.Google Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Spedding, G. R., Browand, F. K. & Fincham, A. M. 1996 Turbulence, similarity scaling and vortex geometry in the wake of a sphere in a stably-stratified fluid. J. Fluid Mech. 314, 53–103.CrossRefGoogle Scholar
Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. Fluid Mech. 517, 281–308.CrossRefGoogle Scholar