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

Journal of Fluid Mechanics

Volumn 660, Issue , 2010, Pages 396-429

  Billant, P ^a   Deloncle, A ^a   Chomaz, J M ^a   Otheguy, P ^a  

^a ECOLE POLYTECHNIQUE   (France)

Author keywords

geophysical and geological flows; instability; vortex flows

Indexed keywords

BENDING STRENGTH; FROUDE NUMBER; NUMERICAL ANALYSIS; PLASMA STABILITY; ROTATIONAL FLOW; STABILITY;

ANTI-CYCLONIC VORTICES; CO-ROTATING VORTICES; COUNTER-ROTATING VORTEX PAIR; GEOPHYSICAL AND GEOLOGICAL FLOWS; NUMERICAL STABILITY ANALYSIS; SYMMETRIC INSTABILITIES; THEORETICAL SOLUTIONS; THREE DIMENSIONAL EFFECT;

VORTEX FLOW;

FROUDE NUMBER; INSTABILITY; MATHEMATICAL ANALYSIS; ROSSBY NUMBER; ROTATING FLUID; STRATIFIED FLUID; VORTEX FLOW; WAVELENGTH;

EID: 77957113511     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S002211201000282X     Document Type: Article

References (40)

1. Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
2. Billant, P. 2010 Zigzag instability of vortex pairs in stratified and rotating fluid. Part 1. Generalstability equations. J. Fluid Mech. doi:10.1017/S0022112010002818.
3. Billant, P. & Chomaz, J.-M. 2000a Experimental evidence for a new instability of a verticalcolumnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 418, 167-188.
4. Billant, P. & Chomaz, J.-M. 2000b Theoretical analysis of the zigzag instability of a verticalcolumnar vortex pair in a strongly stratified fluid. J. Fluid Mech. 419, 29-63.
5. Billant, P. & Chomaz, J.-M. 2000c Three-dimensional stability of a vertical columnar vortex pairin a stratified fluid. J. Fluid Mech. 419, 65-91.
6. Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids13, 1645-1651.
7. Billant, P., Dritschel, D. G. & Chomaz, J.-M. 2006 Bending and twisting instabilities of columnarelliptical vortices in a rotating strongly stratified fluid. J. Fluid Mech. 561, 73-102.
8. Boulanger, N., Meunier, P. & Le Dizès, S. 2008 Tilt-induced instability of a stratified vortex.J. Fluid Mech. 596, 1-20.
9. Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulationof strongly stratified turbulent flows. J. Fluid Mech. 585, 343-368.
10. 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.
11. Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172-2179.
12. Deloncle, A., Billant, P. & Chomaz, J.-M. 2008 Nonlinear evolution of the zigzag instability instratified fluids: a shortcut on the route to dissipation. J. Fluid Mech. 599, 229-239.
13. Dritschel, D. G. & de la Torre Juárez, M. 1996 The instability and breakdown of tall columnarvortices in a quasi-geostrophic fluid. J. Fluid Mech. 328, 129-160.
14. Fincham, A. M., Maxworthy, T. & Spedding, G. R. 1996 Energy dissipation and vortex structurein freely decaying, stratified grid turbulence. Dyn. Atmos. Oceans 23, 155-169.
15. 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.
16. Hebert, D. A. & deBruynKops, S. M. 2006 Relationship between vertical shear rate and kineticenergy dissipation rate in stably stratified flows. Geophys. Res. Lett. 33, L06602.
17. Herring, J. R. & Métais, O. 1989 Numerical experiments in forced stably stratified turbulence.J. Fluid Mech. 202, 97-115.
18. Jimenez, J. 1975 Stability of a pair of co-rotating vortices. Phys. Fluids 18, 1580-1582.
19. Klein, R., Majda, A. J. & Damodaran, K. 1995 Simplified equations for the interaction of nearlyparallel vortex filaments. J. Fluid Mech. 288, 201-248.
20. Leblanc, S. 2003 Internal wave resonances in strain flows. J. Fluid Mech. 477, 259-283.
21. Leweke, T. & Williamson, C. H. K. 1998 Cooperative elliptic instability of a vortex pair. J. FluidMech. 360, 85-119.
22. Lilly, D. K. 1983 Stratified turbulence and the mesoscale variability of the atmosphere. J. Atmos.Sci. 40, 749-761.
23. Lilly, D. K., Bassett, G., Droegemeier, K. & Bartello, P. 1998 Stratified turbulence in theatmospheric mesoscales. Theor. Comput. Fluid Dyn. 11 (3-4), 139-153.
24. Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207-242.
25. Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergentmodes. J. Fluid Mech. 586, 83-108.
26. Meunier, P. & Leweke, T. 2005 Elliptic instability of a co-rotating vortex pair. J. Fluid Mech. 533,125-159.
27. 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.
28. Otheguy, P., Billant, P. & Chomaz, J. M. 2006a The effect of planetary rotation on the zigzaginstability of co-rotating vortices in a stratified fluid. J. Fluid Mech. 553, 273-281.
29. Otheguy, P., Billant, P. & Chomaz, J. M. 2007 Theoretical analysis of the zigzag instability of avertical co-rotating vortex pair in a strongly stratified fluid. J. Fluid Mech. 584, 103-123.
30. Otheguy, P., Chomaz, J. M. & Billant, P. 2006b Elliptic and zigzag instabilities on co-rotatingvertical vortices in a stratified fluid. J. Fluid Mech. 553, 253-272.
31. Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids:layer formation and energetics. J. Fluid Mech. 279, 279-311.
32. Praud, O., Fincham, A. M. & Sommeria, J. 2005 Decaying grid turbulence in a strongly stratifiedfluid. J. Fluid Mech. 522, 1-33.
33. Riley, J. J. & deBruynKops, S. M. 2003 Dynamics of turbulence strongly influenced by buoyancy.Phys. Fluids 15 (7), 2047-2059.
34. Riley, J. J. & Lelong, M.-P. 2000 Fluid motions in the presence of strong stable stratification.Annu. Rev. Fluid Mech. 32, 617-657.
35. Riley, J. J. & Lindborg, E. 2008 Stratified turbulence: a possible interpretation of some geophysicalturbulence measurements. J. Atmos. Sci. 65 (7), 2416-2424.
36. Riley, J. J., Metcalfe, W. & Weissman, M. A. 1981 Direct numerical simulations of homogeneousturbulence in density-stratified fluids. In Proceedings of the AIP Conference on NonlinearProperties of Internal Waves (ed. B. J. Bruce), pp. 79-112. American Institute of Physics.
37. Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.
38. Waite, M. L. & Bartello, P. 2004 Stratified turbulence dominated by vortical motion. J. FluidMech. 517, 281-308.
39. Waite, M. L. & Smolarkiewicz, P. K. 2008 Instability and breakdown of a vertical vortex pair ina strongly stratified fluid. J. Fluid Mech. 606, 239-273.
40. Widnall, S. E., Bliss, D. & Zalay, A. 1971 Theoretical and experimental study of the stability of a vortex pair. In Aircraft Wake Turbulence (ed. A. Goldburg, J. H. Olsen & M. Rogers),pp. 305-329. Plenum.