A New Large Scale Instability in Rotating Stratified Fluids Driven by Small Scale Forces


In this paper, we find a new large scale instability displayed by a stratified rotating flow in forced turbulence. The turbulence is generated by a small scale external force at low Reynolds number. The theory is built on the rigorous asymptotic method of multi-scale development. There is no other special constraint concerning the force. In previous papers, the force was either helical or violating parity invariance. The nonlinear equations for the instability are obtained at the third order of the perturbation theory. In this article, we explain a detailed study of the linear stage of the instability.

Share and Cite:

A. Tur, M. Chabane and V. Yanovsky, "A New Large Scale Instability in Rotating Stratified Fluids Driven by Small Scale Forces," Open Journal of Fluid Dynamics, Vol. 3 No. 4, 2013, pp. 340-351. doi: 10.4236/ojfd.2013.34041.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. Sommeria, S. P. Meyers and H. L. Swinney, “Laboratory Simulation of Jupiter’s Great Red Spot,” Nature (London), Vol. 331, 1988, pp. 689-693. http://dx.doi.org/10.1038/331689a0
[2] G. Dritschel and B. Legras, “Modeling Oceanic and Atmospheric Vortices,” Physics Today, Vol. 46, No. 3, 1993, p. 44. http://dx.doi.org/10.1063/1.881375
[3] J. C. McWilliams, “The Emergence of Isolated Coherent Vortices in Turbulent Flow,” Journal of Fluid Mechanics, Vol. 146, 1984, pp. 21-43. http://dx.doi.org/10.1017/S0022112084001750
[4] J. Sommeria, “Experimental Study of the Two-Dimensional Inverse Energy Cascade in a Square Box,” Journal of Fluid Mechanics, Vol. 170, 1986, pp. 139-168. http://dx.doi.org/10.1017/S0022112086000836
[5] R. H. Kraichnan, “Inertial Ranges in Two-Dimensional Turbulence,” Physics of Fluids, Vol. 10, 1967, p. 1417. http://dx.doi.org/10.1063/1.1762301
[6] M. Chertkov, C. Connaughton, I. Kolokolov and V. Lebedev, “Dynamics of Energy Condensation in Two-Dimensional Turbulence,” Physical Review Letters, Vol. 99, 2007, Article ID: 084501. http://dx.doi.org/10.1103/PhysRevLett.99.084501
[7] D. Byrne, H. Xia and M. Shats, “Robust Inverse Energy Cascade and Turbulence Structure in Three-Dimensional Layers of Fluid,” Physics of Fluids, Vol. 23, 2011, Article ID: 095109. http://dx.doi.org/10.1063/1.3638620
[8] Y. Couder and C. Basdevant, “Experimental and Numerical Study of Vortex Couples in Two-Dimensional Flows,” Journal of Fluid Mechanics, Vol. 173, 1986, pp. 225-251. http://dx.doi.org/10.1017/S0022112086001155
[9] J. Paret and P. Tabeling, “Intermittency in the Two-Dimensional Inverse Cascade of Energy: Experimental Observations,” Physics of Fluids, Vol. 10, No. 12, 1998, p. 3126. http://dx.doi.org/10.1063/1.869840
[10] D. Molenaar, H. J. H. Clercx and G. J. F. van Heijst, “Angular Momentum of Forced 2D Turbulence in a Square No-Slip Domain,” Physica D, Vol. 196, No. 3-4, 2007, pp. 329-340. http://dx.doi.org/10.1016/j.physd.2004.06.001
[11] U. Frisch, Z. S. She and P. L. Sulem, “Large-Scale Flow Driven by the Anisotropic Kinetic Alpha Effect,” Physica D, Vol. 28, No. 3, 1987, pp. 382-392. http://dx.doi.org/10.1016/0167-2789(87)90026-1
[12] P. L. Sulem, Z. S. She, H. Scholl and U. Frisch, “Generation of Large-Scale Structures in Three-Dimensional Flow Lacking Parity-Invariance,” Journal of Fluid Mechanics, Vol. 205, 1989, p. 341. http://dx.doi.org/10.1017/S0022112089002065
[13] G. Rudiger, “On the α-Effect for Slow and Fast Rotation,” Astronomische Nachrichten, Vol. 299, No. 4, 1978, pp. 217-222. http://dx.doi.org/10.1002/asna.19782990408
[14] F. Krause and K.-H. Radler, “Mean-Field Magnetohydrodynamics and Dynamo Theory,” Akademie-Verlag, Berlin, 1980.
[15] H. K. Moffatt and A. Tsinober, “Helicity in Laminar and Turbulent Flow,” Annual Review of Fluid Mechanics, Vol. 24, 1992, pp. 281-312. http://dx.doi.org/10.1146/annurev.fl.24.010192.001433
[16] S. S. Moiseev, R. Z. Sagdeev, A. V. Tur, G. A. Khomenko and V. V. Yanovsky, “A Theory of Large-Scale Structure Origination in Hydrodynamic Turbulence,” Soviet Physics—JETP, Vol. 58, 1983, p. 1149.
[17] S. S. Moiseev, P. B. Rutkevich, A. V. Tur and V. V. Yanovsky, “Vortex Dynamos in a Helical Turbulent Convection,” Soviet Physics—JETP, Vol. 67, 1988, p. 294.
[18] E. A. Lupyan, A. A. Mazurov, P. B. Rutkevich and A. V. Tur, “Generation of Large-Scale Vortices through the Action of Spiral Turbulence of a Convective Nature,” Soviet Physics—JETP, Vol. 75, 1992, p. 833.
[19] G. A. Khomenko, S. S. Moiseev and A. V. Tur, “The Hydrodynamic Alpha-Effect in a Compressible Fluid,” Journal of Fluid Mechanics, Vol. 225, 1991, pp. 355-369. http://dx.doi.org/10.1017/S0022112091002082
[20] R. Marino, P. D. Mininni, D. Rosenberg and A. Pouquet, “Emergence of Helicity in Rotating Stratified Turbulence,” Physical Review E, Vol. 87, 2013, Article ID: 033016. http://dx.doi.org/10.1103/PhysRevE.87.033016
[21] Y. A. Berezin and V. P. Zhukov, “An Influence of Rotation on Convective Stability of Large Scale Distorbances in Turbulent Fluid,” Izv. AN SSSR, Mech. Zhidk. Gaza, Vol. 4, 1989, p. 3.
[22] P. B. Rutkevich, “Equation for Vortex Instability Caused by Convective Turbulence and the Coriolis Force,” JETF, Vol. 77, 1993, p. 933.
[23] A. V. Tur and V. V. Yanovsky, “Non Linear Vortex Structures in Stratified Fluid Driven by Small-Scale Helical Force,” Open Journal of Fluid Dynamics, Vol. 3, No. 2, 2013, pp. 64-74. http://dx.doi.org/10.4236/ojfd.2013.32009
[24] L. M. Smith and F. Waleffe, “Transfer of Energy to Two-Dimensional Large Scales in Forced, Rotating Three-Dimensional Turbulence,” Physics of Fluids, Vol. 11, No. 6, 1999, p. 1608. http://dx.doi.org/10.1063/1.870022
[25] L. M. Smith and F. Waleffe, “Generation of Slow Large Scales in Forced Rotating Stratified Turbulence,” Journal of Fluid Mechanics, Vol. 451, 2002, pp. 145-168. http://dx.doi.org/10.1017/S0022112001006309
[26] B. Galanti and P. L. Sulem, “Inverse Cascades in Three-Dimensional Anisotropic Flows Lacking Parity Invariance,” Physics of Fluids, Vol. A3, 1991, p. 1778.
[27] U. Frisch, “Turbulence: The Legacy of A. N. Kolmogorov,” Cambridge University Press, Cambridge, 1995.
[28] F. Krause and G. Rudiger, “On the Reynolds Stress in Mean Field Hydrodynamics. 1. Incompressible Homogeneous Isotropic Turbulence,” Astronomische Nachrichten, Vol. 295, No. 2, 1974, pp. 93-99. http://dx.doi.org/10.1002/asna.19742950205
[29] H. K. Moffat, “Magnetic Field Generation in Electrically Conducting Fluids,” Cambridge University Press, Cambridge, 1978.
[30] G. V. Levina, S. S. Moiseev and P. B. Rutkevich, “Hydrodynamic Alpha-Effect in a Convective System,” Advances in Fluid Mechanics, Vol. 25, 2000, p. 111.

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.