Non Linear Vortex Structures in Stratified Fluid Driven by Small-Scale Helical Force

Affiliation(s)

Université de Toulouse [UPS], CNRS, Institut de Recherche en Astrophysique et Planétologie,Toulouse Cedex, France.

Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkov, Ukraine.

Université de Toulouse [UPS], CNRS, Institut de Recherche en Astrophysique et Planétologie,Toulouse Cedex, France.

Institute for Single Cristals, Nat. Academy of Science Ukraine, Kharkov, Ukraine.

ABSTRACT

In this work, we consider the effect of a small-scale helical driving force on fluid with a stable temperature gradient with Reynolds number . At first glance, this system does not have any instability. However, we show that a large scale vortex instability appears in the fluid despite its stable stratification. In a non-linear mode this instability becomes saturated and gives a large number of stationary spiral vortex structures. Among these structures there is a stationary helical soliton and a kink of the new type. The theory is built on the rigorous asymptotical method of multi-scale development.

Cite this paper

A. Tur and 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. doi: 10.4236/ojfd.2013.32009.

A. Tur and V. Yanovsky, "Non Linear Vortex Structures in Stratified Fluid Driven by Small-Scale Helical Force,"

References

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[2] J. C. McWilliams, “The Emergence of Isolated Coherent Vortices in Turbulent Flow,” Journal of Fluid Mechanics, Vol. 146, 1984, pp. 21-43. doi:10.1017/S0022112084001750

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[9] 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, 2004, pp. 329-340. doi:10.1016/j.physd.2004.06.001

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[13] U. Frisch, “Turbulence: The Legacy of A. N. Kolmogorov,” Cambridge University Press, Cambridge, 1995.

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[15] 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. doi:10.1017/S0022112001006309

[16] 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. doi:10.1016/0167-2789(87)90026-1

[17] 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. doi:10.1017/S0022112089002065

[18] G. Rudiger, “On the α-Effect for Slow and Fast Rotation,” Astronmische Nachrichten, Vol. 299, No. 4, 1978, pp. 217-222. doi:10.1002/asna.19782990408

[19] A. Pouquet and P. D. Mininni, “The Interplay between Helicity and Rotation in Turbulence: Implications for Scaling Laws and Small-Scale Dynamics,” Philosophical Transactions of the Royal Society A, Vol. 368, No. 1916, 2010, pp. 1635-1662.

[20] H. K. Moffatt and A. Tsinober, “Helicity in Laminar and Turbulent Flow,” Annual Reviews of Fluid Mechanics, Vol. 24, 1992, pp. 281-312. doi:10.1146/annurev.fl.24.010192.001433

[21] 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, Vol. 58, 1983, pp. 1149-1157.

[22] S. S. Moiseev, P. B. Rutkevich, A. V. Tur and V. V. Yanovsky, “Vortex Dynamos in a Helical Turbulent Convection,” Soviet Physics, Vol. 67, 1988, pp. 263-294.

[23] 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, Vol. 75, 1992, pp. 829-833.

[24] G. A. Khomenko, S. S. Moiseev and A. V. Tur, “The Hydrodynamic Alpha-Effect in a Compressible Fluid,” Journal of Fluid Mechanics, Vol. 225, No. 1, 1991, p. 355. doi:10.1017/S0022112091002082

[25] F. Krause and G. Rudiger, “On the Reynolds Stress in Mean Field Hydrodynamics. 1. Incompressible Homogeneous Isotropic Turbulence,” Astronmische Nachrichten, Vol. 295, No. 2, 1974, pp. 93-99. doi:10.1002/asna.19742950205

[26] H. K. Moffat, “Magnetic Field Generation in Electrically Conducting Fluids,” Cambridge University Press, Cambridge, 1978.

[27] G. V. Levina, S. S. Moiseev and P. B. Rutkevich, “Hydrodynamic Alpha-Effect in a Convective System,” Advances in Fluid Mechanics, Vol. 25, 2000, pp. 111-162.

[28] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Dover Publishers, New York, 1961.

[1] J. Jimenez, “The Role of Coherent Structures in Modeling Turbulence and Mixing,” Lecture Notes in Physics, Vol. 136, 1981.

[2] J. C. McWilliams, “The Emergence of Isolated Coherent Vortices in Turbulent Flow,” Journal of Fluid Mechanics, Vol. 146, 1984, pp. 21-43. doi:10.1017/S0022112084001750

[3] 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. doi:10.1017/S0022112086000836

[4] R. H. Kraichnan, “Inertial Ranges in Two-Dimensional Turbulence,” Physics of Fluids, Vol. 10, No. 7, 1967, pp. 1417-1423. doi:10.1063/1.1762301

[5] M. Chertkov, C. Connaughton, I. Kolokolov and V. Lebedev, “Dynamics of Energy Condensation in Two-Dimensional Turbulence,” Physical Review Letters, Vol. 99, No. 8, 2007, Article ID: 084501. doi:10.1103/PhysRevLett.99.084501

[6] 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, No. 9, 2011, Article ID: 095109. doi:10.1063/1.3638620

[7] 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. doi:10.1017/S0022112086001155

[8] J. Paret and P. Tabeling, “Intermittency in the Two-Dimensional Inverse Cascade of Energy: Experimental Observations,” Physics of Fluids, Vol. 10, No. 12, 1998, pp. 3126-3136. doi:10.1063/1.869840

[9] 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, 2004, pp. 329-340. doi:10.1016/j.physd.2004.06.001

[10] J. Sommeria, S. P. Meyers and H. L. Swinney, “Laboratory Simulation of Jupiter’s Great Red Spot,” Nature, Vol. 331, No. 6158, 1988, pp. 689-693. doi:10.1038/331689a0

[11] G. Dritschel and B. Legras, “Modeling Oceanic and Atmospheric Vortices,” Physics Today, Vol. 46, No. 3, 1993, pp. 44-51. doi:10.1063/1.881375

[12] B. Galanti and P. L. Sulem, “Inverse Cascades in Three-Dimensional Anisotropic Flows Lacking Parity Invariance,” Physics of Fluids A, Vol. 3, No. 7, 1991, p. 1778-1784.

[13] U. Frisch, “Turbulence: The Legacy of A. N. Kolmogorov,” Cambridge University Press, Cambridge, 1995.

[14] 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, pp. 1608-1623. doi:10.1063/1.870022

[15] 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. doi:10.1017/S0022112001006309

[16] 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. doi:10.1016/0167-2789(87)90026-1

[17] 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. doi:10.1017/S0022112089002065

[18] G. Rudiger, “On the α-Effect for Slow and Fast Rotation,” Astronmische Nachrichten, Vol. 299, No. 4, 1978, pp. 217-222. doi:10.1002/asna.19782990408

[19] A. Pouquet and P. D. Mininni, “The Interplay between Helicity and Rotation in Turbulence: Implications for Scaling Laws and Small-Scale Dynamics,” Philosophical Transactions of the Royal Society A, Vol. 368, No. 1916, 2010, pp. 1635-1662.

[20] H. K. Moffatt and A. Tsinober, “Helicity in Laminar and Turbulent Flow,” Annual Reviews of Fluid Mechanics, Vol. 24, 1992, pp. 281-312. doi:10.1146/annurev.fl.24.010192.001433

[21] 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, Vol. 58, 1983, pp. 1149-1157.

[22] S. S. Moiseev, P. B. Rutkevich, A. V. Tur and V. V. Yanovsky, “Vortex Dynamos in a Helical Turbulent Convection,” Soviet Physics, Vol. 67, 1988, pp. 263-294.

[23] 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, Vol. 75, 1992, pp. 829-833.

[24] G. A. Khomenko, S. S. Moiseev and A. V. Tur, “The Hydrodynamic Alpha-Effect in a Compressible Fluid,” Journal of Fluid Mechanics, Vol. 225, No. 1, 1991, p. 355. doi:10.1017/S0022112091002082

[25] F. Krause and G. Rudiger, “On the Reynolds Stress in Mean Field Hydrodynamics. 1. Incompressible Homogeneous Isotropic Turbulence,” Astronmische Nachrichten, Vol. 295, No. 2, 1974, pp. 93-99. doi:10.1002/asna.19742950205

[26] H. K. Moffat, “Magnetic Field Generation in Electrically Conducting Fluids,” Cambridge University Press, Cambridge, 1978.

[27] G. V. Levina, S. S. Moiseev and P. B. Rutkevich, “Hydrodynamic Alpha-Effect in a Convective System,” Advances in Fluid Mechanics, Vol. 25, 2000, pp. 111-162.

[28] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Dover Publishers, New York, 1961.