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Quasi-static magnetohydrodynamic turbulence at high Reynolds number

Published online by Cambridge University Press:  21 June 2011

B. FAVIER ,
F. S. GODEFERD ,
C. CAMBON ,
A. DELACHE  and
W. J. T. BOS
B. FAVIER
Affiliation:
LMFA UMR 5509 CNRS, École Centrale de Lyon, Université de Lyon, F-69134 Lyon, France
F. S. GODEFERD*
Affiliation:
LMFA UMR 5509 CNRS, École Centrale de Lyon, Université de Lyon, F-69134 Lyon, France
C. CAMBON
Affiliation:
LMFA UMR 5509 CNRS, École Centrale de Lyon, Université de Lyon, F-69134 Lyon, France
A. DELACHE
Affiliation:
Université de Lyon, F-42023 Saint-Étienne, LMFA@UJM St-Étienne, CNRS UMR 5509, Université de St-Étienne, 23 rue Docteur Paul Michelon, F-42023 Saint-Étienne CEDEX 2, France
W. J. T. BOS
Affiliation:
LMFA UMR 5509 CNRS, École Centrale de Lyon, Université de Lyon, F-69134 Lyon, France
*
†Email address for correspondence: Fabien.Godeferd@ec-lyon.fr
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Abstract

We analyse the anisotropy of homogeneous turbulence in an electrically conducting fluid submitted to a uniform magnetic field, for low magnetic Reynolds number, in the quasi-static approximation. We interpret contradictory earlier predictions between linearized theory and simulations: in the linear limit, the kinetic energy of transverse velocity components, normal to the magnetic field, decays faster than the kinetic energy of the axial component, along the magnetic field (Moffatt, J. Fluid Mech., vol. 28, 1967, p. 571); whereas many numerical studies predict a final state characterized by dominant energy of transverse velocity components. We investigate the corresponding nonlinear phenomenon using direct numerical simulation (DNS) of freely decaying turbulence, and a two-point statistical spectral closure based on the eddy-damped quasi-normal Markovian (EDQNM) model. The transition from the three-dimensional turbulent flow to a ‘two-and-a-half-dimensional’ flow (Montgomery & Turner, Phys. Fluids, vol. 25, 1982, p. 345) is a result of the combined effects of short-time linear Joule dissipation and longer time nonlinear creation of polarization anisotropy. It is this combination of linear and nonlinear effects which explains the disagreement between predictions from linearized theory and results from numerical simulations. The transition is characterized by the elongation of turbulent structures along the applied magnetic field, and by the strong anisotropy of directional two-point correlation spectra, in agreement with experimental evidence. Inertial equatorial transfers in both DNS and the model are presented to describe in detail the most important equilibrium dynamics. Spectral scalings are maintained in high-Reynolds-number turbulence attainable only with the EDQNM model, which also provides simplified modelling of the asymptotic state of quasi-static magnetohydrodynamic (MHD) turbulence.

JFM classification

MHD and Electrohydrodynamics: MHD turbulence Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 681 , 25 August 2011 , pp. 434 - 461
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Alemany, A., Moreau, R., Sulem, P. L & Frisch, U. 1979 Influence of an external magnetic field on homogeneous MHD turbulence. J. Mécan. 18, 277–313.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. J. Fluid Mech. 5, 113–133.CrossRefGoogle Scholar
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave-turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.CrossRefGoogle Scholar
Bos, W. J. T., Kadoch, B., Schneider, K. & Bertoglio, J.-P. 2009 Inertial range scaling of the scalar flux spectrum in two-dimensional turbulence. Phys. Fluids 21 (11), 115105.CrossRefGoogle Scholar
Burattini, P., Kinet, M., Carati, D. & Knaepen, B. 2008 a Anisotropy of velocity spectra in quasistatic magnetohydrodynamic turbulence. Phys. Fluids 20, 065110.CrossRefGoogle Scholar
Burattini, P., Kinet, M., Carati, D. & Knaepen, B. 2008 b Spectral energetics of quasi-static MHD turbulence. Physica D 237, 2062–2066.Google Scholar
Cambon, C. 1990 Homogeneous MHD turbulence at weak magnetic Reynolds numbers: approach to angular-dependent spectra. In Advances in Turbulence Studies: Progress in Astronautics and Aeronautics (ed. Branover, H. & Unger, Y.), vol. 149, pp. 131–145. AIAA.Google Scholar
Cambon, C. & Godeferd, F. S. 1993 Inertial transfers in freely decaying rotating, stably stratified, and MHD turbulence. In Progress in Turbulence Research: Progress in Astronautics and Aeronautics (ed. Branover, H. & Unger, Y.), vol. 162, pp. 150–168. AIAA.Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.CrossRefGoogle Scholar
Cambon, C., Mansour, N. N. & Godeferd, F. S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332.CrossRefGoogle Scholar
Caperan, P. & Alemany, A. 1985 Turbulence homogène MHD à faible nombre de Reynolds magnétique: étude de la transition vers la phase quasi bidimensionnelle et caractérisation de son anisotropie. J. Méca. Théor. Appl. 4 (2), 175–200.Google Scholar
Favier, B., Godeferd, F. S., Cambon, C. & Delache, A. 2010 On the two-dimensionalization of quasi-static MHD turbulence. Phys. Fluids 22, 075104.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, 115–150.CrossRefGoogle Scholar
Herring, J. R. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859–872.CrossRefGoogle Scholar
Ishida, T. & Kaneda, Y. 2007 Small-scale anisotropy in magnetohydrodynamic turbulence under a strong uniform magnetic field. Phys. Fluids 19, 075104.CrossRefGoogle Scholar
Jacquin, L., Leuchter, O., Cambon, C. & Mathieu, J. 1990 Homogeneous turbulence in the presence of rotation. J. Fluid Mech. 125, 505–534.Google Scholar
von Kármán, T. & Lin, C. C. 1949 On the concept of similarity in the theory of isotropic turbulence. Rev. Mod. Phys. 21 (3), 516–519.CrossRefGoogle Scholar
Knaepen, B., Kassinos, S. & Carati, D. 2004 Magnetohydrodynamics turbulence at moderate Reynolds number. J. Fluid Mech. 513, 199–220.CrossRefGoogle Scholar
Knaepen, B. & Moreau, R. 2008 Magnetohydrodynamics turbulence at low magnetic Reynolds number. Annu. Rev. Fluid Mech. 40, 25–45.CrossRefGoogle Scholar
Kraichnan, R. H. 1972 Test-field model for inhomogeneous turbulence. J. Fluid Mech. 56, 287–304.CrossRefGoogle Scholar
Laporta, A. 1995 Spectral study and modelisation of an inhomogeneous turbulence (in French). PhD thesis, Ecole Centrale de Lyon, Lyon, France.Google Scholar
Leith, C. E. 1971 Atmospheric predictability and two-dimensional turbulence. J. Atmos. Sci. 28 (2), 145–161.2.0.CO;2>CrossRefGoogle Scholar
Lesieur, M. & Herring, J. 1985 Diffusion of a passive scalar in two-dimensional turbulence. J. Fluid Mech. 161, 77–95.CrossRefGoogle Scholar
Lesieur, M. & Ossia, S. 2000 3D isotropic turbulence at very high Reynolds numbers: EDQNM study. J. Turbul. 1, 7.CrossRefGoogle Scholar
Matsumoto, T. 2009 Anomalous scaling of three-dimensional Rayleigh–Taylor turbulence. Phys. Rev. E 79, 055301.Google ScholarPubMed
Moffatt, H. K. 1967 On the suppression of turbulence by a uniform magnetic field. J. Fluid Mech. 28, 571–592.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1982 Two-and-a-half-dimensional magnetohydrodynamic turbulence. Phys. Fluids 25 (2), 345–349.CrossRefGoogle Scholar
Okamoto, N., Davidson, P. A. & Kaneda, Y. 2010 On the decay of low-magnetic-Reynolds-number turbulence in an imposed magnetic field. J. Fluid Mech. 651, 295–318.CrossRefGoogle Scholar
Pouquet, A., Lesieur, M., Andre, J. C. & Basdevant, C. 1975 Evolution of high Reynolds number two-dimensional turbulence. J. Fluid Mech. 72 (2), 305–319.Google Scholar
Schumann, U. 1976 Numerical simulation of the transition from three- to two-dimensional turbulence under a uniform magnetic field. J. Fluid Mech. 74, 31–58.CrossRefGoogle Scholar
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525.CrossRefGoogle Scholar
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81–105.CrossRefGoogle Scholar
Turner, L. 2000 Using helicity to characterize homogeneous and inhomogeneous turbulent dynamics. J. Fluid Mech. 408, 205–238.CrossRefGoogle Scholar
Vorobev, A., Zikanov, O., Davidson, P. A. & Knaepen, B. 2005 Anisotropy of MHD turbulence at low magnetic Reynolds number. Phys. Fluids 17, 125105.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4, 350–363.CrossRefGoogle Scholar
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids A 5, 677–685.CrossRefGoogle Scholar
Zhou, Y. 2010 Renormalization group theory for fluid and plasma turbulence. Phys. Rep. 488, 1–49.CrossRefGoogle Scholar
Zhou, Y. & Matthaeus, W. H. 2005 Phenomenology treatment of magnetohydrodynamic turbulence with nonequipartition and anisotropy. Phys. Plasmas 12, 056503.CrossRefGoogle Scholar
Zhou, Y., Matthaeus, W. H. & Dmitruk, P. 2004 Magnetohydrodynamic turbulence and time scales in astrophysical and space plasmas. Rev. Mod. Phys. 76, 1015–1035.CrossRefGoogle Scholar