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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: [email protected]
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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

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