Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T09:52:36.307Z Has data issue: false hasContentIssue false
  • English
  • Français

On the measurement of the turbulent diffusivity of a large-scale magnetic field

Published online by Cambridge University Press:  01 February 2013

S. M. Tobias  and
F. Cattaneo
S. M. Tobias*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
F. Cattaneo
Affiliation:
Department of Astronomy and Astrophysics and The Computation Institute, University of Chicago, Chicago, IL 60637, USA
*
†Email address for correspondence: smt@amsta.leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We argue that a method developed by Ångström (Ann. Phys. Chem., vol. 114, 1861, pp. 513–530) to measure the thermal conductivity of solids can be adapted to determine the effective diffusivity of a large-scale magnetic field in a turbulent electrically conducting fluid. The method consists of applying an oscillatory source and measuring the steady-state response. We illustrate this method in a two-dimensional system. This geometry is chosen because it is possible to compare the results with independent methods that are restricted to two-dimensional flows. We describe two variants of this method: one (the ‘turbulent Ångström method’) that is better suited to laboratory experiments and a second (the ‘method of oscillatory sines’) that is effective for numerical experiments. We show that, if correctly implemented, all methods agree. Based on these results we argue that these methods can be extended to three-dimensional numerical simulations and laboratory experiments.

JFM classification

MHD and Electrohydrodynamics: MHD and Electrohydrodynamics MHD and Electrohydrodynamics: MHD turbulence Mixing: Turbulent mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 347 - 360
Copyright
©2013 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ångström, Å. J. 1861 Neue methode, das wärmeleitungsvermögen der körper zu bestimmen. Ann. Phys. Chem. 114, 513–530.Google Scholar
Brandenburg, A., Rädler, K.-H., Rheinhardt, M. & Subramanian, K. 2008 Magnetic quenching of $\alpha $ and diffusivity tensors in helical turbulence. Astrophys. J. Lett. 687, L49–L52.Google Scholar
Brandenburg, A., Svedin, A. & Vasil, G. M. 2009 Turbulent diffusion with rotation or magnetic fields. Mon. Not. R. Astron. Soc. 395, 1599–1606.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1987 Spectral Methods in Fluid Dynamics. Springer.Google Scholar
Cattaneo, F. 1994 On the effects of a weak magnetic field on turbulent transport. Astrophys. J. 434, 200–205.Google Scholar
Cattaneo, F. & Hughes, D. W. 2009 Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers. Mon. Not. R. Astron. Soc. 395, L48–L51.Google Scholar
Cattaneo, F. & Vainshtein, S. I. 1991 Suppression of turbulent transport by a weak magnetic field. Astrophys. J. Lett. 376, L21–L24.Google Scholar
Courvoisier, A., Hughes, D. W. & Tobias, S. M. 2009 Mean induction and diffusion: the influence of spatial coherence. J. Fluid Mech. 627, 403.Google Scholar
Eyink, G. L. 2009 Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50 (8), 083102.Google Scholar
Frick, P., Noskov, V., Denisov, S. & Stepanov, R. 2010 Direct measurement of effective magnetic diffusivity in turbulent flow of liquid sodium. Phys. Rev. Lett. 105 (18), 184502.Google Scholar
Gruzinov, A. V. & Diamond, P. H. 1994 Self-consistent theory of mean-field electrodynamics. Phys. Rev. Lett. 72, 1651–1653.Google Scholar
Krause, F. & Raedler, K. H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon.Google Scholar
Moffatt, H. K. 1974 The mean electromotive force generated by turbulence in the limit of perfect conductivity. J. Fluid Mech. 65, 1–10.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, 353 p.Google Scholar
Monchaux, R., Berhanu, M., Aumaître, S., Chiffaudel, A., Daviaud, F., Dubrulle, B., Ravelet, F., Fauve, S., Mordant, N., Pétrélis, F., Bourgoin, M., Odier, P., Pinton, J.-F., Plihon, N. & Volk, R. 2009 The von Kármán sodium experiment: turbulent dynamical dynamos. Phys. Fluids 21 (3), 035108.Google Scholar
Noskov, V., Denisov, S., Stepanov, R. & Frick, P. 2012 Turbulent viscosity and turbulent magnetic diffusivity in a decaying spin-down flow of liquid sodium. Phys. Rev. E 85 (1), 016303.Google Scholar
Parker, E. N. 1979 Cosmical Magnetic Fields: Their Origin and their Activity. Clarendon; Oxford University Press, 858 p.Google Scholar
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Schrinner, M., Rädler, K.-H., Schmitt, D., Rheinhardt, M. & Christensen, U. 2005 Mean-field view on rotating magnetoconvection and a geodynamo model. Astron. Nachr. 326, 245–249.Google Scholar
Taylor, G. I. 1921 Experiments with rotating fluids. In Proc. R. Soc. Lond., 100, 114–121.Google Scholar
Zel’dovich, Ya. B. 1957 The magnetic field in the two-dimensional motion of a conducting turbulent fluid. Sov. Phys. J. Expl Theor. Phys. 4, 460–462.Google Scholar