Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T17:02:51.210Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean field electrodynamics: triumphs and tribulations

Part of: 50 years of Mean Field Electrodynamics

Published online by Cambridge University Press:  13 August 2018

David W. Hughes
David W. Hughes*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of mean field electrodynamics, now celebrating its fiftieth birthday, has had a profound influence on our modelling of cosmical dynamos, greatly enhancing our understanding of how such dynamos may operate. Here I discuss some of its undoubted triumphs, but also some of the problems that can arise in a mean field approach to dynamos in fluids (or plasmas) that are both highly turbulent and also extremely good electrical conductors, as found in all astrophysical settings.

Keywords

astrophysical plasmas
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 4 , August 2018 , 735840407
Copyright
© Cambridge University Press 2018 

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

Berger, M. A. 1999 Introduction to magnetic helicity. Plasma Phys. Control. Fusion 41, B167–B175.Google Scholar
Bhat, P., Subramanian, K. & Brandenburg, A. 2016 A unified large/small-scale dynamo in helical turbulence. Mon. Not. R. Astron. Soc. 461, 240–247.Google Scholar
Blackman, E. G. 2003 Recent developments in magnetic dynamo theory. In Turbulence and Magnetic Fields in Astrophysics (ed. Falgarone, E. & Passot, T.), Lecture Notes in Physics, vol. 614, pp. 432–463. Springer.Google Scholar
Blackman, E. G. & Field, G. B. 2000 Constraints on the magnitude of $\unicode[STIX]{x1D6FC}$ in dynamo theory. Astrophys. J. 534, 984–988.Google Scholar
Blackman, E. G. & Field, G. B. 2002 New dynamical mean-field dynamo theory and closure approach. Phys. Rev. Lett. 89, 265007.Google Scholar
Boldyrev, S., Cattaneo, F. & Rosner, R. 2005 Magnetic field generation in helical turbulence. Phys. Rev. Lett. 95, 255001.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. 1996 Nonlinear saturation of the turbulent $\unicode[STIX]{x1D6FC}$ effect. Phys. Rev. E 54, 4532.Google Scholar
Cattaneo, F. & Hughes, D. W. 2006 Dynamo action in a rotating convective layer. J. Fluid Mech. 553, 401–418.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., Hughes, D. W. & Thelen, J.-C. 2002 The nonlinear properties of a large-scale dynamo driven by helical forcing. J. Fluid Mech. 456, 219–237.Google Scholar
Cattaneo, F. & Tobias, S. M. 2014 On large-scale dynamo action at high magnetic Reynolds number. Astrophys. J. 789, 70.Google Scholar
Cattaneo, F. & Vainshtein, S. I. 1991 Suppression of turbulent transport by a weak magnetic field. Astrophys. J. 376, L21–L24.Google Scholar
Childress, S. & Gilbert, A. D. 1995 Stretch, Twist, Fold: The Fast Dynamo. Springer.Google Scholar
Courvoisier, A., Hughes, D. W. & Proctor, M. R. E. 2010a Self-consistent mean-field magnetohydrodynamics. Proc. R. Soc. Lond. A 466, 583–601.Google Scholar
Courvoisier, A., Hughes, D. W. & Proctor, M. R. E. 2010b A self-consistent treatment of the electromotive force in magnetohydrodynamics for large diffusivities. Astron. Nachr. 331, 667.Google Scholar
Courvoisier, A., Hughes, D. W. & Tobias, S. M. 2006 $\unicode[STIX]{x1D6FC}$ -effect in a family of chaotic flows. Phys. Rev. Lett. 96, 034503.Google Scholar
Cowling, T. G. 1933 The magnetic field of sunspots. Mon. Not. R. Astron. Soc. 94, 39–48.Google Scholar
Du, Y. & Ott, E. 1993 Growth rates for fast kinematic dynamo instabilities of chaotic fluid flows. J. Fluid Mech. 257, 265–288.Google Scholar
Frisch, U., She, Z. S. & Sulem, P. L. 1987 Large-scale flow driven by the anisotropic kinetic alpha effect. Physica D 28, 382–392.Google Scholar
Galloway, D. J. & Proctor, M. R. E. 1992 Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion. Nature 356, 691–693.Google Scholar
Gruzinov, A. V. & Diamond, P. H. 1994 Self-consistent theory of mean-field electrodynamics. Phys. Rev. Lett. 72, 1651–1653.Google Scholar
Herreman, W. & Lesaffre, P. 2011 Stokes drift dynamos. J. Fluid Mech. 679, 32–57.Google Scholar
Herzenberg, A. 1958 Geomagnetic dynamos. Phil. Trans. R. Soc. Lond. A 250, 543–583.Google Scholar
Hubbard, A., Del Sordo, F., Käpylä, P. J. & Brandenburg, A. 2009 The $\unicode[STIX]{x1D6FC}$ effect with imposed and dynamo-generated magnetic fields. Mon. Not. R. Astron. Soc. 398, 1891–1899.Google Scholar
Hughes, D. W. & Cattaneo, F. 2008 The alpha-effect in rotating convection: size matters. J. Fluid Mech. 594, 445–461.Google Scholar
Hughes, D. W., Cattaneo, F. & Kim, E.-J. 1996 Kinetic helicity, magnetic helicity and fast dynamo action. Phys. Lett. A 223, 167–172.Google Scholar
Hughes, D. W., Mason, J., Proctor, M. R. E. & Rucklidge, A. M.2018 Nonlinear mean field MHD: how far can you get? (in preparation).Google Scholar
Hughes, D. W. & Proctor, M. R. E. 2009 Large-scale dynamo action driven by velocity shear and rotating convection. Phys. Rev. Lett. 102, 044501.Google Scholar
Hughes, D. W. & Proctor, M. R. E. 2010 Turbulent magnetic diffusivity tensor for time-dependent mean fields. Phys. Rev. Lett. 104, 024503.Google Scholar
Hughes, D. W. & Proctor, M. R. E. 2013 The effect of velocity shear on dynamo action due to rotating convection. J. Fluid Mech. 717, 395–416.Google Scholar
Hughes, D. W., Proctor, M. R. E. & Cattaneo, F. 2011 The $\unicode[STIX]{x1D6FC}$ -effect in rotating convection: a comparison of numerical simulations. Mon. Not. R. Astron. Soc. 414, L45–L49.Google Scholar
Jepps, S. A. 1975 Numerical models of hydromagnetic dynamos. J. Fluid Mech. 67, 625–646.Google Scholar
Krause, F. & Rädler, K.-H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon.Google Scholar
Kulsrud, R. & Anderson, S. 1992 The spectrum of random magnetic fields in the mean field dynamo theory of the galactic magnetic field. Astrophys. J. 396, 606–630.Google Scholar
Larmor, J. 1919 How could a rotating body such as the Sun become a magnet? Rep. British Assoc. Adv. Sci. 159–160.Google Scholar
Marconi, U. M. B., Puglisi, A., Rondoni, L. & Vulpiani, A. 2008 Fluctuation dissipation: response theory in statistical physics. Phys. Rep. 461, 111–195.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Moffatt, H. K. & Proctor, M. R. E. 1982 The role of the helicity spectrum function in turbulent dynamo theory. Geophys. Astrophys. Fluid Dyn. 21, 265–283.Google Scholar
Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293.Google Scholar
Pouquet, A., Frisch, U. & Léorat, J. 1976 Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77, 321–354.Google Scholar
Proctor, M. R. E. 2003 Dynamo processes: the interaction of turbulence and magnetic fields. In Stellar Astrophysical Fluid Dynamics (ed. Thompson, M. J. & Christensen-Dalsgaard, J.), pp. 143–158. Cambridge University Press.Google Scholar
Roberts, G. O. 1970 Spatially periodic dynamos. Phil. Trans. R. Soc. Lond. A 266, 535–558.Google Scholar
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A 271, 411–454.Google Scholar
Roberts, P. H. 1994 Fundamentals of dynamo theory. In Lectures on Solar and Planetary Dynamos (ed. Proctor, M. R. E. & Gilbert, A. D.), p. 1. Cambridge University Press.Google Scholar
Roberts, P. H. & Stix, M.1971 The turbulent dynamo: a translation of a series of papers by F. Krause, K.-H. Rädler, and M. Steenbeck. Tech. Rep. 60. NCAR Tech. Note.Google Scholar
Rogachevskii, I. & Kleeorin, N. 2003 Electromotive force and large-scale magnetic dynamo in a turbulent flow with a mean shear. Phys. Rev. E 68, 036301.Google Scholar
Rüdiger, G. 1989 Differential Rotation and Stellar Convection: Sun and Solar-type Stars. Taylor and Francis.Google Scholar
Rüdiger, G. & Hollerbach, R. 2004 The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory. Wiley.Google Scholar
Shumaylova, V., Teed, R. J. & Proctor, M. R. E. 2017 Large- to small-scale dynamo in domains of large aspect ratio: kinematic regime. Mon. Not. R. Astron. Soc. 466, 3513–3518.Google Scholar
Stix, M. 1972 Non-linear dynamo waves. Astron. Astrophys. 20, 9.Google Scholar
Tao, L., Cattaneo, F. & Vainshtein, S. I. 1993 Evidence for the suppression of the alpha-effect by weak magnetic fields. In Solar and Planetary Dynamos (ed. Proctor, M. R. E., Matthews, P. C. & Rucklidge, A. M.), pp. 303–310. Cambridge University Press.Google Scholar
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139–1141.Google Scholar
Vainshtein, S. I. & Rosner, R. 1991 On turbulent diffusion of magnetic fields and the loss of magnetic flux from stars. Astrophys. J. 376, 199–203.Google Scholar
Vainshtein, S. I. & Zeldovich, Y. B. 1972 Origin of magnetic fields in astrophysics. Sov. Phys. Uspekhi 15, 159–172.Google Scholar
Vishniac, E. T. & Cho, J. 2001 Magnetic helicity conservation and astrophysical dynamos. Astrophys. J. 550, 752–760.Google Scholar
Vladimirov, V. A. 2012 Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo? J. Fluid Mech. 698, 51–61.Google Scholar
Yousef, T. A., Heinemann, T., Schekochihin, A. A., Kleeorin, N., Rogachevskii, I., Iskakov, A. B., Cowley, S. C. & McWilliams, J. C. 2008 Generation of magnetic field by combined action of turbulence and shear. Phys. Rev. Lett. 100, 184501.Google Scholar