Skip to main content Accessibility help

The electromotive force in multi-scale flows at high magnetic Reynolds number

  • Steven M. Tobias (a1) and Fausto Cattaneo (a2)


Recent advances in dynamo theory have been made by examining the competition between small- and large-scale dynamos at high magnetic Reynolds number $\mathit{Rm}$ . Small-scale dynamos rely on the presence of chaotic stretching whilst the generation of large-scale fields occurs in flows lacking reflectional symmetry via a systematic electromotive force (EMF). In this paper we discuss how the statistics of the EMF (at high  $\mathit{Rm}$ ) depend on the properties of the multi-scale velocity that is generating it. In particular, we determine that different scales of flow have different contributions to the statistics of the EMF, with smaller scales contributing to the mean without increasing the variance. Moreover, we determine when scales in such a flow act independently in their contribution to the EMF. We further examine the role of large-scale shear in modifying the EMF. We conjecture that the distribution of the EMF, and not simply the mean, largely determines the dominant scale of the magnetic field generated by the flow.


Corresponding author

Email address for correspondence:


Hide All
Augustson, K., Brun, A. S., Miesch, M. & Toomre, J. 2015 Grand minima and equatorward propagation in a cycling stellar convective dynamo. Astrophys. J. 809, 149.
Boldyrev, S., Cattaneo, F. & Rosner, R. 2005 Magnetic-field generation in helical turbulence. Phys. Rev. Lett. 95 (25), 255001.
Brandenburg, A. & Subramanian, K. 2005 Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417, 1209.
Cattaneo, F. & Tobias, S. M. 2005 Interaction between dynamos at different scales. Phys. Fluids 17 (12), 127105,1–6.
Cattaneo, F. & Tobias, S. M. 2014 On large-scale dynamo action at high magnetic Reynolds number. Astrophys. J. 789, 70.
Childress, S. & Gilbert, A. D. 1995 Stretch, twist, fold. In The Fast Dynamo, XI, Lecture Notes in Physics, vol. 37, p. 406. Springer.
Courvoisier, A., Hughes, D. W. & Tobias, S. M. 2006 ${\it\alpha}$ effect in a family of chaotic flows. Phys. Rev. Lett. 96 (3), 034503.
Courvoisier, A. & Kim, E.-J. 2009 Kinematic ${\it\alpha}$ effect in the presence of a large-scale motion. Phys. Rev. E 80 (4), 046308.
Cowling, T. G. 1933 The magnetic field of sunspots. Mon. Not. R. Astron. Soc. 94, 3948.
Finn, J. M. & Ott, E. 1988 Chaotic flows and fast magnetic dynamos. Phys. Fluids 31, 29923011.
Galloway, D. J. & Proctor, M. R. E. 1992 Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion. Nature 356, 691693.
Hargittai, I. 2013 Buried Glory: Portraits of Soviet Scientists. Oxford University Press.
Käpylä, P. J., Korpi, M. J. & Brandenburg, A. 2010 The ${\it\alpha}$ effect in rotating convection with sinusoidal shear. Mon. Not. R. Astron. Soc. 402, 14581466.
Krause, F. & Raedler, K. H. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon.
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.
Plunian, F. & Rädler, K.-H. 2002 Subharmonic dynamo action in the Roberts flow. Geophys. Astrophys. Fluid Dyn. 96, 115133.
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A 271, 411454.
Steenbeck, M., Krause, F. & Rädler, K.-H. 1966 Berechnung der mittleren LORENTZ-Feldstärke für ein elektrisch leitendes medium in turbulenter, durch CORIOLIS-Kräfte beeinflußter Bewegung. Z. Naturforsch. Teil A 21, 369.
Sunyaev, R. A.(Ed.) 2004 Zeldovich Reminiscences. Chapman & Hall.
Tobias, S. M. & Cattaneo, F. 2008a Dynamo action in complex flows: the quick and the fast. J. Fluid Mech. 601 (1), 101122.
Tobias, S. M. & Cattaneo, F. 2008b Limited role of spectra in dynamo theory: coherent versus random dynamos. Phys. Rev. Lett. 101 (12), 125003.
Tobias, S. M. & Cattaneo, F. 2013 Shear-driven dynamo waves at high magnetic Reynolds number. Nature 497, 463465.
Tobias, S. M., Cattaneo, F. & Boldyrev, S. 2013 Ten Chapters in Turbulence (ed. Davidson, P. A., Kaneda, Y. & Sreenivasan, K. R.), pp. 351404. Cambridge University Press.
Tobias, S. M., Cattaneo, F. & Brummell, N. H. 2008 Convective dynamos with penetration, rotation, and shear. Astrophys. J. 685, 596605.
Vainshtein, S. I. & Kichatinov, L. L. 1986 The dynamics of magnetic fields in a highly conducting turbulent medium and the generalized Kolmogorov–Fokker–Planck equations. J. Fluid Mech. 168, 7387.
Zel’dovich, Y. B. 1957 The magnetic field in the two-dimensional motion of a conducting turbulent fluid. Sov. Phys. JETP 4, 460462.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

The electromotive force in multi-scale flows at high magnetic Reynolds number

  • Steven M. Tobias (a1) and Fausto Cattaneo (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.