Skip to main content Accessibility help

Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes

  • Kannabiran Seshasayanan (a1) and Basile Gallet (a1)


We present analytical examples of fluid dynamos that saturate through the action of the Coriolis and inertial terms of the Navier–Stokes equation. The flow is driven by a body force and is subject to global rotation and uniform sweeping velocity. The model can be studied down to arbitrarily low viscosity and naturally leads to the strong-field scaling regime for the magnetic energy produced above threshold: the magnetic energy is proportional to the global rotation rate and independent of the viscosity $\unicode[STIX]{x1D708}$ . Depending on the relative orientations of global rotation and large-scale sweeping, the dynamo bifurcation is either supercritical or subcritical. In the supercritical case, the magnetic energy follows the scaling law for supercritical strong-field dynamos predicted on dimensional grounds by Pétrélis & Fauve (Eur. Phys. J. B, vol. 22, 2001, pp. 271–276). In the subcritical case, the system jumps to a finite-amplitude dynamo branch. The magnetic energy obeys a magneto-geostrophic scaling law (Roberts & Soward, Annu. Rev. Fluid Mech., vol. 4, 1972, pp. 117–154), with a turbulent Elsasser number of the order of unity, where the magnetic diffusivity of the standard Elsasser number appears to be replaced by an eddy diffusivity. In the absence of global rotation, the dynamo bifurcation is subcritical and the saturated magnetic energy obeys the equipartition scaling regime. We consider both the vicinity of the dynamo threshold and the limit of large distance from threshold to put these various scaling behaviours on firm analytical ground.


Corresponding author

Email address for correspondence:


Hide All
Aubert, J. 2005 Steady zonal flows in spherical shell dynamos. J. Fluid Mech. 542, 5367.
Aubert, J., Gastine, T. & Fournier, A. 2017 Spherical convective dynamos in the rapidly rotating asymptotic regime. J. Fluid Mech. 813, 558593.
Calkins, M. A., Julien, K., Tobias, S. M. & Aurnou, J. M. 2015 A multiscale dynamo model driven by quasi-geostrophic convection. J. Fluid Mech. 780, 143166.
Calkins, M. A., Long, L., Nieves, D., Julien, K. & Tobias, S. M. 2016 Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers. Phys. Rev. Fluids 1, 083701.
Cameron, A. & Alexakis, A. 2016 Fate of alpha dynamos at large Rm. Phys. Rev. Lett. 117, 205101.
Campagne, A., Machicoane, N., Gallet, B., Cortet, P.-P. & Moisy, F. 2016 Turbulent drag in a rotating frame. J. Fluid Mech. 794, R5.
Cattaneo, F. & Hughes, D. W. 1996 Nonlinear saturation of the turbulent 𝛼 effect. Phys. Rev. E 54, 5.
Cattaneo, F. & Hughes, D. W. 2017 Dynamo action in rapidly rotating Rayleigh–Bénard convection at infinite Prandtl number. J. Fluid Mech. 825, 385411.
Courvoisier, A., Hughes, D. W. & Proctor, M. R. E. 2010 Self-consistent mean-field magnetohydrodynamics. Proc. R. Soc. Lond. A 466, 583601.
Dormy, E. 2016 Strong-field spherical dynamos. J. Fluid Mech. 789, 500513.
Dormy, E., Oruba, L. & Petitdemange, L. 2018 Three branches of dynamo action. Fluid Dyn. Res. 50, 1.
Fauve, S., Herault, J., Michel, G. & Pétrélis, F. 2017 Instabilities on a turbulent background. J. Stat. Mech. Theory Exp. 6, 064001.
Frisch, U., She, Z. S. & Sulem, P.-L. 1987 Large-scale flow driven by the anisotropic kinetic alpha effect. Physica D 28, 382392.
Gailitis, A., Lielausis, O., Dement’ev, S., Platacis, E., Cifersons, A., Gerbeth, G., Gundrum, T., Stefani, F., Christen, M., Hänel, H. & Will, G. 2000 Detection of a flow induced magnetic field eigenmode in the Riga dynamo facility. Phys. Rev. Lett. 84, 43654368.
Gallet, B., Berhanu, M. & Mordant, N. 2009 Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal. Phys. Fluids 21, 085107.
Gallet, B., Herault, J., Laroche, C., Pétrélis, F. & Fauve, S. 2012 Reversals of a large-scale field generated over a turbulent background. Geophys. Astrophys. Fluid Dyn. 106, 468492.
Gallet, B. 2015 Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows. J. Fluid Mech. 783, 412447.
Gilbert, A. D. & Sulem, P.-L. 1989 On inverse cascades in alpha effect dynamos. Geophys. Astrophys. Fluid Dyn. 51, 243261.
Gómez-Pérez, N. & Heimpel, M. 2007 Numerical models of zonal flow dynamos: an application to the ice giants. Geophys. Astrophys. Fluid Dyn. 101, 371388.
Hughes, D. W. & Cattaneo, F. 2016 Strong-field dynamo action in rapidly rotating convection with no inertia. Phys. Rev. E 93, 061101(R).
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47, 163193.
Moffatt, H. K. 1972 An approach to a dynamic theory of dynamo action in a rotating conducting fluid. J. Fluid Mech. 53, 385399.
Monchaux, R., Berhanu, M., Bourgoin, M., Moulin, M., Odier, Ph., Pinton, J.-F., Volk, R., Fauve, S., Mordant, N., Pétrélis, F., Chiffaudel, A., Daviaud, F., Dubrulle, B., Gasquet, C., Marié, L. & Ravelet, F. 2007 Generation of magnetic field by dynamo action in a turbulent flow of liquid sodium. Phys. Rev. Lett. 98, 044502.
Morin, J., Dormy, E., Schrinner, M. & Donati, J.-F. 2011 Weak- and strong-field dynamos: from the Earth to the stars. Mon. Not. R. Astron. Soc. 418, 133137.
Nunez, A., Pétrélis, F. & Fauve, S. 2001 Saturation of a Ponomarenko type fluid dynamo. In Dynamo and Dynamics, pp. 6774. Kluwer Academic Publishers.
Oruba, L. & Dormy, E. 2014 Predictive scaling laws for spherical rotating dynamos. Geophys. J. Intl 198 (2), 828847.
Pétrélis, F. & Fauve, S. 2001 Saturation of the magnetic field above the dynamo threshold. Eur. Phys. J. B 22, 271276.
Pétrélis, F., Mordant, N. & Fauve, S. 2007 On the magnetic fields generated by experimental dynamos. Geophys. Astrophys. Fluid Dyn. 101, 289323.
Plumley, M., Calkins, M. A., Julien, K. & Tobias, S. M. 2018 Self-consistent single mode investigations of the quasi-geostrophic convection-driven dynamo model. J. Plasma Phys. 84, 4.
Ponty, Y. & Plunian, F. 2011 Transition from large-scale to small-scale dynamo. Phys. Rev. Lett. 106, 154502.
Roberts, G. O. 1972 Dynamo action of fluid motions with two-dimensional periodicity. Phil. Trans. R. Soc. Lond. A 271, 411454.
Roberts, P. H. & Soward, A. M. 1972 Magnetohydrodynamics of the Earth’s core. Annu. Rev. Fluid Mech. 4, 117154.
Roberts, P. H. 1978 Magnetoconvection in a rapidly rotating fluid. In Rotating Fluids in Geophysics (ed. Roberts, P. H. & Soward, A. M.), pp. 421435. Academic Press.
Roberts, P. H. 1988 Future of dynamo theory. Geophs. Astrophys. Fluid Dyn. 44, 331.
Roberts, P. H. & Soward, A. M. 1992 Dynamo theory. Annu. Rev. Fluid Mech. 24, 459512.
Schaeffer, N., Jault, D., Nataf, H.-C. & Fournier, A. 2017 Turbulent geodynamo simulations: a leap towards Earth’s core. Geophys. J. Intl 211, 129.
Schrinner, M., Petitdemange, L. & Dormy, E. 2012 Dipole collapse and dynamo waves in global direct numerical simulations. Astrophys. J. 752, 121.
Seshasayanan, K. & Alexakis, A. 2016 Turbulent 2.5-dimensional dynamos. J. Fluid Mech. 799, 246264.
Seshasayanan, K., Gallet, B. & Alexakis, A. 2017 Transition to turbulent dynamo saturation. Phys. Rev. Lett. 119, 204503.
Sivashinsky, G. & Yakhot, V. 1985 Negative viscosity effect in large-scale flows. Phys. Fluids 28, 10401042.
Soward, A. M. 1974 A convection-driven dynamo. I. The weak-field case. Phil. Trans. R. Soc. Lond. A 275, 611646.
Stieglitz, R. & Müller, U. 2001 Experimental demonstration of a homogeneous two-scale dynamo. Phys. Fluids 13, 561564.
Tilgner, A. 2008 Dynamo action with wave motion. Phys. Rev. Lett. 100, 128501.
Vainshtein, S. I. & Cattaneo, F. 1992 Nonlinear restrictions on dynamo action. Astrophys. J. 393, 165171.
Yadav, R. K., Gastine, T., Christensen, U. R., Wolk, S. J. & Poppenharger, K. 2016 Approaching a realistic force balance in geodynamo simulations. Proc. Natl Acad. Sci. USA 113 (43), 1206512070.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Dynamo saturation down to vanishing viscosity: strong-field and inertial scaling regimes

  • Kannabiran Seshasayanan (a1) and Basile Gallet (a1)


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.