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

Published online by Cambridge University Press:  13 February 2019

Kannabiran Seshasayanan Open the ORCID record for Kannabiran Seshasayanan [Opens in a new window]  and
Basile Gallet Open the ORCID record for Basile Gallet [Opens in a new window]
Kannabiran Seshasayanan
Affiliation:
Service de Physique de l’État Condensé, CNRS UMR 3680, CEA Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
Basile Gallet*
Affiliation:
Service de Physique de l’État Condensé, CNRS UMR 3680, CEA Saclay, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
*
†Email address for correspondence: basile.gallet@cea.fr
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Abstract

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.

JFM classification

MHD and Electrohydrodynamics: Dynamo theory Geophysical and Geological Flows: Geodynamo
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 971 - 994
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Copyright
© 2019 Cambridge University Press 

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