The elemental shear dynamo

Journal of Fluid Mechanics

Journal of Fluid Mechanics / Volume 699 / May 2012, pp 414-452
Copyright © Cambridge University Press 2012 The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence <http://creativecommons.org/licenses/by-nc-sa/2.5/>. The written permission of Cambridge University Press must be obtained for commercial re-use.
DOI: http://dx.doi.org/10.1017/jfm.2012.120 (About DOI), Published online: 17 April 2012
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The elemental shear dynamo

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mcwilliams jc [Google Scholar]

James C. McWilliams^a1 ^c1

^a1Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA 90095-1565, USA

Abstract

A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially growing, large-scale (mean) magnetic dynamo in the presence of a uniform parallel shear flow. It is a ‘kinematic’ theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magnetohydrodynamics, and it is rigorously derived in the limit of small magnetic Reynolds number, ${\mathit{Re}}_{\eta } \ll 1$ . Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with a small seed wavenumber in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $\gamma$ can occur for arbitrary values of ${\mathit{Re}}_{\eta }$ , viscous Reynolds number ${\mathit{Re}}_{\nu }$ , and random-force correlation time ${t}_{f}$ and orientation angle ${\theta }_{f}$ in the shearing plane. The value of $\gamma$ is independent of the domain size. The shear dynamo is ‘fast’, with finite $\gamma \gt 0$ in the limit of ${\mathit{Re}}_{\eta } \gg 1$ . Averaged over random realizations of the forcing history, the ensemble-mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (magnetic energy). In the limit of small ${\mathit{Re}}_{\eta }$ and ${\mathit{Re}}_{\nu }$ , the dynamo behaviour is related to the well-known alpha–omega ansatz when the force is slowly varying ( $\gamma {t}_{f} \gg 1$ ) and to the ‘incoherent’ alpha–omega ansatz when the force is more rapidly fluctuating.