Self-consistent mean field MHD

Abstract

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (2009a), the 2D basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength $\mathcal{B}$. By extending to the nonlinear regime the kinematic analysis of Roberts (1970), we show that it is possible to predict the growth rate of these perturbations by applying mean field theory to \textit{both} the momentum and the induction equations. If $\mathcal{B}=0$, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic $\alpha$-effect and the AKA instability respectively. However, if $\mathcal{B}\ne 0$, the momentum and induction equations are coupled by the Lorentz force; in this case, we show that four transport tensors are now necessary to determine the growth rate of the perturbations. We illustrate these situations by numerical examples; in particular, we show that a mean field description of the nonlinear regime based solely on a quenched $\alpha$ coefficient is incorrect.

Submitted to Proc R. Soc., 22/07/09 Accepted subject to minor revisions, 11/08/09. Revised version resubmitted, 25/09/09

References in corpus (5)

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• Problems with kinematic mean field electrodynamics at high magnetic Reynolds numbers
• A growing dynamo from a saturated Roberts flow dynamo
• The Turbulent Magnetic Diffusivity Tensor for Time-Dependent Mean Fields
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• The mean electromotive force resulting from magnetic buoyancy instability
• A Self-Consistent Treatment of the Electromotive Force in Magnetohydrodynamics for Large Diffusivities
• Mean Field Electrodynamics: Triumphs and Tribulations
• Dissipative structures in a nonlinear dynamo
• Nonlinear dynamos at infinite magnetic Prandtl number
• Path integrals for mean-field equations in nonlinear dynamos
• Mode analysis of numerical geodynamo models