Self-consistent mean-field magnetohydrodynamics

We consider the linear stability of two-dimensional nonlinear magnetohydrodynamic basic states to long-wavelength three-dimensional perturbations. Following Hughes & Proctor (Hughes & Proctor 2009 Proc. R. Soc. A 465, 1599-1616 (doi:10.1098/rspa.2008.0493)), the two-dimensional basic states are obtained from a specific forcing function in the presence of an initially uniform mean field of strength B. By extending to the nonlinear regime the kinematic analysis of Roberts (Roberts 1970 Phil. Trans. R. Soc. Lond. A 266, 535 -558 (doi:10.1098/rsta.1970.0011)), we show that it is possible to predict the growth rate of these perturbations by applying mean-field theory to both the momentum and the induction equations. If B = 0, these equations decouple and large-scale magnetic and velocity perturbations may grow via the kinematic alpha-effect and the anisotropic kinetic alpha instability, respectively. However, if B not equal 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.