  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 $\mathcal{B} \neq 0$, 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.
