We perform numerical experiments to calculate the kinematic alpha effect for a family of maximally helical, chaotic flows with a range of correlation times. We find that the value of depends on the structure of the flow, on its correlation time and on the magnetic Reynolds number in a nontrivial way. Furthermore, it seems that there is no clear relation between alpha and the helicity of the flow, contrary to what is often assumed for the parametrization of mean-field dynamo models.

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 α-effect and the anisotropic kinetic alpha instability, respectively. However, if B = 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 α coefficient is incorrect.

Within the same framework we calculate the mean induction of a magnetic field and the mean diffusivity of a passive scalar, for two families of flows in which the degree of spatial decorrelation can be systematically adjusted. We investigate the dependence of these quantities both on the spatial decoherence and on the molecular diffusivity. We demonstrate that for flows with similar global properties, the mean induction is dramatically reduced as the flows become less spatially correlated; the mean diffusivity, on the other hand, shows no significant or systematic variation.

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The coupled equations that describe the effect of large-scale magnetic and velocity fields on forced high-diffusivity magnetohydrodynamic flows are investigated through an extension of mean field electrodynamics. Our results generalise those of Rädler & Brandenburg (2010), who consider a similar situation but assume that the effect of the Lorentz force on the momentum equation can be neglected. New mean coupling terms are shown to appear, which can lead to large-scale growth of magnetic and velocity fields even when the usual α-effects are absent.