Self-consistent theory of mean-field electrodynamics

Mean-field electrodynamics, including both α and β effects while accounting for the effects of small-scale magnetic fields, is derived for incompressible magnetohydrodynamics. The principal result is α=(${\mathrm{\alpha }}_0$+${\mathrm{\beta }}_0$ R⋅∇×R)/(1+${\mathit{R}}^2$), β=${\mathrm{\beta }}_0$; where ${\mathrm{\alpha }}_0$,${\mathrm{\beta }}_0$ are conventional kinematic dynamo parameters, the reduction factor is proportional to the mean magnetic field R=${\mathit{R}}_{\mathit{m}}^{1/2}$ B/(ρ${\mathit{V}}^2$ ${)}^{1/2}$, ${\mathit{R}}_{\mathit{m}}$ is the magnetic Reynolds number, and V is the characteristic turbulent velocity. This result follows from a generalization of the Zeldovich theorem to three dimensions, exploiting magnetic helicity balance.