Self-consistent theory of mean-field electrodynamics

Mean-field electrodynamics, including both \ensuremath{\alpha} and \ensuremath{\beta} effects while accounting for the effects of small-scale magnetic fields, is derived for incompressible magnetohydrodynamics. The principal result is \ensuremath{\alpha}=(${\mathrm{\ensuremath{\alpha}}}_{0}$+${\mathrm{\ensuremath{\beta}}}_{0}$R\ensuremath{\cdot}\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}R)/(1+${\mathit{R}}^{2}$), \ensuremath{\beta}=${\mathrm{\ensuremath{\beta}}}_{0}$; where ${\mathrm{\ensuremath{\alpha}}}_{0}$,${\mathrm{\ensuremath{\beta}}}_{0}$ are conventional kinematic dynamo parameters, the reduction factor is proportional to the mean magnetic field R=${\mathit{R}}_{\mathit{m}}^{1/2}$B/(\ensuremath{\rho}${\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.