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}=(α₀+β₀R{·}{∇}×{}R)/(1+R²), {β}=β₀; where α₀,β₀ are conventional kinematic dynamo parameters, the reduction factor is proportional to the mean magnetic field R=Rₘ1/2B/({ρ}V²{)}1/2$, ${R}ₘ$ 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.