  The connection between helically isotropic MHD turbulence and mean-field
dynamo theory is reviewed. The nonlinearity in the mean-field theory is not yet
well established, but detailed comparison with simulations begin to help select
viable forms of the nonlinearity. The crucial discriminant is the magnetic
helicity, which is known to evolve only on a slow resistive time scale in the
limit of large magnetic Reynolds number. Particular emphasis is put on the
possibility of memory effects, which means that an additional explicitly
time-dependent equation for the nonlinearity is solved simultaneously with the
mean-field equations. This approach leads to better agreement with the
simulations, while it would also produce more favorable agreement between
models and stellar dynamos.

Astrophysics