Fluctuations in Quasi-Two-Dimensional Fast Dynamos

The ratio R₁ between the average magnetic energy and the square-averaged flux plays an important role in the study of nonlinear dynamos, as a measure of the efficiency of a dynamo at generating flux. For large values of Rₘ, R₁ displays a scaling behavior of the type R₁~Rₘⁿ, where Rₘ is the magnetic Reynolds number. We show by direct numerical evaluation that n depends sensitively on the flow complexity for small-scale dynamos. Furthermore, by relating n to the cancellation exponent and the correlation dimension of the magnetic field, we argue that n is not likely to be close to zero in general.