Fluctuations in Quasi-Two-Dimensional Fast Dynamos

The ratio $R_1$ 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 $R_m$, $R_1$ displays a scaling behavior of the type $R_1\sim R_m^n$, where $R_m$ 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.