Transition from large-scale to small-scale dynamo

The dynamo equations are solved numerically with a helical forcing corresponding to the Roberts flow. In the fully turbulent regime the flow behaves as a Roberts flow on long time scales, plus turbulent fluctuations at short time scales. The dynamo onset is controlled by the long time scales of the flow, in agreement with the former Karlsruhe experimental results. The dynamo mechanism is governed by a generalized $\alpha$-effect which includes both usual $\alpha$-effect and turbulent diffusion, plus all higher order effects. Beyond the onset we find that this generalized $\alpha$-effect scales as $O(Rm^{-1})$ suggesting the take-over of small-scale dynamo action. This is confirmed by simulations in which dynamo occurs even if the large-scale field is artificially suppressed.