An optimal scale separation for a dynamo experiment

Scale separation between the flow and the magnetic field is a common feature of natural dynamos. It has also been used in the Karlsruhe sodium experiment in which the scale of the magnetic field is roughly seven times larger than the scale of the flow [R. Stieglitz and U. Müller, “Experimental demonstration of the homogeneous two-scale dynamo,” Phys. Fluids 13, 561 (2001)]. Recently, Fauve and Pétrélis [Peyresq Lectures on Nonlinear Phenomena, edited by J. Sepulchre (World Scientific, Singapore, 2003), p. 1] have shown that the power needed to reach the dynamo threshold in a dynamo experiment increases with the scale separation in the limit of large scale separation. With a more elaborate method based on subharmonic solutions [F. Plunian and K.-H. Rädler, “Subharmonic dynamo action in the Roberts flow,” Geophys. Astrophys. Fluid Dyn. 96, 115 (2002)], we show, for the Roberts flow, the existence of an optimal scale separation for which this power is minimum. Previous results obtained by Tilgner [“A kinematic dynamo with a small scale velocity field,” Phys. Lett. A 226, 75 (1997)] with a completely different numerical method are also reconsidered here. Again, we find an optimal scale separation in terms of minimum power for dynamo action. In addition we find that this scale separation compares very well with the one derived from the subharmonic solutions method.