Critical Magnetic Prandtl Number for Small-Scale Dynamo

We report a series of numerical simulations showing that the critical magnetic Reynolds number Rmc for the nonhelical small-scale dynamo depends on the Reynolds number Re. Namely, the dynamo is shut down if the magnetic Prandtl number Prₘ=Rm/Re is less than some critical value Prm,c1 even for Rm for which dynamo exists at Prₘ≥1. We argue that, in the limit of Re→∞, a finite Prm,c may exist. The second possibility is that Prm,c→0 as Re→∞, while Rmc tends to a very large constant value inaccessible at current resolutions. If there is a finite Prm,c, the dynamo is sustainable only if magnetic fields can exist at scales smaller than the flow scale, i.e., it is always effectively a large-Prₘ dynamo. If there is a finite Rmc, our results provide a lower bound: Rmc220 for Prₘ≤1/8. This is larger than Rm in many planets and in all liquid-metal experiments.