Critical Magnetic Prandtl Number for Small-Scale Dynamo

We report a series of numerical simulations showing that the critical magnetic Reynolds number Rm_c for the nonhelical small-scale dynamo depends on the Reynolds number Re. Namely, the dynamo is shut down if the magnetic Prandtl number Pr_m=Rm/Re is less than some critical value Pr_m,c≲1 even for Rm for which dynamo exists at Pr_m≥1. We argue that, in the limit of Re→∞, a finite Pr_m,c may exist. The second possibility is that Pr_m,c→0 as Re→∞, while Rm_c tends to a very large constant value inaccessible at current resolutions. If there is a finite Pr_m,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-Pr_m dynamo. If there is a finite Rm_c, our results provide a lower bound: Rm_c≳220 for Pr_m≤1/8. This is larger than Rm in many planets and in all liquid-metal experiments.