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)< approximately 1 even for Rm for which dynamo exists at Pr(m)> or =1. We argue that, in the limit of Re-->infinity, a finite Pr(m,c) may exist. The second possibility is that Pr(m,c)-->0 as Re--> infinity, 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) greater, similar 220 for Pr(m)< or =1/8. This is larger than Rm in many planets and in all liquid-metal experiments.