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)less than or similar to1 even for Rm for which dynamo exists at Pr(m)greater than or equal to1. We argue that, in the limit of Re-->infinity, a finite Pr-m,Pr-c may exist. The second possibility is that Pr-m,Pr-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,Pr-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 than or similar to220 for Pr(m)less than or equal to1/8. This is larger than Rm in many planets and in all liquid-metal experiments.