The onset of a small-scale turbulent dynamo at low magnetic Prandtl numbers

We study numerically the dependence of the critical magnetic Reynolds number for the turbulent small-scale scale dynamo on the hydrodynamic Reynolds number. The turbulence is statistically homogeneous, isotropic, Re and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number, which Pm = Rm/Re < 1 is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rm(c) -> const as Re -> infinity (a small-scale dynamo exists at low) or Rm(c)/Re = Pm-c -> const Re -> infinity (no small-scale dynamo exists at low). Results obtained in two independent sets of simulations of MHD turbulence using grid and spectral codes are brought together and found to be in quantitative agreement. We find that at currently accessible resolutions, grows with with no sign of approaching a Rm Re constant limit. We reach the maximum values of for. By comparing simulations with Rm(c) similar to 500 Re similar to 3000 Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that is not sensitive to the particular form of the viscous cutoff. This work represents a significant Rm c extension of the studies previously published by Schekochihin et al. (2004a) and Haugen et al. (2004a) and the first detailed scan of the numerically accessible part of the stability curve. Rm(c) (Re).