The Onset of a Small-Scale Turbulent Dynamo at Low Magnetic Prandtl Numbers

We study numerically the dependence of the critical magnetic Reynolds number Rm c for the turbulent small-scale dynamo on the hydrodynamic Reynolds number Re. The turbulence is statistically homogeneous, isotropic, and mirror-symmetric. We are interested in the regime of low magnetic Prandtl number Pm = Rm/Re < 1, which is relevant for stellar convective zones, protostellar disks, and laboratory liquid-metal experiments. The two asymptotic possibilities are Rm c → const as Re → ∞ (a small-scale dynamo exists at low Pm) or Rm c /Re = Pm c → const as Re → ∞ (no small-scale dynamo exists at low Pm). 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, Rm c grows with Re with no sign of approaching a constant limit. We reach the maximum values of Rm c ~ 500 for Re ~ 3000. By comparing simulations with Laplacian viscosity, fourth-, sixth-, and eighth-order hyperviscosity, and Smagorinsky large-eddy viscosity, we find that Rm c is not sensitive to the particular form of the viscous cutoff. This work represents a significant 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).