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

We study numerically the dependence of the critical magnetic Reynolds numberRmc for the turbulent small-scale dynamo on the hydrodynamic Reynolds numberRe. The turbulence is statistically homogeneous, isotropic, andmirror--symmetric. We are interested in the regime of low magnetic Prandtlnumber Pm=Rm/Re<1, which is relevant for stellar convective zones, protostellardisks, and laboratory liquid-metal experiments. The two asymptoticpossibilities are Rmc->const as Re->infinity (a small-scale dynamo exists atlow Pm) or Rmc/Re=Pmc->const as Re->infinity (no small-scale dynamo exists atlow Pm). Results obtained in two independent sets of simulations of MHDturbulence using grid and spectral codes are brought together and found to bein quantitative agreement. We find that at currently accessible resolutions,Rmc grows with Re with no sign of approaching a constant limit. We reach themaximum values of Rmc~500 for Re~3000. By comparing simulations with Laplacianviscosity, fourth-, sixth-, and eighth-order hyperviscosity and Smagorinskylarge-eddy viscosity, we find that Rmc is not sensitive to the particular formof the viscous cutoff. This work represents a significant extension of thestudies previously published by Schekochihin et al. 2004, PRL 92, 054502 andHaugen et al. 2004, PRE, 70, 016308 and the first detailed scan of thenumerically accessible part of the stability curve Rmc(Re).Comment: 4 pages, emulateapj aastex, 2 figures; final version as published in ApJL (but with colour figures