Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers

This paper is a detailed report on a programme of direct numerical simulations of incompressible nonhehcal randomly forced magnetohydrodynamic (MHD) turbulence that are used to settle a long-standing issue in the turbulent dynamo theory and demonstrate that the fluctuation dynamo exists in the limit of large magnetic Reynolds number Rm ≫1 and small magnetic Prandtl number Pm ≪ 1. The dependence of the critical Rmc for dynamo versus the hydrodynamic Reynolds number Re is obtained for 1 ≲ Re ≲ 6700. In the limit Pm ≪ 1, Rmc is at most three times larger than for the previously well established dynamo at large and moderate Prandtl numbers: Rmc ≲ 200 for Re ≳ 6000 compared to Rmc ∼ 60 for Pm ≥ 1. The stability curve Rmc(Re) (and, it is argued, the nature of the dynamo) is substantially different from the case of the simulations and liquid-metal experiments with a mean flow. It is not as yet possible to determine numerically whether the growth rate of the magnetic energy is ∝Rm1/2 in the limit Re ≫ Rm ≫ 1, as should be the case if the dynamo is driven by the inertial-range motions at the resistive scale, or tends to an Rm -independent value comparable to the turnover rate of the outer-scale motions. The magnetic-energy spectrum in the low-Pm regime is qualitatively different from the Pm ≥ 1 case and appears to develop a negative spectral slope, although current resolutions are insufficient to determine its asymptotic form. At Rm e (1, Rmc), the magnetic fluctuations induced via the tangling by turbulence of a weak mean field are investigated and the possibility of a k-1 spectrum above the resistive scale is examined. At low Rm < 1, the induced fluctuations are well described by the quasistatic approximation; the k-11/3 spectrum is confirmed for the first time in direct numerical simulations. Applications of the results on turbulent induction to understanding the nonlocal energy transfer from the dynamo-generated magnetic field to smaller-scale magnetic fluctuations are discussed. The results reported here are of fundamental importance for understanding the genesis of small-scale magnetic fields in cosmic plasmas. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft