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 nonhelical randomly forced 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 vs. 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 ∝ Rm 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 Rmindependent 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 ∈ (1,Rmc), the magnetic fluctuations induced via the tangling by turbulence of a weak mean field are investigated and the possibility of a k spectrum above the resistive scale is examined. At low Rm < 1, the induced fluctuations are well described by the quasistatic approximation; the k 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 field in cosmic plasmas. PACS numbers: 91.25.Cw, 47.65.-d, 95.30.Qd, 96.60.Hv, 47.27.ek Submitted to: New J. Phys. Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers 2