Abstract
This paper is a detailed report on a programme of direct numerical simulations of incompressible nonhelical 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 . 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 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 , 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.
Highlights
DEUTSCHE PHYSIKALISCHE GESELLSCHAFT and appears to develop a negative spectral slope, current resolutions are insufficient to determine its asymptotic form
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
The most important of these is whether the dynamo we have found is driven by the inertial-range motions at the resistive scale—if it is, its growth rate should be proportional to Rm1/2, which would make it a dominant field-amplification effect
Summary
We use the standard pseudospectral method to solve in a periodic cube the equations of incompressible MHD:. It is the effective viscosity given by (3) that is used for calculating Re and Pm in the hyperviscous runs. All other runs were done using another code, written by J Maron, which was the code used in our earlier papers (Schekochihin et al 2004a, 2004b, 2005) Both codes are pseudospectral, solve the same equations (1) and (2) and use the same units of length and time, but different time-stepping, fast Fourier transforms and parallelisation algorithms, as well as slightly different implementations of the random forcing. 1.34 1, 3(a) 1.38 1, 3(a) 1.42 1, 3(a) 1.38 1, 3(a) 1.41 1, 3(a), 4, 5 1.41 1, 3(b), 6(a) 1.43 1, 6(b) 1.37 1, 3(b), 6(a) 1.44 1, 6(d) 1.45 1, 3(b), 4, 5, 6(e), 8(a) 1.40 1, 3(b), 6(f)
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