Simulations of the Small‐Scale Turbulent Dynamo

Abstract

We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Pr m , which is relevant for hot low-density astrophysical plasmas. A Pr m parameter scan is given for the model case of viscosity-dominated (low-Reynolds-number) turbulence. We concentrate on three topics: magnetic-energy spectra and saturation levels, the structure of the magnetic-field lines, and intermittency of the field-strength distribution. The main results are as follows: (1) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime; (2) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; and (3) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic-energy spectrum becomes much shallower. We propose an analytical model of saturation based on the idea of partial two-dimensionalization of the velocity gradients with respect to the local direction of the magnetic folds. The model-predicted saturated spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Pr m runs are carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower-than-exponential magnetic-energy growth accompanied by an increase of the resistive scale and partial suppression of the kinetic-energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed. Subject headings: magnetic fields -MHD -plasmas -turbulence -methods: numerical 1 Present-time address: DAMTP/CMS,

Exact scaling laws and the local structure of isotropic magnetohydrodynamic turbulence

“…However, one can study the subviscous fields in numerical simulations with Pm ≫ 1 and Re ∼ 1 (the viscosity-dominated limit; see Schekochihin et al 2004). This means that the inertial range physics is sacrificed and we study the saturated state of a small-scale dynamo with Rm ≫ 1 and a single-scale (∼ ℓ 0 ) randomly forced flow.…”

“…The structure of the turbulence at small scales is expected to be the same for the decaying and forced cases. This appears not to be true for MHD, probably because of a tendency of the magnetic field to decay towards large-scale forcefree states (compare, e.g., the decaying simulations of Biskamp & Müller 2000 with the forced ones of Schekochihin et al 2004or Haugen et al 2004. We consider the case in which the velocity field is forced, while the magnetic field is not and can receive energy only via interaction with the velocity field.…”

“…In the presence of this mean field, the turbulence at the small scales is (globally) anisotropic -a distinct case that will not be discussed here (see review by Schekochihin & Cowley 2006). Turbulence produced by a spatially homogeneous isotropic nonhelical random forcing does not generate a mean field, but, at least when the magnetic Prandtl number Pm = ν/η ≥ 1, does lead to amplification of small-scale (r < ℓ 0 ) magnetic energy (e.g., Schekochihin et al 2004;Haugen et al 2004), a process known as small-scale dynamo.…”

JITTER SELF-COMPTON PROCESS: GeV EMISSION OF GRB 100728A

“…As the magnetic field may have a subLarmor length scale, the jitter radiation in this sub-Larmor scale magnetic field was fully studied by Medvedev et al (2011). The small-scale turbulent dynamo with large Reynolds numbers at a saturated state in a fluid flow was simulated by Schekochihin et al (2004). The simulation identified a power-law turbulent energy spectrum.…”

Saturation of Zeldovich stretch–twist–fold map dynamos

“…Besides showing features similar to the two-strip maps, the eigenfunctions of the four-strip map have fine-scale reversals of magnetic field. These could be thought of as an analogue of the field reversals seen in the DNS of fluctuation dynamos (Schekochihin et al 2004;Brandenburg & Subramanian 2005). An interesting case is when we use a random initial seed field, instead of a uniform one.…”