Nonlinear turbulent dynamo during gravitational collapse

Abstract

Via amplification by turbulent dynamo, magnetic fields can be potentially important for the formation of the first stars. To examine the dynamo behavior during the gravitational collapse of primordial gas, we extend the theory of nonlinear turbulent dynamo to include the effect of gravitational compression. The relative importance between dynamo and compression varies during contraction, with the transition from dynamo-to compression-dominated amplification of magnetic fields with the increase of density. In the nonlinear stage of magnetic field amplification with the scale-by-scale energy equipartition between turbulence and magnetic fields, reconnection diffusion of magnetic fields in ideal magnetohydrodynamic (MHD) turbulence becomes important. It causes the violation of flux-freezing condition and accounts for (a) the small growth rate of nonlinear dynamo, (b) the weak dependence of magnetic energy on density during contraction, (c) the saturated magnetic energy, and (d) the large correlation length of magnetic fields. The resulting magnetic field structure and the scaling of magnetic field strength with density are radically different from the expectations of flux-freezing.

Magnetic fields in the formation of the first stars.--II Results

“…It should be noted that the actual slope of the B − nH relation varies with density: It is closer to 2 3 than to 0.59 for nH < ∼ 8 × 10 7 cm −3 and also for a short range around nH = 10 10 cm −3 . The fact that the slope of the B − nH relation is less than 2/3 is consistent with the result of Xu & Lazarian (2020) that flux-freezing is violated in the nonlinear dynamo (see sections 6.1 and 6.2).…”

“…Furthermore, we have followed Schleicher et al (2010) and Schober et al (2012b) in assuming that compression obeys flux-freezing, so that B ∝ ξ 2/3 in equation ( 24). In section 6.2 below, we shall see that our simulation is consistent with B ∝ ξ 0.54−58 for the nonlinear dynamo, which violates fluxfreezing (Xu & Lazarian 2020). It is possible to get good fits to the data for the kinematic stage with exponents that differ from 2/3; for example, setting B ∝ ξ 0.5 in equation ( 24) gives agreement with the data to within a factor 1.7 for CΓ = 0.088.…”

“…I.43). Xu & Lazarian (2020) have given a self-consistent derivation of the equation for a nonlinear dynamo in a timedependent medium in which they drop this assumption. We generalize their treatment and follow the evolution of the dy- Red depicts the result in the hydro case without magnetic fields, while blue shows the result in the MHD case when they are included.…”

“…where the subscript "ref" denotes a reference value and kp is the wavenumber at which the turbulent eddies in the inertial range are in equipartition with the field (Xu & Lazarian 2016). In the absence of dynamo action, EB = B 2 /(8πρ) would increase as ρ 1/3 in an isotropic compression, so that E B,ref = EB1(ξ/ξ1) 1/3 (Xu & Lazarian 2020). They couple the dynamo to the compression by assuming that the reference wavenumber scales inversely with the Jeans length, k ref ∝ λJ −1 ∝ ξ 1/2 for isothermal contraction.…”

“…where We now follow the derivation of Xu & Lazarian (2020). The growth rate of field is Γ = vp/ℓp, where vp is the eddy velocity on the scale ℓp.…”