  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.
