Microphysical dissipation, turbulence and magnetic fields in hyper-accreting discs

ABSTRACT Hyper-accreting discs occur in compact-object mergers and in collapsed cores of massive stars. They power the central engine of γ-ray bursts in most scenarios. We calculate the microphysical dissipation (the viscosity and resistivity) of plasma in these discs, and discuss the implications for their global structure and evolution. At the temperatures (k_BT > m_ec²) and densities (ρ ∼ 10⁹-10¹² g cm^-3) characteristic of the neutrino-cooled innermost regions, the viscosity is provided mainly by mildly degenerate electrons, while the resistivity is modified from the Spitzer value due to the effects of both relativity and degeneracy. Under these conditions the magnetic Reynolds number is very large (Re_M ∼ 10¹⁹) and the plasma behaves as an almost ideal magnetohydrodynamic (MHD) fluid. Among the possible non-ideal MHD effects the Hall term is relatively the most important, while the magnetic Prandtl number, Pm (the ratio of viscosity to resistivity), is typically larger than unity: 10 ≲ Pm ≲ 6 × 10³. Inspection of the outer radiatively inefficient regions indicates similar properties, with magnetic Prandtl numbers as high as ∼10⁴. Numerical simulations of the magnetorotational instability (MRI) indicate that the saturation level and angular momentum transport efficiency may be greatly enhanced at high Prandtl numbers. If this behaviour persists in the presence of a strong Hall effect we would expect that hyper-accreting discs should be strongly magnetized and highly variable. The expulsion of magnetic field that cannot be dissipated at small scales may also favour a magnetic outflow. We note that there are limited similarities between hyper-accreting discs and X-ray binary discs - which also have a high magnetic Prandtl number close to the black hole - which suggests that a comparison between late-time activity in γ-ray bursts and X-ray binary accretion states may be fruitful. More generally, our results imply that the possibly different character of high Prandtl number MHD flows needs to be considered in studies and numerical simulations of hyper-accreting discs.