Magnetic field dragging in accretion discs

TLDR

A thin accretion disc of half‑thickness H is threaded by a magnetic field produced by both the disc current and an external uniform field. The authors derive an integro‑differential equation for the field evolution that depends on magnetic diffusivity η, radial accretion velocity νr, and a single dimensionless parameter D = 2η/(3H|νr|), with field lines bent by an angle i satisfying tan i = 1.52 D⁻¹ at the disc surface. Numerical integration shows the system reaches a steady state, and for D slightly below unity the field is strongly concentrated toward the disc centre due to substantial inward dragging of the field lines.

Abstract

We consider a thin accretion disc of half-thickness H, vertically threaded by a magnetic field. The field is due to contributions from both the disc current and an external current (giving rise to a uniform external field). We derive an integro-differential equation for the evolution of the magnetic field, subject to magnetic diffusivity η and disc accretion with radial velocity νr. The evolution equation is solved numerically, and a steady state is reached. The evolution equation depends upon a single, dimensionless parameter |$D=2\eta/(3H\ |\nu_r|)=(R/H(eta/nu)$|⁠, where the latter equality holds for a viscous disc having viscosity ν. At the disc surface, field lines are bent by angle i from the vertical, such that tan |$i=1.52D^{-1}$|⁠. For values of D somewhat less than unity, the field is strongly concentrated towards the disc centre, because the field lines are dragged substantially inwards.