Advection/Diffusion of large scale magnetic field in accretion disks

Abstract

Winds and jets of proto-stellar systems are thought to arise from disk accretion involving (1) a small-scale turbulent magnetic field in the disk (due to the magneto-rotational instability or MRI) and (2) a large-scale magnetic field which gives rise to the winds and/or jets. An important problem with this picture is that the enhanced viscosity is accompanied by an enhanced magnetic diffusivity which acts to prevent the build up of a significant large-scale field. Recent work has pointed out that the surface layers of the disk are non-turbulent and thus highly conducting (or non-diffusive). This is because the MRI is suppressed in the surface layers where the magnetic and radiation pressures are larger than the thermal pressure. Here, we calculate the vertical (z) profiles of the stationary accretion flows (with radial and azimuthal components), and the profiles of the large-scale, magnetic field taking into account the turbulent viscosity and diffusivity due to the MRI and the fact that the turbulence vanishes at the surface of the disk. We derive a sixth-order differential equation for the radial flow velocity vr(z) which depends mainly on the midplane thermal to magnetic pressure ratio β > 1 and the magnetic Prandtl number of the turbulence P viscosity/diffusivity. Boundary conditions at the disk surfaces take into account possible magnetic winds or jets and allow for a surface current flow in the highly conducting surface layers. The stationary solutions we find indicate that a weak (β > 1) large-scale field does not diffuse away as suggested by earlier work.