Activity of the nuclei of galaxies and stellar mass systems involving disk accretion to black holes is thought to be due to ( 1 ) a small-scale turbulent magnetic field in the disk ( due to the magneto-rotational instability or MRI ) which gives a large viscosity enhancing accretion , and ( 2 ) a large-scale magnetic field which gives rise to matter outflows and/or electromagnetic jets from the disk which also enhances accretion .
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 disk ’ s surface layers are non-turbulent and thus highly conducting ( or non-diffusive ) because the MRI is suppressed high in the disk 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 v _ { r } ( z ) which depends mainly on the midplane thermal to magnetic pressure ratio \beta > 1 and the Prandtl number of the turbulence { \cal P } = viscosity/diffusivity .
Boundary conditions at the disk surface take into account a possible magnetic wind or jet and allow for a surface current in the highly conducting surface layer .
The stationary solutions we find indicate that a weak ( \beta > 1 ) large-scale field does not diffuse away as suggested by earlier work .
For a wide range of parameters \beta > 1 and { \cal P } \geq 1 , we find stationary channel type flows where the flow is radially outward near the midplane of the disk and radially inward in the top and bottom parts of the disk .
Channel flows with inward flow near the midplane and outflow in the top and bottom parts of the disk are also found .
We find that Prandtl numbers larger than a critical value ( estimated to be 2.7 ) are needed in order for there to be magnetocentrifugal outflows from the disk ’ s surface .
For smaller { \cal P } , electromagnetic outflows are predicted .