The Shakura-Sunyaev Viscosity Prescription with Variable alpha(r)
  Almost all hydrodynamic accretion disk models parametrize viscosity with the dimensionless parameter alpha. There is no detailed model for alpha, so it is usually taken to be a constant. However, global simulations of magnetohydrodynamic disks find that alpha varies with distance from the central object. Also, Newtonian simulations tend to find smaller alpha's than general relativistic simulations. We seek a one-dimensional model for alpha that can reproduce these two observations. We are guided by data from six general relativistic magnetohydrodynamic accretion disk simulations. The variation of alpha in the inner, laminar regions of the flow results from stretching of mean magnetic field lines by the flow. The variation of alpha in the outer, turbulent regions results from the dependence of the magnetorotational instability on the dimensionless shear rate. We give a one-dimensional prescription for alpha(r) that captures these two effects and reproduces the radial variation of alpha observed in the simulations. For thin disks, the prescription simplifies to the formula alpha(r)=0.025[q(r)/1.5]^6, where the shear parameter, q(r), is an analytical function of radius in the Kerr metric. The coefficient and exponent are inferred from our simulations and will change as better simulation data becomes available. We conclude that the alpha-viscosity prescription can be extended to the radially varying alpha's observed in simulations. It is possible that Newtonian simulations find smaller alpha's than general relativistic simulations because the shear parameter is lower in Newtonian flows. 
Robert F. Penna, Aleksander Sadowski, Akshay K. Kulkarni, and Ramesh   Narayan
astro-ph.HE