It has recently been demonstrated that the Shakura-Sunyaev prescription for the kinematic viscosity in an advection-dominated accretion disk yields physically reasonable solutions for the structure of the inflow close to the event horizon. In particular, no violations of relativistic causality occur at the horizon. This is somewhat surprising considering the diffusive nature of the angular momentum transport in the Shakura-Sunyaev scenario, and it is therefore natural to ask whether one can also obtain acceptable solutions for the disk structure based on the various alternative models for the viscosity that have been proposed, including the "deterministic" forms. In this paper we perform a rigorous asymptotic analysis of the structure of an advection-dominated accretion disk close to the event horizon of a nonrotating black hole based on three of the alternative prescriptions for the viscosity that have been suggested in the literature. We constrain the physical disk model by stipulating that the stress must vanish at the horizon, which is the fundamental inner boundary condition imposed by general relativity. Surprisingly, we find that none of the three alternative viscosity prescriptions yield physically acceptable disk structures close to the horizon when the zero-torque condition is applied, whether the flow is in vertical hydrostatic equilibrium or free fall. Hence we conclude that the original Shakura-Sunyaev prescription is the only one proposed so far that is physically consistent close to the event horizon. We argue that, somewhat ironically, it is in fact the diffusive nature of the Shakura-Sunyaev form that is the reason for its success in this application. Our focus here is on advection-dominated accretion disks, but we expect that our results will also apply to generalized disks provided that losses of matter and energy become negligible as the gas approaches the event horizon.
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