We consider an accretion flow model originally proposed by Bisnovatyi-Kogan & Ruzmaikin (1974), which has been confirmed in recent 3D MHD simulations. In the model, the accreting gas drags in a strong poloidal magnetic field to the center such that the accumulated field disrupts the axisymmetric accretion flow at a relatively large radius. Inside the disruption radius, the gas accretes as discrete blobs or streams with a velocity much less than the free-fall velocity. Almost the entire rest mass energy of the gas is released as heat, radiation and mechanical/magnetic energy. Even for a non-rotating black hole, the efficiency of converting mass to energy is of order 50% or higher. The model is thus a practical analog of an idealized engine proposed by Geroch and Bekenstein.

We consider height-integrated equations of an advection-dominated accretion flow (ADAF), assuming that there is no mass outflow. We include convection through a mixing length formalism. We seek self-similar solutions in which the rotational velocity and sound speed scale as R −1/2 , where R is the radius, and consider two limiting prescriptions for the transport of angular momentum by convection. In one limit, the transport occurs down the angular velocity gradient, so convection moves angular momentum outward. In the other, the transport is down the specific angular momentum gradient, so convection moves angular momentum inward. We also consider general prescriptions which lie in between the two limits.When convection moves angular momentum outward, we recover the usual self-similar solution for ADAFs in which the mass density scales as ρ ∝ R −3/2 . When convection moves angular momentum inward, the result depends on the viscosity coefficient α. If α > α crit1 ∼ 0.05, we once again find the standard ADAF solution. For α < α crit , however, we find a non-accreting solution in which ρ ∝ R −1/2 . We refer to this as a "convective envelope" solution or a "convectiondominated accretion flow." Two-dimensional numerical simulations of ADAFs with values of α < ∼ 0.03 have been reported by several authors. The simulated ADAFs exhibit convection. By virtue of their axisymmetry, convection in these simulations moves angular momentum inward, as we confirm by computing the Reynolds stress. The simulations give ρ ∝ R −1/2 , in good agreement with the convective envelope solution. The R −1/2 density profile is not a consequence of mass outflow. The relevance of these axisymmetric low-α simulations to real accretion flows is uncertain.

We present three-dimensional MHD simulations of rotating radiatively inefficient accretion flows onto black holes. We continuously inject magnetized matter into the computational domain near the outer boundary and run the calculations long enough for the resulting accretion flow to reach a quasi-steady state. We have studied two limiting cases for the geometry of the injected magnetic field: pure toroidal field and pure poloidal field. In the case of toroidal field injection, the accreting matter forms a nearly axisymmetric, geometrically-thick, turbulent accretion disk. The disk resembles in many respects the convection-dominated accretion flows found in previous numerical and analytical investigations of viscous hydrodynamic flows. Models with poloidal field injection evolve through two distinct phases. In an initial transient phase, the flow forms a relatively flattened, quasi-Keplerian disk with a hot corona and a bipolar outflow. However, when the flow later achieves steady state, it changes in character completely. The magnetized accreting gas becomes two-phase, with most of the volume being dominated by a strong dipolar magnetic field from which a thermal low-density wind flows out. Accretion occurs mainly via narrow slowly-rotating radial streams which 'diffuse' through the magnetic field with the help of magnetic reconnection events.