Self-similar accretion flows with convection

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 selfsimilar 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 that 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 rho proportional to R-3/2. When convection moves angular momentum inward, the result depends on the viscosity coefficient alpha. If alpha > alpha(crit1) similar to 0.05, we once again find the standard ADAF solution. For alpha < alpha(crit2) similar to alpha(crit1), however, we find a nonaccreting solution in which rho proportional to R-1/2. We refer to this as a convective envelope solution or a convection-dominated accretion flow. Two-dimensional numerical simulations of ADAFs with values of alpha less than or similar to 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 rho proportional to 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-alpha simulations to real accretion flows is uncertain.