Self-Similar Accretion Flows with Convection

We consider height-integrated equations of an advection-dominated accretionflow (ADAF), assuming that there is no mass outflow. We include convectionthrough a mixing length formalism. We seek self-similar solutions in which theangular velocity and sound speed scale as R^{-1/2}, where R is the radius, andconsider two limiting prescriptions for the transport of angular momentum byconvection. In one limit, the transport occurs down the angular velocitygradient, so convection moves angular momentum outward. In the other, thetransport is down the specific angular momentum gradient, so convection movesangular momentum inward. We also consider general prescriptions which lie inbetween the two limits. When convection moves angular momentum outward, we recover the usualself-similar solution for ADAFs in which the mass density scales asrho~R^{-3/2}. When convection moves angular momentum inward, we again find thissolution if the viscosity coefficient alpha>alpha_{crit1)~0.05. For smallvalues of alpha, we find a non-accreting solution, which we call a `convectiveenvelope', in which rho~R^{-1/2}. Two-dimensional numerical simulations of ADAFs with values of alpha<0.03 havebeen reported by several authors. The simulated ADAFs exhibit convection. Byvirtue of their axisymmetry, convection in these simulations moves angularmomentum inward, as we confirm by computing the Reynolds stress. Thesimulations give rho~R^{-1/2}, in good agreement with the convective envelopesolution. The R^{-1/2} density profile is not a consequence of mass outflow.The relevance of these axisymmertic low-alpha simulations to real accretionflows is uncertain.