  The principles underlying a proposed class of black hole accretion models are
examined. The flows are generally referred to as ``convection-dominated,'' and
are characterized by inward transport of angular momentum by thermal convection
and outward viscous transport, vanishing mass accretion, and vanishing local
energy dissipation. We examine the viability of these ideas by explicitly
calculating the leading order angular momentum transport of axisymmetric modes
in magnetized, differentially rotating, stratified flows. The modes are
destabilized by the generalized magnetorotational instability. There are no
inward transporting modes at all unless the magnitude of the (imaginary) Brunt
frequency is comparable to the epicyclic frequency, a condition requiring
substantial levels of dissipation. When inward transporting modes do exist,
they appear at long wavelengths, unencumbered by magnetic tension. Moreover,
very general thermodynamic principles prohibit the complete recovery of
irreversible dissipative energy losses, a central feature of
convection-dominated models. Dissipation of the free energy of differential
rotation inevitably requires outward angular momentum transport. Our results
are in good agreement with global MHD simulations, which find significant
levels of outward transport and energy dissipation, whether or not
destabilizing entropy gradients are present.
