  In this paper we consider two outstanding intertwined problems in modern
high-energy astrophysics: (1) the vertical thermal structure of an optically
thick accretion disk heated by the dissipation of magnetohydrodynamic (MHD)
turbulence driven by the magneto-rotational instability (MRI), and (2)
determining the fraction of the accretion power released in the corona above
the disk. For simplicity, we consider a gas-pressure-dominated disk and assume
a constant opacity. We argue that the local turbulent dissipation rate due to
the disruption of MRI channel flows by secondary parasitic instabilities should
be uniform across most of the disk, almost up to the disk photosphere. We then
obtain a self-consistent analytical solution for the vertical thermal structure
of the disk, governed by the balance between the heating by MRI turbulence and
the cooling by radiative diffusion. Next, we argue that the coronal power
fraction is determined by the competition between the Parker instability,
viewed as a parasitic instability feeding off of MRI channel flows, and other
parasitic instabilities. We show that the Parker instability inevitably becomes
important near the disk surface, leading to a certain lower limit on the
coronal power. While most of the analysis in this paper focuses on the case of
a disk threaded by an externally imposed vertical magnetic field, we also
discuss the zero-net-flux case, in which the magnetic field is produced by the
MRI dynamo itself, and show that most of our arguments and conclusions should
be valid in this case as well.
