  The equations governing the vertical structure of a stationary keplerian
accretion disc are presented. The model is based on the alpha-viscosity,
includes self-gravity, convective transport and turbulent pressure. A few
properties of the model are discussed for circumstellar and AGN discs.
  We show the strong sensitivity of the disc structure to the viscous energy
deposition towards the vertical axis, specially when entering inside the
self-gravitating part of the disc. The local version of the alpha-prescription
leads to a "singular" behavior which is also predicted by the vertically
averaged model. With respect, a much softer transition is observed with the
  "alpha-P" formalism. Turbulent pressure is important only for alpha > 0.1. It
lowers vertical density gradients, significantly thickens the disc, tends to
wash out density inversions and pushes the self-gravitating region to slightly
larger radii. Curves localizing the inner edge of the self-gravitating disc as
functions of the viscosity parameter and accretion rate are given. The lower
alpha, the closer to the center the self-gravitating regime, and the
sensitivity to the accretion rate is generally weak, except for alpha < 0.1.
This study suggests that models aiming to describe T-Tauri discs beyond about a
few to a few tens astronomical units from the central protostar using the
alpha-theory should consider vertical self-gravity. The Primitive Solar Nebula
was probably a bit (if not strongly) self-gravitating at the actual orbit of
giant planets. Alpha-discs hosted by active galaxies are self-gravitating
beyond about a thousand Schwarzchild radii. The inferred surface density
remains too high to lower the accretion time scale. More efficient mechanisms
driving accretion are required.
