Convection in axially symmetric accretion discs with microscopic   transport coefficients
  The vertical structure of stationary thin accretion discs is calculated from the energy balance equation with heat generation due to microscopic ion viscosity {\eta} and electron heat conductivity {\kappa}, both depending on temperature. In the optically thin discs it is found that for the heat conductivity increasing with temperature, the vertical temperature gradient exceeds the adiabatic value at some height, suggesting convective instability in the upper disc layer. There is a critical Prandtl number, Pr = 4/9, above which a Keplerian disc become fully convective. The vertical density distribution of optically thin laminar accretion discs as found from the hydrostatic equilibrium equation cannot be generally described by a polytrope but in the case of constant viscosity and heat conductivity. In the optically thick discs with radiation heat transfer, the vertical disc structure is found to be convectively stable for both absorption dominated and scattering dominated opacities, unless a very steep dependence of the viscosity coefficient on temperature is assumed. A polytropic-like structure in this case is found for Thomson scattering dominated opacity. 
K.L. Malanchev, K.A. Postnov, N.I. Shakura
astro-ph.HE physics.flu-dyn