Vertically averaged alpha closure for a mixed-pressure layer ($\bar{\nu }=\alpha \Omega H^2/9$)

For the constant gas-radiation ratio vertical disc structure, the integrated alpha prescription $\bar{\nu }\Sigma =\int \mu dz=(\alpha /\Omega )\int pdz$ gives the displayed mean kinematic viscosity and $\lambda /(1+\lambda )=\alpha H\Omega \kappa \Sigma /(8c)$. Uniform dynamic viscosity and nonuniform pressure are compatible because this prescription is an integrated equality. Imposing the corresponding pointwise equality would not preserve the exact vertical solution.

For small positive radiation-to-gas ratio, constant opacity, fixed alpha and molecular constants, eliminate $\lambda$ between $\lambda \propto H\Omega \Sigma$ and $\lambda \Sigma \propto H^7{\Omega }^6$. This gives $H^6\propto {\Sigma }^2{\Omega }^{-5}$ and $\bar{\nu }\propto {\Sigma }^{2/3}{\Omega }^{-2/3}$. Keplerian rotation supplies the displayed radial power. Since $\bar{\nu }\Sigma \propto {\Sigma }^{5/3}$, its derivative with respect to surface density is positive and local viscous perturbations diffuse rather than amplify.