Vertically averaged viscous disk equations (${\partial }_t\Sigma +r^{-1}{\partial }_r(r\Sigma v_r)=0$)

In an axisymmetric thin disc with no vertical mass or torque flux, integrate mass and conservation of angular momentum over height. With fixed height-independent $l=r^2\Omega$, the equations are ${\partial }_t\Sigma +r^{-1}{\partial }_r(r\Sigma v_r)=0$ and ${\partial }_t(\Sigma l)+r^{-1}{\partial }_r(r\Sigma v_{r}l-r^3\bar{\nu }\Sigma {\Omega }^{\prime })=0$. Their difference gives $r\Sigma v_r{l}^{\prime }=(r^3\bar{\nu }\Sigma {\Omega }^{\prime }{)}^{\prime }$. The density-weighted viscosity of a disk supplies the integrated stress.

Use $l=\sqrt{G{M}_{\ast }r}$ and ${\Omega }^{\prime }=-(3/2)\Omega /r$ in the vertically averaged viscous disk equations. The radial drift is $v_r=-3(\Sigma \sqrt{r}{)}^{-1}{\partial }_r(\sqrt{r}\bar{\nu }\Sigma )$. Conservation of mass gives this nonlinear diffusion equation. It assumes the central rotation law remains fixed and excludes wind/surface torque terms.

For a steady zero-mass-flux background and constant viscous transport response exponent, this change removes the first derivative in the radial diffusion operator. If ${\Sigma }_0\propto r^{-p}$, the kinematic viscosity scales as $r^{p-1/2}$, so the transformed diffusivity scales as $x^{2p-3}$. It is constant at $p=3/2$.

Two integrations of the steady Keplerian viscous diffusion equation give this general transported quantity. $A$ fixes the mass accretion rate, while $B$ fixes the additive angular-momentum flux or inner torque. A zero-torque inner boundary condition gives $B=-A\sqrt{r_{\mathrm{in}}}$; a nonaccreting constant-torque disk has $A=0$.