 Solution

Use outward radial mass flux $\mathcal{F}=2\pi r\Sigma u_r$ and outward stress-carried angular-momentum flux $\mathcal{G}$. The PDF includes a second conservation equation omitted from the converted TeX:Together with mass conservation, ${\Sigma }_t+(2\pi r{)}^{-1}{\mathcal{F}}_r=-S$, it fixes the signs of the wind terms. Subtract h times the mass equation, noting that the prescribed $h(r)$ is independent of time. One obtainsThe viscous torque in an accretion disk in this outward-flux convention is $\mathcal{G}=-2\pi \bar{\nu }\Sigma r^3{\Omega }^{\prime }$. Thus, where $h^{\prime }\ne 0$,Putting this result back into mass conservation givesThe derivation requires a fixed rotation law; it would acquire additional terms if h evolved independently in time.

For Keplerian rotation, $h=\sqrt{GMr}$, $h^{\prime }=h/(2r)$, and $r^3{\Omega }^{\prime }=-(3/2)h$. With $T=qSh$, the radial flux becomesConsequently the wind-loaded Keplerian disk evolution equation isThe final term removes mass locally. The extra flux term describes the disc's radial response when the outflow removes more or less than the local specific angular momentum. In particular, $q=1$ cancels this additional transport term without canceling the local mass sink.