Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 65 / 2 / i / Solution

Assume an axisymmetric thin disk in the fixed Newtonian potential of a point mass, negligible disk self-gravity, and slow evolution compared with an orbital period. Radial pressure and radial inertia are small compared with gravity and centrifugal acceleration, so $u_{\varphi }=r\Omega (r)$ with $\Omega =(GM/r^3{)}^{1/2}$, independent of height and time. Assume no mass loss or torque through the upper and lower disk faces. Define the surface density of a disk, mass-weighted radial velocity and integrated shear stress byVertical integration of the continuity equation givesThe azimuthal momentum equation, including the geometric term $u_r{u}_{\varphi }/r$, gives conservation of angular momentum in terms of $j=r^2\Omega$:Subtracting $j$ times mass conservation eliminates the time derivative and leaves $\Sigma {\bar{u}}_r{j}^{\prime }=r^{-1}(r^{2}W{)}^{\prime }$.

Define the effective density-weighted viscosity of a disk through the actual stress:If the local viscous stress is $T_{r\varphi }=\rho \nu r{\Omega }^{\prime }$, this reduces to $\bar{\nu }={\Sigma }^{-1}\int \rho \nu dz$. The definition can also describe an effective turbulent transport stress, provided the same shear closure is appropriate. Since ${\Omega }^{\prime }<0$, outward transport corresponds to $W<0$ and positive $\bar{\nu }$.

For Keplerian rotation, $j=\sqrt{GMr}$ and $r^{2}W=-(3/2)\sqrt{GM}{r}^{1/2}\bar{\nu }\Sigma$. ThereforeSubstitution into mass conservation yields the Keplerian viscous diffusion equationThe absence of surface fluxes, fixed rotation law and leading thin-disk approximation are essential: a wind, appreciable radial inertia or evolving gravitational potential would alter this reduction.