Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 2

The azimuthal momentum equation with $u_y=-3\Omega x/2$ isHence the slow radial drift isSubstitution into mass conservation, ${\partial }_t\Sigma +{\partial }_x(\Sigma u_x)=0$, gives the nonlinear diffusion equationIt describes slow axisymmetric viscous spreading while the velocity remains close to the imposed Kepler shear. It omits epicyclic and acoustic waves, rapid transients, self-gravity, nonaxisymmetric structure, and edge dynamics for which the assumed shear and secular force balance fail.

Let $F(\Sigma )=\nu (\Sigma )\Sigma$ and perturb the homogeneous state by $\delta \Sigma \propto e^{st+ikx}$. Linearization givesThus $d(\nu \Sigma )/d\Sigma <0$ produces the viscous instability of an accretion disk: a density enhancement transports angular momentum less effectively, loses material more slowly, and grows. The ring separates into denser narrow ringlets and lower-density gaps until nonlinear effects regularize the backward diffusion.

Inside the ring, $\nu \Sigma =A{\Sigma }^3=A{\sigma }^3(1-x^2/w^2{)}^{3/2}$. Part a therefore givesThe edge is material, so $\dot{w}=u_x(w)$ andDirect substitution of the profile into the diffusion equation giveshence $\sigma w={\sigma }_0{w}_0$. This is exactly mass conservation, sinceUsing $\sigma ={\sigma }_0{w}_0/w$ in the width equation and integrating,ThereforeIn particular, $w\propto t^{1/4}$ at late times.