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

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.

Take $A>0$ and a nontrivial nonnegative profile. On $0<r<R(t)$ put $X=(R/r{)}^a$ and $\Sigma =\sigma (X-1)$. The diffusion operator acts on $r^{1/2}\bar{\nu }\Sigma =A{\sigma }^2[r^{5/2-2a}{R}^{2a}-2r^{5/2-a}{R}^a+r^{5/2}]$, givingBut differentiating the trial profile in time gives onlyFor $a\ne 0$, the functions $1,X,X^2$ are independent on the interior. Their coefficients can match only ifMatching the remaining coefficients givesThus $\sigma =[15A(t+t_0){]}^{-1}$, with $R=C(t+t_0{)}^{2/5}$ when $a=1$, and $R=C(t+t_0{)}^{1/2}$ when $a=5/4$. A nonzero solution whose support shrinks to zero at $t=0$ has $t_0=0$. The two compact self-similar disk with density-linear viscosity profiles are consequentlySet $\Sigma =0$ for $r>R$. Both $r^{1/2}\bar{\nu }\Sigma$ and its first radial derivative vanish at the moving edge, because $\bar{\nu }\Sigma$ vanishes quadratically there. Together with $\Sigma (R)=0$, this makes the cutoff consistent with the diffusion equation without an added singular edge flux.

These are singular-origin self-similar solutions: $R(0)=0$ is a limiting support condition, while $\sigma$ diverges as $t\downarrow 0$. It is not ordinary finite pointwise initial data. A finite nonzero $\sigma (0)$ would instead make $R=0$ an invariant solution of the radius equation, precluding an expanding nontrivial profile from that initial radius.

Using the radial-drift expression already derived, the factor $X-1$ cancels and givesFor $a=1$, $X=R/r$ and $\sigma =1/(15At)$, soThe instantaneous flow is inward for $r<R/5$ and outward for $R/5<r<R$. Nevertheless outward motion need not prevent eventual accretion. The trajectories in accretion trajectories in a compact viscous similarity disk satisfyMultiplication by $t^{-1/2}$ and integration giveA parcel at $r_i\in (0,R_i)$ at time $t_i>0$ has $K=(r_i-R_i)/\sqrt{t_i}<0$. It reaches the origin atBefore arrival, $r/R=1+K{t}^{1/10}/C$ decreases monotonically. Even a parcel initially moving outward eventually crosses $r/R=1/5$, turns inward, and is accreted. Only $K=0$, the outer-edge trajectory, never arrives; its surface density of a disk is zero. Thus almost every mass-carrying parcel is accreted in finite time.

For $a=5/4$, the coefficient $5/2-2a$ vanishes, leavingThe trajectories are $r(t)=r_i\sqrt{t/t_i}$, so each parcel keeps a fixed ratio $r/R$. This is an expanding nonaccreting similarity disk supplied by an inner torque, not an inward accretion solution.

Both profiles have integrable central cusps. Directly integrating the surface density of a disk and its specific angular momentum $j=\sqrt{GMr}$ givesFor $a=1$, substitution of the similarity functions givesThe inward accretion rate is $\dot{M}(r)=-2\pi r\Sigma {\bar{u}}_r$. At the origin it tends to $M_d/(5t)=-d{M}_d/dt$. The outward viscous torque in an accretion disk isFor this profile $\mathcal{G}\to 0$ as $r\to 0$, and the advected angular-momentum flux $\dot{M}j$ also vanishes there. Both outer fluxes vanish at $R$. Thus disk mass decreases through central accretion while disk angular momentum is conserved. Including the accreted central mass restores total mass conservation for the disk-plus-sink system.

For $a=5/4$, instead,Here the mass flux tends to zero at the origin, despite positive radial velocity everywhere else. The inner torque has the nonzero limitThus the nonaccreting similarity disk supplied by an inner torque conserves disk mass but gains angular momentum from an external central torque. Its vanishing edge flux cannot supply that gain. The disk is not globally isolated in angular momentum; the combined disk and torque source may of course conserve it.

These profiles illustrate different boundary-controlled similarity classes of the Keplerian viscous diffusion equation. The $a=1$ solution is compatible with a zero-torque inner boundary condition and conserved finite angular momentum, and can describe a late-time spreading accretion disk after the detailed initial profile has been forgotten. Its mass diverges as $t\downarrow 0$, so the formal zero-radius origin is not a finite-mass initial disk. The $a=5/4$ solution has finite initial mass concentrated at the origin and initially zero disk angular momentum, but requires the specified inner torque for $t>0$; that torque's $t^{-3/4}$ singularity is time-integrable. It may describe a spreading nonaccreting disk for compatible torque-driven boundary conditions, but cannot be an isolated angular-momentum-conserving disk's generic asymptotic state. Similarity scaling suggests their possible asymptotic roles; it does not establish convergence for arbitrary initial and boundary data.