Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 (Dynamics of Astrophysical Discs)

In the Toomre stability criterionthe disk's gravitational potential supplies the destabilizing attraction, gas pressure represented by $c_s$ suppresses short wavelengths, and the Keplerian shear represented by the angular frequency $\Omega$ prevents coherent collapse on long scales through epicyclic motion. Axisymmetric perturbations are stable for $Q>1$ and gravitationally unstable for $Q<1$.

The state $Q\sim 1$ is a self-regulating fixed point. If cooling lowers $c_s$ until $Q<1$, gravitational instability produces shocks and turbulence that heat the disk and raise its effective velocity dispersion. If $Q$ becomes appreciably larger than one, the instability and its heating switch off, while turbulent dissipation and radiative cooling reduce $c_s$. A sustained gravito-turbulent state therefore remains close to marginal stability.

The first equation is vertical hydrostatic equilibrium: the pressure gradient balances the vertical gravity of the central mass, ${\Omega }^{2}z$, and the disk's own potential ${\Phi }_d$. The second is the plane-parallel Poisson equation for disk self-gravity. The third balances the vertical increase of radiative flux against local viscous heating in a Keplerian alpha disk, and the fourth is the optically thick radiative diffusion law.

With $P/\rho \sim c_s^2$, $\Sigma \sim \rho H$, and all terms in hydrostatic balance comparable,The first comparison gives $H\sim c_s/\Omega$; inserting it into the second gives $\Omega c_s\sim G\Sigma$. Henceup to the order-one constants deliberately omitted by the scaling argument.

For a perfect gas, $c_s^2\sim kT/({\mu }_m{m}_p)$. At the margin of the Toomre stability criterion,Integrating viscous heating through the disk and applying radiative diffusion givesEquating these fluxes yieldsThe kinematic viscosity is $\bar{\nu }\sim \alpha c_s^2/\Omega$, soBecause a Keplerian accretion disk has $\Omega \propto r^{-3/2}$,

For $P=K{\rho }^{1+1/n}$,The pseudo-enthalpy definition therefore integrates towhere the constant makes $w=0$ at the surface. Thus

Hydrostatic balance is $c_0^{2}dw/dz=-{\Omega }^{2}z-d{\Phi }_d/dz$. Differentiate it, use the Poisson equation, and set $\zeta =\Omega z/c_0$:Since the definitions in the question give $Q_0={\Omega }^2/(\pi G{\rho }_0)$,

For $n=1$, put $\lambda =2/\sqrt{Q_0}$. The solution isThe surface $z=H$ is the first zero, so, using $c_0^2=2K{\rho }_0$,

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.

The shearing sheet is a local Cartesian expansion about a reference circular orbit in a differentially rotating disk. One works in a frame rotating at $\Omega$, identifies $x,y,z$ with radial, azimuthal, and vertical directions, retains the linear background shear and radial tidal acceleration, and neglects curvature and background variation across a patch much smaller than the orbital radius. The incompressible version also takes constant reference density and imposes $\nabla \cdot \mathbf{u}=0$, filtering sound waves.

It represents local three-dimensional vortical motion, epicyclic motion, shear dynamics, inertial waves, and, after adding buoyancy, internal gravity waves and convection. It cannot represent global geometry or boundaries, order-radius structures, density-changing compressible motion, or acoustic waves.

Dot the momentum equation with $\mathbf{u}$. The Coriolis acceleration does no work, while the buoyancy-variable equation giveswhich cancels the buoyancy work $-N^2\theta u_z$. ThusUsing incompressibility,Therefore

For ${\mathbf{u}}_0=-3\Omega x{\mathbf{e}}_y/2$, advection vanishes and the Coriolis acceleration exactly cancels $3{\Omega }^{2}x{\mathbf{e}}_x$, so constant $P_0$ and ${\theta }_0=0$ complete the equilibrium.

Write perturbations $(\mathbf{v},p,\vartheta )e^{i\mathbf{k}\cdot \mathbf{x}-i\omega t}$ with $\mathbf{k}=(k_x,0,k_z)$. Incompressibility gives $\mathbf{k}\cdot \mathbf{v}=0$. It also makes the quadratic terms $\mathbf{v}\cdot \nabla \mathbf{v}=i(\mathbf{k}\cdot \mathbf{v})\mathbf{v}$ and $\mathbf{v}\cdot \nabla \vartheta$ vanish exactly; the background-advection terms vanish because $k_y=0$. The amplitude equations areEliminating the amplitudes yields the inertia-gravity wave dispersion relation

For $N^2=0$, $\omega =\pm \Omega k_z/k$, so the group velocity isIt is perpendicular to $\mathbf{k}$ and hence to the phase velocity. An inertial wave packet transports energy along beams lying in its phase surfaces.

If $N^2>0$ and $k_x/k_z\to 0$, incompressibility suppresses vertical motion and $\omega \to \Omega$: horizontal epicyclic motion and rotation dominate. If $k_z/k_x\to 0$, radial motion is suppressed and $\omega \to N$: vertical buoyancy oscillations dominate. Intermediate ratios give hybrid inertia-gravity waves.

If $N^2<0$, exponential growth occurs exactly whenOnly modes with sufficiently large radial wavenumber permit enough vertical displacement for unstable buoyancy to overcome rotational restoration; the other orientations remain stabilized by the Coriolis force.