Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 57 / 2

Write the perturbation velocity as $(u,v)$, the density amplitude as ${\sigma }_1$, and ${\gamma }_d=ϵ\Omega$. Axisymmetry removes advection by the background azimuthal flow, but the radial perturbation advects the Keplerian shear: $u{\partial }_x{U}_{g,y}=-3\Omega u/2$. Combining this with the Coriolis force gives the linearized shearing sheet equationsThe razor-thin disk Poisson kernel supplies the self-gravity term. The $\Omega /2$ coefficient, rather than $2\Omega$, is essential: it includes the perturbed advection of the background velocity.

Put $A=c^2{k}^2-2\pi G{\sigma }_0|k|$ and ${\omega }^2={\Omega }^2+A$. The determinant of the three amplitude equations isExpanding it gives the dust gravitational dispersion relation with gas dragUsing a determinant avoids division by $s$ and retains the neutral/secular branch. The radial epicyclic frequency of this Keplerian shearing sheet is $\Omega$, so the three terms in ${\omega }^2$ represent rotational support, self-gravity and dust pressure.

Without gas drag, the dispersion relation factorizes as $s(s^2+{\omega }^2)=0$. There is a neutral branch and two density-wave branches. If ${\omega }^2>0$, the waves have $s=\pm i\omega$; if ${\omega }^2<0$, one root is $\boxed{s=\sqrt{-{\omega }^2}>0}$ and the dust layer is gravitationally unstable.

For $c>0$, complete the square in the wavenumber magnitude:The minimum occurs at $|k|=\pi G{\sigma }_0/c^2$. Thus the Toomre stability criterion isAt $Q=1$ the minimizing mode is marginal. For $Q>1$ there is no growing axisymmetric wave in this razor-thin, pressure-supported, drag-free model. The unstable band for $Q<1$ isIn a finite layer, this band must contain an admissible mode; the continuum statement assumes an adequately large domain.

For fixed ${\omega }^2>0$ and weak gas drag, expand a wave root as $s=s_0+ϵs_1+O({ϵ}^2)$ with $s_0=\pm i\omega$. At order $ϵ$, the dispersion relation yieldsSince $s_0^2=-{\omega }^2$, this reduces to $-2{\omega }^2{s}_1-\Omega ({\omega }^2+{\Omega }^2)=0$. The weak-drag dust density waves therefore haveThe first-order real part is negative: the two density waves are weakly damped, with their frequencies unchanged to first order. Gas drag removes their perturbation energy relative to the fixed gas flow. The expansion is an asymptotic result at fixed positive ${\omega }^2$; it is not uniform close to ${\omega }^2=0$, where the nominal damping can become comparable to the oscillation frequency and a different scaling is needed.

The branch originating at $s=0$ has $s=ϵs_1+O({ϵ}^2)$. Keeping first-order terms in the cubic dispersion relation gives ${\omega }^2{s}_1+\Omega ({\omega }^2-{\Omega }^2)=0$. Hence the weak-drag secular gravitational growth rate isCarrying forward the positive-${\omega }^2$ regime of the preceding part, this mode grows when $\boxed{0<{\omega }^2<{\Omega }^2}$. This specifies the scope of the source's abbreviated condition. If ${\omega }^2<0$, the small root written here is instead damped, while the separate dynamical branch already grows; the dust is still unstable, but growth is not identified with this particular small-root continuation. The expansion also fails at ${\omega }^2=0$.

At sufficiently small nonzero $|k|$, ${\omega }^2$ is positive and just below ${\Omega }^2$, sinceTherefore an infinite dust layer with nonzero self-gravity and nonzero drag has a long-wavelength secular gravitational instability of an astrophysical disk, even when $Q>1$. Its small-$k$ growth rate tends to $2\pi ϵG{\sigma }_0|k|/\Omega$, and vanishes at $k=0$; the uniform-density mode itself is neutral. Drag transfers perturbation angular momentum to the gas, weakening rotational support while self-gravity exceeds pressure on long wavelengths.

The instability is not limited to weak drag as an existence statement. For any $ϵ>0$, the cubic has $P(0)=ϵ\Omega ({\omega }^2-{\Omega }^2)<0$ and $P(s)\to +\infty$ on the positive real axis. Thus it has a positive real root whenever ${\omega }^2<{\Omega }^2$. This exact positive-root criterion for dust self-gravity with drag distinguishes the growth criterion from the approximation used to compute its slow rate.

The secular gravitational instability needs $|k|<k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2$. A finite radial extent $L$ removes arbitrarily small wavenumbers; the smallest available one is $k_{\min }\sim \alpha /L$, where $\alpha$ is a boundary-dependent constant of order unity. With $H=c/\Omega$ and the dust Toomre parameter,Thus the finite-size secular gravitational criterion is $\boxed{Q\lesssim L/H}$ at the rough accuracy requested. For example, a periodic radial interval has $k_{\min }=2\pi /L$ and gives $Q<L/(\pi H)$ in this idealized model. The numerical prefactor is not universal, but the scaling is. Finite size therefore restores a practical threshold for the otherwise long-wavelength instability. Growth must additionally occur within the disk lifetime, and a global wavelength comparable to disk radius lies beyond the strictly local shearing sheet approximation.

Turbulence usually makes the onset of dust gravitational instability of an astrophysical disk harder in three complementary ways. It increases the dust random velocity dispersion, raising $c$ and the Toomre parameter; it stirs dust vertically, reducing the self-gravity enhancement of a razor-thin disk; and it mixes density enhancements through turbulent diffusion. The last effect is particularly important for a slowly growing secular gravitational instability, which can be erased before it amplifies appreciably.

A simple diffusion model adds $D{\nabla }^2\sigma$ to the dust continuity equation, giving a damping scale of order $D{k}^2$. Growth then has to compete with mixing as well as pressure and rotational support. In the long-wavelength weak-drag regime where ${\omega }^2\simeq {\Omega }^2$, the indicative competition iswith further pressure and finite-thickness corrections. This is a model-dependent long-wave estimate, not a substitution into the original cubic without changing its continuity equation. In an infinite domain, diffusion proportional to $k^2$ need not remove growth proportional to $|k|$ at every arbitrarily long wavelength; in a finite disk, the remaining wavelengths and their growth times may be inadequate. The conclusion is therefore a higher practical collapse threshold and slower growth, rather than guaranteed stability for all turbulent flows.

Turbulence can also concentrate dust through coherent structures or pressure maxima, increasing local surface density and encouraging collapse. Its net quantitative effect requires a model for stirring, diffusion, thickness and concentration. The deterministic fixed gas flow used in the preceding parts does not specify those statistics. This competition is turbulent mixing of a secularly unstable dust layer.