Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 61 / 3

Let ${\Sigma }_0$ be the uniform background surface density, ${\mathbf{u}}_0=-Sx{\mathbf{e}}_y$, and write $\Sigma ={\Sigma }_0+{\Sigma }^{\prime }$, $\mathbf{u}={\mathbf{u}}_0+\mathbf{v}$, ${\Phi }_d={\Phi }_{d0}+{\Phi }_d^{\prime }$. The background sheet has no horizontal gravitational gradient; its vertical field is irrelevant to the stated planar equations. Put $D_0={\partial }_t-Sx{\partial }_y$. Keeping first-order terms givesThe $-S{v}_x$ contribution comes from $\mathbf{v}\cdot \nabla {\mathbf{u}}_0$, while the Coriolis acceleration supplies the remaining coupling. The prescribed tidal potential has no perturbation, and its background force already balances the background Coriolis term. The isothermal pressure linearizes to $c_s^2\nabla {\Sigma }^{\prime }/{\Sigma }_0$.

Apply the shearing-wave ansatz to each disturbance. Acting on its phase givesThus a spatially uniform amplitude system exists whenFor a nonzero horizontal wavenumber $k=|\mathbf{k}|$, the razor-thin disk Poisson kernel follows from $({\partial }_z^2-k^2){\tilde{\Phi }}_d^{\prime }=4\pi G{\tilde{\Sigma }}^{\prime }\delta (z)$. The potential decays above and below the sheet as ${\tilde{\Phi }}_{d,m}^{\prime }{e}^{-k|z|}$. Integrating across the midplane gives the jump $-2k{\tilde{\Phi }}_{d,m}^{\prime }=4\pi G{\tilde{\Sigma }}^{\prime }$, henceSubstitute this potential and the phase evolution into the linearized equations. The barotropic shearing-wave gravitational amplitude system isThe self-gravity sign is attractive and opposes the pressure response. The uniform $k=0$ mass density shift requires its own gravitational reference and is not obtained by dividing by $k$.

Write $B=2\Omega -S$, $A(t)=c_s^2-2\pi G{\Sigma }_0/k(t)$, and $w=i{k}_x{\tilde{v}}_y-i{k}_y{\tilde{v}}_x$. Differentiation, including ${\dot{k}}_x=S{k}_y$, givesThe two pressure/self-gravity cross terms cancel. Continuity gives ${\dot{\tilde{\Sigma }}}^{\prime }=-i{\Sigma }_0(k_x{\tilde{v}}_x+k_y{\tilde{v}}_y)$, so $\dot{w}=B{\dot{\tilde{\Sigma }}}^{\prime }/{\Sigma }_0$. ConsequentlyThe conserved amplitude is linearized vortensity: the perturbation of the absolute vertical vorticity divided by surface density,A two-dimensional inviscid barotropic fluid materially conserves this potential vorticity. Both pressure enthalpy and gravitational forces are gradients, so they create no baroclinic vorticity source. Since $q_0$ is uniform, its linear perturbation obeys $D_0{q}^{\prime }=0$, explaining why a shearing wave has constant vortensity amplitude.

For $k_y=0$ and nonzero $k_x$, the wavenumber is constant. Continuity gives ${\dot{\tilde{\Sigma }}}^{\prime }=-i{k}_x{\Sigma }_0{\tilde{v}}_x$, and the conserved vortensity relation givesDifferentiate continuity and substitute radial momentum and this identity. The forced axisymmetric density mode with vortensity obeysThe constant forcing represents the stationary vortical or balanced component, which is lost if every mode is assumed to have nonzero frequency. Let $k_{\ast }=\pi G{\Sigma }_0/c_s^2=\kappa /(Q{c}_s)$. Completing the square givesFor $Q>1$, every nonzero radial wavenumber has positive ${\omega }_k^2$. The solution is a constant balanced offset $-2\Omega {\Sigma }_0^2{\tilde{q}}^{\prime }/{\omega }_k^2$ plus bounded sine and cosine oscillations. Pressure stabilizes short wavelengths, rotation stabilizes long wavelengths, and self-gravity is strongest at intermediate wavelengths.

For $Q<1$, the Toomre stability criterion fails in the bandThere ${\omega }_k^2<0$ and the homogeneous solutions grow or decay exponentially. The fastest rate is $s_{\max }=\kappa \sqrt{Q^{-2}-1}$ at $|k_x|=k_{\ast }$. Outside the band the modes oscillate; at its edges the frequency vanishes.

At $Q=1$, ${\omega }_k^2\ge 0$ and it vanishes only at $|k_x|=\kappa /c_s$. At that critical wavenumber,This Toomre marginal algebraic growth is spectrally marginal, not bounded for all data: a nonzero conserved vortensity gives quadratic secular growth, and even zero vortensity can allow the linear term. Other wavenumbers still oscillate about a balanced offset. $Q>1$ gives bounded axisymmetric modes, $Q<1$ allows exponential instability, and $Q=1$ has a zero-frequency marginal mode with possible algebraic growth. These statements are axisymmetric; nonaxisymmetric shearing disturbances can have transient swing behavior. For $k_x=k_y=0$, continuity instead fixes the uniform mass density shift, while the velocities execute epicyclic motion; no $1/k$ gravitational formula is used.