Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3

Letbe the vertical absolute vorticity. Taking the vertical curl of the momentum equation removes both the tidal potential and the isothermal pressure force, because each is a gradient field. The two-dimensional vorticity equation isThe surface-density equation isThe quotient rule now givesSince $\zeta =W/\Sigma$ is the vortensity,

For $u_x=0$ and $u_y=u_y(x)$, the material acceleration vanishes. The radial component of force balance isThereforeThe first term is the background Keplerian shear; the second is the geostrophic balance correction produced by the surface-density gradient.

Differentiating the equilibrium velocity givesOn the other hand, the definition of vortensity and $\Sigma ={\Sigma }_0{e}^{\sigma }$ giveMultiplying by $2/\Omega$, and introducing the dimensionless variablesyields

Linearizing $e^{\sigma }=1+\sigma +O({\sigma }^2)$ givesThe prescribed $q$ and the decay conditions are even in $\xi$, so the localized solution is even. Continuity of $\sigma$ and ${\sigma }^{\prime }$ at $\xi =\pm d$ determines all constants:Thus $\Sigma ={\Sigma }_0{e}^{\sigma }\simeq {\Sigma }_0(1+\sigma )$ is a symmetric density bump: it has a concave parabolic core, reaches its maximum at $\xi =0$, and has exponential tails approaching ${\Sigma }_0$. The linearization is self-consistent when $d+d^2/2\ll 1$.

Let $\delta ={\Sigma }^{\prime }/{\Sigma }_0$ and define the derivative following the background Keplerian shear byThe linearized continuity and momentum equations areThe coefficient $\Omega /2$ combines the background shear with the Coriolis acceleration.

Applying ${\mathcal{D}}_0$ to the shearing-wave ansatz produces an extra phase termThe perturbation equations have spatially uniform amplitude coefficients only when this term vanishes. HenceFor $k_y\ne 0$, the radial wavenumber changes linearly. A leading disturbance with $k_x/k_y<0$ first opens until $k_x=0$, then becomes an increasingly tightly wound trailing disturbance with $k_x/k_y>0$. This is the geometric swing of a shearing wave.

The linearized vortensity amplitude isSince the exact vortensity obeys $D_t\zeta =0$ and the time-dependent $k_x$ cancels the background shear in the phase, linearization gives

On the zero-vortensity branch,Substitution in the two momentum equations yields the linear system of differential equationswithwhere $k_x=k_x(t)$ is the result from part (ii).

When $k_y=0$, $k_x$ is constant and the system becomesThusFor time dependence $e^{-i\omega t}$, the dispersion relation isExplicitly,and $\tilde{\delta }=2i{k}_x{\tilde{u}}_y/\Omega$. This is an axisymmetric inertial-acoustic wave: pressure supplies the $c_s^2{k}_x^2$ restoring term and epicyclic motion supplies the ${\Omega }^2$ term.