Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 321 / 3

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.