Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 321 / 3

For the Keplerian shearing sheet velocity $\mathbf{U}=-(3/2)\Omega x{\mathbf{e}}_y$, its material acceleration vanishes because $\mathbf{U}\cdot \nabla \mathbf{U}=0$. The $x$ component of the Coriolis acceleration is $-3{\Omega }^{2}x$, which is cancelled by $-\nabla {\Phi }_t=+3{\Omega }^{2}x{\mathbf{e}}_x$ with the signs in the stated equation; constant $P_0$ supplies no force. The flow is also incompressible.

Let $\mathbf{u}=\mathbf{U}+{\mathbf{u}}^{\prime }$ and $h^{\prime }=P^{\prime }/\rho$. Axisymmetry removes $\mathbf{U}\cdot \nabla {\mathbf{u}}^{\prime }$, while ${\mathbf{u}}^{\prime }\cdot \nabla \mathbf{U}=-(3/2)\Omega u_x^{\prime }{\mathbf{e}}_y$. Keeping first-order terms givesDifferentiate the momentum equations in time and use incompressibility to eliminate $h^{\prime }$. The radial velocity obeysFor a plane wave proportional to $e^{i(k_{x}x+k_{z}z-\omega t)}$, this gives the inertial wave dispersion relation

When $k_x=0$ and $k_z\ne 0$, incompressibility forces $u_z^{\prime }=0$, and the vertical momentum equation then forces $h^{\prime }=0$. The remaining motion has $\omega =\pm \Omega$ and satisfies $u_y^{\prime }=\mp i{u}_x^{\prime }/2$: each horizontal layer executes an epicyclic motion, with the phase varying vertically but no pressure or vertical-velocity perturbation.

Write the vortex aspect ratio as $\chi$ to distinguish it from cylindrical radius. The Kida vortex core flow isAs $\chi \to \infty$, $u_x\to 0$ and $u_y\to -(3/2)\Omega x$, recovering Keplerian shear.

For a fluid particle,It therefore circulates around an ellipse with angular frequency $3\Omega /[2(\chi -1)]$ and period

For a perturbation depending only on $z,t$, horizontal pressure gradients vanish. Linearization givesTaking ${\mathbf{u}}^{\prime }\propto e^{-i\omega t}$ yieldsThe first bracket is positive for $\chi >3/2$, while the second is negative for $1<\chi <4$. Their product is therefore negative, so $\omega$ is imaginary and the mode grows precisely when