Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 3 / d / Solution

Let $K=(k^2+l^2{)}^{1/2}$ and $c=\sqrt{g^{\prime }{h}_0}$. The three branches areFor nonzero $K$ and $f$, the zero-frequency mode is stationary geostrophic flow, with $f\mathbf{k}\times \mathbf{u}=-g^{\prime }\nabla \eta$ and zero horizontal divergence. It carries the linear potential-vorticity disturbance. There is no Rossby-wave propagation on a constant-$f$ plane; a varying Coriolis parameter would be needed for that.

The other two branches are oppositely phased inertia-gravity waves. Their phase-speed magnitude and radial group velocity are $|\omega |/K=\sqrt{c^2+f^2/K^2}$ and $|\partial \omega /\partial K|=c^{2}K/\sqrt{f^2+c^2{K}^2}$. When $K{L}_R\gg 1$, gravity dominates: $|\omega |\simeq cK$ and the waves are approximately nondispersive, with both speeds approaching $c$. When $K{L}_R\ll 1$, rotation dominates: $|\omega |\simeq |f|+c^2{K}^2/(2|f|)$, the frequency approaches the inertial frequency and the group velocity tends to zero. The large long-wave phase speed is not a large signal speed.

Without rotation the propagating branches become nondispersive gravity waves, $\omega =\pm cK$, and the stationary branch is a nondivergent vortical velocity disturbance rather than a finite-pressure geostrophic balance. At $K=0$, the modes reduce to constant interface height and spatially uniform inertial oscillations. The wave branches have zero linear potential-vorticity perturbation; it is the stationary branch that retains the initial potential vorticity.