The linearised equations governing the horizontal components of flow $\mathbf{u}(x, y, t)$ in a rapidly rotating shallow layer of depth $h=h_{0}+\eta(x, y, t)$, where $\eta \ll h_{0}$, are

$$\begin{gathered} \frac{\partial \mathbf{u}}{\partial t}+\mathbf{f} \times \mathbf{u}=-g \nabla \eta \\ \frac{\partial \eta}{\partial t}+h_{0} \nabla \cdot \mathbf{u}=0 \end{gathered}$$

where $\mathbf{f}=f \mathbf{e}_{z}$ is the constant Coriolis parameter, and $\mathbf{e}_{z}$ is the unit vector in the vertical direction.

Use these equations, either in vector form or using Cartesian components, to show that the potential vorticity

$$\mathbf{Q}=\zeta-\frac{\eta}{h_{0}} \mathbf{f}$$

is independent of time, where $\zeta=\nabla \times \mathbf{u}$ is the relative vorticity.

Derive the equation

$$\frac{\partial^{2} \eta}{\partial t^{2}}-g h_{0} \nabla^{2} \eta+f^{2} \eta=-h_{0} \mathbf{f} \cdot \mathbf{Q}$$

In the case that $\mathbf{Q} \equiv 0$, determine and sketch the dispersion relation $\omega(k)$ for plane waves with $\eta=A e^{i(k x+\omega t)}$, where $A$ is constant. Discuss the nature of the waves qualitatively: do long waves propagate faster or slower than short waves; how does the phase speed depend on wavelength; does rotation have more effect on long waves or short waves; how dispersive are the waves?