(i) In a reference frame rotating about a vertical axis with constant angular velocity $f / 2$ the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible fluid of constant density $\rho$ are

$$\begin{aligned} &\frac{D u}{D t}-f v=-\frac{1}{\rho} \frac{\partial P}{\partial x} \\ &\frac{D v}{D t}+f u=-\frac{1}{\rho} \frac{\partial P}{\partial y} \end{aligned}$$

where $u, v$ and $P$ are independent of the vertical coordinate $z$.

Define the Rossby number $R o$ for a flow with typical velocity $U$ and lengthscale $L$. What is the approximate form of the above equations when $R o \ll 1$ ?

Show that the solution to the approximate equations is given by a streamfunction $\psi$ proportional to $P$.

Conservation of potential vorticity for such a flow is represented by

$$\frac{D}{D t} \frac{\zeta+f}{h}=0$$

where $\zeta$ is the vertical component of relative vorticity and $h(x, y)$ is the thickness of the layer. Explain briefly why the potential vorticity of a column of fluid should be conserved.

(ii) Suppose that the thickness of the rotating, shallow-layer flow in Part (i) is $h(y)=H_{0} \exp (-\alpha y)$ where $H_{0}$ and $\alpha$ are constants. By linearising the equation of conservation of potential vorticity about $u=v=\zeta=0$, show that the stream function for small disturbances to the state of rest obeys

$$\frac{\partial}{\partial t}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) \psi+\beta \frac{\partial \psi}{\partial x}=0$$

where $\beta$ is a constant that should be found.

Obtain the dispersion relationship for plane-wave solutions of the form $\psi \propto$ $\exp [i(k x+l y-\omega t)]$. Hence calculate the group velocity.

Show that if $\beta>0$ then the phase of these waves always propagates to the left (negative $x$ direction) but that the energy may propagate to either left or right.