In a reference frame rotating about a vertical axis with angular velocity $f / 2$, the horizontal components of the momentum equation for a shallow layer of inviscid, incompressible, fluid of uniform 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 $u$ and $v$ are independent of the vertical coordinate $z$, and $p$ is given by hydrostatic balance. State the nonlinear equations for conservation of mass and of potential vorticity for such a flow in a layer occupying $0<z<h(x, y, t)$. Find the pressure $p$.

By linearising the equations about a state of rest and uniform thickness $H$, show that small disturbances $\eta=h-H$, where $\eta \ll H$, to the height of the free surface obey

$$\frac{\partial^{2} \eta}{\partial t^{2}}-g H\left(\frac{\partial^{2} \eta}{\partial x^{2}}+\frac{\partial^{2} \eta}{\partial y^{2}}\right)+f^{2} \eta=f^{2} \eta_{0}-f H \zeta_{0}$$

where $\eta_{0}$ and $\zeta_{0}$ are the values of $\eta$ and the vorticity $\zeta$ at $t=0$.

Obtain the dispersion relation for homogeneous solutions of the form $\eta \propto \exp [i(k x-$ $\omega t)$ ] and calculate the group velocity of these Poincaré waves. Comment on the form of these results when $a k \ll 1$ and $a k \gg 1$, where the lengthscale $a$ should be identified.

Explain what is meant by geostrophic balance. Find the long-time geostrophically balanced solution, $\eta_{\infty}$ and $\left(u_{\infty}, v_{\infty}\right)$, that results from initial conditions $\eta_{0}=A \operatorname{sgn}(x)$ and $(u, v)=0$. Explain briefly, without detailed calculation, how the evolution from the initial conditions to geostrophic balance could be found.