The equations governing the flow of a shallow layer of inviscid liquid of uniform depth $H$ rotating with angular velocity $\frac{1}{2} f$ about the vertical $z$-axis are

$$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-g \frac{\partial \eta}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-g \frac{\partial \eta}{\partial y} \\ \frac{\partial \eta}{t}+H\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) &=0 \end{aligned}$$

where $u, v$ are the $x$ - and $y$-components of velocity, respectively, and $\eta$ is the elevation of the free surface. Show that these equations imply that

$$\frac{\partial q}{\partial t}=0, \quad \text { where } \quad q=\omega-\frac{f \eta}{H} \text { and } \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$$

Consider an initial state where there is flow in the $y$-direction given by

$$\begin{aligned} u &=\eta=0, \quad-\infty<x<\infty \\ v &= \begin{cases}\frac{g}{2 f} e^{2 x}, & x<0 \\ -\frac{g}{2 f} e^{-2 x}, & x>0\end{cases} \end{aligned}$$

Find the initial potential vorticity.

Show that when this initial state adjusts, there is a final steady state independent of $y$ in which $\eta$ satisfies

$$\frac{\partial^{2} \eta}{\partial x^{2}}-\frac{\eta}{a^{2}}= \begin{cases}e^{2 x}, & x<0 \\ e^{-2 x}, & x>0\end{cases}$$

where $a^{2}=g H / f^{2}$.

In the case $a=1$, find the final free surface elevation that is finite at large $|x|$ and which is continuous and has continuous slope at $x=0$, and show that it is negative for all $x$.