The linear shallow-water equations governing the motion of a fluid layer in the neighbourhood of a point on the Earth's surface in the northern hemisphere are

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

where $u(x, y, t)$ and $v(x, y, t)$ are the horizontal velocity components and $\eta(x, y, t)$ is the perturbation of the height of the free surface.

(a) Explain the meaning of the three positive constants $f, g$ and $h$ appearing in the equations above and outline the assumptions made in deriving these equations.

(b) Show that $\zeta$, the $z$-component of vorticity, satisfies

$$\frac{\partial \zeta}{\partial t}=-f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)$$

and deduce that the potential vorticity

$$q=\zeta-\frac{f}{h} \eta$$

satisfies

$$\frac{\partial q}{\partial t}=0$$

(c) Consider a steady geostrophic flow that is uniform in the latitudinal $(y)$ direction. Show that

$$\frac{d^{2} \eta}{d x^{2}}-\frac{f^{2}}{g h} \eta=\frac{f}{g} q .$$

Given that the potential vorticity has the piecewise constant profile

$$q= \begin{cases}q_{1}, & x<0 \\ q_{2}, & x>0\end{cases}$$

where $q_{1}$ and $q_{2}$ are constants, and that $v \rightarrow 0$ as $x \rightarrow \pm \infty$, solve for $\eta(x)$ and $v(x)$ in terms of the Rossby radius $R=\sqrt{g h} / f$. Sketch the functions $\eta(x)$ and $v(x)$ in the case $q_{1}>q_{2}$.