Let’s consider the Shallow Water Equations linearized about a state of rest, on an $$f$$-plane: $$\newcommand{\sgn}{\text{sgn}}$$

\begin{align*} u_t – f_0 v &= -g \eta_x \\
v_t + f_0u &= -g\eta_y \\
\eta_t + H(u_x+v_y) &= 0
\end{align*}

Consider an initial scenario where we are initially at rest, and $$\eta$$ at $$t=0$$ is defined by a step function:

\begin{align*}
\eta(x,t=0) &= -\eta_0 \sgn(x) =
\begin{cases}
\eta_0 & x < 0\\
-\eta_0 & x > 0
\end{cases}
\end{align*}

What will happen when we let the system run? We will see that the inclusion of rotation affects the results significantly.

1. When rotation is turned off, we get dispersionless waves. Write an explicit solution for the zonal velocity field.
2. Now turn rotation on. Use potential vorticity conservation to find a steady state solution for the height and meridional velocity fields.

From Wikipedia.