### Shallow Water PV

The rotating shallow water equations are

\begin{align*} u_t + \vec{u} \cdot\nabla u -fv &= -gh_x \\
v_t + \vec{u} \cdot\nabla v +fu &= -gh_y \\
h_t + \nabla \cdot (\vec{u}h) &= 0
\end{align*}

where $$\vec{u}=(u,v)$$ is the horizontal velocity, $$h=h(x,y,t)$$ is the depth of the fluid, $$g$$ is the gravitational acceleration, and $$f$$ is the Coriolis parameter.

1. Define the potential vorticity for the rotating shallow water equations (SWPV).
2. Show that the SWPV is conserved.
3. Show that for a parcel initially at rest in the Northern hemisphere, there is an absolute lower bound on its relative vorticity if it remains in the same hemisphere, but no upper bound.

From Wikipedia.

From Wikipedia.