Potential Vorticity Conservation: Shallow-Water Theory
Using mass conservation \(\frac{dH}{dt} + H \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0\)
\[\frac{d\zeta}{dt} = \frac{\zeta + f}{H} \frac{dH}{dt}\]
It expresses the vortex-tube stretching (or squeezing). For stretching \(dH/dt > 0\), the vorticity \(\zeta\) is intensified by an amount proportional to the product of the column stretching and the absolute vorticity, \(\zeta + f\), which is already present. Note that \(\zeta\) cannot change by vortex tilting.
Since \(f\) is a constant
\[
\boxed{
\frac{d}{dt} \left( \frac{\zeta + f}{H} \right) = 0
}
\]
Following the motion of each fluid column
\[
\Pi_s = \frac{\zeta + f}{H}
\]
is conserved. If \(H\) increases, the absolute (and hence relative) vorticity must increase to keep \(\Pi_s\) constant for the column
.
Note that if \(\zeta\) is originally zero, it will remain so only if \(H\) remains constant; relative vorticity is produced by column stretching in the field of planetary vorticity \(f\).
1 Pedlosky, J. (1982). Geophysical Fluid Dynamics. Springer study edition. Springer, Berlin, Heidelberg.