Potential Vorticity Conservation in Shallow-Water Theory
In the approximate compact form that \(\beta y \ll f_0\) has been assumed
\[\frac{D}{Dt} \left(
\underbrace{
\frac{\zeta + \overbrace{f_0 + \beta y}^{\text{planetary vorticity } f}}{h}
}_{\text{potential vorticity}}
\right) = 0\]
The ratio \((\zeta + f)/h\) is called the potential vorticity in shallow-water theory, and the equation above shows that the potential vorticity is conserved along the horizontal trajectory of a fluid particle, an important principle in geophysical fluid dynamics
In the ocean, outside regions of strong current vorticity such as coastal boundaries,
the magnitude of \(\zeta\) is much smaller than that of \(f\).
In such a case \(\zeta + f\) has the sign of \(f\).
The principle of conservation of potential vorticity means that an increase in \(h\) must
make \(\zeta + f\) more positive in the northern hemisphere and more negative in the southern hemisphere.
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