Potential Vorticity Conservation in Shallow-Water Theory
Vorticity Equation
\[\frac{\partial}{\partial t} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right)
+ \frac{\partial}{\partial x} \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right)
- \frac{\partial}{\partial y} \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right)
+ f_0 \left( \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \right)
+ \beta v = 0\]
Following the customary β-plane approximation, \(f\) has been treated as constant (and replaced by an average value \(f_0\)) except when \(df/dy\) appears
\[
\zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}
\]
as the vertical component of relative vorticity, the vorticity measured relative to the rotating earth
\[
\underbrace{
\underbrace{u \frac{\partial \zeta}{\partial x}}_{\text{advection by } u}
+
\underbrace{v \frac{\partial \zeta}{\partial y}}_{\text{advection by } v}
}_{\text{material derivative of } \zeta}
+
\underbrace{
\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) \zeta
}_{\text{divergent stretching of } \zeta}
\]
\[
\underbrace{
\frac{\partial \zeta}{\partial t} + u \frac{\partial \zeta}{\partial x} + v \frac{\partial \zeta}{\partial y}
}_{= \frac{D \zeta}{D t}}
+ (\zeta + f_0)\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \beta v = 0
\text{ or }
\frac{D \zeta}{Dt} + (\zeta + f_0)\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) + \beta v = 0
\]
For this section, \(D/Dt\) is the derivative following only the horizontal motion of the layer \(\frac{D}{Dt} = \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y}\)
\[
\frac{D\zeta}{Dt}
+ (\zeta + f_0)
\underbrace{
\cancel{
\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right)
}
}_{\text{from continuity: } \frac{D h}{Dt} + h \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) = 0 \Rightarrow \frac{D\zeta}{Dt}
+ (\zeta + f_0)
\left( -\frac{1}{h} \frac{D h}{D t} \right)
+ \beta v = 0}
+ \beta v = 0
\]
\[
\Rightarrow
\frac{D\zeta}{Dt}
= \underbrace{
\left( \frac{\zeta + f_0}{h} \right) \frac{D h}{D t}
}_{\text{divergence replaced using continuity}}
- \beta v
\text{ or }
\underbrace{
\frac{D(\zeta + f)}{Dt}
}_{\substack{f = f_0 + \beta y \\
\frac{Df}{Dt} = \frac{\partial f}{\partial t} + u \frac{\partial f}{\partial x} + v \frac{\partial f}{\partial y}
= 0 + 0 + \beta v
}}
=
\left( \frac{\zeta + f}{h} \right) \frac{D h}{Dt} \Rightarrow \frac{D}{Dt} \left( \frac{\zeta + f}{h} \right) = 0
\]
◀
▶