\[ \frac{d}{dt} \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \lambda \right) = \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \Psi + \nabla \lambda \cdot \left[ \nabla \rho \times \frac{\nabla p}{\rho^3} \right] + \frac{\nabla \lambda}{\rho} \cdot \left( \nabla \times \frac{\mathbf{F}}{\rho} \right)\]

\(\lambda\) is a conserved quantity for each fluid element, i.e., \(\Psi = 0\)
the frictional force is negligible, i.e., \(\mathbf{F} = 0\)

and either

the fluid is barotropic, i.e.,\(\nabla \rho \times \nabla p = 0\)
or \(\lambda\) can be considered a function only of \(p\) and \(\rho\)

Then the potential vorticity \[\Pi = \frac{\boldsymbol{\omega} + 2\boldsymbol{\Omega}}{\rho} \cdot \nabla \lambda \] is conserved by each fluid element, i.e., \[\boxed{\frac{d \Pi}{d t} = 0}\] Note that the scalar \(\Pi\) involves the component of \(\boldsymbol{\omega}_a\) parallel to the gradient of \(\lambda\).
In the examples listed above for the choice of \(\lambda\), the condition that \(\lambda\) is conserved is equivalent to the condition that the motion is adiabatic, but this is not a general requirement. It is important to emphasize that the theorem does not require the fluid to be barotropic, for if \(\lambda\) can be considered a function only of \(p\) and \(\rho\) obtains [i.e., if \(\lambda = \lambda(p,\rho)\), \(\lambda\) is a thermodynamic function], then \[ \nabla \lambda = \frac{\partial \lambda}{\partial p} \nabla p + \frac{\partial \lambda}{\partial \rho} \nabla \rho \]

¹ Pedlosky, J. (1982). Geophysical Fluid Dynamics. Springer study edition. Springer, Berlin, Heidelberg.

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