Kelvin's theorem is more powerful, but it is an integral theorem dealing with a scalar and requires a knowledge of the detailed evolution of material
surfaces in the fluid. Moreover, it is valid only in the absence of baroclinic effects, which substantiallt restricts its application.
Since \(\boldsymbol{\Omega}\) is a constant vector, the vorticity equation can be written as
\[\frac{d\boldsymbol{\omega}_a}{dt} = \boldsymbol{\omega}_a \cdot \nabla \mathbf{u} - \boldsymbol{\omega}_a \nabla \cdot \mathbf{u} + \frac{\nabla \rho \times \nabla p}{\rho^2} + \nabla \times \frac{\mathbf{F}}{\rho}\]
Continuity equation for compressible flow
\[\nabla \cdot \mathbf{u} = -\frac{1}{\rho} \frac{d\rho}{dt}\]
Eliminating divergence using continuity
\[\frac{d}{dt} \left( \frac{\boldsymbol{\omega}_a}{\rho} \right)
= \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \right) \mathbf{u}
+ \nabla \rho \times \frac{\nabla p}{\rho^3}
+ \left( \nabla \times \frac{\mathbf{F}}{\rho} \right) \frac{1}{\rho}\]
Consider some scalar fluid property λ which satisfies an equation of the form
\[\frac{d\lambda}{dt} = \Psi ,\quad \frac{\boldsymbol{\omega}_a}{\rho} \cdot \frac{d}{dt} \nabla \lambda
= \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \right) \frac{d\lambda}{dt}
- \left[ \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \mathbf{u} \right) \cdot \nabla \lambda \right]\]
If the dot product of \(\nabla \lambda\) is taken
\[\nabla \lambda \cdot \frac{d}{dt} \left( \frac{\boldsymbol{\omega}_a}{\rho} \right)
= \left[ \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \right) \mathbf{u} \right] \cdot \nabla \lambda
+ \nabla \lambda \cdot \left( \nabla \rho \times \frac{\nabla p}{\rho^3} \right)
+ \frac{\nabla \lambda}{\rho} \cdot \left( \nabla \times \frac{\mathbf{F}}{\rho} \right)\]
\[\frac{d}{dt} \left( \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \lambda \right)
= \frac{\boldsymbol{\omega}_a}{\rho} \cdot \nabla \Psi
+ \nabla \lambda \cdot \left[ \nabla \rho \times \frac{\nabla p}{\rho^3} \right]
+ \frac{\nabla \lambda}{\rho} \cdot \left( \nabla \times \frac{\mathbf{F}}{\rho} \right)\]
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