In a Cartesian frame, the components of relative vorticity are
\( \omega_x = \frac{\partial w}{\partial y} - \frac{\partial v}{\partial z} \)
\( \omega_y = \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x} \)
\( \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \)

Assuming \(u\) and \(v\) are independent of \(z\), and using aspect ratio \(\delta = \frac{H}{L}\)
\( \omega_x = \frac{\partial w}{\partial y} = \mathcal{O} \left( \frac{W}{L} \right) = \mathcal{O} \left( \delta \frac{U}{L} \right) \)
\( \omega_y = -\frac{\partial w}{\partial x} = \mathcal{O} \left( \frac{W}{L} \right) = \mathcal{O} \left( \delta \frac{U}{L} \right) \)
\( \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = \mathcal{O} \left( \frac{U}{L} \right) \)
So that the horizontal components of relative vorticity are \(\mathcal{O}(\delta)\) smaller than the vertical component of vorticity

Define the vertical vorticity \(\zeta \equiv \omega_z\), and taking the curl of the momentum equations
\( \frac{d\zeta}{dt} = \frac{\partial \zeta}{\partial t} + u \frac{\partial \zeta}{\partial x} + v \frac{\partial \zeta}{\partial y} = -(\zeta + f)\left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) \)
with the definition \( \zeta \equiv \omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \)

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