Vorticity
The vorticity of the fluid as observed from an inertial, nonrotating frame is called the absolute vorticity, \(\boldsymbol{\omega}_a\), and it is simply defined as the curl of the velocity observed in the nonrotating frame, \[ \boldsymbol{\omega}_a = \nabla \times \left\{ \mathbf{u} + \boldsymbol{\Omega} \times \mathbf{r} \right\} = \boldsymbol{\omega} + 2\boldsymbol{\Omega} \] where \(\boldsymbol{\omega}\) is the relative vorticity, the curl of the relative velocity
The absolute vorticity of each fluid element is the sum of the planetary vorticity \(2\boldsymbol{\Omega}\) and the relative vorticity \(\boldsymbol{\omega}\). The component of the planetary vorticity normal to the Earth's surface is the Coriolis parameter: \[ f = 2\Omega \sin \theta \] where \(\theta\) is the latitude of the fluid element

The corresponding estimate of the vertical component of the relative vorticity is given by the characteristic value of the velocity tangent to the Earth's surface (the horizontal velocity) divided by \(L\) \[ \omega_n = \mathcal{O}\left( \frac{U}{L} \right) \] whose ratio to \(f\) is \[ \frac{\omega_n}{f} = \frac{U}{fL} = \frac{U}{2\Omega L \sin \theta} = \frac{\varepsilon}{\sin \theta} \equiv R_0 \] where \(R_0\) is defined as the local Rossby number. In regions excluding the equator, \(\sin \theta\) is \(\mathcal{O}(1)\), so that low-Rossby-number flows have the property that their relative vorticity is small compared to the planetary vorticity. One immediate consequence of this fact is that large-scale flows are hardly ever free of vorticity, and their vorticity is primarily the planetary vorticity When \(R_0 \ll 1\), the flow is largely governed by Earth's rotation (planetary effects dominate) When \(R_0 \sim 1\), both planetary and relative vorticity are comparable When \(R_0 \gg 1\), planetary rotation can be neglected (e.g., small-scale turbulence)

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

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