F-plane

Remarks

Only the vertical component of the rotation vector is kept, so the horizontal Coriolis acceleration is $f_0 \, \mathbf{k} \times \mathbf{u}$, with $\mathbf{k}$ the upward unit vector and $\mathbf{u}$ the horizontal velocity. The f-plane is the beta plane with $\beta = 0$.

Example: at $\varphi_0 = 45^\circ$ on the Earth, $\Omega = 7.292 \times 10^{-5}$ s$^{-1}$ gives $f_0 \approx 1.03 \times 10^{-4}$ s$^{-1}$. Unforced frictionless flow $\partial_t \mathbf{u} = -f_0 \mathbf{k} \times \mathbf{u}$ on the f-plane gives inertial oscillations, in which $\mathbf{u}$ rotates with period $2 \pi / \abs{f_0}$, about $17$ hours at $45^\circ$.

Non-example: a model with $f = f_0 + \beta y$ and $\beta \ne 0$ is not an f-plane; it supports Rossby waves, which the f-plane does not.