Beta plane

Remarks

The formula is the first-order Taylor expansion of $f = 2 \Omega \sin \varphi$ in $y = a (\varphi - \varphi_0)$, so it is accurate when $\abs{\beta y} \ll \abs{f_0}$. The case $\beta = 0$ is the f-plane. At $\varphi_0 = 0$ one gets the equatorial beta plane $f = \beta y$ with $\beta = 2 \Omega / a$.

Example: for the Earth, $\Omega = 7.292 \times 10^{-5}$ s$^{-1}$ and $a = 6.371 \times 10^6$ m. At $\varphi_0 = 45^\circ$ this gives $f_0 \approx 1.03 \times 10^{-4}$ s$^{-1}$ and $\beta \approx 1.62 \times 10^{-11}$ m$^{-1}$ s$^{-1}$.

Non-example: keeping $f = 2 \Omega \sin(\varphi_0 + y / a)$ exactly in flat coordinates is not a beta plane, since $f$ is not linear in $y$. Such models lack the simple constant-coefficient wave solutions of the beta plane.

The beta-plane equations use flat geometry and drop all other curvature terms. This is consistent only at leading order in $L / a$, where $L$ is the scale of the motion.