Coriolis Parameter from Latitude

The Rossby and Ekman numbers both divide by $f$, and it is easy to treat $f$ as an abstract parameter that arrives from somewhere. It does not. It is a place on the Earth.

\[f = 2\Omega\sin\varphi\]

The sine, not the cosine, and it is worth understanding why. The Coriolis acceleration in a rotating frame is $-2\boldsymbol\Omega \times \mathbf{u}$. What turns a horizontal flow is the vertical component of the rotation vector, and at latitude $\varphi$ that component is $\Omega\sin\varphi$. At the pole the rotation axis points straight up and the whole of $\Omega$ is available; at the equator it lies flat along the ground and none of it is. So $f$ is maximum at the poles, zero at the equator, and negative in the southern hemisphere, where the vertical component points downward and the deflection reverses.

The factor of two is not a fudge factor. It falls out of the coordinate transformation into a rotating frame, where it appears alongside the centrifugal term, and half of it comes from the change in the velocity vector and half from the change in position.

The most useful way to feel $f$ is as a period. $2\pi/f$ is the inertial period, the time a frictionless parcel takes to complete one circle under the Coriolis force alone. At the pole it is half a sidereal day, 11.97 hours. At 30° latitude, where $\sin\varphi = \tfrac12$, it is a full sidereal day, 23.93 hours. At 45° it is about 16.9 hours. Towards the equator it lengthens without limit. Current-meter records from the ocean are full of oscillations at precisely this period, which is as direct a confirmation as one could ask for that the effect is real physics rather than a bookkeeping device of the rotating frame.

At mid-latitude $f \approx 10^{-4}$ s⁻¹, and that number is worth memorizing because it makes every Rossby and Ekman estimate a piece of mental arithmetic.

Use the sidereal rate. $\Omega = 7.2921\times10^{-5}$ rad/s, which is one rotation in 23 h 56 m 4 s — not in 24 hours. The solar day is longer because the Earth has also moved about a degree along its orbit and must turn that extra degree to bring the Sun back to the meridian. Using 24 hours gives an $\Omega$ low by 0.27 %. That is a small error, but it is a real one and it costs nothing to avoid.

Where $f$ is nearly zero, the whole edifice changes. Within about five degrees of the equator there is not enough planetary vorticity to organize a rotating storm, which is why tropical cyclones do not form there however warm the sea is — they need a seed of rotation to concentrate, and near the equator there is none to concentrate. Equatorial dynamics is a separate discipline with its own wave families, precisely because $f$ changes sign there.

Two approximations get built on this. The f-plane treats $f$ as a constant over a limited region, which is a good enough model for a single storm and turns the equations into something tractable. The β-plane keeps the first term of its variation with northward distance:

\[\beta = \frac{df}{dy} = \frac{2\Omega\cos\varphi}{R}\]

and that gradient, rather than $f$ itself, is responsible for two of the largest features in geophysical fluid dynamics. It gives rise to Rossby waves, the great meanders of the jet stream, which propagate westward relative to the mean flow because a displaced column of fluid conserving its potential vorticity is restored towards its original latitude. And it produces westward intensification: the reason every ocean basin has a narrow, fast, deep current on its western side — the Gulf Stream, the Kuroshio, the Agulhas — and a broad slow drift on its eastern side. Henry Stommel showed in 1948 that the asymmetry comes entirely from $\beta$. Without the variation of $f$ with latitude the gyres would be symmetric and there would be no Gulf Stream.

Finally, the same arithmetic travels. Mars turns once in 24 h 37 m, giving $\Omega = 7.088\times10^{-5}$ rad/s — within three percent of Earth's, which is a large part of why Martian weather systems have a family resemblance to ours. Jupiter turns in under ten hours at $\Omega = 1.76\times10^{-4}$ rad/s, and its enormous Coriolis parameter is much of the reason its atmosphere organizes into zonal bands rather than into travelling storms.