One Spin, Written Across Every Latitude

1 · Three models make three different pictures

Call the planet's rotation rate Ω, one turn per sidereal day. The local Coriolis parameter, f, is twice the part of that rotation that points along your local vertical, and it sets both the rate and the handedness of the loops. The three models do not disagree by a little:

• — A stationary Earth. Nothing rotates, so f = 0 everywhere: no inertial loops at any latitude. This is the book's position, a stationary disc under a turning sky.
• — A flat turntable spinning about an axis through the pole. The axis is perpendicular to the surface everywhere on the disc, so f = 2Ω at every location: loops of identical period, roughly 12 hours, turning the same way from centre to rim, and nothing happening at the equator. Nobody defends this model; it is here as a logical bracket, the only way a flat plane can rotate at all.
• — A spinning sphere. Only the vertical component of the rotation counts, so f = 2Ω sin φ. Loops are fastest at the poles, slow to nothing at the equator, and reverse handedness across it, because the local vertical points the opposite way relative to the spin.

Why the equator is the null and not the maximum, when the surface there moves fastest: the Coriolis acceleration is a = −2Ω × v, and only the vertical component of Ω deflects horizontal motion. At the equator the axis lies flat in the local horizontal plane, so the cross product for motion along it vanishes and motion across it is pushed up or down, not sideways. The same sign flip is the reason hurricanes spin counter-clockwise north of the equator and clockwise south of it, and the reason none form within about five degrees of the line, where f is too small to organise a rotating storm. Most readers already believe that one. The drifters are the same fact, measured rather than watched.

2 · Nobody deployed these buoys to prove anything

These buoys are not instruments for measuring the Earth. A Global Drifter Program float is a plastic sphere with a satellite transmitter and a drogue — a cloth sea-anchor — hanging fifteen metres below it, released by people who want to know about currents and sea-surface temperature. It has no sensitive axis. It is not aimed at anything. All it does is report where it is.

The rotation isn't in what the buoys measure. It's in how they move.

The signature is a property of the trajectory, and it emerges from a list of positions when someone else, later, asks a different question. The oceanographers who collected this data were not testing Earth's spin; many actively did not want these loops, because near-inertial motion obscures the mean flow their studies are about and is routinely filtered out. The part of the record that demonstrates the rotation is the part the original users throw away.

That provenance closes a door that is otherwise always open. The standard objection to any rotation experiment is about the instrument — it drifts, it was calibrated on a rotating-Earth assumption. A float has no bias term and no calibration chain; the sea takes it in circles whose direction reverses at the equator.

3 · The measurements

One buoy per latitude rung, January to April 2019, each analysed independently. “Confidence” is the ratio of clockwise to counter-clockwise energy in the search band around the predicted frequency — how cleanly one handedness dominates. Two rows, 9.9°N and 22.7°S, fall outside 10% of prediction and are shaded.

Predicted period is 11.967 h ÷ sin φ, computed from the buoy's own mean latitude with no fitted parameters.

4 · The buoys measure the length of the day

The law has exactly one constant in it. Write it as T = A ÷ sin φ and ask what value of A the fifteen buoys imply, rather than assuming it:

That is roughly the accuracy Arthur Compton got in 1915 from a ring of water on a bench, built for the purpose. These fifteen were built to track ocean currents. Drifters are not a precision clock — a ring laser does this ten million times better — the point is that the number falls out of data collected for something else, by instruments that cannot see the sky, and lands on the right answer. Whether the unresolved 25.4°N buoy stays in the fit, and how the rows are weighted, moves it by less than one per cent either way (method notes, below).

5 · Where this page could be wrong

Fifteen buoys is a ladder, not a distribution. The airtight version of this claim is statistical — hemisphere energy asymmetry across hundreds of drifters, binned by latitude, crossing zero at the equator. One buoy per rung traces the curve; it does not pin down the scatter. That larger run has been done, by Elipot & Lumpkin (2008) across the whole array, and it returns the sin φ law and the hemisphere asymmetry — but it is their work, not this page's. The selection rule here: latitude rungs were fixed first, roughly every 5–10°, and for each rung the first buoy in the January–April 2019 window with a continuous track long enough to hold several loop periods was taken and analysed. No buoy was analysed and then discarded. Three rows carry short windows (887, 951 and 1,029 samples against 2,881 for a full window) because those tracks ended, not because they were trimmed. Anyone can pull a different fifteen; the interesting reply would be a set that does not reverse at the equator.

Several periods are a few per cent off, and one is about 20% off. That is the buoy at 22.7°S, and two known effects push measured periods off f. Near-inertial waves are intrinsically super-inertial — their frequency sits slightly above f — which is why measured periods skew short. And the spin of the background flow, its vorticity ζ, shifts the effective frequency to roughly f + ζ/2: cyclonic spin (the same sense as the local Earth rotation) shortens the loop, anticyclonic spin lengthens it, which is why near-inertial energy pools in anticyclones. So the fast buoy at 22.7°S wants cyclonic vorticity or wave propagation; the slow one at 9.9°N fits the anticyclonic story. These periods are deliberately left uncorrected: correcting them would tighten the fit, and would also be the easiest place to bias it.

“These are tides, not inertial loops.” The 63.1°N buoy invites it: 12.4 h sits close to the M2 tide — the twice-daily lunar tide — at 12.42 h, and northern shelf seas have strong tidal currents, so that row may well be partly tidal — treat it as the weakest in the table. But it cannot be the general explanation. Tidal periods are fixed — 12.42, 12.00, 23.93, 25.82 hours — and do not scale with sin φ; tidal ellipses put energy into both rotary senses rather than one; and most of the table sits at periods (17 to 80 hours) with no tidal line anywhere near. A tide that reversed handedness at the equator and stretched toward infinity approaching it would not be a tide.

The confound is sharpest near 30°, where the inertial period is 11.967 h ÷ sin 30° = 23.93 h — one full sidereal day, on top of the diurnal tidal lines — so a single loop there cannot be told from a diurnal tide by its period alone. Five of the fifteen sit between 22° and 34°, where the predicted periods (31.1 down to 21.5 h) bracket the diurnal band, and two deserve suspicion: the 22.7°S buoy reads 24.3 h, nearer the 23.93 h tidal line than its own 31.1 h prediction, and the 25.4°N buoy at 29.7 h splits its energy evenly between the two senses, which is what a tidal ellipse looks like — one more reason it is out of the handedness count. The other three — 31.5°N at 23.4 h, 27.6°S at 26.0 h, 33.8°S at 22.7 h — load one rotary sense by 6×, 23× and 19×, which a tide does not do, and they step through the diurnal band in the order sin φ says rather than sitting at one fixed period. Near 30° the discriminator is not the period but the one-sidedness of the rotation and the ladder either side.

“The fixes assume a globe, so the loops are a coordinate artefact.” Satellite positioning does use a WGS-84 ellipsoid, and it does correct for Earth's rotation during signal transit — the Sagnac term — so an assumption about rotation does touch the data. But the Sagnac correction is a static offset of order tens of metres, and coordinate transformations are smooth; neither can manufacture a twelve-to-eighty-hour oscillation whose period tracks latitude and whose handedness flips at a line. The loops are kilometres across in raw latitude and longitude, before any projection.

“The loops are wind-driven, not rotation-driven.” Half right, and the half that is right does not help. Wind is what excites an inertial oscillation, which is why loop amplitude is erratic and is not plotted here. The frequency is set by f alone, and frequency is what this page tests.

“A cosmos turning around a fixed Earth would make the same loops.” It would. In a Machian reading — Ernst Mach's proposal that inertia is set by the distant matter of the universe — a cosmos turning once per sidereal day about a fixed Earth drags the local inertial frame with it, the drifters loop exactly as they do now, and nothing on this page can tell the two descriptions apart. But that is a point for geocentrism, not for a flat Earth. The sin φ law needs the local vertical to tilt away from the rotation axis by exactly the latitude, and to point the opposite way on the two sides of the equator — the geometry of a sphere and of nothing else. Hand the same Machian frame to a flat plane and you get the turntable panel: f = 2Ω everywhere, one handedness, no reversal. Nor does the book's own sky do the job: the Machian reading needs the mass of the whole cosmos turning about an axis through the poles, not a set of lights over a dome. Either way, the drifters record a relative rotation of Earth and the fixed stars, one turn per sidereal day, about an axis through the poles — the rotation Q10 and Q11 dispute.

Velocities are a derived product, and the band-pass filter (which keeps only frequencies near the predicted loop rate) is for display. The ve/vn fields are computed by interpolating satellite position fixes and differentiating them, with quality control applied. The inertial signal is in the raw fixes too, but anyone who wants to argue “interpolation artefact” is entitled to, and the clean answer — redo one buoy from raw fixes — is not done here. The unfiltered rotary spectrum — the velocity record split into clockwise and counter-clockwise turning components — already shows the hemisphere asymmetry at roughly ten to one; filtering makes the loops legible on a chart, it does not create them.

Sources & further reading

• NOAA AOML Global Drifter Program — roughly 1,300 active buoys; public domain.
• ERDDAP tabledap, dataset drifter_hourly_qc, erddap.aoml.noaa.gov/gdp/erddap. Analysis window January–April 2019.
• S. Elipot, R. Lumpkin, R. C. Perez, J. M. Lilly, J. J. Early & A. M. Sykulski, “A global surface drifter data set at hourly resolution,” J. Geophys. Res. Oceans 121, 2937–2966 (2016).
• S. Elipot & R. Lumpkin, “Spectral description of oceanic near-surface variability,” Geophys. Res. Lett. 35, L05606 (2008) — the whole-array version of §5's ladder.
• E. Kunze, “Near-inertial wave propagation in geostrophic shear,” J. Phys. Oceanogr. 15, 544 (1985) — the f + ζ/2 shift in §5.
• A. H. Compton, “A Laboratory Method of Demonstrating the Earth's Rotation,” Science 37, 803 (1913) — the water ring in §4.
• Companion page: gyroscopes and Earth's rotation — the same rotation by an entirely different route.