The forces that are not there

A puck is slid across a spinning roundabout. Filmed from a gantry above it, the puck travels in a straight line at constant speed, because nothing is pushing it sideways. Filmed by a camera bolted to the roundabout, the same puck sweeps out a curve. Both films are correct, they are of the same puck, and the difference between them is the whole subject.

The right-hand panel is not a drawing of a different experiment. Every point in it is a point from the left-hand panel with its angle shifted. So whatever is bending the path on the right is arithmetic — and it still needs a name, because the right-hand panel is the one everybody lives in. A person standing on a planet is a camera bolted to the roundabout.

What the transformation actually produces #

Rewriting $\mathbf{F}=m\mathbf{a}$ in coordinates that rotate at angular velocity $\boldsymbol\Omega$ produces exactly three extra terms and no others:

$$m\mathbf{a}_{\text{rot}} = \mathbf{F} - m\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf{r}) - 2m\boldsymbol\Omega\times\mathbf{v}_{\text{rot}} - m\dot{\boldsymbol\Omega}\times\mathbf{r}.$$

The first is centrifugal, pointing away from the axis, depending on position and not on velocity. The second is Coriolis, perpendicular to the velocity, depending on velocity and not on position. The third is Euler, which appears only while the rotation rate is changing and which is why a merry-go-round throws people about when it starts and not when it is running.

They are called fictitious for one precise reason: no other body exerts them, so they violate the third law and cannot be traced to a source. That is the whole content of the word. It does not mean small, and it does not mean optional.

The cleanest way to keep the two apart is to ask what the object is doing. A stone on a string, seen from the room, accelerates inward and the string supplies the force. Seen from the frame turning with the stone, the stone is at rest, so the forces must sum to zero — and they do, once the outward centrifugal term is included alongside the string’s tension. Neither description contains an outward force on anything: the outward force the hand feels is the string pulling back on it, which is the third-law partner of the tension and is present in both descriptions.

The term that flattens a planet #

The centrifugal term does not depend on velocity, so it is felt by everything on a rotating body whether it moves or not, and its consequences are permanent rather than dynamic.

That second curve is worth dwelling on, because it is the one that gets left out. A plumb line does not point at the centre of the Earth anywhere except at the poles and the equator, and “down” as everyone experiences it is the direction of the sum of gravity and the centrifugal term. The tenth of a degree is small and it is not negligible: it is the reason the Earth is an oblate spheroid rather than a sphere. Rock is a fluid on geological timescales, and a fluid settles until its surface is perpendicular to the effective gravity — so the planet took the shape that makes the plumb-line deflection vanish, which is a bulge of 21 km at the equator.

The same term is why a satellite in a geostationary orbit is not weightless in some special sense and why a rotating space station provides an ersatz gravity of exactly $\Omega^2 R$. It is also why an athlete swinging a hammer round at the equator has a marginally easier time of it than one at the poles, by three parts in a thousand, which is smaller than the difference between two throws and larger than the resolution of the record book.

The term that turns the weather #

The Coriolis term is the one that depends on velocity, so it does nothing at all to a stationary object and acts on everything that moves.

The deflection grows as the square of the time, which is the single most useful thing to know about it.

Over a second it is microns; over an hour it is kilometres; over a day it is the whole circulation of the atmosphere. A parcel of air moving toward a low-pressure centre is deflected, misses, and ends up circling the low rather than filling it — anticlockwise in the northern hemisphere, clockwise in the southern — and the resulting balance between the pressure gradient and the Coriolis term is called geostrophic flow, and is why a weather map’s isobars are very nearly streamlines.

The same reasoning at sea gives the ocean gyres, and at the scale of a bathroom sink gives nothing whatever. The comparison that decides it is the Rossby number, the ratio of the ordinary acceleration to the Coriolis one: large for a sink, of order one for a cyclone. The sink question is not settled by declaring the force fictitious; it is settled by putting numbers into it and finding that the answer is a hundred thousand times smaller than the swirl left over from filling the basin.

The deflection depends on latitude because only one component of the planet’s rotation counts. A horizontal motion is turned horizontally by the component of the angular velocity about the local vertical, and that component is the full rotation rate at the pole and nothing at all at the equator. So the Coriolis parameter carries a sine of the latitude, and a cyclone cannot form within a few degrees of the equator because there is nothing there to turn it.

The instrument #

An effect whose size depends only on the rotation of the Earth can be turned around and used to measure it, which is what Foucault did in 1851 by hanging a 28 kg bob on a 67-metre wire from the dome of the Panthéon.

What makes the demonstration good is that the pendulum has no way of knowing about the Earth. Its plane of swing is maintained by nothing but the absence of any horizontal force to change it, and the building turns underneath. It was the first laboratory-scale evidence that the Earth rotates, three hundred years after the argument was settled astronomically and philosophically, and it was popular for exactly that reason: the sun going round the sky is compatible with either arrangement, and a pendulum in a basement is not.

The modern version of the same measurement is a ring laser gyroscope, which compares the travel times of two beams going opposite ways round a loop and reads $\Omega$ off the difference. The best of them measure the Earth’s rotation rate to a part in $10^9$ and detect the wobble of the pole. The physics is different — the Sagnac effect rather than an inertial plane — but the principle is the one Foucault used: rotation is not relative, and any sufficiently isolated thing can be asked about it.

The trade winds, which were explained twice #

The historical route to the Coriolis term did not run through mechanics. It ran through the question of why the trade winds blow from the north-east in the northern tropics and the south-east in the southern ones — a fact of enormous commercial importance and no obvious cause, since the air is clearly being drawn toward the equator, which is a north-south motion.

Hadley’s answer in 1735 was that air moving toward the equator arrives over ground that is moving east faster than the ground it left, and therefore lags behind: conservation of the eastward velocity gives an easterly wind. That is the right shape of argument and the wrong conserved quantity. What is conserved is angular momentum about the Earth’s axis, not eastward velocity, and using the correct one gives a deflection three times larger. Coriolis, in 1835, derived the general term for machinery rather than for weather, and the meteorologists found it later.

The trade winds themselves are the lower branch of a circulation cell in which air rises at the equator, where it is warm and therefore less dense, travels poleward aloft, sinks around 30°, and returns along the surface. The cell exists because of heating; its east-west character is entirely the term on this page. Take away the rotation and there would still be a circulation and there would be no trades — and no westerlies, no jet stream, and no reason for a storm to spin.

What a trajectory looks like with the rotation left out is the baseline every ballistic calculation on this page is a correction to. That is worth stating plainly, because the corrections are small and the temptation is to treat them as a different subject: a shell fired north in the northern hemisphere lands to the right of where the non-rotating calculation puts it, by an amount that is a fraction of a per cent over a kilometre and several hundred metres over a hundred.

An accidental instrument #

The Coriolis term is exploited industrially in a way that has nothing to do with planets, and it is the cleanest demonstration that the term is about frames rather than about the Earth.

A Coriolis mass flow meter is a U-shaped tube driven into vibration at a few hundred hertz. Fluid flowing along the tube is, in the tube’s own oscillating frame, a mass with a velocity in a frame that is rotating — so it experiences $2m\boldsymbol\Omega\times\mathbf{v}$, and the two arms of the U are pushed in opposite senses because the flow in them runs in opposite directions. The tube twists, the twist is proportional to the mass flow rate, and the reading is independent of density, viscosity, temperature and what the fluid actually is. Nothing else in the instrument catalogue measures mass flow directly; everything else measures a volume and multiplies by a density that has to be assumed. The viscosity of the fluid does not appear anywhere in the answer.

It is worth noticing what has happened to the vocabulary here. The tube is not on a planet, nothing about the Earth’s rotation enters, and the frame doing the rotating is a few centimetres of steel oscillating through a fraction of a degree. The term arrived because a coordinate system was chosen, and it would have been just as legitimate to describe the whole instrument in the laboratory frame and never mention Coriolis at all — at the cost of an equation nobody could solve by hand.

The same term with the sign turned around is what a gyroscope resists with: a spinning wheel forced to change its axis is a body being made to move in a rotating frame, and the torque needed is the angular-momentum statement of exactly the Coriolis force on its rim.

The gyroscope in every pocket #

The flow meter above uses the Coriolis term in a vibrating tube. The same idea with the fluid removed is a rotation sensor, and it is now manufactured by the billion.

Drive a small proof mass into oscillation along one axis of a silicon chip. If the chip is not rotating, the mass simply goes back and forth. If the chip rotates about a perpendicular axis, the oscillating velocity puts $2m\boldsymbol\Omega\times\mathbf{v}$ into the third direction — an oscillation at the drive frequency, perpendicular to the drive, whose amplitude is proportional to the rotation rate. Measure that with a pair of capacitor plates and the chip reports how fast it is turning.

That is a MEMS vibratory gyroscope. It is a few square millimetres, it costs less than a cup of coffee, and one is in every phone, every drone, every camera with image stabilisation and every car with stability control. The quantity it measures is a rate of rotation, sensed by an object that has no view of anything outside itself — which is the same claim Foucault’s pendulum made, at a hundredth of the size and a millionth of the price.

Evolution got there first, by about three hundred million years. A fly’s hindwings are not wings; they are halteres, small clubs on stalks that beat in antiphase with the forewings at a couple of hundred hertz. When the fly rotates, the oscillating club experiences exactly the Coriolis force above, bending out of its beat plane, and strain receptors at the base of the stalk detect the bending. The fly reads its own rotation rate from it, in about five milliseconds — far faster than vision — and a fly with its halteres removed cannot fly.

The architecture is identical in all three cases: a mass driven to oscillate, a Coriolis response perpendicular to the drive, and a detector for the perpendicular motion. What is being sensed is not a force anybody exerts and not a property of the Earth; it is the extra term that appears when Newton’s law is written in coordinates that turn. The word fictitious survives in the textbooks and describes an industry.

The drift that is not downwind #

There is one more consequence of the term that decides where fish are, and it began as an observation nobody could explain.

Nansen, drifting with the Fram in the Arctic ice through the 1890s, noticed that the ice did not move in the direction of the wind. It moved twenty to forty degrees to the right of it, consistently. Wind drags on ice, ice drags on water, and neither of those explains a systematic deflection.

Ekman worked it out in 1905. The wind applies a stress to the surface layer; that layer is dragged along, and once it is moving the Coriolis term acts on it, pushing it to the right in the northern hemisphere. The layer beneath is dragged by the one above and deflected further; the layer beneath that further still. The result is a spiral: the current direction rotates steadily with depth while its speed falls away exponentially, and the surface current ends up about forty-five degrees to the right of the wind.

The elegant part is what happens when the whole spiral is added up. The depth-integrated transport comes out exactly ninety degrees to the right of the wind stress — and it is independent of the eddy viscosity, which is the one quantity in the problem nobody can measure. The spiral’s shape depends on the friction; the total transport does not.

That independence is why the result is used and the spiral is not. A steady wind blowing along a coast with the land on its left drives the surface water directly offshore, and water must rise from below to replace it. That is coastal upwelling, it brings cold nutrient-rich water into the sunlit layer, and the four great upwelling systems it produces — off Peru, California, north-west Africa and Namibia — occupy about one per cent of the ocean and supply something like a fifth of the world’s fish catch.

Reverse the wind and the upwelling stops, the nutrients stop arriving, and the fishery collapses within weeks. That is what an El Niño does to the Peruvian anchoveta, and the chain from a change in the wind to a change in the catch runs entirely through a term that is not exerted by anything.

What the picture cannot show #

The two panels at the top of this page are a snapshot of a puck, and three things about them are misleading in ways worth naming.

The spiral is not caused by anything. It is tempting to read the curved path as the puck being pushed, and the strength of the figure is precisely that no push exists on the left. Anyone reading the right-hand panel as the record of a force is reading a coordinate change as a mechanism.

The Coriolis and centrifugal terms are drawn separately and are not separate. They are two pieces of one transformation and a frame carries both; a figure that shows the deflection of a moving object without also showing the weight it has lost is showing half of the change of coordinates.

Nothing here is relativistic and the third term is missing entirely. The Euler term appears only while $\Omega$ is changing, which for the Earth means the tidal slowing — two milliseconds per century — and is therefore invisible in every figure on this page while being the reason the length of the day is not a constant.

The domain of validity is narrow in one direction and wide in another. The three terms are exact for any rotation rate in Newtonian mechanics, with no approximation anywhere. What is approximate is the smallness used to derive the simple deflection formula above, which assumes the deflection is a small correction to a straight path; for a trajectory lasting a substantial fraction of a rotation period, the full equations have to be integrated and the answer is not a sideways drift but an epicycle. Rotating frames also stop being ordinary in general relativity, where a rigidly rotating disc has no consistent global slicing and the Sagnac effect is the surviving fragment.

Why the argument matters beyond the weather #

Once the three terms are written down, the interesting question stops being what they are and becomes why the transformation had to be made at all. Newton’s laws hold in inertial frames; nothing in mechanics identifies which frames those are, and Newton’s own answer was absolute space. Foucault’s pendulum keeps its plane relative to the fixed stars, and no mechanical experiment says why the distant matter of the universe should be the thing an unforced pendulum agrees with.

That question is what the equivalence principle takes up, where a uniform gravitational field and a linearly accelerating frame are declared indistinguishable and the fictitious force becomes the model for gravity itself. The rotating case is harder and was never resolved that way: rotation is detectable locally, by exactly the experiments on this page, and a theory in which acceleration is relative has to explain why.

The ladder from here #

Later rungs on this anchor: the full integration of a trajectory in a rotating frame, where the deflection becomes an inertial circle of period half a pendulum day; the Eötvös effect and the vertical Coriolis term; geostrophic and gradient wind balance, and the Rossby number as the quantity that decides whether the rotating terms matter; the Euler term and the mechanics of spin-up; and the tidal deformation of a rotating planet, where the centrifugal term and self-gravity are solved together to give the flattening.

The neighbouring ladders are circular motion in an inertial frame, which is the statement all of this is a change of coordinates away from, and the equivalence principle, which takes the fictitious force seriously enough to build a theory of gravity out of it.