The deflection that closes on itself

The rung below this one writes Newton’s law in a turning frame and reads off three extra terms. It computes what each of them does — the centrifugal term flattens the planet, the Coriolis term deflects the weather — and it stops at describing them.

Description is where almost every treatment of the Coriolis force stops, and the description is the phrase deflects to the right. That phrase is true and it hides the shape of the answer, because a deflection applied continuously does not produce a bent line. It produces a circle.

Two lines, integrated #

Take the equations the rung below derived, drop everything except the Coriolis term, and write them out for horizontal motion at latitude $\varphi$:

$$\frac{\mathrm{d}u}{\mathrm{d}t} = f v, \qquad \frac{\mathrm{d}v}{\mathrm{d}t} = -f u, \qquad f = 2\Omega\sin\varphi.$$

That is the whole system. It has two obvious properties. The speed $u^2 + v^2$ is constant, because the force is perpendicular to the velocity — the same reason a magnetic force does no work, and the same structure. And the velocity vector rotates at a fixed rate $f$, so it returns to itself after a time $2\pi/f$.

A velocity that returns to itself with constant magnitude traces a closed curve, and it is the same construction that makes turning an acceleration run backwards: there, a circle implied a centripetal acceleration; here, a perpendicular acceleration implies a circle of radius

$$r = \frac{U}{f}.$$

Nothing about that is a hidden subtlety; it is what the equations say if they are integrated instead of read. What makes it worth a figure is that the answer contradicts the mental picture the usual description installs. Deflects to the right suggests a trajectory that curves and goes somewhere. The actual trajectory goes nowhere: after one period the parcel is where it started, moving as it was, with zero net displacement.

The size, which is small, and the period, which is not #

Both numbers surprise people who have only met the qualitative version.

At 45 degrees, $f$ is $1.03 \times 10^{-4}$ per second. An ocean current of half a metre a second therefore circles with a radius of five kilometres. A wind of twenty metres a second circles with a radius of two hundred. Those are not planetary scales — an inertial circle in the ocean is smaller than a city.

The period is where the connection to the rung below becomes exact.

A Foucault pendulum’s plane of swing turns once in $23.93/\sin\varphi$ hours, which is called a pendulum day. An inertial oscillation’s period is exactly half of that, and the factor of two is not a coincidence: a pendulum’s plane is an undirected line, so it returns to itself after half a turn, while a velocity is a vector and needs a whole one. Two quantities turning at the same rate, counted differently.

That is worth pausing on because it makes the pendulum and the ocean current the same measurement. The rung below treats the Foucault pendulum as a demonstration that the Earth turns. An inertial oscillation is the identical demonstration performed by a piece of seawater, and it happens continuously, everywhere, without anybody hanging anything from a ceiling.

What is actually seen #

The oscillations are not a theoretical construction. They are the largest single feature in most long records of ocean current.

Drop a drifting buoy in the open ocean, record its position, and the trajectory is a slow meandering with small loops superimposed. Take the frequency spectrum of the velocity and there is a sharp peak, and the peak is at the local inertial frequency — moving as the buoy moves north or south, which is the strongest possible evidence that it is what it appears to be.

Nansen noticed the effect during the Fram expedition in the 1890s, from ice-drift measurements in the Arctic, where the period is close to twelve and a half hours. Ekman explained it in 1905 in the same paper that produced the Ekman spiral, and the modern version is a global array of several thousand drifters returning the loops continuously.

What generates them is a sudden push: a squall, a storm front, an abrupt change in wind stress. The wind accelerates the surface layer over a few hours, then stops, and what is left is a slab of water moving with no force on it — which is precisely the initial condition of the figure above. The oscillation then persists for days to weeks, decaying as the energy leaks downward into internal waves, which is one of the routes by which the wind’s energy reaches the deep ocean — and where it goes after that is decided by the shape of the basin.

The circle in a laboratory, and why it is hard #

The result is clean enough to invite a bench demonstration, and building one is instructive because of what it takes.

The inertial period on Earth is between twelve hours and never, so a laboratory version needs a faster rotation. Put a tank on a turntable spinning at one revolution every ten seconds and the effective $f$ is $2\Omega = 1.26$ per second, giving an inertial period of five seconds. A float released at a centimetre a second then circles with a radius of eight millimetres, which is visible.

The difficulty is everything else. The tank’s water must be brought to solid-body rotation first, which takes many spin-up times and a spin-up time in a shallow tank of water is minutes. The free surface is a paraboloid, so a float drifts down the slope unless it is neutrally buoyant at the right depth. Friction on the bottom drives an Ekman layer that drains the interior and damps the oscillation within a few periods. And the tank has walls a few radii away, which reflect.

Every one of those problems is the same problem in different clothing: the inertial oscillation is a free motion, and a laboratory is a place where nothing is free. That is why the clean demonstrations of this effect are oceanographic rather than laboratory, and why a several-thousand-buoy array is a more practical instrument for it than any tank. The ocean supplies a rotating frame ten thousand kilometres across with no walls anywhere near.

The one laboratory arrangement that does work well is a rotating tank many metres across, and there are perhaps a dozen in the world. They are used mostly for the next rung’s subject rather than this one — the balanced flows, which are steady and therefore survive the damping that kills a free oscillation in a few periods.

The equator, where the circle stops fitting #

The radius is $U/f$ and $f$ vanishes at the equator, so the circles grow without bound as the equator is approached.

At 45 degrees a one-metre-per-second current circles in ten kilometres. At five degrees the same current needs eighty. At one degree it needs four hundred, which is comparable with the distance to the equator itself, and the whole construction has broken: the parcel would leave the latitude at which $f$ was evaluated long before completing a circle.

That failure is not a nuisance to be patched. It is the reason equatorial dynamics is a separate subject with its own set of waves — Kelvin waves, Yanai waves, equatorially trapped Rossby waves — none of which exists at mid-latitude, all of which depend on $f$ changing sign at the equator rather than merely being small there — a channel with no walls made by a change of sign. The equator is a waveguide, and it is a waveguide because the restoring effect that makes inertial circles reverses direction across it.

The circles that do not close #

The clean result assumed $f$ constant, which is to say that the whole circle happens at one latitude. It does not.

The control matters more than the result, and the reason is worth recording. The first version of that figure used a first-order integrator, which on a rotation gains amplitude at every step. It produced a smooth, entirely plausible spiral drifting three quarters of a radius to the north — the wrong direction, at ten times the right size, with nothing in the drawing to indicate that any of it was arithmetic. The figure now integrates to fourth order, checks that the speed is conserved to a part in a hundred million, and runs the identical code with $\beta$ set to zero to confirm that the drift disappears.

The drift itself is measured. Drifter loops in the real ocean migrate westward and slightly poleward-of-nothing at a rate consistent with $\beta$, and separating that signal from the background currents the buoy is also riding is a real analysis problem — which is why the confirmation came from long records rather than from single trajectories.

What the term is doing to a real flow #

An inertial circle is what happens when the Coriolis term acts alone, and it never does. Its role in a real flow is to be one term among several, and the rung below’s figures show two of the others.

That is the reconciliation between the two pictures. A projectile in flight for forty seconds completes $40/60{,}930$ of an inertial period — a fraction of a degree of arc — so its trajectory looks like a straight line with a small sideways displacement growing as $t^2$, which is exactly how ballistic tables present it, and is why a projectile’s range depends on its launch angle and not detectably on its heading. Nothing is wrong with that description; it is the beginning of the circle, and the circle is invisible because the flight ends far too soon.

The two regimes are separated by how long the motion lasts compared with $1/f$. A bullet, a thrown ball and a falling stone are all far inside one inertial period, so they see a small deflection. A parcel of ocean disturbed by a storm lasts for days, so it sees the whole circle. Nothing changes about the physics between the two; only the duration does.

What is conserved, and how it becomes something larger #

There is a conserved quantity behind the closed circle, and naming it is how this rung joins the rest of geophysical fluid dynamics rather than staying a curiosity.

The first integral of the two-line system is that $u$ and $y$ are tied together: differentiating $u$ gives $f$ times $\mathrm{d}y/\mathrm{d}t$, so $u - f y$ is constant along the trajectory. On an $f$-plane that is the statement that the circle’s centre stays put. Written differently, it says the parcel’s absolute momentum in the east–west direction — its own plus what the rotating frame supplies at its latitude — does not change.

That is the same bookkeeping an isolated system’s momentum obeys, applied in a frame that is itself moving. And it generalises. Allow $f$ to vary and the conserved combination becomes $u - \int f\,\mathrm{d}y$, which is the β-plane result the drifting figure showed. Allow the fluid layer to change thickness as well and the conserved quantity becomes potential vorticity, the ratio of absolute rotation to layer depth — the central invariant of the whole subject, from which Rossby waves, the westward intensification of ocean currents and the behaviour of a cyclone crossing a mountain range all follow.

So the closed circle is the simplest member of a family. It is what conservation of absolute momentum looks like when nothing else is happening, in the same way that a straight line is what conservation of ordinary momentum looks like. Adding a pressure gradient, a varying $f$ or a varying depth each turns the circle into something else, and each of those somethings is a recognisable feature of a real ocean or atmosphere.

The same motion, in the frame where there is no force #

The circle exists in the rotating frame. It is worth asking once what the same motion looks like from outside, because the answer settles what kind of object the Coriolis force is.

From outside, the parcel travels in a straight line at constant speed, indefinitely, because nothing is acting on it. The circle appears because the observer is turning underneath it. That is not a diminishment — a rotating frame is the only frame in which the Earth’s surface is at rest, and every measurement anybody makes of an ocean current is made in it — but it fixes what is being described.

It also explains why the period is what it is. In the inertial frame the parcel goes straight; the frame rotates once per sidereal day about the Earth’s axis; and the horizontal plane at latitude $\varphi$ rotates about its own vertical at $\Omega\sin\varphi$, which is the projection of the Earth’s rotation onto the local vertical. The inertial period is the time for that local rotation to complete a turn relative to the straight line, which is $2\pi/(2\Omega\sin\varphi)$ — the factor of two arriving because the velocity’s rotation rate in the moving frame is twice the frame’s own, a fact the algebra supplies and the geometry does not make obvious.

Where this stops being right #

The motion has been horizontal throughout. The full Coriolis force has a vertical component, $2\Omega\cos\varphi \times u_{\text{east}}$, which makes an eastward-moving body slightly lighter and a westward-moving one slightly heavier — the Eötvös effect, worth about 0.1 per cent of $g$ for a fast aircraft, and measurable in gravity surveys carried on moving ships. Dropping it is a good approximation for a thin fluid layer and not for a projectile.

The frame has been a plane tangent to a sphere. The circles drawn are small enough for that to hold, and the β-plane figure is the first correction to it. The next corrections involve the curvature of the surface itself and matter for trajectories spanning many degrees of latitude.

Friction has been ignored entirely. Real inertial oscillations decay, in days to weeks in the ocean and in hours in the atmospheric boundary layer, and what they decay into — internal waves, turbulence, or a mean flow — is not settled in detail.

And $f$ was treated as a constant of the place rather than of the parcel. A parcel that moves in latitude carries its own angular momentum, and the conserved quantity in a real flow is potential vorticity rather than $f$ — which is the whole basis of large-scale dynamics and is a subject this rung does not reach.

What the pictures cannot show #

Every figure here is drawn in the rotating frame, which is the frame in which the circles exist. In an inertial frame there is no circle at all: the parcel travels in a straight line at constant speed, and what is turning is the coordinate system. Neither picture is more true, and the drawing has no way to indicate which one it is in except by the axes being labelled east and north.

Nor can any of them show that nothing is pushing. The Coriolis term is a consequence of describing motion in a turning frame, and a figure that draws a curved path invites the reading that something curved it. What curved it is the paper.

Where this ladder goes next #

Three rungs stand on circular-motion. The first found that turning is an acceleration. The second wrote Newton’s law in a turning frame and produced three extra terms. This one integrates the smallest of them and finds that it closes.

The habit worth carrying away is about the difference between a rate and a trajectory. A statement of the form “X deflects Y” describes a rate of change and says nothing about where the motion ends up; integrating it is a separate step and frequently produces a different-shaped answer. Here a continuous deflection produces no displacement at all. The same distinction separates a drift from a diffusion, and a force from an impulse.

What is left on this ladder is the question of when any of this matters. An inertial circle exists whenever a parcel is left alone, and parcels are almost never left alone — there is a pressure gradient, and the balance between it and the Coriolis term is what a weather map is a picture of. Whether the rotating terms dominate that balance or are a correction to it is decided by a single dimensionless number, and it spans ten orders of magnitude between a teacup and an ocean.