Illustrated Physics
physics, drawn
Mechanics

The ratio that decides whether the planet is turning

Whether the rotating terms matter is not a question about size. It is one dimensionless ratio, U over fL, and it runs from ten thousand in a teacup to a millionth in the Earth's core. Where it is small the pressure gradient stops accelerating the fluid and starts balancing a force on fluid already moving across it — so the flow runs along the pressure contours instead of down them, and a weather map is a streamline plot.
14 min read 5 figures Approximations that lieThe same equation again

Assumes: The deflection that closes on itself · The forces that are not there

The rung below this one integrated the Coriolis term with nothing else acting and found a closed circle. Nothing in the ocean or the atmosphere is ever in that state, because there is always a pressure gradient. What that rung leaves open is when the rotating terms matter at all — and the answer is a single ratio, not a size.

Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 2.0e-4 metres per second squared pointing east, integrated for 36 hours at 60°, 30°, 10°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.
Fig. 1 A parcel released from rest under a steady eastward pressure-gradient acceleration, integrated for thirty-six hours at three latitudes. Without rotation it would accelerate east for ever. With it, the motion is an inertial circle about a mean velocity at right angles to the push — northward, for an eastward gradient — and the mean over whole inertial periods matches the geostrophic value to two per cent, measured off each path. Near the equator the loops are so slow that the parcel travels a long way down the gradient first.

The ratio, and what is in it

Compare the two accelerations in the equation. The advective one is of order U2/LU^2/L, where UU is a typical speed and LL a typical size. The Coriolis one is fUfU. Their ratio is

Ro=UfL.\text{Ro} = \frac{U}{fL}.

Above one the flow’s own accelerations dominate and the planet’s rotation is a small correction. Below one the Coriolis term is the larger and the flow is organised by it, in the way a stationary mountain wave is organised by a wind.

The expression has three things in it and none of them is a property of the fluid. Not its density, not its viscosity, not what it is made of. A rotating flow of treacle and a rotating flow of hydrogen at the same UU, ff and LL have the same Rossby number and the same answer to this question.

Whether the planet's turning matters, in one ratio. The Rossby number U/fL for 8 rotating flows, each computed from its own speed, size and latitude. Below one the Coriolis term dominates the acceleration and the flow is shaped by the Earth's rotation; above one it is a negligible correction. 4 of 8 sit below one, and the range across the set is 4.85·10⁻⁶ to 4.85·10⁴. A draining bath is at four thousand: the Coriolis force on the water is smaller than the asymmetry of the plug hole by orders of magnitude, and a bath that drains one way is doing so for a reason having nothing to do with the hemisphere. A mid-latitude cyclone is at a fifth, and cannot be anything other than what it is.
Fig. 2 The Rossby number for eight rotating flows, each computed from its own speed, size and latitude. Ten orders of magnitude separate a stirred teacup from flow in the Earth’s outer core, and the line at one is where the character of the dynamics changes. Everything above it spins for reasons of its own; everything below it spins because the planet does.

What the bath is not

The figure settles an old question, and it settles it by four orders of magnitude rather than by an argument.

A draining bath is thirty centimetres across with water moving at perhaps ten centimetres a second, so its Rossby number is about four thousand. The Coriolis acceleration on that water is four thousand times smaller than the accelerations it already has, which come from the shape of the basin, from the residual circulation left by filling it, and from the same three terms a turning frame always produces, from the asymmetry of the plug hole, and from whatever the last person to use it did.

That does not make the effect zero, and the honest statement is more interesting than the dismissal. The experiment can be made to work. Shapiro did it in Boston in 1962 and Trefethen in Sydney the following year: a symmetric circular tank two metres across, filled carefully, covered, and left to stand for eighteen to twenty-four hours so that the residual motion decays below the Coriolis signal. Drained slowly through a central hole, the vortex is counter-clockwise in the north and clockwise in the south, reliably.

The eighteen hours are the whole content of the result. They are how long it takes the initial circulation to decay to something below fUfU, which is to say how long it takes to reduce the effective Rossby number below one — and a bath, filled minutes earlier and drained in a minute, is nowhere near that. So the claim is not that the Coriolis force is absent from a bath. It is that it is present and beaten.

What happens when the ratio is small

Take the Rossby number to zero and the acceleration term drops out of the equation entirely. What is left is a balance between two forces:

f v×z^=−1ρ∇p.f\,\mathbf{v} \times \hat{z} = -\frac{1}{\rho}\nabla p.

The pressure gradient is not accelerating anything. It is balancing the Coriolis force on fluid that is already moving, and since the Coriolis force is perpendicular to the velocity, the velocity must be perpendicular to the pressure gradient.

So the flow runs along the isobars rather than across them, with low pressure on the left in the northern hemisphere. That is geostrophic balance, and it is the reason a weather map is legible: the contours drawn on it are not merely a scalar field, they are the streamlines of the wind, and their spacing is its speed.

The wind a pressure map already contains. The geostrophic wind against latitude, for pressure gradients of 2, 4, 8 hectopascals per hundred kilometres. In the limit of small Rossby number the pressure gradient does not accelerate the air down the gradient; it balances the Coriolis force on air already moving across it, so the wind blows along the isobars with the low pressure on its left in the northern hemisphere. The speed is then the gradient divided by density and by the Coriolis parameter — 32 metres a second for four hectopascals per hundred kilometres at 45 degrees, which is the calibration a forecaster carries. The curves run away below about fifteen degrees of latitude, and that divergence is the approximation announcing its own failure: near the equator there is no geostrophic wind, and tropical meteorology is a different subject for that reason alone.
Fig. 3 The geostrophic wind against latitude for three pressure gradients. Four hectopascals per hundred kilometres at 45 degrees is 32 metres a second, which is the calibration a forecaster carries. The curves run away below fifteen degrees of latitude because f vanishes at the equator — the approximation announcing its own failure in its own expression, which is the most useful thing an approximation can do.

The arithmetic is worth doing once. A gradient of four hectopascals per hundred kilometres is a perfectly ordinary mid-latitude situation. Dividing by the density and by ff gives 32 metres a second, which is a strong wind and is what such a chart shows. Double the gradient and the wind doubles; halve the latitude and it nearly doubles again.

Getting from one to the other

The hero figure is the transition, integrated rather than asserted, and it repays a second look.

At sixty degrees the inertial period is under fourteen hours, so within the thirty-six hours drawn the parcel completes nearly three loops. Its mean motion is almost exactly geostrophic — perpendicular to the push — and the loops are a small wobble on top of it. That is what the atmosphere does after a pressure system establishes itself: it adjusts within a day, and the adjustment radiates away as inertia–gravity waves — the same family as the internal waves a stratified fluid carries, with rotation putting a floor under their frequency.

At ten degrees the inertial period is nearly three days, so within thirty-six hours the parcel has completed less than half a loop. It is still mostly accelerating down the gradient, and it has travelled a long way doing so. Geostrophy has not had time to establish itself, and the geostrophic speed the balance would imply is enormous.

The two panels are the same equation with one number changed, and they are the difference between mid-latitude and tropical meteorology. Every forecasting method that starts from a pressure field and infers a wind is using the first; none of them works in the second, where the wind has to be measured rather than inferred and where the dominant balances involve convection and the divergence field rather than rotation, and whether a column overturns at all is the question that replaces this one.

Pushed one way, travelling another. A parcel released from rest under a steady pressure-gradient acceleration of 5.0e-4 metres per second squared pointing east, integrated for 60 hours at 45°, 20°. Without rotation it would accelerate east indefinitely. With it, the motion is an inertial circle about a mean velocity at right angles to the push, and the mean over a whole number of inertial periods is measured off each path and matches the geostrophic value a/f to two per cent. At high latitude the loops are tight and the drift is almost purely along the isobars; near the equator the parcel travels a long way down the gradient before the rotation has had time to turn it, which is why geostrophic balance is a high-latitude statement and the tropics need a different set of approximations.
Fig. 4 A stronger gradient over a longer time, at two latitudes. The geostrophic speed is now two and a half times larger, so the mean drift is faster and the loops are wider, but the structure is unchanged: an oscillation at the inertial period about a mean flow at right angles to the forcing. Changing the strength of the push changes the size of the answer and nothing about its direction.

The size the ratio implies

The Rossby number contains a length, and that invites a question it does not answer: what sets LL? For a laboratory flow the apparatus does. For an atmosphere or an ocean, nothing external does, and the length has to come from the physics.

It does, and the answer is one of the more satisfying numbers in the subject. A disturbance in a rotating stratified fluid spreads by gravity waves at speed cc and is turned by rotation at rate ff, so it spreads a distance c/fc/f before rotation takes over. That length is the Rossby radius of deformation, and it is the natural size of everything the balance describes.

For the atmosphere, cc is the speed of a long internal gravity wave, of order thirty metres a second, and ff at mid-latitude is 10−410^{-4}: the deformation radius is three hundred kilometres, and the systems on a weather chart are a few times that. For the ocean the stratification is much weaker and cc is about two metres a second, so the radius is twenty to thirty kilometres — and ocean eddies are a few times that, which is why they are a hundred times smaller than atmospheric ones and why satellites rather than ships were needed to find them.

The same arithmetic run at the equator, where ff vanishes and the relevant scale becomes c/β\sqrt{c/\beta}, gives about fifteen hundred kilometres in the ocean — the width of the equatorial waveguide, and the reason El Niño’s signal is a basin-spanning object rather than a local one.

So the length in the Rossby number is not arbitrary after all. Left to itself, a rotating stratified fluid produces disturbances at the deformation radius, and evaluating the ratio there gives a Rossby number of order the flow’s speed over cc — small, for both fluids, which is why both are geostrophic. The scale and the regime select each other.

The teacup, which is not a counterexample

The teacup sits at the far end of the Rossby figure and it does something rotational anyway, which is worth explaining because the mechanism is the one that damps every flow in this essay.

Stir a cup and the tea leaves gather in a heap at the centre. That is the wrong way round for the obvious argument — a centrifuge throws heavy things outward — and Einstein wrote the explanation in 1926.

The rotating liquid has a pressure gradient balancing its own centrifugal term, higher at the rim. At the bottom, friction slows the liquid in a thin layer, so the centrifugal term there is weaker while the pressure gradient, being set by the fluid above, is not. The layer is therefore pushed inward, drags the leaves with it, and returns upward at the centre. What is drawn is a secondary circulation driven by a boundary layer failing to balance, and a boundary layer whose thickness is set by a frequency is exactly the object involved.

The same circulation with the Earth’s rotation supplying ff instead of the stirring is the Ekman layer, and it does the same thing to an ocean: the surface layer is pushed to the right of the wind, the interior is pumped up or down where that transport converges, and the whole large-scale ocean circulation is driven by it. A teacup and an ocean gyre are the same secondary flow, at Rossby numbers ten orders of magnitude apart, which is what makes the ratio a classification rather than a threshold.

What the balance leaves out, and why that is where the weather is

Geostrophic balance is exact only in the limit, and a flow exactly in it does nothing: it is steady, it neither rises nor falls, and no weather happens. Everything interesting is in the departure.

The first correction is the gradient wind, which restores the centripetal acceleration around a curved isobar. It matters where the curvature is tight, and it produces an asymmetry worth knowing: around a low, the centripetal acceleration adds to the Coriolis force and the wind is slower than geostrophic; around a high, it opposes it and the wind is faster. There is also a limit on how deep a small high can be, because past a certain gradient no balanced solution exists at all — which is why intense highs are always large and intense lows can be small — an asymmetry with the same origin as the one a rotating frame’s extra terms produce.

The second correction is friction, which near the surface turns the wind across the isobars toward low pressure by twenty or thirty degrees over land. That cross-isobar component is what fills a low in, and it is why weather systems decay rather than persisting indefinitely.

The third is the ageostrophic circulation, which is the small part of the flow that is not balanced and is responsible for essentially all vertical motion — and therefore for all cloud and all rain. The balanced flow is a hundred times larger and produces no weather. Numerical weather prediction spends most of its effort on the residual.

The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.4 m/s at 60°, 0.4 m/s at 20°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 3.17 km, 8.02 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all.
Fig. 5 The free circles at two of the latitudes above, at the same speed. What the trajectory figures draw as a wobble on a geostrophic drift is exactly this motion superimposed on it — and the wobble’s size, U/f, is why the loops are tight at high latitude and enormous near the equator. Rotation supplies both the balance and the oscillation about it, out of the same term.

The other ratios that go with it

One dimensionless number rarely settles a problem on its own, and the rotating case needs three. Naming them together is how the regimes get sorted.

The Ekman number, ν/(fL2)\nu/(fL^2), compares viscosity with rotation. It is minute in every geophysical flow — around 10−1210^{-12} for an ocean basin — which is why friction appears only in thin layers at the boundaries and never in the interior. Its square root sets the thickness of those layers: the Ekman depth in the ocean is tens of metres, in the atmosphere a kilometre, and in a stirred teacup a fraction of a millimetre.

The Burger number, the square of the deformation radius over the flow’s own size, compares stratification with rotation. Near one, the flow is at the scale the physics prefers and both effects matter equally, which is the usual case for weather systems and ocean eddies alike. Far from one, one of the two is doing all the work.

And the Rossby number itself has two versions that are worth keeping apart. The one used here is the advective Rossby number, U/fLU/fL, comparing steady accelerations with rotation. There is also a temporal one, 1/fT1/fT, comparing how fast the flow changes with how fast the frame turns — and it is the temporal number that decides whether an inertial oscillation is excited. A pressure system that establishes itself over a week has a small temporal Rossby number and provokes almost no oscillation; a squall lasting an hour has a large one and leaves the ocean ringing for a fortnight.

That last distinction resolves an apparent tension between this rung and the one below. The rung below’s free circles look like a large-Rossby-number phenomenon and they happen in exactly the flows this rung calls geostrophic, because the two numbers being small and large refer to different comparisons. A flow can be perfectly balanced in its mean and vigorously oscillating about it, and most of the real ocean is.

Where this stops being right

The Rossby number is a scale estimate, not a boundary. Calling a flow geostrophic at Ro = 0.1 means the departures are of order ten per cent, and ten per cent of a thirty-metre-per-second wind is three metres a second — which is more than enough to matter. The number says which term is largest; it does not say the others are absent.

One flow has many Rossby numbers. A hurricane’s eyewall is at fifty and its outer circulation at a tenth, in the same storm at the same moment, which is why the eyewall is a different dynamical object from the rest of it.

The balance was written for a shallow layer. A deep rotating fluid is subject to the Taylor–Proudman theorem, which says slow steady motion cannot vary along the rotation axis at all — a constraint far stronger than geostrophy and the reason flow in the Earth’s core is organised into columns.

And the pressure gradient was taken as given. In a real fluid the pressure field is produced by the flow, so the balance is a statement of consistency rather than a calculation — which is why diagnosing a wind from a chart works and predicting the chart does not.

What the pictures cannot show

The trajectory figures draw a parcel, and there is no parcel. What is being integrated is the velocity a fluid element would have if the pressure gradient were externally imposed and unchanged, which is a fiction: real gradients are made by the fluid and adjust as it moves. The picture is a picture of a term in an equation rather than of anything that happens.

Nor can the Rossby number figure show that its rows are distributions. A hurricane has no single speed and no single length, and the point plotted is a representative choice. Where two rows are within an order of magnitude of one another, that ordering should not be trusted.

Where the ladder stands

Four rungs stand on circular-motion. Turning is an acceleration; a turning frame produces three extra terms; the smallest of them integrates to a closed circle; and whether any of it matters is one ratio.

The habit worth carrying away is what a dimensionless number is for. A ratio of two terms in one equation says which of them is being neglected and by how much, which is a stronger statement than any argument about whether an effect is important. The Rossby number does that here; the Reynolds number does it for viscosity, the Mach number for compressibility, and the Péclet number for diffusion against advection. In each case the useful move is the same: write both terms, divide, and read off the regime rather than debating it.

What is left on this ladder is the constraint one level up. Geostrophic balance says the flow follows the contours; it says nothing about what sets the contours, and the answer involves a quantity that is conserved as a parcel moves — its absolute rotation divided by the depth of the layer carrying it. That quantity is what makes a westward-moving disturbance in a rotating fluid behave differently from an eastward one, and it is the beginning of a subject this ladder has only been supplying the terms for.

Part 4 of 4

This essay is one argument about Circular motion. The others:

The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all. The deflection that closes on itself Part 3 — The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that.

Where this goes next

The deflection is a circle, not a bend. Trajectories integrated from Newton's law in a rotating frame with the Coriolis term and nothing else — no pressure gradient, no friction, no force of any kind. 0.5 m/s at 45°, 0.3 m/s at 30°. Each path closes on itself after one inertial period, checked to a millionth of its own radius, and the radius is the speed divided by the Coriolis parameter: 4.85 km, 4.11 km. The deflection usually described as a curving of the path is a complete circle, traversed clockwise in the northern hemisphere in half a pendulum day, and a body left alone in a rotating frame goes nowhere at all. The deflection that closes on itself The Coriolis term is usually described as bending a path to the right. Integrated rather than described, it does not bend the path — it closes it. A body left alone in a rotating frame travels a circle of radius U/f and comes back to where it started in half a pendulum day, having gone nowhere at all, and drifting buoys in every ocean draw exactly that. One puck, two frames. A puck slides outward from the centre of a turntable at 1.4 m/s across a table of radius 1 m, which turns through 0.55 of a revolution in the 0.71 seconds the crossing takes. On the left, in the room: a straight line, because no force acts along it. On the right, on the turntable: a spiral, because the coordinates turned underneath it. The two panels are the same numbers with the same clock — the second is the first with the angle rotated by −Ωt — so whatever is bending the path on the right is arithmetic, and it is still worth a name, because on a turning planet the right-hand panel is the one everybody lives in. The forces that are not there Writing Newton's law in a frame that is turning produces three extra terms. Nothing was added to the world to make them appear and nothing is removed by calling them fictitious — one of them flattens the planet, one of them turns the weather, and both are computable to four figures. The pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream. The wave that is required to stand still A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

What links here

Essays that reach for this one mid-argument — the half of a link its own author cannot write down.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

ApproximationCoriolis forceDimensionless numberEquilibriumFictitious forceGeostrophic balanceInertial oscillationLatitudePressure gradientRossby numberRotating frameScaling