The wave that can only travel west

The ratio that decides whether the planet is turning ends with a balance and a gap. Where the Rossby number is small, the pressure gradient on a rotating planet stops accelerating the fluid and steers it instead: the wind blows along the isobars rather than across them. That geostrophic balance says which way the flow goes once the pressure field is given. It says nothing about how the pressure field changes, and so nothing about how weather moves.

The missing piece is a conservation law, and it concerns spin. A column of fluid in a thin layer — the troposphere, the ocean above its thermocline — spins about the vertical at two rates added together: the planet’s own spin about the local vertical, the Coriolis parameter $f = 2\Omega\sin\varphi$, which is largest at the poles and zero at the equator; and the column’s spin relative to the ground, the relative vorticity $\zeta$. In the absence of friction the sum does not change, except in proportion to the column’s depth:

$$q = \frac{f + \zeta}{h}.$$

This is Kelvin’s circulation theorem applied to a column: a column squashed to half its depth is twice as wide, and a spinning body that widens must spin more slowly, exactly as a skater’s arms trade spin for reach. Carl-Gustaf Rossby used the simplest form of it, with the depth fixed, to explain in 1939 why the great waves in the upper westerlies move as they do, and Hans Ertel gave the general form in 1942. It is called the potential vorticity, and dynamical meteorology and oceanography are, to a large extent, the study of what keeping it forces.

The spin a column must take on #

The two panels are the two ways of changing a column’s potential vorticity from outside, and both are large. Weather systems spin relative to the ground at around $10^{-5}$ per second. Carrying a column a thousand kilometres north at mid-latitudes, or changing its depth by a fifth, demands a relative spin of the same size. Neither is a small correction to anything; each is enough to make a cyclone or an anticyclone out of nothing.

The left-hand effect is the one that matters most, because it needs no mountains and no convergence — only motion across latitudes. The rate at which $f$ changes northward, $\beta = \mathrm{d}f/\mathrm{d}y = 2\Omega\cos\varphi/R$, is $1.62\times10^{-11}$ per metre per second at 45°, and it is the whole of what distinguishes a spherical rotating planet from a rotating flat table, for motions smaller than the planet. The deflection that closes on itself found that a body coasting on a rotating flat table goes round a closed circle and that a small $\beta$ makes the circle drift. Applied to a fluid, the same $\beta$ does something more consequential: it makes waves.

The right-hand effect matters where the layer’s depth changes — over mountains, over the continental slope, and wherever the layer’s own thickness is pushed up and down by the flow. Air crossing the Rockies from the west is squashed on the way up, turns clockwise, and is stretched on the way down the lee side, turns anticlockwise; that is why a trough of low pressure forms reliably to the east of the range, and why depressions are so often born there.

Why the meander has to move west #

The drawing is the argument, with no equation in it that has not already been stated. Start with a line of fluid columns along a circle of latitude, at rest relative to the ground, and push them into a meander: some north, some south. Each column pushed north now sits where the planet spins faster about the vertical, so to keep its total it must spin clockwise relative to the ground. Each column pushed south spins anticlockwise.

A clockwise-spinning column pushes fluid southward on its east side and northward on its west side, the way a turning wheel carries the air around it. An anticlockwise column does the reverse. Add up the effect of all the spinning columns along the meander and the result is a north–south flow that is northward just west of each crest and southward just east of it — exactly a quarter wavelength out of step with the displacement. The columns west of each crest are carried north, and become the new crest. The columns east of it are carried south, away from the old one. The whole pattern moves west.

Nothing in that argument depends on which way the columns were pushed. Reverse every displacement and every spin reverses, every induced flow reverses, and the troughs move west instead of the crests — which is the same thing. The wave has one direction of travel, set by the sign of $\beta$, and on Earth $\beta$ is positive in both hemispheres because $f$ always increases towards the pole from the equator’s zero, whatever its sign. A Rossby wave’s crests move west in both hemispheres.

Writing the argument as an equation gives the speed. For a meander of wavenumber $k$ with no structure north–south, $\zeta$ is $-\beta$ times the displacement, the induced flow is $\zeta$ integrated along the line, and the pattern moves at

$$c = -\frac{\beta}{k^2}.$$

For the meander in the drawing that is 3.69 metres per second westward. It is a wave with no restoring force in the ordinary sense. What restores the columns is the conservation law, operating through the gradient of the planet’s spin with latitude — a background gradient doing for vorticity what a stratified column’s density gradient does for buoyancy.

Long waves go west fastest #

The speed depends on the wavelength, and steeply: doubling the wavelength quadruples the speed. The waves are therefore dispersive in the sense of the speed that depends on the length, and a disturbance made of many wavelengths spreads, its long components running ahead westward. Their energy, which travels at the group velocity rather than the phase velocity, goes east for short waves and west for long ones; the drawing shows only the crests.

In the atmosphere the waves live in a westerly wind, which carries every wave east at the wind’s speed while the wave moves west through the air at its own speed. Short waves are carried east: they are the ordinary travelling weather systems, the troughs and ridges that cross a mid-latitude country every few days. Long waves move west against the wind. In between is one wavelength that does neither, whose westward speed through the air exactly matches the wind’s eastward speed, and which stands still over the ground.

For a 15 metre-per-second westerly at 45° that wavelength is about 6,000 kilometres, and a latitude circle there holds between four and five of them. The jet stream does carry four to six great meanders around each hemisphere, and they are persistent in a way the smaller weather systems are not: a ridge over the western mountains of North America and a trough over its east coast, for example, show up in the average of any winter. They are stationary Rossby waves, forced by the mountains and by the contrast of land and sea and held still by the wind — the planetary cousin of the lee wave that is required to stand still behind a ridge, where the wind picks out the one wavelength that keeps pace with it. When one of these patterns locks in place for weeks — a blocking pattern — the result is a heatwave or a cold spell that refuses to move.

How long an ocean takes to hear about the wind #

The ocean has Rossby waves too, and the ones that matter most for climate live on the thermocline — the sharp boundary between the warm upper ocean and the cold water beneath. Here the right-hand panel of the first drawing does the work alongside the left: a column moved north also has its upper layer thickened or thinned, and the stretching contributes as much as the change in $f$. The long waves then move west at $\beta c_0^2/f^2$, where $c_0$, two or three metres per second, is the speed of the thermocline’s own gravity waves.

The factor $1/f^2$ makes these waves extraordinarily slow away from the equator. Near the equator the ocean hears about a change in the winds within a year or two, because the adjustment is carried across the basin by fast waves trapped along the equator, where $f$ changes sign and the equator becomes a channel with no walls. At 30 degrees the crossing takes about a decade, at 50 degrees several decades. A change in the wind over the eastern North Pacific is still arriving at Japan years later, carried as a slow undulation of the thermocline a few tens of centimetres high at the surface. This is one of the reasons the ocean gives the climate a memory measured in decades.

Satellite altimeters have measured these waves since the 1990s, as sea-surface bumps a few centimetres high and hundreds of kilometres across, marching steadily west. Their measured speeds agree with the theory at low latitudes and run up to about twice as fast as the simplest theory at mid-latitudes — a discrepancy attributed to the mean currents and the sloping thermocline the simple theory ignores, and a good reminder that the formula in the drawing is a first estimate and not a prediction.

The return flow crowds against the western wall #

The same conservation law explains the most conspicuous asymmetry in the world’s oceans. The Gulf Stream, the Kuroshio, the Agulhas and the Brazil Current are all fast, narrow currents on the western sides of their ocean basins; the eastern sides have broad, slow, sluggish flows. The winds that drive the gyres are not that asymmetric. In 1948 Henry Stommel solved the simplest model that could show why, a square ocean driven by westerlies in the north and trade winds in the south, with a little friction on the bottom, and found that the answer was $\beta$.

The wind puts clockwise spin into the subtropical ocean. In the interior, the only way a column can shed that spin is to move south, towards smaller $f$, where the conservation law lets it hold less; so the whole interior drifts slowly south. The water has to come back north somewhere, and a column moving north must gain anticlockwise spin relative to the ground, which the wind is not supplying. Only friction against a coast can supply it — and friction against a coast supplies anticlockwise spin to a northward current only if the coast is on its western side, with the fast water shearing past the still water at the wall. So the return flow is forced against the western boundary, into a current narrow and fast enough for friction to balance the spin budget. With $\beta$ set to zero the argument has nothing to act on, and Stommel’s gyre is symmetric.

The drawing shows the two cases side by side. The same winds and the same friction give a symmetric gyre when $f$ is constant and a crowded western current when it is not. Real western boundary currents are narrower still, about a hundred kilometres against an ocean five thousand wide, and inertia rather than bottom friction sets their final width; but the side they are on is Stommel’s result, and it is the conservation of potential vorticity that puts them there.

The same wave on a sloping sea floor #

The right-hand panel of the first drawing says that changing a column’s depth changes its spin just as moving it across latitudes does. So a sloping bottom should make Rossby waves on its own, with no $\beta$ at all, and it does. A column carried up a slope into shallower water is squashed and must turn clockwise; carried down, it is stretched and turns anticlockwise. Replace “north” in the meander argument by “towards shallow water” and every step goes through unchanged: the induced flow is a quarter wavelength out of step with the displacement, and the pattern moves along the depth contours in one direction only — with the shallow water on its right in the northern hemisphere, and on its left in the southern.

These topographic Rossby waves are how a continental shelf carries disturbances along a coast. Storms that push water across the shelf off one stretch of coastline launch shelf waves that travel along it, days later raising and lowering the sea level hundreds of kilometres away, always in the direction that keeps the coast on the wave’s right north of the equator. Tide gauges along the coasts of Australia and the eastern United States record them as slow sea-level oscillations of a few tens of centimetres, travelling at a few metres per second in exactly the predicted direction.

The same equivalence is how Rossby waves are studied in a laboratory. A tank of water on a turntable has no $\beta$, since the rotation rate is the same everywhere; give the tank a sloping bottom, or let the water’s free surface take its parabolic shape in rotation, and the depth varies across the tank as $f$ varies across latitudes. Seen from a camera rotating with the table — the frame in which the forces that are not there act — dye streaks in such a tank meander and drift in one direction round it, and the drift speed follows the dispersion relation with the tank’s slope in place of the planet’s curvature. The laboratory tank is a planet whose latitude is measured with a ruler.

Where the thin-layer picture stops #

A thin layer. Potential vorticity in this form belongs to a layer much wider than it is deep, with the motion nearly horizontal. It fails in a thunderstorm, whose updrafts are as tall as they are wide, and in the deep convection that sinks water in the Labrador Sea.

No friction and no heating. Friction near the ground and latent heat released in clouds both change a column’s potential vorticity. The Rossby waves of the drawings are free waves in an inviscid layer; real ones are forced, damped and, in the case of blocking patterns, strongly nonlinear.

No north–south structure. The westward speed $\beta/k^2$ is for a wave infinitely long north to south. A wave of finite width has $k^2 + l^2$ in the denominator and moves more slowly; the stationary wavelengths in the drawing are therefore upper bounds, and the counts of four to six meanders round a hemisphere are consistent with them rather than derived from them.

A flat β-plane. Treating $\beta$ as constant is fair over a few thousand kilometres. Over a whole hemisphere the waves are spherical harmonics, and the planetary-scale ones feel the sphere’s curvature directly.

What the pictures cannot show #

The drawings are all in plan or on a graph. What they cannot show is that the waves are three-dimensional: an atmospheric Rossby wave extends up through the troposphere and, if long enough, into the stratosphere, where waves forced by mountains break in winter and can reverse the polar vortex within days — a sudden stratospheric warming, which is felt at the ground weeks later as a cold spell. The vertical propagation of Rossby waves is a subject of its own, and the plan view shows none of it.

Nor do they show the waves as they are usually seen — on a weather map, as the undulations of the jet stream, or on a satellite map of sea-surface height, as a slow westward drift of bumps. Recognising the drawings in those maps takes practice, because every real map contains many wavelengths at once, each moving at its own speed.

Still open: when a jet stream locks in place #

Blocking — a large meander that stops moving for a week or more, diverting the storms around it and holding a heatwave or a cold spell in place — is one of the most consequential features of mid-latitude weather and one of the least well forecast. The stationary Rossby wave explains why a wave of the right length can stand still; it does not explain why a block forms at a particular time, why it persists far longer than the linear theory allows, or why it ends. Theories treat blocks as nonlinear Rossby-wave packets that have broken, as states in which the jet’s own potential-vorticity gradient traps the wave, or as a kind of traffic jam in the flow of wave activity along the jet. Weather models still under-predict how often blocks occur, and whether a warming climate will make them more frequent, longer or neither is disputed, because the answer depends on exactly the parts of the dynamics the theories disagree about.

The habit worth carrying away is to ask what a moving parcel keeps before asking what pushes it. On a rotating planet a column keeps its total spin over its depth, and the gradient of the planet’s own spin with latitude turns that bookkeeping into a restoring force with a direction — which is why weather’s largest waves go one way only, why an ocean remembers its winds for decades, and why the fastest currents in every ocean basin are on its western side.