Dimples on the Rhine

1A dimple is the top of a whirlpool

Water going round in a circle needs a push toward the centre, and the only thing inside water that can push is pressure. So the pressure must drop toward the middle of a whirlpool. At the surface the pressure is pinned by the air above, and the only way to lower it underneath is to have less water on top: the surface sags into a small funnel, a millimetre or two deep.

That funnel is a weak lens. It spreads the sunlight passing through its middle and gathers it toward the rim, so the riverbed shows a dark disk with a bright edge, a caustic. The sun isn't straight overhead, so the shadow lands a little to one side of its dimple. Seen from the boat, the simulation's first view, a dimple shows up mostly by how it bends the reflections of the sky and the far bank, with its shadow faint on the bed below; switch to From above to look straight down through clear shallows at the shadows themselves.

For a whirlpool of circulation Γ with a core of radius a, balancing pressure against swirl gives a dip η = −Γ² / 8π²g(r² + a²): about 1.5 mm at the centre for Γ = 60 cm²/s and a = 6 mm.

2Measuring spin: walk around a loop

How much does a whirlpool spin? Walk around a loop and add up how hard the water pushes you along the way. That total is the circulation, Γ.

Any loop that goes around the whirlpool's core gives nearly the same answer, however big or lopsided, and a loop that misses the core gives almost nothing. The spin is concentrated in the core: outside it the water swirls, but a tiny paddle wheel carried along there hardly turns. Add a second whirlpool spinning the other way, and a loop around both gives zero.

In symbols, with u the water's velocity and dl a small step along the loop, the circulation is Γ = ∮ u · dl. Take the loop to be a circle of radius r around a whirlpool: the water runs along it at a steady speed v, so the sum is just v times the length, Γ = v · 2πr. Turned around, that gives the swirl a whirlpool makes around itself:

the swirl of one whirlpoolv = Γ / 2πr

It falls off as one over the distance, and it is the seed of everything below. (The spin per unit area is called the vorticity, ω; the circulation is the total vorticity inside the loop, Γ = ∫ω dA. That is why only loops around the core pick it up.)

3Spin lives on lines

From above, a whirlpool looks like a point. But the water is three-dimensional, and what spins is a line, the whirlpool's axis, carrying on down into the water. Thicken the line into a tube: its circulation is the same at every depth. The dimple is just where the line meets the surface. A line (one dimension) and a plane (two) in space (three) meet in 1 + 2 − 3 = 0 dimensions, a point. That's why the surface shows whirlpools as points.

4A vortex line can't just stop

Helmholtz noticed in 1858 that vortex lines behave like the field lines of a magnet: they cannot end in the middle of the water. The argument takes a single loop. Suppose a tube of spin did stop somewhere. Put a loop around it: circulation Γ. Slide the loop along the tube and past its end. The circulation around a loop can only change when the loop sweeps across spin, and no spin crosses the side of a tube, so it is still Γ. But now the loop surrounds water that isn't spinning at all, and it can be shrunk to a point, where the circulation is plainly zero. So Γ = 0, a contradiction. (Press Ends in the water? above.)

In symbols, the spin is the curl of the velocity, ω = ∇ × u, and a curl never has sources or sinks: ∇ · ω = 0. It is the same rule, ∇ · B = 0, that keeps a magnet's field lines from ending.

A vortex line therefore has three options: close into a ring, like a smoke ring; end on the riverbed; or end on the surface. The last two are the ones we see.

5Why whirlpools come in pairs

Take an arch, a tube with both feet on the surface. Follow the spin along it, down one leg and up the other. The same spin looks counterclockwise from above at one foot and clockwise at the other. One arch, two dimples, opposite spins.

Kelvin found a second reason in 1869. In water without friction, the circulation around any loop that moves with the water never changes: dΓ/dt = 0. Draw a big loop in still water: its circulation is zero, and it stays zero while you stir inside it, as long as the loop never touches the paddle. So whatever spin a paddle makes, it makes equal amounts both ways; the arch is what gathers it into two dimples. Drag a paddle through a pool and you see exactly this: two dimples, spinning opposite ways, sailing off together. Each is carried along by the other's swirl, so the pair moves in a straight line, crosswise to the line joining them, and section 2 says how fast: at distance d, each feels the other's swirl v = Γ/2πd, so the pair travels at

V = Γ / 2πd

That is the whole law of motion for whirlpools: each one is carried by the swirl of all the others. It sounds too simple to be deep. Section 10 shows it is Hamilton's mechanics in disguise.

Drag across the water in the simulation to make a paddle stroke: it leaves a pair that sails off the way you pushed, and the total spin readout doesn't move. Two pairs travelling together take turns passing through each other (press Leapfrog).

6Where the Rhine's dimples come from

The dimples that started this page probably weren't made by paddles: there were hardly any boats about, and they showed up beside and ahead of the boat more than behind it. Most likely the river makes them itself.

Kelvin's rule has one loophole, friction, and the river's friction is at its bed. Water touching the bed is held still, while a hand's width above it the current is racing. That shear is spin: a paddle wheel lying on the bed would turn. So the bed is where the river's spin is born, as a sheet of vortex lines lying across the current. The sheet is unstable. It wrinkles into arches, hairpin vortices, whose heads lift into faster water and get dragged up and downstream.

When an arch reaches the surface, the surface cuts through it and its two legs become a pair of dimples with opposite spins. A leg can also keep its other end on the bed, and then you see a single dimple, which Helmholtz allows. The rising water spreads out at the surface as a smooth, glassy patch that river people call a boil, and dimples gather at its edges, where surface water converges and sinks. Bridge piers shed whirlpools from alternate sides, and the shear between the main current and slow water along a bank or behind a groyne makes rows of whirlpools that all spin the same way.

The pattern seen from the boat makes sense too. A dimple is only a millimetre or two deep, about as steep as a small wind ripple, so ripples hide it: you need smooth water. Behind the boat the propeller's wash churns the surface; ahead and beside it, the river's own dimples float on undisturbed water. With Boils on, the simulation brings up a pair now and then, the way the river does, and the current carries them off.

7Why they can live so long: flat flow

The surface does two things to a whirlpool. First, it stands it upright. Air is far too thin to drag on water, so right at the surface there is no sideways pull, and that forces vortex lines to meet the surface at right angles: sideways spin can't survive there, only upright whirlpools can. Second, water near the surface can't move up through it, so the flow there is mostly sideways. From here on we go one step further and use a model: the flow is exactly flat, two-dimensional, the same at every depth. Real surfaces aren't quite that simple. Where surface water converges and sinks, it stretches an upright whirlpool and spins it up, which is one way dimples form and strengthen.

That matters because of stretching. Turbulence in three dimensions runs on stretched vortex tubes: a stretched tube gets thinner and spins faster, like a skater pulling in their arms. (Its circulation can't change, so as its cross-section shrinks its spin must grow.) Stretch, spin up, break into smaller eddies, and within seconds friction eats the smallest ones. In two dimensions there is no third direction to stretch into, and each drop of water simply keeps its spin as it is carried along.

In symbols, the spin of a drop changes as Dω/Dt = (ω · ∇)u, where D/Dt means "following the drop" and the right side measures how the flow stretches the drop along its spin axis. In flat flow the spin points straight up and nothing changes in that direction, so the right side vanishes:

flat flowDω / Dt = 0

So in flat flow a whirlpool can only fade by friction slowly smearing out its core. The time that takes grows like the core size squared divided by the viscosity ν (0.01 cm²/s for water); for the smooth Lamb–Oseen core the width grows exactly as a² → a² + 4νt, and the dip shrinks roughly like 1/a². A dimple with a 6 mm core fades to a quarter of its depth in about half a minute; one with a 2 cm core takes about five minutes. In the real river, turbulence from below frays them sooner, and ripples hide them, so the long-lived ones are easiest to see on smooth water. Flat flow has another quirk: whirlpools spinning the same way tend to merge into bigger ones instead of shattering into small ones.

8Hamilton's rule, from scratch

To see what flatness buys, we need one idea from mechanics. Describe a pendulum by two numbers, its angle q and its momentum p. Each moment of its life is then a point in a plane, the pendulum's phase space, and as time runs the point moves. Hamilton saw that a single landscape tells it how to move: the energy H(q, p). The point walks along the contour lines of H, faster where they crowd together:

Hamilton's ruledq/dt = ∂H/∂p,    dp/dt = −∂H/∂q

Read it as a recipe: the speed across is the slope up the page, and the speed up the page is minus the slope across. In the figure the white dot's velocity is split into those two parts. Two consequences come for free. Walking along a contour keeps H fixed, so energy is conserved. And the flow never squeezes: a patch of starting points can be stretched thin, but its area stays exactly the same. (That is Liouville's theorem; the non-squeezing camel of episode 1 is its sharper cousin in higher dimensions.)

9The river surface is a phase space

Now the payoff, inside the flat model of section 7. There, water can't be squashed sideways, so the flow keeps areas, just as Hamilton's flow does. That is no coincidence. In two dimensions, any flow that keeps areas comes with a landscape of its own, the stream function ψ, and the water moves by exactly Hamilton's rule:

every drop on the surfacedx/dt = ∂ψ/∂y,    dy/dt = −∂ψ/∂x

Line it up with the pendulum: x plays the angle, y plays the momentum, ψ plays the energy. In the model, every drop on the surface moves like a point in phase space; the surface is its own phase space. (Press Two whirlpools in the figure above: same rule, new landscape.) For a single whirlpool the landscape is a funnel,

ψ = −(Γ/2π) log r

whose contours are circles and whose steepness is Γ/2πr: the swirl of section 2, back again. With one steady whirlpool, a speck of water follows a contour. When whirlpools move, the landscape changes beneath it: its velocity still runs along the current contours, but it needn't stay on one contour value. The same rule still sets its motion. Phase spaces come as position–momentum pairs, so they always have an even number of dimensions. A plane qualifies; the three-dimensional inside of a river doesn't. Conserving volume in three dimensions does not, by itself, give us this two-coordinate stream-function description. Toggle Stream function in the simulation (which uses this model) to see ψ for the whirlpools on the water, then drop the Camel dye: it is stretched into filaments and its outline grows longer and longer, but its area stays at 100%.

A real river is less tidy, and more interesting. A patch of real surface water doesn't have to keep its area: it spreads where water wells up from below and shrinks where water sinks away. (Incompressibility keeps volume; only in the flat model does it keep area.) That spreading and shrinking is exactly what researchers now read from dimples: in simulations by Babiker, Ellingsen and colleagues, the number of dimples in a region tracks how strongly its surface spreads and converges, and a 2026 laboratory tank experiment found the same link at much higher Reynolds numbers.

10The whirlpools obey it too

One step remains. By section 5, each whirlpool is carried by the swirl of the others, so whirlpool k moves by the others' stream function, taken where it sits (write z = (x, y) for a point on the water):

dx_k/dt = ∂ψ_others/∂y,   dy_k/dt = −∂ψ_others/∂x,   ψ_others = −Σ_j≠k (Γ_j/2π) log|z − z_j|

Kirchhoff noticed in 1876 that these N separate recipes are really one. Collect the whole system's energy, one term for each pair of whirlpools:

H = −(1/2π) Σ_{pairs j<k} Γ_jΓ_k log|z_j − z_k|

Differentiating H by y_k keeps only the terms that involve whirlpool k, and what is left is Γ_k times the others' ∂ψ/∂y. The same goes for x_k. So, multiplying through by Γ_k:

Kirchhoff's equationsΓ_k dx_k/dt = ∂H/∂y_k,    Γ_k dy_k/dt = −∂H/∂x_k

That is Hamilton's rule once more, with position x_k and momentum Γ_ky_k: each whirlpool's own spot on the river is a position–momentum pair. A single speck of water has the surface itself as its phase space; N whirlpools together have a 2N-dimensional one, N copies of the surface. The rule hands us conserved quantities, each from a symmetry: the energy H (walking along contours), the centre of spin ΣΓ_kz_k/ΣΓ_k, a Γ-weighted average like a centre of mass, which makes sense when the spins don't cancel (the river looks the same if everything slides), and the angular impulse ΣΓ_k|z_k|² (it looks the same if everything turns). The readout under the simulation tracks them. That is just enough to keep three whirlpools regular (press Three) and not enough for four (press Four): four can be chaotic, and in this example two starts a micron apart end up a centimetre apart within about six seconds. These all spin the same way, so their other ends must be on the riverbed; the Single whirlpool tool makes such bed-anchored whirlpools one at a time, and it is the only thing that changes the total spin. (The simulation smooths each whirlpool's core over a few millimetres, which keeps every formula here but tames the log at zero distance.)