The Earth bends every river a little

The bath does not know the hemisphere finds that the Coriolis term in a draining bath is present, computable and hundreds of times smaller than the motion a hand left in the water an hour before, and the bath that was only ever a wait finds what it takes to see the planet’s effect anyway. Both settle a claim about the planet’s rotation by comparing it with something else in the same flow.

Rivers are the other famous claim. In 1860 the naturalist Karl Ernst von Baer, surveying the rivers of Russia, argued that those flowing north or south had steep, eroding right banks and low left ones, and that the Earth’s rotation was the cause. The claim — Baer’s law — was argued over for decades, and in 1926 Albert Einstein wrote a short paper on it, explaining the helical flow in river bends by the same reasoning that explains why tea leaves gather at the centre of a stirred cup, and adding that the Earth’s rotation should drive a weak helix of the same kind. This essay computes that helix and sets it beside the one a river’s bends already drive.

A tilt of a millimetre #

The straightforward effect is a tilt of the water surface. A river flowing at speed $V$ in the northern hemisphere is pushed to the right by a Coriolis acceleration $fV$, where $f = 2\Omega\sin\phi$ is the Coriolis parameter at latitude $\phi$. Nothing across the river can supply a force to balance it except a slope of the surface, so the water stands higher on the right bank by $fVB/g$ across a width $B$.

The number is small. For a gravel-bed river forty metres wide running at 1.5 metres a second at 50°N, the right bank’s water stands 0.68 millimetres higher than the left’s. A bend of 120 metres’ radius in the same river tilts the surface by 77 millimetres, a hundred times more. For the middle Volga, a kilometre wide and running slowly at 0.8 metres a second, rotation tilts the surface by ten millimetres and a typical bend by twenty-two: the two are comparable. A tilt is not erosion, but it is the first sign that the comparison depends on the river.

A helix, from a force that varies with depth #

The tilt balances the Coriolis force on the depth-averaged flow. It cannot balance it at every depth, because the water near the surface flows faster than the water near the bed, and the Coriolis force is proportional to the local speed. Near the surface the force exceeds the uniform push-back of the tilted surface, and the water is driven right; near the bed the push-back wins, and the water is driven left. The two together form a transverse circulation — a helix superimposed on the downstream flow.

The current that runs across the river solves exactly this problem for a bend, where the depth-varying force is the centripetal acceleration $u^2/R$ in place of $fu$. The same model is used here: a logarithmic velocity profile, the eddy viscosity the bed shear implies, no slip at the bed, no stress at the surface, and the surface tilt set so that no net water crosses the river. With the Coriolis force in place of the bend’s, the circulation closes to two parts in $10^{17}$ of its own scale, and it runs right at the surface and left at the bed.

Rotation is a bend of twenty kilometres #

The comparison with a bend is more than an analogy. The force that drives a bend’s helix is the part of $u^2/R$ that differs from its depth average; to first order in the velocity’s variation with depth, $u^2 - V^2 \approx 2V(u - V)$, so the bend’s driving force is $(2V/R)(u - V)$. The Earth’s is $f(u - V)$. They are the same force when $R = 2V/f$. A straight river on the rotating Earth circulates exactly as a non-rotating river would round a bend of radius $2V/f$, turning to the left in the northern hemisphere so that the right bank is the outer one.

For a river running at 1.2 metres a second at 50°N that radius is 21.5 kilometres. For a faster river it is larger, for a slower one smaller, and at the equator, where $f$ vanishes, it is infinite; south of the equator the equivalent bend turns right and the left bank is the outer one.

The equivalence is checked by solving both: the rotating straight river and the non-rotating bend of radius $2V/f$. Their circulations agree in shape and strength to 6.8 per cent, the size of the neglected term $(u - V)^2$ for a logarithmic profile, and they agree at every latitude from ten to seventy degrees. The rotating river’s circulation is exactly linear in $f$ — doubling the Coriolis parameter doubles it to the last digit — because its driving force is.

How strong the Earth’s helix is, in a river’s own units #

The circulation the model computes is a velocity profile across the depth, and its size is worth stating in the units a river engineer would use. In the straight river of the figure — four metres deep, running at 1.2 metres a second, with a bed roughness that gives a Chézy coefficient of 45 — the Earth’s rotation drives a transverse flow whose largest value, at the surface, is 0.09 per cent of the downstream speed: a little over a millimetre a second. A bend of 120 metres’ radius in the same river drives one of fifteen per cent, eighteen centimetres a second.

The model behind both numbers is the same four lines. The downstream profile is logarithmic, fixed by the bed friction the Chézy coefficient implies; the eddy viscosity rises from the bed and falls to the surface as the mixing length does; the transverse force is whatever part of the Coriolis or centripetal acceleration differs from its depth average; and a single constant, the surface tilt’s correction, is chosen so that no net water crosses the section. Nothing in it is tuned to rivers in particular, which is why it can compare two driving forces on equal terms.

Why the tilt and the helix give different answers #

The two effects of rotation have different shares, and the difference is a factor of two that the equivalence explains. The tilt balances the whole transverse acceleration — $fV$ for rotation, $V^2/R$ for a bend — so the tilt from rotation is $fR/V$ of the bend’s. The helix responds only to how that acceleration varies with depth, and for the bend that variation is $2V(u - V)/R$ where for rotation it is $f(u - V)$: the bend’s is doubled because its force goes as the square of the speed. So the helix from rotation is only $fR/2V$ of the bend’s, half the tilt’s share.

On the middle Volga the Earth tilts the surface by 45 per cent as much as a typical bend does, but drives only 22 per cent as much helix; on a gravel-bed river the figures are 0.9 and 0.45 per cent. It is the helix, not the tilt, that moves bed sediment and attacks banks, so the smaller number is the one Baer’s law turns on.

River by river #

The equivalence turns Baer’s question into a comparison of two radii. A river’s helix is driven by its bends, of radius $R$, and by the Earth, a bend of radius $2V/f$; the Earth’s share is $fR/2V$. That is half the inverse of the Rossby number $V/fR$ — the ratio of inertia to rotation that decides whether a flow notices the planet at all, and the same number that, set against the height of a hill on the sea floor, decides whether an ocean current climbs over it or circles it.

For a mountain stream, fast and tightly curved, the share is 0.07 per cent. For a gravel-bed river, 0.45. For a lowland river 150 metres wide, 2.9. For the lower Mississippi, 7.7, because its bends are wide and its latitude moderate. For the middle Volga, Baer’s own river, 22 per cent, and for the lower Ob, wide, slow and at 63°N, 39. For the Amazon near Manaus, three degrees south of the equator, minus two: the equator removes the effect and turns it round.

The map puts the whole question on two axes. What matters is not the size of the river as such but the time the water takes to go round a bend, $R/V$, set against the planet’s time, $1/f$ — about three hours at mid-latitudes. A mountain stream goes round a bend in ten seconds and never feels the planet. The lower Ob takes nearly two hours, a good fraction of the planet’s time, and feels it strongly. Baer drew his conclusion from exactly the rivers where it has some chance of being true.

Bends alternate; the Earth does not #

A share of a fifth still sounds too small to decide which bank a river erodes, and here Einstein’s point matters.

A meandering river’s bends turn alternately left and right, so the helix they drive throws surface water alternately against the right bank and the left, and averaged over a meander wavelength it throws it against neither. The Earth’s contribution does not alternate. Along the whole river, in every bend and every straight, it adds a steady push towards the right bank. After the bends have cancelled, the rotation is all that is left. Over the many centuries a river takes to migrate across its floodplain, a steady bias of a fifth of the bend-driven attack can plausibly show in which banks end up steep.

That is the version of Baer’s law the arithmetic supports: not that the Coriolis force erodes right banks directly, but that it biases the erosion a river’s own bends do, by a fraction that is negligible for most rivers and not negligible for a few. Whether the Volga’s high right bank owes anything to it is a different question, because the same bank also has a geological history — uplift, differences in the rock, the retreat of ice — that can produce the same asymmetry by itself, and the arithmetic cannot separate them.

The hemisphere gives the one test the arithmetic suggests. Everything about a river’s helix except the Earth’s contribution is indifferent to the equator, while the Earth’s contribution reverses there. A river at 60°S has the same share as one at 60°N with the opposite sign, so if a bias in bank steepness exists on the great northern rivers because of rotation, the comparably large and slow rivers of the far south should show the mirror image. There are few of them — most of the southern hemisphere’s land at high latitude is Patagonia and Antarctica — which is one reason the question has stayed open for a century and a half: the natural control experiment is mostly ocean.

The same arithmetic in a teacup and an ocean #

Einstein’s paper was titled after both halves of the argument, meanders and Baer’s law, and its first half was the teacup. Stir a cup of tea and the leaves gather at the centre of the bottom, because the rotating tea near the bottom is slowed by friction, the inward pressure gradient that holds the faster tea above in its circle then drives the slower bottom water inward, and the leaves go with it. That is the bend’s helix in a cup, and how long a fluid takes to forget it was not rotating is the same secondary flow doing the work of spinning the cup’s contents up or down.

The ocean is the other end of the scale. There the Rossby number is small, the Earth’s rotation dominates, and the depth-varying Coriolis force in a friction layer produces the Ekman spiral, in which the transport turns a full right angle from the wind. A river lives between the teacup and the ocean, and which end it resembles is decided by the single number the map above plots.

Where the same bias is well established #

In one kind of channel the rotational helix is not a subtle bias but a measured, first-order part of the flow: the wide tidal estuary. An estuary such as the Chesapeake Bay is tens of kilometres wide, its tidal currents are slow and reverse every six hours, and the time the water takes to cross it is comparable with the planet’s. There, observations show lateral circulations driven by the Earth’s rotation that are as strong as those driven by the channel’s curvature, and that change direction with the tide because the current does. The share $fR/2V$ that is a fraction of a per cent for a stream is of order one for an estuary, and oceanographers treat rotation there as a matter of course.

The river and the estuary sit on one line of the map above, separated only by how long their water takes to go round a bend. That is also the line along which the geostrophic balance becomes the dominant one: a flow that feels the planet over its own turning time is a flow on its way to being steered by it.

What the picture cannot show #

A wide, uniform channel. The model ignores the banks, which in a real river concentrate the secondary flow near them, and it takes the depth and roughness as uniform across the section.

The helix, not the erosion. A bank erodes when the flow’s stress on it exceeds what the soil can take, which is a threshold, and the helix’s contribution to that stress depends on bank height, vegetation and sediment in ways this calculation does not attempt. A threshold can make a small bias either decisive or irrelevant.

Round values for real rivers. The speeds, depths and meander radii in the table are illustrative, and a real river’s vary along its length and with its flow.

The convention the numbers depend on #

The Coriolis parameter is $f = 2\Omega\sin\phi$ with $\Omega = 7.29 \times 10^{-5}\,\mathrm{s^{-1}}$. Transverse velocity is positive to the right looking downstream. The Earth’s share of the helix is $fR/2V$, with $R$ a typical meander radius; the surface tilts are across the full width, $fVB/g$ from rotation and $V^2B/gR$ from a bend. The river values are round numbers chosen to represent each kind of river, not measurements. The circulation’s strength is its largest transverse velocity divided by the depth-averaged downstream speed, and the equivalent bend is the one whose linearised forcing equals the rotation’s.

Who found it, and when #

Karl Ernst von Baer set out the law in 1860, after Jacques Babinet had suggested the mechanism the year before; the ensuing argument drew in Russian and French geographers for decades. Einstein’s 1926 paper, “The cause of the formation of meanders in the courses of rivers and of the so-called Baer’s law”, explained the bend helix by the teacup argument and estimated the rotational effect as weak but cumulative. Quantitative bend-flow models of the kind used here came later, with Rozovskii’s work in the 1950s. Surveys of bank asymmetry since have found the effect hard to separate from geology except, if at all, on the largest northern rivers.

Still open: a meander that migrates #

The helix biases which bank a river attacks, and the next calculation asks what that does to a river’s course. Linear models of meander migration make a bank’s erosion rate proportional to the near-bank velocity excess, which the helix controls; adding the Earth’s constant curvature to such a model, with its $2V/f$ radius, should give a meander train that drifts steadily towards its right bank as it migrates. The question is how fast, in widths per century, for a river like the Ob, and whether that drift is large enough to have been noticed by the surveys that have looked for it — which would settle, for the one class of river where it could be true, what Baer’s law actually predicts.