A river drifts right, and only a reach can show it

The Earth bends every river a little took Baer’s law — that rivers of the northern hemisphere cut into their right banks because the Earth turns — and replaced a quarrel with a number. The Coriolis force on a straight river drives the same helix as a bend of radius 2V/f turning left, throwing surface water against the right bank, and the Earth’s share of a river’s helix is therefore fR/2V: seven hundredths of a per cent on a mountain stream, 22 per cent on the middle Volga, 39 on the lower Ob. Because a river’s own bends alternate and the Earth’s contribution does not, what is left after the bends cancel is the Earth’s.

It ended with the question that turns a bias into a prediction. A bias in which bank is attacked is not yet a statement about where the river goes. Rivers move: their bends grow, travel downstream and cut off, at rates that on large rivers are tens of metres a year. What does a steady extra push towards one bank do to that motion, how fast is the result, and could the surveys that have looked for Baer’s law have seen it?

The law a meander migrates by #

The standard account of how a meandering river moves is linear and short. At each point the water near the outer bank of a bend runs faster than the average, because the helix of the current that runs across the river piles fast surface water there and scours the bed. That near-bank excess erodes the bank in proportion to its size, and the opposite bank is built up by deposition at the same rate, so the channel keeps its width and its centreline moves. The excess does not follow the curvature instantly: the flow takes a distance to adjust to a change of bend, the relaxation length $h/(2C_f)$ — a depth divided by twice a friction coefficient, two kilometres on a river like the lower Ob — so the excess lags the curvature downstream.

That is the law of Ikeda, Parker and Sawai in its simplest form: the near-bank excess is a lagged copy of the curvature, and the bank moves in proportion to it. Its constant of proportionality combines an erodibility nobody can compute from first principles with a gain from the flow calculation, and one observation fixes it: how fast a bend of the river’s own typical radius erodes its outer bank. Here that is one per cent of the channel’s width a year, which is a common order of magnitude for large alluvial rivers and is stated rather than derived. Every rate below is proportional to it.

The Earth enters in one place. The previous essay showed that rotation drives a helix exactly as a curvature of f/2V turning left does. In the migration law the helix is what the curvature is standing for, so the Earth’s contribution is a constant curvature added to the river’s own: the near-bank excess becomes a lagged copy of C + f/2V.

A constant does nothing to a wave #

A linear law has a simple property: a constant added to its input produces a constant in its output, and nothing else. The meanders are waves in the centreline — each a sinusoid of some wavelength, growing and travelling downstream at rates the law fixes — and the constant curvature is the one component with no wavelength at all. It cannot change how any meander grows or moves. What it does is move the whole centreline sideways at a steady rate: the migration rate a bend would have at curvature f/2V, which is the bend migration rate times the ratio of the two curvatures, fR/2V. The drift rate is the Earth’s share of the helix, applied to the bank.

The figure draws a meander train like the lower Ob’s, in its own widths, now and a century later, with and without the Earth. In a century its bends grow and travel 0.77 widths downstream, and that motion is identical in the two cases. The Earth’s only effect is a shift of the whole river towards its right bank by 0.39 widths — 780 metres, at a drift of 7.8 metres a year. For the middle Volga it is 0.22 widths a century, for the lower Mississippi 0.08, for a lowland river of ordinary size 0.03, and for a mountain stream seven ten-thousandths. Near the equator it vanishes, and south of it the drift is towards the left bank.

The drift is not a new mechanism. It is the previous essay’s bias, integrated. The bias in which bank the helix attacks is a fifth to two-fifths of the bend-driven attack on the great northern rivers, and because the bends’ contribution reverses from bend to bend while the Earth’s does not, the bends move the river back and forth while the Earth moves it one way.

Why one bend shows nothing #

That one-way motion is slow compared with the back-and-forth. In the same century the lower Ob’s own bends move its centreline about 0.86 widths at a typical point, growing outward and travelling downstream, in a direction set by where that point sits on its meander. A single surveyed bend moves more than twice as far as the drift, in a direction the drift cannot predict. Looking at one bend and asking whether it has cut its right bank is asking whether a 0.39-width drift is visible under a 0.86-width scatter of random sign. It is not, and a survey that looked at bends one at a time and found no consistent preference has not found evidence against Baer’s law. It has found the expected result.

What a survey can do is average. The meanders’ own motion averages to zero over a reach, because a reach contains bends of every phase; the Earth’s drift does not. The figure computes what a survey of N independent points along the lower Ob would report as the mean rightward movement over a century, with the band inside which 95 per cent of such surveys fall. One point cannot distinguish the drift from zero. The band narrows as the square root of the number of points, and by about twenty independent points — twenty bends far enough apart to have unrelated phases — it clears zero. The river averages the meanders away for anyone who measures enough of it.

Waiting does not help #

The instinct is that a longer survey helps, since the drift accumulates. It does not, and the figure shows why. Over short intervals the drift and the meanders’ own movement both grow in proportion to time, so their ratio is fixed and the number of points needed is flat: about ten on the Ob, twenty on the Volga, a hundred on the Mississippi. Over centuries the meanders grow — in the linear law, exponentially — and the scatter they produce outpaces the drift, so the count rises. A century of surveys on the Ob needs twenty points, three centuries a hundred. The information is in the number of bends compared, not in the length of the wait, which is the same lesson a record as long as its integral scales teaches about time series: independent samples, not duration, set the error of a mean.

The count, river by river #

For a fifty-year survey the count runs from about thirteen points on the lower Ob and two dozen on the middle Volga, through a hundred on the lower Mississippi and several hundred on an ordinary lowland river, to tens of thousands on a gravel-bed river and millions on a mountain stream. It scales as the inverse square of the Earth’s share of the helix, which is itself half the inverse of the Rossby number of the bends — the number that decides whether an ocean current climbs a hill or circles it — so the whole calculation comes down, again, to what “of order one” is worth — here, how many orders a ratio sits below one, squared.

That table reads as a plan. The rivers where Baer’s law could possibly be tested are few, and every one of them is a great slow river at high latitude with hundreds of bends along its course. For them the test is feasible with ordinary historical charts and modern imagery: identify the centreline at two dates a few decades apart along a few hundred kilometres, take the mean lateral displacement, and compare its sign with the hemisphere. For every other river the test is hopeless, and a claim of Baer’s law from a stream, or from the bends of a single reach, is not a claim the arithmetic can support.

What a survey would look like #

The count translates directly into kilometres of river. Points count as independent when they sit on different meanders, so they must be at least a meander wavelength apart — about eleven channel widths, some twenty-two kilometres on the lower Ob. Twenty independent points then need about 440 kilometres of river, and the lower Ob runs for well over a thousand below its confluence with the Irtysh. The middle Volga, with thirty points at eleven kilometres apiece, needs a little over three hundred kilometres. Both are within the reach of a single study using the charts of the late nineteenth century and modern satellite imagery, and neither requires anything more exotic than tracing a centreline twice and taking the mean of the difference.

Two things would spoil it. The first is anything else that moves a whole reach sideways in a consistent direction: a tilting of the land, which in the far north after the ice is real and slow, or engineering. The second is a mistake in the sign convention, since the prediction is a mean of small numbers of mixed sign and an error in which side is “right” reverses it. The hemisphere is the defence against the first, and it is the subject of the last section. Nothing defends against the second except care.

This is the lesson of the bath that was only ever a wait in another form: whether the Coriolis effect shows in a draining tank depends not on how large it is but on whether the experiment waited for the memory of the filling to die, and a test run without that wait says nothing about the effect either way. Here the wait is replaced by a reach. A drift of a third of a width a century is a large effect on the right design and an invisible one on the wrong one.

After the ice #

A drift of a few metres a year is invisible in a lifetime and enormous over the time a river has had. Since the last ice sheets retreated, about ten thousand years ago, a lower Ob eroding at one per cent of its width a year would have drifted thirty-nine widths towards its right bank, around eighty kilometres; the middle Volga twenty-two widths. Those are distances of the order of these rivers’ whole floodplains. A river that has meandered freely for that long, with its meanders constantly reworking the floodplain, should on this arithmetic have ended up against the right side of its valley and stayed there, with its meanders pressed against a bluff they keep undercutting.

That is the observation Baer started from — the high right banks of the Volga and the Siberian rivers — reached from the other end. It is not proof. A valley’s asymmetry has other causes, as the previous essay said: tilting of the land, contrasts in the rock, the geometry the ice left, and a river confined by its valley wall does not keep migrating at its free rate. The drift calculation shows only that the Earth’s contribution is large enough over the Holocene to produce the asymmetry on its own, for the rivers Baer named and for almost no others.

What the Earth’s drift is not #

It is worth being exact about what has not been shown. The drift acts on the river’s position, not on the bank’s steepness directly: a river drifting right erodes its right bank everywhere at an extra 7.8 metres a year and builds its left bank at the same rate, but whether that produces a steep right bank depends on what the river is cutting into. On a floodplain of its own sediments the drift moves the channel and leaves no mark; only where it meets older, higher ground does a bluff form. So the testable prediction is about position — the mean lateral movement of the centreline — and not about which bank looks steeper, which is the form in which Baer’s law has usually been argued.

It is also a case of the move the effect that explains nothing warned about, run in reverse. There the mistake was giving something real and small the job of something real and large; here the mistake would be dismissing something real because it is small, when its smallness is compensated by never changing sign. And the drift is small against the meanders on every time scale a survey can span. A meander train is a large, noisy motion with a small steady signal in it. That shape of problem — a real bias under a large random motion — is the same as the bath that does not know the hemisphere, where a real Coriolis effect sits under the far larger memory of how the water was filled. There, the effect was a four-hundredth of the noise and needed a laboratory to see. Here it is a fraction of the noise and needs a reach.

What was checked #

The closed-form meander and drift were checked against the migration law integrated directly on a periodic reach four wavelengths long, the lag equation solved by marching downstream and the centreline stepped in time: after forty years the meander’s shape agrees to 0.8 per cent and the drift to rounding error. The survey’s Monte Carlo — random points along a meander train of random phase — reproduces the closed-form scatter of a random-phase sinusoid’s difference to a quarter of a per cent. The tests also refuse a migration rate of zero and a tolerance of zero.

What the model leaves out #

Finite amplitude. The law is linear and its meanders grow without limit; real meanders saturate, become skewed and cut off. Cutoffs straighten the river abruptly and reset its phase, which adds scatter to a survey without affecting the drift, so it raises the number of points needed rather than changing the conclusion.

An equivalent bend. The Earth’s helix is the same as a bend’s, as the previous essay showed. That the bed and the banks respond to a helix from rotation exactly as to one from curvature — that the same point-bar topography forms — is assumed.

One migration rate. One per cent of the width a year is a round number, and rates vary with bank material, vegetation and discharge; the drift scales with it, and the number of points needed does not, since both drift and scatter are proportional to the same rate.

A free floodplain. A river against its valley wall migrates more slowly on that side, so the drift slows as it arrives, and the Holocene totals are an upper bound for the time before it arrived.

Who worked it out #

Baer’s law is from 1860. The linear theory of meander migration is Ikeda, Parker and Sawai’s, from 1981, with Johannesson and Parker’s extension of 1989; numerical models built on it, such as Howard and Knutson’s of 1984, have simulated meander belts over thousands of years. Einstein’s 1926 note on the meandering of rivers, the source of the helix argument, already pointed out that the Earth’s contribution does not alternate. The drift and the survey statistics here are a direct consequence of combining the two.

Still open: a southern control #

Every number here is northern, and the hemisphere is the one variable that separates the Earth’s drift from everything else a valley has. The drift reverses south of the equator, while tilting, rock contrasts and glacial geometry have no reason to. The next calculation takes the survey count to the large rivers of the southern mid-latitudes — few, among them the Paraná, the Murray and the rivers of Patagonia, each with its own speeds and bend radii — and asks whether any of them has enough bends at a high enough share for a survey of historical and modern imagery to detect a leftward drift, which would be the one observation able to attribute a northern river’s rightward drift to the Earth rather than to its valley.