A river boundary goes where the river goes, or stays where it was

Five essays about boundaries price what a sentence in a treaty leaves unsaid. One sentence, and the ground between its readings finds seven kilometres between readings of “a straight line”; a meridian boundary moves when its datum does finds 129 metres hidden inside “the 141st meridian”; a tripoint defined three times finds three answers to where three states meet. Every one of those boundaries is defined by points or coordinates, and every one of them, once a convention is chosen, stays put.

The other half of the world’s land boundaries are defined by features, and the commonest feature is a river. A river boundary has no convention problem of that kind: nobody doubts which river is meant. It has a different problem, and it is of a different kind. The river moves.

Two doctrines, one word apart #

The law of river boundaries, in the United States between states and in a great many international treaties, rests on a distinction between two ways a river can move.

Accretion is gradual. A meandering river erodes the outside of each bend and builds a bar on the inside, a little every flood, so the channel migrates across its floodplain year by year. Under accretion the boundary moves with the river: it is wherever the river now runs, and land that the river has worked across from one side to the other has changed country with it.

Avulsion is sudden. In a single flood the river breaks through the narrow neck of a loop, takes the short way, and abandons the long way round. Under avulsion the boundary does not move: it stays in the old channel, now a dry or stagnant oxbow, and the land inside the abandoned loop keeps the country it belonged to — while lying, from then on, across the river from it.

The places where the second doctrine has been applied are the places people remember. Kaskaskia, in Illinois, has been on the western side of the Mississippi since a flood in 1881 moved the river east of it. Carter Lake, Iowa, sits on the Nebraska side of the Missouri because a flood in 1877 cut a loop off, and the Supreme Court held in 1892 that the boundary had stayed where the river had been. The Chamizal, between El Paso and Ciudad Juárez, was argued over for a century on exactly the question of whether the Rio Grande had moved one way or the other, and was settled by a treaty in 1963.

The question this essay puts is how much land each doctrine moves, and whether the choice between them is a choice between small and large.

The river here is a stated curve #

No real river is used, because a surveyed channel is somebody’s survey and the point is a mechanism.

The channel’s centreline is a sine-generated curve, the standard idealisation of a meander: walking along the channel, its direction swings from one side of the valley’s direction to the other and back, reaching a greatest deflection ω at the tip of each bend. The valley repeats every five kilometres. As ω grows the channel gets longer and loopier while the valley stays put, which is how a meander grows, and at some ω the neck of each loop closes to the channel’s own width and the loop is cut off.

For a channel 150 metres wide in a five-kilometre valley, that happens at ω = 120.3°. The channel is then six times as long as its valley.

Where in the channel the boundary is #

A treaty that names a river still has to say where in the river the line runs, and there are three answers in use.

The thalweg is the usual choice for a navigable river, because it keeps the navigable water shared, and it is the reading used for the boundary in every figure here. On a river 150 metres wide the thalweg and the median line are never more than a few tens of metres apart, and they cross at every change of bend, so over a long stretch the land between them nets to nothing.

That makes the choice of reading a small matter on this river and the choice of doctrine a large one. What follows is entirely about the doctrine: the thalweg is held fixed as the reading, and the question is whether it moves with the river.

A map cannot see the reading, and can see the doctrine #

The difference between the two choices has a size that decides what a printed map can show of either.

A line on a map is drawn to a tolerance. A tolerance is a promise about the picture is the essay about what that promise means: a river simplified for a map at one to a million is faithful to within the width of its ink and the simplification behind it, a few hundred metres on the ground. The thalweg and the median line of this river are 53 metres apart at their widest, and its two banks are 150 metres apart. At that scale all three readings are the same stroke of ink, and no reader of the map can tell which one the treaty meant.

The doctrine is not like that. The land accretion moves over a loop’s life is twenty-eight square kilometres; the loop an avulsion strands is twenty-six. Both are larger than anything a map at any ordinary scale smudges, and the enclave an avulsion leaves is exactly the kind of feature a political map is drawn to show.

There is a further reason a map is a poor witness to a river boundary. The score is not stable at any scale finds that the length of a wiggly boundary has no limit as it is measured more finely, and a meandering river is the textbook case: measured with a ruler longer than a loop, this river’s three loops at their cut-off run fifteen kilometres down their valley, and measured at the channel’s own scale the same channel is ninety. A treaty that fixed a river boundary’s length, rather than its course, would be fixing a number that depends on the ruler.

Under accretion, the boundary moves with the river #

The hero figure compares two moments of one meander’s life, and the land between them is what accretion has transferred.

Measuring it needs some care, because the two boundaries are very different curves: one is a gentle wave sixteen kilometres long, the other a set of loops ninety kilometres long, and no point of one corresponds to any point of the other. So the measurement does not compare curves at all. It divides the valley into a fine grid and asks, for every cell, which side of each boundary the cell is on. A cell that is on one side at the start and the other at the end has changed country. The count needs no correspondence between the curves, and it is checked two ways before any meander is measured: a line compared with itself moves nothing, and a straight channel moved half a kilometre sideways along thirty moves fifteen square kilometres to five decimal places.

Between gentle bends and the cut-off, 83.5 square kilometres of land end on the other side of the boundary from where they began — 27.8 square kilometres for each loop. That is more than the whole area enclosed by a loop’s own channel at its largest, and it is moved by nothing but the river doing what meandering rivers do.

Land changes country more than once #

The comparison of two moments hides something the river’s history does not.

A growing loop does not simply sweep outward. As a bend swells, its outer bank advances across land on one side while the next bend’s inner bank retreats across land on the other, and as the loop turns back on itself the channel crosses some ground twice — taking a field from one country and, a few decades on, handing it back. Counted at the end, that field never moved. Counted as the river moved, it changed country twice.

The step-by-step count is the one a landholder would recognise, and it is 158 square kilometres over the meander’s life, nearly twice the 84 that a comparison of before and after shows. Under accretion the boundary is not only in a different place at the end. It has been in a great many places, and some of the land it passed over now belongs, once more, to whoever held it first.

The count has settled #

A step-by-step count has an obvious weakness: a parcel swept across and back inside a single step is never seen to move. So the count can only rise as the steps shrink, and it is worth nothing until it stops rising.

The last halving of the step adds eight hundredths of a square kilometre to 158. That is what turns the finest value from a lower bound into a measurement, and it is the same discipline every reach set ever drawn is too small applies to a fan of bearings: know which way an instrument errs, and refine it until the error is smaller than anything being claimed.

Under avulsion, one flood strands a loop #

The other doctrine does nothing while the river migrates slowly, and then, once, a great deal.

At the moment of cut-off the river shortens itself by nearly twenty-two kilometres of channel in a single event. Under avulsion the boundary is left behind in the old loop, and the land inside it — 26.4 square kilometres — becomes an enclave of one country on the far side of a river from the rest of it, reachable by land only through the other.

That is almost exactly the land accretion moved over the loop’s whole slow growth. The ratio in this case is 0.95: the flood strands, as one parcel and in one night, nineteen twentieths of what decades of migration moved across the boundary piece by piece.

The two doctrines move land of one size #

A single case could be a coincidence of the channel’s width against its valley. It is not.

Across valleys from three to eight kilometres and channels from fifty to three hundred metres, accretion over a loop’s life and a single avulsion move land within ten per cent of each other. Each area scales with the square of the valley’s wavelength, as any area of one shape must, so the ratio is a property of the meander’s shape rather than of its size; and it rises a little as the channel gets wider against its valley, because a wider channel closes its neck earlier and strands a smaller loop.

So the choice between the doctrines is not a choice between moving little land and moving much. It is a choice about the form the change takes. Accretion moves the land as a strip spread along the whole length of the river, gradually, and in both directions. Avulsion moves none of it for decades and then leaves one compact piece of one country on the wrong bank.

What each doctrine is protecting #

Put that way, the two doctrines stop looking like arbitrary rules and start looking like answers to different worries.

Accretion protects the river as a boundary. A border that stays in the water is a border everyone can see, police and cross at a bridge; a line in a dry oxbow is not. Under accretion the boundary is always where the river is, and the cost is that nobody’s land is quite certain to stay in its country for a lifetime — about twenty-eight square kilometres a loop changes hands, and half as much again changes hands and back.

Avulsion protects the land. A farm, a town or a county does not change country because a flood took a short cut one night, and the cost is exactly the enclaves the famous cases are made of: a piece of one state across the river from it, sometimes with no bridge of its own.

A map of an accretion boundary is dated #

The first doctrine has a consequence for every map that shows it, and it is easy to miss because it looks like no error at all.

Under accretion the boundary is wherever the river runs now. A map shows where the river ran when it was surveyed. So a map of an accretion boundary is exact on the day of its survey and wrong afterwards by however far the river has since migrated — not because anything was measured badly, but because the thing measured has moved and the definition moved with it. Over a loop’s life that error reaches every piece of land in the strip the river sweeps, twenty-eight square kilometres a loop, and it is not uniform: the land near the tips of the bends goes first.

A published coordinate is a result makes the general point about coordinates: a published number is the output of a particular survey on a particular datum at a particular date, and without those it is not yet a position. A river boundary under accretion is the case where the date is not a detail of the metadata but half the definition. The line on the map is a statement about a year, and the boundary itself is a statement about the river.

An avulsion boundary has the opposite property. Once a cut-off has happened, the boundary is a dry channel that no longer changes, and a map of it stays right for as long as anybody can find the old channel on the ground — which, as oxbows fill with silt and are ploughed, is not forever either.

Neither is the more accurate, since there is no fact about where the “real” boundary is once the river has moved. A boundary that two features share meets a similar object in a dataset — ground that belongs to both of two features or to neither because the line was simplified twice — and there, too, the repair is a rule about which version is authoritative rather than a better measurement.

What the model leaves out #

The river is a stated curve. A real meander is not a sine-generated curve, its loops are not identical, and it does not grow symmetrically. The curve is the standard idealisation and the areas are what that idealisation predicts; a surveyed river would give different numbers and no reason to expect a different ratio.

The growth is a sequence of shapes, not a process. Rivers migrate at rates that vary with flood frequency, bank material and vegetation, and the time a loop takes to reach its cut-off is anything from years to centuries. The land measured here is land per loop’s life, whatever that life lasts.

The thalweg is a rule, not a survey. The line of deepest water is drawn as swinging towards the outside of each bend in proportion to how sharply the channel turns. Real thalwegs wander more than that, and a river with a wandering thalweg moves its accretion boundary more often than this one does.

And the law is simplified. Real cases turn on whether a particular change was gradual enough to count as accretion, and courts have divided the same shift of the same river into both. The measurement takes the two doctrines at their cleanest, which is where their difference is easiest to see.

Still open: a boundary on a watershed, where nothing moves and nothing is visible #

A river boundary is at least a line anybody can find: it is in the water. The other great natural boundary is the watershed — the line from which water runs one way on one side and the other way on the other — and it is the opposite kind of object. It does not move on any timescale a treaty cares about, and it cannot be seen at all: it is a property of which way every drop of rain would flow, decided by the whole shape of the land around it.

Whether such a line is as well defined as it sounds, how much it depends on the detail at which the land’s shape is known, and what happens where a treaty names it alongside the highest peaks and the two turn out not to be the same line, is a question a river cannot ask.