Why Rivers Wind

Rivers on Plains Are Rarely Straight

Where the slope is gentle and the banks are sand and mud, a river goes from bend to bend. The word meander comes from a famously winding river in western Turkey, the Maeander of the ancient Greeks, today the Büyük Menderes.

The channel’s length divided by the straight distance between its ends is its sinuosity. A straight river has 1. Leopold and Wolman called rivers with a sinuosity of 1.5 or more meandering. The river on the model is about 22 m wide; once its bends have grown, its sinuosity is mostly between 2 and 3.5.

Model: a valley about 900 m long. Heights are exaggerated about 15 times

In a Bend, the Water Spirals

Water entering a bend is thrown outward. The surface rises a little against the outer bank, and its slope pushes the water back in. On average the two balance, but not at every depth.

Near the surface the water is fastest and is thrown hardest, so it drifts outward. Near the bed, friction slows it, the slope wins, and it flows inward. The water sinks at the outer bank and rises over the inner one, turning as it goes downstream, like a stretched spiral.

Tea leaves gather in the middle of a stirred cup for the same reason: the water at the bottom, slowed by friction, flows to the centre and takes the leaves with it. In a short paper in 1926, Einstein used the teacup to describe this circulation, and with it explained why river bends grow ever more curved.

A bend in section. The flow across is drawn enlarged; in a river it is about a tenth of the speed downstream

The Outer Bank Goes, the Inner Bank Grows

Under the outer bank is the deep pool. The sinking water scours the foot of the bank, and the undercut bank falls in, block by block. This is the cut bank, often a steep scarp.

The water flowing inward along the bed carries sand to the inner bank and leaves it there as a gentle slope: the point bar. As the outer bank retreats, the point bar grows, so the river stays about as wide and the whole bend moves outward.

Each step of the bar leaves a low ridge of sand. Over the years the inner bank fills with curved ridges and swales: scroll bars. In section, the bar’s sand layers dip towards the channel; each one was the bank in some year.

A Small Bend Grows

Suppose the channel starts nearly straight, off by less than a metre. Water flowing past these tiny bends still scours their outer banks and fills their inner ones, so each bend gets a little bigger; a bigger bend scours harder.

The scour is strongest not at the apex of a bend but a little downstream of it, because the flow takes a distance to turn. So bends do not only grow, they travel downstream as they grow.

Very short bends are smoothed away and very long ones grow slowly. The bends that grow fastest are about ten channel widths long. On the model, after about five hundred years the valley is full of them.

The model starts from a nearly straight channel

Cutoffs and Oxbow Lakes

As a loop grows, its two ends come closer until only a narrow neck is left. A flood spills over the neck, or the banks on either side wear through, and the river takes the short way. This is a neck cutoff.

No current runs through the abandoned loop. Its ends silt up first, and a crescent lake is left in the middle: an oxbow lake. Over the following decades to centuries mud and plants fill it in, it becomes a marsh, and in the end woodland.

A cutoff makes the river shorter and straighter at a stroke, but it soon starts to bend again. So a meandering river’s sinuosity rises and falls, staying within a range over the long run.

Bends Scale with the River

Leopold and Wolman compared many reaches, from streams a few feet wide to rivers a mile wide. In their 1957 data the meander wavelength was 7 to 12 times the channel width; in 1960 they measured 50 reaches and found an average of about 11 times, with bend radii mostly 2 to 3 channel widths.

So on a map without a scale bar, a creek is hard to tell from a great river. On the model the fastest-growing bends are about 11 widths long. The number depends on the friction of the bed and on how wide the river is for its depth; the model uses values usual for sand-bed rivers on plains.

The ruler along the front edge: one division is one channel width (22 m)

The Same River from Orbit

This is the Ucayali in Peru, one of the headwaters of the Amazon, a far bigger river than the one on the model. Both pictures are from Landsat satellites and show the same reach, one in 1985 and one in 2023, about 50 km across.

In 38 years several loops were cut off. The river carries a lot of silt and is light brown; the loops cut off are now dark oxbow lakes, still water where the silt has settled. The curved stripes either side are old channels and scroll bars.

Satellite images: NASA Scientific Visualization Studio, Landsat (NASA / USGS)