Lee

A barchan is what a dune becomes where there is not enough sand to cover the ground: a crescent with a long back the wind climbs, a steep face it drops over, and two horns reaching downwind. It walks. North of Laayoune, on the Atlantic edge of the Sahara, a corridor of them crosses a limestone plateau toward the south-south-west at tens of metres a year, the small ones faster than the large. This is that corridor from 370 metres up, looking straight down at dawn, with the years running: the wind blows from left to right, and the dunes go with it.

Nothing here is drawn as a crescent. Each dune is a height of sand on a grid of one-metre cells and four rules from the continuum model Klaus Kroy, Gerd Sauermann and Hans Herrmann published in 2002. The wind over the sand is computed from the sand's own shape: it speeds up over a rise, and the speed-up peaks a little upwind of the crest, so the back erodes and the front grows. The sand the wind carries does not respond at once; it needs a saturation length to catch up, and Hicham Elbelrhiti, Bruno Andreotti and Philippe Claudin measured it in this corridor: 1.7 metres. Where the lee drops more steeply than 14 degrees the wind leaves the surface, and in that bubble the sand falls out of the air. Where any slope passes 34 degrees it slides back to 33. Crescents, horns and slip faces come out of those rules, or they do not appear.

The first version did not make barchans. Every heap became a tall ridge with a slip face and no horns, because I had left out the force that moves a grain sideways: gravity on a slope. Models of a single slice of a dune can drop it; the three-dimensional ones keep it, in the balance that sets each grain's velocity (Orencio Durán, Veit Schwämmle and Hans Herrmann, 2005). Put back, a four-metre heap became a crescent 53 metres wide and 3.3 metres high, with horns.

Then the model was held against the corridor, dune by dune, fed sand the way a corridor feeds them. Five steady barchans between 40 and 95 metres wide moved at speeds that fit the corridor's own law, a speed proportional to one over the width plus a constant: the corridor's constant is 16.6 metres, the model's 17.3. The model's dunes run about a tenth faster at every width, as if the wind moved 83 square metres of sand a metre a year instead of the 75 I chose from the 60 to 90 measured there. They also stand taller than the corridor's dunes of the same width, a sixth taller in those runs and a quarter taller once the wind's wandering, described below, is added, as it is here. Their slip faces lie at 33 degrees, and sand streams from both horns, four and a half times the background in the rows behind them and none behind the slip face, as the photographs of horns trailing sand show.

Two things the corridor has, the model at this wind does not. Its commonest dunes are about 25 metres wide; here a heap of 20 or 30 metres blows away within two years. Somewhere between 30 and 40 metres a heap stops keeping a slip face, and without one it cannot hold the sand that blows over it. And I predicted that a small dune catching a large one would seem to pass through it, the exchange Durán and his colleagues found in this model at a stronger wind. At this wind a dune of four tenths the volume merged with the larger one, and the dune that left the front carried about an eighth of the sand: not a crossing. A dune a twentieth the size never arrived; it blew away on the way.

Some of it is chosen, not solved. The wind blows at one strength, set so the model's saturation length is the corridor's 1.7 metres, from a direction that wanders by eight degrees, which spreads the moving sand sideways the way the corridor's wind does; because a real year has calms and gusts, the clock converts this steady wind's work into calendar years at 75 square metres a year. The field is a window 512 metres long onto a corridor hundreds of kilometres long: sand blows in across the left edge at a fifth of what the wind could carry, dunes walk out past the right one, and when the window runs short of sand a dune comes in from upwind, one of four sizes the same model made steady first. The light is the sun as it stood over the corridor at 08:36 local time on the morning this was made, 10 degrees up and just south of east, held there while the years run: a composite no camera could take. The plateau's stones and the grain are invented.

Press and hold on the ground to pour sand. A short pour is a heap the wind takes apart: too small for a slip face, it leaks away downwind. Hold for several seconds and the heap grows a slip face, then horns, and runs to the right, faster than the large dunes and into them.

What the board, registered before any code, did not get: a calendar year in thirty seconds (on the machine this was made on the sand ran a year in about a minute, and since the second pass in about fifty seconds; the readout says how long it takes on yours); a black point at 5 per cent (it is 5.5); and the same dune at half-metre cells, which in the one run there was time for stood an eighth taller, so the one-metre grid is not converged in height.

7 October, the day after. A film of the first version had a mark at its right edge that never moved, a thin hook I could not explain. It was sand the model could not move: a ridge 10 to 20 centimetres high, shaped like the outline of a dune, in the lee of another. A dune traps the sand blowing into it, so the rows behind it carry almost nothing (at one point I measured about 10⁻⁴⁵ of what the wind could carry), and the model's rule for moving sand multiplies what arrives: grains in motion knock more grains into motion. From almost nothing that takes about a hundred saturation lengths, some 170 metres of sand, and a ridge two metres wide never gets there. Kroy, Sauermann and Herrmann wrote the remedy into their paper in one sentence: rather than model how the wind first lifts a grain off a still bed, allow a small residual influx even where there is nominally none. Now, on sand, the moving sand never starts below a hundredth of what the wind could carry there. The hundredth is mine, a twentieth of what blows in at the left edge. The ridge was gone in nine days of the model's calendar, and a single steady dune moves and stands within about one per cent of where it did. I expected a visitor's heap poured behind a dune to have been stuck the same way; it was not. A heap that size is wide enough to wake its own sand from the trace that leaks in from the side.

The same pass made the brinks sharp. The ground was drawn as one surface stretched between the heights of one-metre cells, so a brink, where the gentle back meets the steep slip face, could only fall on a cell edge: it showed as a staircase, and the low sun drew the stairs down the slip faces as rays. Near a brink the ground is now the lower of two surfaces, the back continued past the brink and the slip face continued up past it, each fitted to its own cells; they meet where the brink is. The light on the backs comes from slopes that run on smoothly from cell to cell, which took away a fine banding at the grid's pitch, and the shadows are cast by a smoothed copy of the ground. The field the page opens on was grown again under the changed rules, from the same start for the same three years: the same dunes, less the ridge. Under these rules the same start gives the window three dunes at three years instead of four, two of them merging. The rule that stops a cell losing sand it does not have now settles each cell's donors before the cell itself, which is where the extra speed came from; it moves the dunes less than nudging every cell of sand by a hundred-billionth of a metre does.