The grain the wind lifts first

The sandcastle a spoonful of water holds found that adding a little water gives grains a strength out of proportion to the amount. It took for granted what sand is. Why are there so many grains around a fifth of a millimetre across, in deserts and beaches and on Mars? Rock breaks into every size. Grinding makes more of the small ones than the large. So wind and water, sorting all that material, have to be concentrating this range, and a natural first guess about what they do is simple: wind picks up the small grains and leaves the big ones. A beach would then be whatever the wind could not move, and the fine material would have gone to sea.

That guess turns out to be half right, and the half that is wrong is more interesting than the half that is right. Ralph Bagnold, a British Army officer who spent the years between the wars driving modified Fords across the Libyan desert, measured how wind moves sand with a bench-top wind tunnel he built himself and published the results in 1941 in a book that is still the place to start. He found, among much else, that very fine powders are hard to raise, that a smooth layer of flour resists a wind that shifts coarse sand. The fine material stays on the ground.

Two ways to hold a grain down #

Wind on the ground has a logarithmic profile, with the speed varying with height in a way set by the friction speed $u_*$. That speed is a measure of the shear stress on the ground, $\tau = \rho u_*^2$. It is about a twentieth of what a weather station reports at ten metres, depending on the roughness. Grains on the surface feel this shear, as a drag proportional to their exposed area, of order $\rho u_*^2 d^2$ for a grain of diameter $d$. For a grain to be started, drag has to overcome what holds it.

Two things hold it. One is weight, going as the volume, so as $\rho_s g d^3$. So the balance between drag and weight gives

$$u_*^2 \propto \frac{\rho_s}{\rho}\, g\, d.$$

The large grain has more weight per unit of area than the small grain. This is the half of the guess that is right, and if it were the whole story, the threshold would fall forever as grains shrink. The other holding force is cohesion. All surfaces attract at close range through van der Waals forces, and a real sand grain usually has some adsorbed water and may be a bit sticky through its roughness. The pull-off force between two small spheres, in the simplest theory, is proportional to their size, going as $\gamma d$ with a measure of the surface energy $\gamma$. Weight goes as $d^3$, drag as $d^2$ and cohesion as $d$. So as a grain gets smaller, cohesion falls least quickly, and at some size it must dominate over the other two.

Yutaka Shao and Hua Lu put the two together in 2000 in a compact expression fitted to wind-tunnel measurements:

$$u_{*t} = \sqrt{A\left(\frac{\rho_s}{\rho}\, g\, d + \frac{\gamma}{\rho\, d}\right)},$$

with $A = 0.0123$ and $\gamma = 3 \times 10^{-4}$ N/m, which it should be said is not a measured property of quartz but a fit. Now there is a minimum, and it has to be there.

So the wind’s favourite is about a tenth of a millimetre across, a fact that does not sit in the same relation to size as the outlet that jams by chance, where the critical thing about a grain is how big it is next to a hole.

Move away in either direction and it takes a stronger wind. Below the minimum, the curve rises as $d^{-1/2}$, so flour-fine material is a much harder start than fine sand. It is the reason a dry dust surface looks so stable in a wind: nothing moves until something else strikes it.

Set the derivative of the expression to zero and the minimum lies at

$$d_{\min} = \sqrt{\frac{\gamma}{\rho_s g}}.$$

The air density is not there. It multiplies both terms in the threshold, so it is a common factor and cannot move the position of the minimum. This has a consequence worth drawing.

Four atmospheres, and almost the same answer #

Mars has an atmosphere a hundred times thinner than Earth’s. Venus has one about fifty times denser. Titan has a thick cold atmosphere and low gravity. Each has dunes. If the favoured grain depended strongly on the air, those dunes would be made of grains of quite different sizes.

The thresholds are very different. On Mars the wind has to be about six times as fast at the surface to start the same grain, and that has been a puzzle in itself since Mars’s dune fields were discovered to be active with winds seldom reaching this value. Yet the position of the minimum, on three of the four worlds, varies by less than a factor of two. Earth, Venus and Mars have minima near a tenth of a millimetre to a sixth of a millimetre, because they differ mainly in the air, and only gravity and material show up in the formula. Titan is the outlier here because its gravity is low and its grains are expected to be organic material of low density; cohesion may also be very different for that material, and the assumption of a common value is certainly wrong to some degree. Titan’s dunes are thought to be made of grains a few hundred micrometres across, broadly in line, and with the uncertainty on what they are made of the agreement is suggestive only.

The Curiosity rover and others have now taken pictures of Martian dune sand, finding grains typically between about 100 and 300 µm. That was predicted in outline before any measurement. It is one of those results in planetary science where a formula from a laboratory tunnel crossed interplanetary space and got the size to within a factor of two, in an air it had never been tried in.

What a moved grain does next #

Being started is only the first step. A grain that lifts off into a wind is now in moving air, and it is accelerated horizontally, and falls back under gravity. Its path is a long, low hop.

Bagnold called this saltation, after the Latin for leaping, and the word has stuck. On landing, a hopping grain is much faster than it left, because for half a metre it has been in much faster air. That impact is what matters. A grain landing at shallow angle at a few metres per second carries much more momentum than the wind applies to a grain at rest on the surface, and it can splash up several neighbours. So sand on the move does not need the wind to lift each grain. The wind only needs to accelerate grains already in the air, and they do the lifting themselves.

Two thresholds follow, and they differ.

This is a hysteresis loop with a physical cause, of a kind familiar from the curve that is really a staircase in a very different setting. A wind that has fallen to 0.22 m/s can keep a flow of sand going, even though it could never start one. So a gusting wind can move a surprising amount, because a gust gets the sand going and the lulls between gusts do not stop it. Transport also grows as nearly the cube of the friction speed once it has begun, and it follows that a few windy days a year move most of a dune. Average wind speed is nearly useless as a guide. That is a difficulty for anyone predicting what a desert will do from monthly weather statistics, as well as a reason storms do almost all the work.

The hopping layer also has a feedback on itself. Grains in the air take momentum from the wind near the ground, and with enough of them the wind at the surface falls until it is just at the impact threshold. So the wind near the bed is pinned at about the threshold value no matter how hard it blows overhead, and an increase upstairs just lofts more sand and makes the layer thicker. Bagnold found a fixed height near the ground at which the wind’s speed was the same whatever was happening above. The result is a self-regulating system, and it explains why sand transport is as predictable as it is. It also gives a surface layer that rarely extends above about knee height, though the grains that fly highest reach a metre or more, on hard pebble surfaces, and every driver who has spent time in deserts knows how paint is stripped off the lowest part of a car and left alone higher up.

Why dust is a different substance #

There is a third line to add. Once a grain is up in the air, how long can it stay? Smaller grains settle very slowly, because the viscous force dominates: in Stokes’s regime the fall speed goes as $d^2$. Air in motion has turbulent eddies whose vertical speeds are of order $u_*$ itself. So a grain falling more slowly than $u_*$ can be carried indefinitely by the turbulence.

The crossing is at about 62 µm, and it is an interesting coincidence that the conventional geological boundary between silt and sand has for a long time been placed at about the same value, 63 µm, for reasons that were originally about sieves. Two regimes meet there with the same wind. Above it, a grain can be started and cannot stay up, so it hops. Below it a grain, if somehow released, would stay in the air for a long time, kilometres up and thousands of kilometres from home. But such a grain resists being started. This is the paradox of dust: the wind that could carry it cannot pick it up.

The solution is what Bagnold saw and what modern work has confirmed. Dust gets into the air mainly because saltating sand hits it. Hopping grains land on a dusty surface and blast out fine particles at speeds no wind shear could give them. So the sand, of a size that can be started, liberates the dust, of a size that can be kept. Experiments show that a pure dust layer in a tunnel barely moves, and when a stream of sand is added upwind it emits abundantly. So when a major dust storm takes a million tonnes of soil from the Sahara to the Amazon, in the first step sand grains did the work of shaking it loose. A dust source needs both, which is why the world’s great dust sources are mostly old dry lakes and riverbeds with fine sediment and sand nearby. The grain size of a beach is a similar sum. Sand survives where finer material has been carried off and coarser material has not been moved, and the minimum of the threshold curve is the doorway through which it passed.

Why a dune has a smallest size #

Sand in motion responds to what the wind does, but only after a delay. A wind that speeds up does not immediately carry its full load, since the hopping population takes some distance to reach the number that wind can sustain. The distance is called the saturation length, and it is about a metre on Earth. It has a simple meaning: a grain has to be accelerated up to the wind’s speed, and that takes a length set by the density ratio of grain to air, multiplied by the grain size.

That one delay decides what can be a dune. A small pile on a flat bed has wind speeding up over its upwind side. Sand is carried up, and, because of the lag, the maximum transport is reached past the crest, where it can no longer take sand away and thus deposits it. So the pile grows, unless it is so small that the lag covers its whole length. A heap much smaller than the saturation length cannot form a dune. It is simply eroded. The result, tested with care by Hersen, Andreotti and Douady and others from about 2002, is a minimum dune length of perhaps ten saturation lengths. On Earth there are no real dunes much smaller than about ten metres. That is why children on a beach can pile up sand but it will not move as a dune, and why dunes have a well-defined scale of metres and not of centimetres. Because the saturation length depends on the density ratio, Martian dunes should have a much larger minimum size, of hundreds of metres, and they do. Similar results have been obtained on the bed of a stream, where water’s far greater density brings the scale down to centimetres. Sand’s slope on the lee side is then set by the angle that does not know the size of the heap, the same on all of these.

That is to say, a dune’s dimensions come from three numbers. The threshold curve fixes the grain, the density ratio gives the saturation length and the angle of repose gives the slope. What emerges from all of this is a landscape made from a single scale. There are other things going on in a dune too, including the flow behaviour that the sand whose friction depends on how fast it flows described, for avalanches on its lee side, a sorting by size related to the big one comes to the top, and even a hum from those avalanches that has made some dunes famous. They sit on top of a size set by how long a grain takes to catch up with the air. A boundary layer’s thin bottom part, which the shear that only reaches so far studied, supplies the shear that all of this depends on.

Where this curve is wrong #

The threshold formula is an empirical fit. Its cohesion constant covers a real variation of more than a factor of three in reported wind-tunnel data, and the real cohesion depends on humidity, on roughness of the grains, on packing and on electric charges. Damp sand is much harder to start, as the sandcastle would predict, and crusts of salt or biological origin can raise the threshold many times over. A disturbed crust in a desert can lift material that had been held for decades, and that is the reason a track across a dry lake bed may raise more dust than the wind ever did.

On the figure, a grain of one size lies among identical grains. A real surface has a mixture, in which small particles hide behind large ones, sheltered from the wind, and large ones protrude further and are easier to start than a uniform bed of them. Those effects shift the threshold noticeably. The trajectories above use simple drag with no rotation, no lift, no effect of the other grains on the wind and no turbulence. Spinning grains land further, and real launch speeds have a spread, so a real hop is not a single path. The same goes for the 0.8 ratio between thresholds, which is a typical figure from tunnel work and varies. For Mars, the ratio of thresholds may be much smaller, and that is a leading explanation for how sand moves so readily at the low winds measured there.

The two-term formula also treats the drag force as a simple average. Real turbulence fluctuates, with short gusts several times stronger than the mean, and a grain may be started by one of them. A threshold is therefore not a sharp line but a statistical band. Calling it a threshold, with a definite number, is a tidy way of expressing how likely motion becomes as the mean wind rises.

Still open: how sand moves where the air is almost absent #

Mars is the sharpest test. Its measured winds rarely exceed what the threshold formula says should be needed, and yet Mars’s dunes have been caught moving in orbiter images taken a few years apart, at rates much greater than expected. Some of that gap can be attributed to the low impact threshold: under low gravity and thin air, a grain takes a very long hop and picks up much speed, so a flow once begun can go on at winds about a tenth of what was needed to start it. How it starts in the first place is less clear. Gusts, dust devils, slopes and processes that loosen the surface without wind have all been proposed. Instruments on Mars have now recorded grains moving during a passing wind, and on a surface with those conditions, it is possible for the first time to test a model of sand transport in an atmosphere it was not fitted to. Similar questions await Titan, whose vast dunes will be visited by the Dragonfly mission in the 2030s, carrying a means of finding what its grains actually are and how sticky.

Weight falls faster than drag as a grain shrinks and cohesion falls more slowly than drag, so the wind’s easiest grain sits at $\sqrt{\gamma/\rho_s g}$, about 107 µm on Earth with air density cancelled; below about 62 µm a grain that gets up can stay up and cannot get up by itself, so sand is what the wind can lift and cannot keep, and dust gets into the air only when sand knocks it there. The smallest thing is not the easiest thing to move. The habit worth carrying is to ask, of any size that nature prefers, which two opposing scalings have crossed there, since a preferred size is rarely the result of one effect alone.