Why are some sand dunes crescent shaped?

The recent book, the Martian, and the corresponding film, have sparked pretty much everyone's imagination when it comes to space, and Mars in particular. In an article for Slate magazine, Phil Plait makes a case for how the film could have been  improved if it had showcased some of the beatufiul landscapes that the HiRISE camera has observed from its perch aboard the Mars Reconnaissance Orbiter. I recommend the article. It's a good introduction to why Mars is interesting to some scientists, especially geomorphologists. Unfortunately, he gets a key fact about sand dunes wrong.

This image of a large, crescent-shaped sand dune, known as a barchan dune, is striking.  A quick Google search, or better yet, an exploration through some of the beautiful images captured by HiRISE, will reveal more observations of these phenomena, often arrayed in large fields of relatively similarly-shaped dunes. What gives these dunes their characteristic shape? Unlike Plait claims in the Slate article, it doesn't require the wind to deflect around an obstacle. It's actually much more fundamental to the dune itself.

 NASA/JPL/University of Arizona/HiRISE

NASA/JPL/University of Arizona/HiRISE

First, consider that individual sand grains are transported by wind in a series of long hops, a process called saltation. When a saltating grain impacts a surface, it has some probability of being deposited there (versus rebounding and making another hop). Rough patches in the landscape increase this probability, so small patches of sand quickly grow into larger piles as new sand grains preferentially "stick" there.

The physics of wind flow over the sand pile causes it to eventually take on the characteristic barchanoid form. The flow is forced to accelerate up and over the stoss side of the pile, which increases the shear stress there. The pile is a relatively bluff object in the path of the wind. As the flow moves over the crest of the pile, a turbulent separation zone of lower velocity forms on the lee side of the pile. The stoss side thus becomes a zone of net sediment transport, and the lee side a zone of deposition. This sets up a positive feedback loop between dune height, windward erosion, and deposition on the lee side. The feedback loop is halted when the dune reaches a critical height at which any more increases in height are unstable under the dominant shear stress conditions. As grains build up on the lee side, they avalanche down when the slope becomes too steep, giving the dune its characteristic cross-sectional shape (gentle stoss side, steep lee side). At this point, the dune is in a dynamic equilibrium.

At dynamic equilibrium, the dune migrates downwind through time. But it doesn't move at a uniform speed. The migration speed depends on the flux of the sediment across the crest, but it's also inversely proportional to the dune height. This makes sense because for a constant sediment flux, it will take more time to pile up a larger amount of sediment in the downwind direction. In other words, larger sections of the sand pile migrate more slowly than smaller sections. So the edges migrate faster than the middle, which leads to the crescent shape.

Sometimes the horns of the barchan will grow so long that they spawn new, smaller dunes that proceed to migrate away at a quick pace. And small dunes migrate faster than larger ones, which leads to some cool interactions when small dunes catch up to big ones.