Desert Dune Migration Lab

When sand starts to move

Wind drags on the bed with shear velocity u* = κ U / ln(z/z0), with κ = 0.4, the height z of the wind measurement and a roughness length z0 ≈ d/30. Grains start to hop (saltate) once u* exceeds the threshold u*t = A √((ρs − ρa) g d / ρa) with A ≈ 0.1. Above it the saturated flux follows a Bagnold-Lettau law, qs = C (ρa/g) √(d/D) u* (u*² − u*t²).

How a dune moves

The flux needs a distance to adapt to the local wind, dq/dx = (qs − q)/ℓs. Over the crest the wind speeds up, so erosion exceeds deposition on the windward side. The sand balance is the Exner equation ρb ∂h/∂t = −∂q/∂x. In the sheltered lee the flux stops and sand piles up until the slope exceeds the angle of repose (about 34 degrees) and avalanches. The result is a steady march of the dune, roughly c ≈ q / (ρb H): small dunes move faster than tall ones.

Honest simplifications

The model is two-dimensional (real dunes such as barchans are three-dimensional), the wind is steady and unidirectional, the crest speed-up is a simple linear law in the height above the mean bed, the lee shelter is a slope rule rather than a flow-separation calculation, there is a single grain size and no vegetation or moisture. The displayed migration speeds are illustrative.

Checks in the page

Check 1 tests that the sand volume stays constant over many flux steps. Check 2 tests the saturated flux ratio against the Lettau formula and the absence of transport below the threshold. Check 3 tests that a steep pile avalanches to below the repose slope with no loss of sand.

• Load Light breeze (below threshold): the wind is weaker than the threshold and the dune does not move at all.
• Load Storm wind and compare the migration speed: the flux grows roughly with the cube of the wind.
• Compare Small fast dune and Tall slow dune: the same flux moves a smaller dune much farther.
• Raise the crest speed-up factor and watch the crest sharpen while the lee face avalanches at about 34 degrees.