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The science
Wind-blown sand and dune migration
Sand moves in hops (saltation) once the friction velocity u* exceeds a threshold u*t. Bagnold's law gives the sand flux q = C (u* − u*t)3 (here C = 1.8 ρair/(g ρbulk), about 1.4×10−4 in SI units, with u*t = 0.2 m/s). Air speeds up over a crest, so u* = u0(1 + s h) grows with the local height h. Sand conservation, ∂h/∂t + ∂q/∂x = 0, is a kinematic wave equation: a point of height h moves at c(h) = dq/dh = 3 C s u0 (u* − u*t)2. Taller parts move faster, catch up with the lee side and steepen until the slope hits the 34° angle of repose, where avalanches form the slip face that real dunes show.
Numerics
The profile is advanced by a second-order flux-limited (minmod) scheme with Heun time stepping on a periodic 120 m domain, followed by a mass-conserving avalanche relaxation. The checks verify mass conservation, the repose limit, Bagnold's exponent and the crest speed against c(h).
Honest simplifications: a one-dimensional cross-section (no barchan horns, no lateral flow, no vegetation), a local speed-up law instead of the full hydrodynamic stress perturbation (so no separation bubble behind the slip face), unlimited sand supply and a single grain size. The 3D page itself only shows procedural noise dunes with drifting particles; this companion gives the physics behind such a scene. Time is a time-lapse: the slider converts seconds of animation to days of real wind.
What to try
- Press Single dune and watch the lee side steepen to the 34° line on the slope graph while the crest travels downwind.
- Compare Strong wind and Weak wind: the crest speed follows (u* − u*t)2, so doubling the excess wind quadruples it.
- Try Low wide swell: low dunes are slow and never reach the repose angle.
- Run the four model checks in the left panel.