Dune Cross-Section: Exner Sand-Continuity Model

Wind & Grain

Shear velocity u* 0.45 m/s

Grain diameter D 0.25 mm

Initial dune height H₀ 4.0 m

Time acceleration 2.0e5×

below threshold — no transport

Live Readouts

Threshold u*t

0.20 m/s

Free-stream flux q

0 kg/m/yr

Celerity (Bagnold)

0 m/yr

Celerity (measured)

0 m/yr

Elapsed (sim)

0 yr

Mass drift

0.00%

How it works

This is a genuine continuum morphodynamic model of the dune's downwind cross-section h(x) — not a rigid shape slid along by a scripted speed. Four coupled real physical mechanisms drive the profile's own evolution, each computed independently every frame:

1) Wind speed-up over topography (linearised hill flow):
   u*(x) = u*0 · clamp(1 + 2·∂h/∂x, 0, 2.5)
2) Lee-side flow separation: behind the brink, wind
   recirculates below a straight separation streamline
   falling at 14° — u*(x) = 0 inside that bubble.
3) Bagnold saturated flux at each point:
   q_sat(x) = C·(ρa/g)·√(D/D₀)·u*(x)²·(u*(x) − u*t)
4) Flux relaxes toward q_sat over a saturation length
   ℓ_sat (grains need a "fetch" to reach equilibrium):
   dq/dx = (q_sat − q) / ℓ_sat

The bed itself then evolves by mass continuity (Exner equation) — sand piles up wherever flux converges and erodes wherever it diverges:

∂h/∂t = −(1/ρ_bulk) · ∂q/∂x

Slip-face grain flow is enforced by repeated local avalanche relaxation clipping any slope past the 34° angle of repose (mass exactly conserved — sand only moves between neighbouring cells, never created or destroyed). The dune's forward creep, its asymmetric shape, and its eventual steady speed all emerge from these four rules — they are not set directly, unlike the single celerity number used to translate the 3D dune. Saltating grains are drawn following true ballistic projectile motion (initial launch speed ∝ u*, parabola under gravity g, landing when it returns to ground) rather than a cosmetic hop curve.

Shear velocity u* — free-stream wind strength; must exceed threshold u*t for any transport.
Grain diameter D — sets both the threshold and the saturation length (coarser grains need a longer fetch to reach equilibrium flux).
Initial dune height H₀ — starting condition only; the model then evolves height on its own via continuity.
Celerity (Bagnold) — the classical closed-form estimate c = q/(ρ_bulk·H) using the current peak flux and height.
Celerity (measured) — the crest's actual tracked speed from the PDE, which can differ from the closed-form estimate as the shape adjusts.

Saltating grains (ballistic) Reference stakes

wind → downwind (+x)