A haboob's leading dust wall behaves like a two-dimensional gravity current: a fixed volume of cool, dust-laden outflow air spreads under warmer desert air because it is denser. The density contrast sets a reduced gravity g′, and a Froude-number closure fixes the head's advance speed for a given local depth h (fixed-volume, rectangular-profile idealisation after Huppert & Simpson 1980):
g′ = g · ΔT / T₀(K)
h(x) = V / x (volume conservation, V = H₀·L₀ fixed)
U_grav = Fr · √(g′·h), Fr ≈ 1.19
Frₓ = C_d · x / h (friction number)
U = U_grav / (1 + Frₓ) (simplified bottom-drag correction)
Both the main current and a frictionless "twin" (C_d = 0) are integrated with RK4 from the same g′ and V, so the amber wall shows the friction-slowed front while the translucent twin shows how far the same storm would have travelled on a perfectly smooth desert floor. Near the source, x is small and the friction number is negligible — the front moves at the classic inertial gravity-current speed. As x grows, Frₓ grows and U falls below the frictionless curve: the storm decelerates faster than volume-conservation alone would predict, which is the friction-dominated regime real haboobs enter once they have travelled several kilometres over rough dune terrain.
- Main pane — side-view cross-section: dune terrain, the advancing dust wedge with its turbulent head bulge, and the frictionless twin front as a dashed line.
- Front-position pane — x_f(t) for both runs; the gap between the solid and dashed curves is the friction lag.
- Front-speed pane — U(t) for both runs, showing the deceleration curve.