Volcanic Ash Fall (2D side view)

Settling velocity

A grain of diameter d and density rho_p reaches the speed where drag balances its buoyant weight: (rho_p - rho_a) g pi d^3/6 = 0.5 rho_a C_D (pi d^2/4) v^2. The drag coefficient follows Schiller-Naumann, C_D = 24/Re (1 + 0.15 Re^0.687), floored at 0.44 for Re > 1000 (Newton regime), with Re = rho_a v d / mu. For fine ash (Re << 1) this reduces to Stokes' law v = (rho_p - rho_a) g d^2 / (18 mu): a 20 micrometre grain falls at only about 2 cm/s, so fine ash stays aloft for days. Air density falls with height as 1.2 exp(-z/8.4 km), so grains fall faster up high.

Drift and deposit

A grain released at plume height H drifts with the local wind while it falls: x_land = integral of u(z)/v_t(z) dz, evaluated numerically for 60 size classes. The wind rises linearly from a ground fraction to its plume-height value (shear). The erupted mass is log-normal in grain size; each class lands at its own distance, smoothed with a 10 % random spread to mimic turbulence, giving the green deposit curve. The orange bars are the live count of simulated grains that have landed, which converges to the curve.

Numerical checks on this page

(1) The solver reproduces Stokes' law for 10 micrometre ash. (2) A time-integrated falling grain approaches the root-solver terminal velocity. (3) For 8 mm lapilli the speed agrees with the quadratic-drag formula. (4) The landing distance in uniform wind lies between the bounds set by the top and ground settling speeds.

Honest simplifications

Grains are spheres with one density; no aggregation of fine ash (which in reality speeds up its fall), no cloud buoyancy or umbrella spreading, no turbulent diffusion beyond a fixed spread, no change of wind direction with height, no particle shape drag, and no plume dynamics: grains are released at a single height from a point. Results show the physics of settling, not a forecast, and are not suitable for hazard assessment.

• Run Moderate eruption: coarse grains drop near the vent and fine ones travel hundreds of kilometres.
• Raise the wind in Strong wind and note the whole footprint stretch out.
• Fine ash only needs a very large map range because the grains settle slowly.
• Compare the first graph with the red Stokes line: the two agree for small grains and split for large ones.