Volcano and ash fallout (3D)

How high does the column go, and where does the ash land?

Plume height. Mastin and co-workers (2009) fitted observed eruptions: the column height in km is about H = 2.0 V0.241 with V the volume rate of dense rock (m³/s). The mass rate is converted with a rock density of 2500 kg/m³. A 1991-Pinatubo-class rate of about 10⁹ kg/s gives roughly 38 km.

Falling grains. A grain of diameter d falls until drag equals its buoyant weight: Cd(Re) ½ ρa v² (πd²/4) = (ρp - ρa) g πd³/6. The drag coefficient C_d follows a standard sphere correlation, solved by bisection for v. For fine grains it reduces to Stokes' law v = (ρp - ρa) g d² / (18 μ); the graph shows the two curves splitting for coarse grains.

Wind carries the ash. Each grain starts high in the column, drifts with the wind (which can grow with height, set by the shear) while it sinks, and spreads by random turbulent steps of size √(2 K dt) along and across the wind. Landing distance in a uniform wind is u z₀ / v_t: fine grains travel hundreds of kilometres, coarse ones fall near the vent. The coloured map on the ground is the mass that has landed in each square kilometre, on a logarithmic scale.

Honest simplifications

The column is a schematic bent-over cloud, not a plume-rise model; grains are spheres of one density and do not clump, aggregate or fall faster as wet clusters; wind is one direction with a linear change in height; the spreading coefficient is a free number. Time is sped up by a large factor so a whole fallout fits in a minute. The 2D page follows only the downwind coordinate; this 3D page adds the crosswind one with the same spreading, so the downwind totals match the 2D numbers.

Model checks (run live)

Mastin scaling for a sixteen-fold change in rate; the Pinatubo-class height; the Stokes limit for a 20 µm grain; drag against weight for a 100 µm grain; the stepper's landing distance against u z₀/v_t; and conservation of mass between emitted, airborne and landed ash (including the ground map).