The science
Integral plume. A steady, axisymmetric column of gas and ash is described by its mass flux m, momentum flux P = m w and total energy. Air is entrained at the edge: dm/dz = 2α ρa w · πb (Morton-Taylor-Turner, α ≈ 0.09). Buoyancy accelerates or brakes the column: dP/dz = g m (ρa/ρ − 1)/w.
Energy. Entrained air is heated by the hot ash, and the rising mixture does work against gravity and expands: dH/dz = caTa dm/dz − m g − w dP/dz + ½w2 dm/dz, where H is enthalpy flux. The mixture density is 1/ρ = xgas RgT/p + xash/ρs. Thickening the column with cold air can turn a dense, heavy jet into a light, buoyant plume - or it cannot, and the column falls back.
Rise vs collapse. If the mixture never becomes lighter than the surrounding air before its momentum is used up, the column forms a fountain and collapses into pyroclastic density currents (here drawn spreading with an energy-line run-out of H/L = 0.2, a typical value). Wider vents and slower, gas-poor jets entrain too little air.
Scaling. For a buoyancy-dominated plume in a stratified atmosphere the height follows H ∝ F1/4 N−3/4 (F buoyancy flux, N the Brunt-Väisälä frequency, about 0.011 /s in the troposphere), so height grows only as mass flux1/4: ten times more magma adds ~80 % to the height. The empirical fit of Mastin et al. (2009) is H = 2.00 V0.241 (km, V in m3/s dense-rock equivalent); this page compares the integral model, that fit and the Sparks et al. (1997) fit 1.67 V0.259.
Simplifications: steady state, no crosswind in the physics (the wind slider only bends the drawing), exit at atmospheric pressure (no overpressured jet), one particle class, no particle fallout, no water condensation or latent heat, no moist atmosphere, isothermal stratosphere above 11 km (real atmosphere warms above ~20 km), vent at the crater height. The umbrella spreading speed and the run-out of the density currents are scaling estimates, not solutions. The airliner is enlarged about 20 times so that it is visible.
Independent educational model; the volcano is a generic stylised stratovolcano, not a specific mountain.