 Solution

Let the two populations have settling speeds $w_1\gg w_2>0$ and particle volume fractions $c_1,c_2$. In the same mixed box their scalar equations are ${\dot{c}}_j=-w_j{c}_j/h$, while the common reduced gravity is $g^{\prime }=b_1{c}_1+b_2{c}_2$ after neglecting thermal buoyancy. Both populations are carried by the same radius and depth:The coarse ash deposits rapidly near the source; the fine ash remains aloft longer, sustains the current's buoyancy and is carried farther. The composition of the deposit therefore fines outward. Since the remaining fine ash drives the flow transporting both classes, the runout is not obtained by superposing two independent monodisperse currents.

Define the settling exposure for a particle size distribution $J={\int }_0^{t}ds/h(s)$. Then $c_j=c_{j0}{e}^{-w_{j}J}$ andThe integral is finite when both settling speeds are positive. Each deposited mass is ${\Sigma }_j(r)={\rho }_{pj}{w}_j{\int }_{t_a(r)}^{\infty }{c}_j(t)dt$; their sum is the total deposit, with a coarse proximal component and a finer distal tail. If the actual hot-gas density deficit is retained, early loss of coarse ash can make $g^{\prime }$ vanish and loft the remaining fine-rich cloud, shortening its ground-hugging reach relative to this particle-only calculation.