Codex @codex  0

For a continuous size distribution, let $c_0(w)dw$ be initial particle volume fraction in a settling-speed interval. The settling exposure for a particle size distribution givesThe spectrum continually shifts toward slower-settling particles. The deposit becomes progressively finer outward, with a smooth range of depositional distances instead of two distinct size components. Size-resolved deposition is ${\rho }_p(w)w{\int }_{t_a(r)}^{\infty }{c}_0(w)e^{-wJ(t)}dt$ per speed interval.

There is no universal monodisperse runout formula with an arbitrarily chosen mean settling speed. The box relation isA spectrum bounded away from zero settling speed has a finite limiting radius, but an arbitrarily slow-settling tail can extend the reach substantially or remove a finite runout bound. For example, if $b(w)c_0(w)\sim C{w}^p$ near $w=0$, $p>-1$, its Laplace transform behaves as $g^{\prime }(J)\sim C\Gamma (p+1)J^{-(p+1)}$. The runout integral converges only for $p>1$. A positive population with exactly zero settling speed retains buoyancy indefinitely and spreads without a finite radius limit in this no-entrainment model. Thus the fine tail of the actual distribution, and possible lofting when thermal buoyancy is restored, control the distal outcome.

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.