Settling exposure for a particle size distribution ($J(t)={\int }_0^{t}ds/h(s)$)

In a mixed particle-laden gravity current, a size class with settling speed $w$ has concentration $c(w,t)=c_0(w)e^{-wJ(t)}$. The shared reduced gravity is $g^{\prime }(J)=\int g_0(w)c_0(w)e^{-wJ}dw$, and $d{R}^4/dJ=4\mathrm{Fr}(\mathcal{V}/\pi {)}^{3/2}\sqrt{g^{\prime }(J)}$. Finite runout therefore requires ${\int }_0^{\infty }\sqrt{g^{\prime }(J)}dJ<\infty$. A positive minimum settling speed guarantees this; a distribution with arbitrarily small settling speeds need not. For $g_0{c}_0(w)\sim C{w}^p$ at zero, with $p>-1$, the integral converges precisely when $p>1$.