= Settling exposure for a particle size distribution
{title2=$J(t)=\int_0^t ds/h(s)$}
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 $g'(J)=\int g_0(w)c_0(w)e^{-wJ}\,dw$, and $dR^4/dJ=4\operatorname{Fr}(\mathcal V/\pi)^{3/2}\sqrt{g'(J)}$. Finite runout therefore requires $\int_0^\infty\sqrt{g'(J)}\,dJ<\infty$. A positive minimum settling speed guarantees this; a distribution with arbitrarily small settling speeds need not. For $g_0c_0(w)\sim Cw^p$ at zero, with $p>-1$, the integral converges precisely when $p>1$.
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