Formal sedimentation runout of a sloping gravity current ($x_{\max }-x_0\sim (h_0/E)[e^{3{\beta }_{\ast }E{B}_o^{1/3}/v_s}-1]$)

Extrapolating the leading weak-settling attenuation of a sloping gravity current to zero flux gives $x_{\max }-x_0=(h_0/E)[\exp (3{\beta }_{\ast }E{B}_o^{1/3}/v_s)-1]$. Since speed tends to zero there, the weak-settling assumption fails in the terminal region; this is a formal leading-model estimate, not a uniformly accurate full-flow runout.