= Solution
The <formal sedimentation runout of a sloping gravity current> is obtained when the bracket in the leading <weak-settling attenuation of a sloping gravity current> reaches zero. With the consistent speed factor,
$$
\boxed{x_{\max}-x_0\sim\frac{h_0}{E}\left[\exp\left(\frac{3\beta_*E B_o^{1/3}}{v_s}\right)-1\right].}
$$
Setting $x_0=0$ gives the distance from the inlet. The corresponding formal expression with a stipulated independent speed coefficient is identical with that coefficient replacing $\beta_*$.
This is an extrapolated leading-model sedimentation distance. Since $u\sim\beta_*B^{1/3}$ tends to zero there, the weak-settling condition fails in the terminal region; the exponential value is not a uniformly controlled prediction of the fully coupled flow. In particular, disappearance of particle buoyancy need not coincide with instantaneous disappearance of every inertial motion.
Back to article page
