 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,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 ${\beta }_{\ast }$.

This is an extrapolated leading-model sedimentation distance. Since $u\sim {\beta }_{\ast }{B}^{1/3}$ 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.