Limiting Behavior of the Probability Distribution in Einstein's Bed-Load Formula and its Improvement

The bed-load transport rates predicted using Einstein's formula in the low and high transport regimes deviate from the experimental data. The deviation, which becomes significant when the transport rate is extremely low or high, may be solely attributed to the folded Gaussian distribution assumed by Einstein. The trends of an Einstein-type formula at the limits can be separately determined by considering the limiting behavior of the embedded probability distribution, which facilitates the derivation of a formula that best fits the bed-load data. By applying a Gamma distribution with the shape parameter equal to 3/2, a modified Einstein's formula is successfully applied to the low and high transport regimes.