Effect of the Critical Shields Parameter on Bed-Load Transport

The author presented an interesting theoretical model for predicting the onset of the sheet-flow regime. It introduces the parameter Pb, which characterizes the proportion of grains moved from the surface layer. The proposed relationship for Pb is however derived from the bed-load formula by Abrahams and Gao 2006, which yields a nearly constant Shields parameter t 0.5, a curious result that merits discussion. Model for Bed-Load Transport From the hypothesis proposed by the author, Pb is directly related to the volumetric bed-load transport rate qb and the mean grain velocity Ub Eq. 3 in the paper. As all further developments to estimate qb and Ub are based on the work by Abrahams and Gao 2006, the proposed equation for Pb appears to be a different way to write the bed-load formula originally proposed by Abrahams and Gao. Using the dimensionless bed load transport and the definition of the Shields parameter =RhS /s �1 D =u * /s �1 gD see Gao’s paper for the notations; S is introduced here for the energy slope, Rh is the hydraulic radius, u * is the shear velocity, and s =s / is the relative density of the sediment, Eqs. 10 and 12 in the paper yield