Quantitative definition of Shields criterion for incipient sediment motion

The Shields diagram has been widely used in the description of the threshold condition for incipient sediment motion. Because the current definition of the Shields criterion is generally subjective, the relevant data are subject to high uncertainty. To date, several efforts have been made to define the threshold condition quantitatively, but they are all based on certain low bedload rates that were empirically chosen. The current study aims to provide a theoretical framework to quantify the Shields criterion based on the concept of the pickup probability of sediment particles. The results show that the Shields criterion can be reasonably described with a low pickup probability (p = 1%), which can be further converted to a low dimensionless bedload rate (i.e., $\varphi /{\theta }^{1.5}=0.13$, where $\varphi$ is the Einstein number and $\theta$ is the Shields number). The new approach is finally validated with experimental data of bedload rates published in the literature.