River channel slope, flow resistance, and gravel entrainment thresholds

River beds are traditionally assumed to become mobile at a fixed value of nondimensional shear stress, but several flume and field studies have found that the critical value is higher in steep shallow flows. Explanations for this have been proposed in terms of the force balance on individual grains. The trend can also be understood in bulk-flow terms if total flow resistance has base and additional components, the latter due to protruding immobile grains as well as any bedforms, and the stress corresponding to additional resistance is not available for grain movement in threshold conditions. A quantitative model based on these assumptions predicts that critical Shields stress increases with slope, critical stream power is near-invariant with slope, and each has a secondary dependence on bed sorting. The proposed slope dependence is similar to what force-balance models predict and consistent with flume data and most field data. Possible explanations are considered for the inability of this and other models to match the very low critical values of width-averaged stress and power reported for some low-gradient gravel bed rivers.