Scaling behavior of bedload transport: what if Bagnold was right?

There is a paradox in the relationship between bedload transport rates and flow variables: laboratory and field studies have reported on how bedload transport rates depend on flow variables through a power law, but none of the empirical laws fitted to the data has managed to provide accurate predictions of bedload transport over a wide range of flow conditions. Inferring bedload transport's scaling behavior from data has remained a stubborn problem because the data are very noisy. It is, therefore, difficult to progress on this problem without some informed speculation about how bedload and flow interact. Ralph Bagnold proposed an original theory to resolve this problem. This paper reviews and updates Bagnold's model by separating the effects of flow resistance and efficiency (energy transfer from water to bedload) on dimensionless transport rates phi. Both variables' contributions to transport rates can be parameterized separately for the three transport regimes that Bagnold defined (no transport, transitional, and sheet flow). We also consider two possible control variables: the dimensionless Shield stress tau* and a dimensionless number related to stream power. In the transitional regime, the dimensionless bedload transport rate scales as phi proportional to tau3, whereas in the sheet-flow regime, it varies as phi proportional to tau5/3. We end up with two Bagnold equations: one based on physical principles and involving Shields stress tau*, flow resistance f, a density ratio, and a bed slope; the other based on non-linear regression and stream power. Compared to a large set of laboratory and field data, predictions from Bagnold's model show reasonable accuracy when the bed is plane.