## 1. Introduction

[2] Attempts to provide a mechanistic description of bed load transport under uniform equilibrium conditions have invariably fallen into one or the other of two camps, one having its origin in the work of *Einstein* [1950] and the other deriving from the work of *Bagnold* [1956].

[3] The centerpiece of the Einsteinean formulation is the specification of an entrainment rate of particles into bed load transport (pick-up function) as a function of boundary shear stress and other parameters. The work of *Nakagawa and Tsujimoto* [1980], *van Rijn* [1984], and *Tsujimoto* [1991], for example, represent formulations of this type.

[4] In the Bagnoldean formulation, however, a relation for the areal concentration of bed load particles as a function of boundary shear stress derives automatically from the imposition of a dynamic condition at the bed, according to which the fluid shear stress drops to the critical value for the onset of sediment motion. This dynamic condition is referred to here interchangeably as the Bagnold hypothesis or Bagnold constraint. The hypothesis was used by *Owen* [1964] to calculate sediment transport by saltation for the case of wind-blown sand. It is inherent in the bed load formulations of *Ashida and Michiue* [1972] and *Engelund and Fredsoe* [1976] for nearly horizontal beds. *Wiberg and Smith* [1989], *Sekine and Kikkawa* [1992], and *Nino and Garcia* [1994a, 1994b], for example, have used the hypothesis to derive models of bed load on nearly horizontal beds based on an explicit calculation of grain saltation. *Sekine and Parker* [1992] used the Bagnold hypothesis to develop a saltation model for bed load on surface with a mild transverse slope, and *Kovacs and Parker* [1994] extended the analysis of *Ashida and Michiue* [1972] to the case of arbitrarily sloping beds. *Bridge and Bennett* [1992] have employed the Bagnold hypothesis to study the bed load transport of size mixtures.

[5] Based on the most recently published formulations of bed load transport, then, it is possible to say that the field as a whole has tended away from the Einsteinean and toward the Bagnoldean formulation. This notwithstanding, doubts have been expressed from time to time concerning the Bagnold hypothesis. For example, the experimental work of *Fernandez Luque and van Beek* [1976] does not support the Bagnold hypothesis. A reanalysis of the data and formulation presented by *Nino and Garcia* [1994a, 1994b] caused *Nino and Garcia* [1999] to cast further doubts on the hypothesis. *Kovacs and Parker* [1994] were forced to modify the hypothesis in order to obtain a well-behaved theory of bed load transport on arbitrarily sloping beds. Most recently, *Schmeekle* [1999] has provided experimental evidence, and *McEwan et al.* [1999] have provided evidence based on a numerical model suggesting that the Bagnold hypothesis yields very poor results at low transport stage.

[6] It is shown here that the straightforward extension of the Bagnold hypothesis to the case of arbitrarily sloping bed proves to be impossible. In particular, a solution for areal concentration of grains in bed load transport fails to exist for the case of transverse bed slopes beyond some modest value that is typically much smaller than the angle of repose. An alternative formulation that neither satisfies nor requires the Bagnold constraint is the subject of a companion contribution (G. Parker et al., Bedload at low Shields stress on arbitrarily sloping beds: Alternative entrainment formulation, submitted to *Water Resources Research*, 2002). The mechanistic basis for that contribution is presented below.