## 1. Introduction

[2] Bed load prediction is of primary importance for river engineering, fluvial geomorphology, eco-hydrology, environmental surveys and management, and hazard prediction. Using similarity principles, *Shields* [1936] established a framework for bed load prediction that is still in use today. He considered bed load a threshold phenomenon and established a diagram relating the dimensionless critical shear stress θ_{c} = *τ*_{c}/[(*ρ*_{s} − *ρ*)*gD*] (where *τ*_{c} = *ρgHS* is the critical shear stress, *ρ*_{s} is the sediment density, *ρ* is the water density, *g* is the acceleration of gravity, *D* is the grain diameter, *H* is the water depth and *S* is the energy slope) to the roughness Reynolds number *Re** = *u***D*/*ν* (where *u** = (*τ*/*ρ*)^{1/2} is the shear velocity). Although this curve is not easy to use (because both θ_{c} and *Re** depend on the shear velocity *u**, which implies an iterative approach), one interesting practical issue is that θ_{c} was hypothesized by Shields to be constant when *Re** > 1000, which is the case for most natural flow conditions (rough and turbulent flows). Thus, knowing the value of this constant, the calculation of threshold flow conditions (characterized by a flow depth *H*) for a given sediment (characterized by its grain-size distribution curve) and a given energy slope *S* should be straightforward.

[3] However, whereas *Shields* [1936] proposed an asymptotic value of 0.06 for θ_{c}, the appropriate value for this constant has been widely and continuously discussed since that time. For instance, the well-known bed load transport equation proposed by *Meyer-Peter and Muller* [1948] considered θ_{c} = 0.047. Values as low as 0.01 were also proposed [*Fenton and Abbott*, 1977; *Carling*, 1983; *Mueller et al.*, 2005] as well as values higher than 0.1 [*Mizuyama*, 1977; *Church*, 1978; *Reid et al.*, 1985; *Mueller et al.*, 2005]. More generally, values were proposed in the range 0.03 [*Parker et al.*, 2003] to 0.07 (an exhaustive review was provided by *Buffington and Montgomery* [1997]), with a mean value at approximately 0.045 [*Gessler*, 1971; *Miller et al.*, 1977; *Yalin and Karahan*, 1979; *Saad*, 1989]. The importance attached to this question can easily be understood when considering that in most natural gravel bed rivers the Shields number θ barely exceeds 20% of the critical value θ_{c} [*Parker*, 1978; *Andrews*, 1983; *Mueller et al.*, 2005; *Ryan et al.*, 2002; *Parker et al.*, 2007] and that for these flow conditions, transport rates increase by several orders of magnitude for very small changes in shear stress, which can lead to very large errors in bed load prediction if θ_{c} is not correct.

[4] Numerous explanations can be given for the uncertainty on θ_{c} [*Buffington and Montgomery*, 1997], including the definition of incipient motion itself, the shear stress definition (mean or instantaneous) and calculation (from the energy slope, the velocity profile or the Reynolds stress profile), sediment specificities (near-uniform or nonuniform) and the general protocol used (measurement techniques, duration, and sidewall correction method). A subject that has received less attention is the natural dependence of θ_{c} on flow parameters. In particular, the shear stress *τ* is usually calculated from the energy slope *S* and the flow hydraulic radius *R*. Calculating it thus, the Shields parameter (equation (1)) indicates dependence with the relative depth *R*/*D* (the ratio between the hydraulic radius and the grain diameter) and the slope *S* (considered here in the streamwise direction):

where *s* = *ρ*_{s}/*ρ* is the sediment's relative density. Shields himself first recognized these dependences [*Shields*, 1936] and observed increasing critical Shields stress with increasing slopes. This result has been confirmed since that time by several researchers, on the basis of both flume and field experiments [*Tabata and Ichinose*, 1971; *Aksoy*, 1973; *Bathurst et al.*, 1982; *Bettess*, 1984; *Bathurst*, 1987; *Graf and Suszka*, 1987; *Tsujimoto*, 1991; *Shvidchenko and Pender*, 2000; *Shvidchenko et al.*, 2001; *Mueller et al.*, 2005; *Armanini and Gregoretti*, 2005; *Vollmer and Kleinhans*, 2007; *Lamb et al.*, 2008] (see also J. Bogardi, Sediment transportation in alluvial streams, lecture notes from an international postgraduate course on hydrological methods for developing water resources management, Research Institute for Water Research Development, UNESCO, Budapest, Hungary, 1980). Other researchers proposed a θ_{c}(*R*/*D*) equation [*Mizuyama*, 1977; *Torri and Poesen*, 1988; *Suszka*, 1991; *Lenzi et al.*, 2006] instead of a θ_{c}(*S*) equation, but both types of equations give similar results. This is illustrated in Figure 1, where *Mizuyama* [1977] and *Suszka* [1991] flume experiment results are compared to other flume results after being rearranged in a θ_{c} (*S*) equation using equation (1).

[5] Since these observations were mostly reported for low relative depth *R*/*D* corresponding to gravel initiation of motion on steep slopes, *Bathurst et al.* [1982] hypothesized that the traditional Shields approach (which assumes a constant value of approximately 0.04–0.06 at a high Reynolds number) could be based on the coincidence that most studies involved values of channel slope small enough that the real variation of θ_{c} with flow conditions has been too small to deserve comment. For instance, Shields himself considered the slope effects negligible for the low slopes (≤1%) examined in his analysis [*Shields*, 1936]. In addition, decreasing critical Shields stress could logically have been expected when the channel slope becomes very steep because of increased gravitational effects. The inverse was observed. This contradictory result was interpreted as the consequence of additional effects. For instance, *Coleman and Nikora* [2008] considered it the consequence of flow momentum exchanges through highly permeable gravel beds. It has also often been interpreted as the consequence of bed form drag [*Buffington and Montgomery*, 1997; *Mueller et al.*, 2005]. This latter hypothesis also holds for flume results with near-uniform sediments, as most studies extrapolated to zero the bed load transport rate data even though associated flows are usually associated with the presence of small bed forms (undulating beds), whatever the initial planar bed surface considered. Recent work [*Recking et al.*, 2008a, 2008b] investigated bed load and flow resistance interactions using a data set consisting of 1551 flume data. In these studies, the critical Shields value was defined by incipient deformation of an initially planar bed (recognized by changes in flow resistance with increasing discharge). The results [*Recking*, 2008] showed a strong correlation between θ_{c} and *S*, similar to those obtained in other studies (Figure 1), despite the exclusion of bed form drag in the approach used by *Recking et al.* [2008b]. Consequently, although bed forms can contribute to increasing Shields values in some circumstances, this does not explain the variations in θ_{c} observed with low relative depth on steep slopes. Another argument in favor of not considering the relative depth was proposed by *Yalin* [1977]. He argued that since the initiation of motion must depend only on the flow in the vicinity of the bed and since the flow in the vicinity of the bed is independent of the relative depth *R*/*D*, initiation of motion must be independent of *R*/*D*. This is true as long as the Nikuradse profile is valid with a well-developed logarithmic velocity profile close to the bed. However, for steep slopes and low relative depth flow conditions over gravel beds, several authors measured a deviation from the logarithmic profile both in flume experiments and in the field [*O'Loughlin and Annambhotla*, 1969; *Christensen*, 1971; *Ashida and Bayazit*, 1973; *Mizuyama*, 1977; *Day*, 1977; *Nowell and Church*, 1979; *Marchand et al.*, 1984; *Nakagawa et al.*, 1988; *Bathurst*, 1988; *Jarrett*, 1990; *Aguirre-Pe and Fuentes*, 1990; *Robert*, 1990; *Wiberg and Smith*, 1991; *Tsujimoto*, 1991; *Pitlick*, 1992; *Ferro and Baiamonte*, 1994; *Byrd et al.*, 2000; *Byrd and Furbish*, 2000; *Nikora et al.*, 2001; *Katul et al.*, 2002; *Franca*, 2005]. Instead, a more complicated pattern was generally described, with a roughness layer close to the bed where the velocity profile was nearly constant, and a second zone (above the former), where the velocity profile was logarithmic. The roughness layer corresponds to zones of intense shear downstream of each roughness element (grains and particle clusters, producing Kelvin-Helmholtz instabilities) where the kinetic energy of the mean flow is transformed into turbulence energy. This turbulence thus produced intensifies the mixing or transfer of momentum, resulting in a continuous adjustment in the velocity profile close to the bed [*O'Loughlin and Annambhotla*, 1969]. Several studies showed that in these flows the mean flow velocity and turbulence intensity near the bed decreased with decreasing relative depth *R*/*D* [*Bayazit*, 1976; *Tsujimoto*, 1991; *Wang et al.*, 1993; *Dietrich and Koll*, 1997; *Carollo et al.*, 2005; *Lamb et al.*, 2008] which contradicts *Yalin* [1977] argument.

[6] Grain movements are governed by the association of drag and lift forces, which are dependent, respectively, on the mean flow velocity and the velocity gradient in the vicinity of the grain. If the mean velocity and the velocity gradient are affected by low relative depths on steep slopes, as suggested above, this would have direct consequences on the force balance and θ_{c}. This led *Tsujimoto* [1991] to postulate that the effect of slope on θ_{c} is composed of two parts (equation (2)): Ψ_{1}(*S*) as an effect of gravity itself and Ψ_{2}(*S*) as an effect of the degeneration of velocity distribution owing to small relative depth.

The former would be a decreasing function of *S* while the latter would be an increasing function of *S*. Increasing θ_{c} with increasing slope implies that Ψ_{2} grows faster than Ψ_{1} when the slope is increased. Considering equation (1), this means that the changing flow hydraulics (mean velocity and turbulence profiles) with increasing slopes results in critical relative depths (i.e., *R*/*D* measured at incipient motion) declining more slowly than the rate of slope increase. This could be demonstrated using an analytical model on the basis of a force balance, but the problem is estimating the appropriate near-bed flow velocity and its variations with relative depth on steep slopes. *Tsujimoto* [1991] first successfully reproduced these opposite influences of decreasing relative depth and simultaneously increasing slope using a near bed flow velocity function fitted on his own data. Despite large uncertainties associated with calibrating the model, it produced increasing critical Shields stress values with increasing slope values. *Armanini and Gregoretti* [2005] successfully reproduced incipient motion experiments [*Gregoretti*, 2000] on steep slopes (21% < *S* < 36%) by replacing the logarithmic profile with a linear profile [*Nikora et al.*, 2001]. Their objective was to study effects of grain protrusion and the linear profile used in the model was fitted from the experimental data set considered. By correcting the logarithmic Nikuradse function with a coefficient (a function of the ratio between flow depth and roughness layer thickness) deduced from a turbulence model [*Bezzola*, 2002] for the mean flow velocity and adding a turbulence-induced pressure fluctuation, *Vollmer and Kleinhans* [2007] also reproduced opposite influences of decreasing relative depth and simultaneously increasing slope but finally concluded that both effects were partly compensated as soon as the relative depth was higher than 2 (slopes less than 6%), producing a constant Shields value. *Lamb et al.* [2008] developed a near-bed mean flow velocity profile by matching *Nikuradse* [1933] logarithmic profile with a quadratic profile obtained by integrating a constant mixing length within the roughness layer (which they assumed proportional to the bed roughness *k*_{s}). They also considered a fluctuating turbulent component from data available in the literature. Thus, they obtained a variation of θ_{c} with increasing slope much similar to the equations presented in Figure 1.

[7] Whatever the approach used, one major difficulty consists in evaluating the model's ability to reproduce the roughness layer velocities and its variations with relative depth because available roughness layer velocity measurements are scarce. On the other hand, a large number of flume and field measurements are available for depth average velocities, covering a wide range of relative depth and slope. These data were analyzed in the work of *Recking et al.* [2008b]. In this paper, they are used to fit a velocity profile incorporating a roughness layer [*Aguirre-Pe and Fuentes*, 1990]. This reproduced a confident variation of the near-bed flow velocities with varying relative depth *R*/*D*. This flow velocity function was incorporated into a critical Shields stress model on the basis of a force balance. Finally, the results were compared with flume and field critical Shields stress data.