## 1. Introduction

[2] The bed load sediment flux, defined as the solid volume of bed load particles crossing a vertical surface per unit time per unit width, figures prominently in descriptions of sediment transport and the evolution of alluvial channels. Translating this definition of the flux into conceptually simple quantities that accurately characterize the collective motions of particles, however, is not necessarily straightforward, and quantitative definitions of the flux have several forms. We note at the outset that, when viewed at the particle scale, the instantaneous, vertically integrated flux *q*_{A}(*t*) [L^{2} t^{−1}] associated with a surface *A* [L^{2}] is precisely defined as the surface integral of surface-normal velocities of the solid fraction, namely

where **u**_{p} [L t^{−1}] is the discontinuous particle velocity field viewed at the surface *A*, **n** is the unit vector normal to *A*, and *b* [L] is the width of *A*, where *A* extends over the vertical domain of moving particles (Figure 1). This precise definition, however, is impractical. Except possibly using high-speed imaging of a small (observable) number of particles [*Drake et al.*, 1988; *Lajeunesse et al.*, 2010; *Roseberry et al.*, 2012] at high resolution, the flux described by (1) is virtually impossible to measure, and we are far from possessing a theory of sediment transport that describes the velocity field **u**_{p}as it responds to near-bed turbulence [*Parker et al.*, 2003]. Conventional descriptions of the flux therefore instead appeal to measures of collective particle behavior, specifically averaged quantities such as the average particle velocity and concentration, to replace the detailed information contained in the particle velocity field **u**_{p} at the surface *A.*

[3] With equilibrium (i.e., quasi-steady and uniform) bed and transport conditions, for example, the sediment flux normally is defined in “flux form” as the product of a mean particle velocity*U*_{p} [L t^{−1}] and a particle concentration, namely, the volume of particles in motion per unit streambed area [e.g., *Bridge and Dominic*, 1984; *Wiberg and Smith*, 1989; *Seminara et al.*, 2002; *Parker et al.*, 2003; *Francalanci and Solari*, 2007; *Wong et al.*, 2007; *Lajeunesse et al.*, 2010], herein referred to as the bed load particle activity *γ*[L]. That is, for one-dimensional transport in the*x* direction the flux *q*_{x} [L^{2} t^{−1}] is

with the caveat that *U*_{p} and *γ* represent macroscopic quantities averaged over stochastic fluctuations [*Wong et al.*, 2007]. Note that this is like the definition of advection associated with a continuous medium. As elaborated below, to describe the sediment flux as the product of a mean velocity and a concentration indeed assumes a continuum behavior where active (moving) particles are uniformly (albeit quasi-randomly) distributed. But as recently noted [*Schmeeckle and Furbish*, 2007; *Ancey*, 2010], the continuum assumption is rarely satisfied for sediment particles transported as bed load, particularly at low transport rates [*Roseberry et al.*, 2012], and the details of the averaging, whether involving ensemble, spatial or temporal averaging [*Coleman and Nikora*, 2009], matter to the physical interpretation as well as the form of the definition of the flux. *Ancey* [2010] notes in his review of several definitions of the flux that it remains unclear how the flux is actually related to the mean particle velocity and the particle concentration.

[4] Another important definition of the bed load sediment flux is the “entrainment form” of this quantity, first introduced by *Einstein* [1950] and recently elaborated by *Wilcock* [1997a], *Parker et al.* [2000], *Seminara et al.* [2002], *Wong et al.* [2007], *Ganti et al.* [2010]and others. By this definition, with quasi-steady bed and transport conditions the flux is equal to the product of the volumetric rate of particle entrainment per unit streambed area,*E* [L t^{−1}], and the mean particle hop distance, [L], measured start to stop. That is,

This is essentially a statement of conservation of particle volume where, assuming spatially uniform transport, rates of entrainment and deposition are steady, uniform and everywhere balanced. The value of this definition is highlighted in treating tracer particles [*Ganti et al.*, 2010], notably involving exchanges between the active and inactive layers of the streambed [*Wong et al.*, 2007]. What is unclear is how the ingredients of (3), notably the distribution of particle hop distances with mean , translate to unsteady and nonuniform conditions [*Lajeunesse et al.*, 2010], and the extent to which this and other definitions overlap or match (2) [*Ancey*, 2010]. On this point we note that any formulation of the flux must be consistent with (1).

[5] At any instant the solid volume of bed load particles in motion per unit area of streambed, the particle activity *γ*, can vary spatially due to short-lived near-bed turbulence excursions as well as longer-lived influences of bed form geometry on the mean flow [*Drake et al.*, 1988; *McLean et al.*, 1994; *Nelson et al.*, 1995; *Schmeeckle and Nelson*, 2003; *Singh et al.*, 2009; *Roseberry et al.*, 2012]. During their motions, particles respond to turbulent fluctuations and interact with the bed and with each other so that, at any instant within a given small area, some particles move faster and some move slower than the average within the area, and fluctuations in velocity are of the same order as the average velocity. Moreover, a hallmark of bed load particles is their propensity to alternate between states of motion and rest over a large range of timescales. These attributes mean that, in relation to the definition (2) above, the bed load particle flux involves an advective part, as is normally assumed, but more generally it also involves a diffusive part associated with variations in particle activity and velocity [*Lisle et al.*, 1998; *Schmeeckle and Furbish*, 2007; *Furbish et al.*, 2009a, 2009b]. In relation to the definition (3) above, because the distribution of particle hop distances may be considered the marginal distribution of a joint probability density function of particle hop distances and associated travel times [*Lajeunesse et al.*, 2010], a more general form of this definition similarly involves a diffusive part as well as a time derivative term that represents a lag effect associated with the exchange of particles between the static and active states.

[6] The purpose of this contribution is to clarify the points above, namely, how variations in particle activity and velocity influence the volumetric bed load flux, **q** = **i***q*_{x} + **j***q*_{y} [L^{2} t^{−1}], with components *q*_{x} and *q*_{y} parallel to the coordinates *x* [L] and *y*[L], selected here to coincide with the streamwise and cross-stream directions, respectively. Our analysis involves a probabilistic formulation wherein particle positions and motions are treated as stochastic quantities, leading to a kinematic description of**q** that illustrates how and why it involves both advective and diffusive terms, borrowing key elements from closely related formulations [*Furbish et al.*, 2009a, 2009b; *Furbish and Haff*, 2010].

[7] It is straightforward using the probabilistic framework of the Master equation [e.g., *Risken*, 1984; *Ebeling and Sokolov*, 2005] to formulate a statement of conservation of particle concentration *c* having the form ∂*c/*∂*t* = −∇ · **q** [*Ancey*, 2010], and then by inspection extract from this statement a kinematic description of the flux **q** [e.g., *Furbish et al.*, 2009a], therein revealing that it has both advective and diffusive parts. More challenging, however, is to formulate a definition of **q** directly from a description of particle motions as *Einstein* [1905] did for Brownian motions. This is particularly desirable inasmuch as a direct formulation more fully clarifies the geometrical and kinematic ingredients of the flux, including its relation to such quantities as particle hop distances and velocities [e.g., *Drake et al.*, 1988; *Wilcock*, 1997b; *Wong et al.*, 2007; *Ancey*, 2010], and variations in particle velocity and activity.

[8] In section 2we formulate a qualitative version of the one-dimensional flux*q*_{x}, with the purpose of clarifying key geometrical and kinematic ingredients in the problem, notably particle size, shape and velocity, and spatial variations in particle concentration. We show how the definitions (1) and (2) are related. This provides the basis for illustrating that, in appealing to averaged particle quantities (specifically the mean particle velocity and concentration) to replace the detailed information contained in the discontinuous particle velocity field **u**_{p} at the surface *A*, the resulting description of the flux must in general involve both advective and diffusive parts. In section 3we provide a more formal, probabilistic description of the one-dimensional flux, and describe the implications of different definitions of the probability distribution of particle displacements versus hop distances. In the final part of this section we generalize to the two-dimensional case. Insection 4we show the flux form of the Exner equation to illustrate how it is like a Fokker-Planck equation, and for comparison we obtain the entrainment form of the Exner equation to illustrate how this form involves a time derivative term (not contained in the Fokker-Planck equation) that represents a “memory” (or lag) effect associated with the exchange of particles between the static and active states. This result has implications for the use of the related entrainment formulation of conservation of tracer particles. Insection 5 we elaborate how ensemble, spatial and temporal averaging matter in defining the flux, and we consider time averaging of the flux to suggest how persistent spatial variations in particle activity associated with bed topography influence the flux. For simplicity throughout, we consider transport of a single particle size, then briefly comment on the problem of generalizing the formulation to mixtures of sizes in section 6.

[9] As noted by *Ancey* [2010]and others, there is no unique way to define the solid volumetric flux. Nonetheless, an unambiguous, probabilistic definition exists. Beyond a definition of the bed load flux, moreover, the formulation highlights that the probability distribution of particle displacements, including details of how this distribution is defined, has a central role in describing particle motions across a range of scales. This is particularly significant in view of a growing interest in the possibility of non-Fickian behavior in the transport of sediment and associated materials [e.g.,*Nikora et al.*, 2002; *Schumer et al.*, 2009; *Foufoula-Georgiou and Stark*, 2010; *Bradley et al.*, 2010; *Ganti et al.*, 2010; *Voller and Paola*, 2010; *Hill et al.*, 2010; *Ball*, 2012; *Martin et al.*, 2012], and in relation to connecting probabilistic descriptions of particle motions with treatments of fluid motion.

[10] In companion papers [*Roseberry et al.*, 2012; *Furbish et al.*, 2012a, 2012b] we present detailed measurements of bed load particle motions obtained from high-speed imaging in laboratory flume experiments. These measurements support key elements of the formulation described here.