2.1 Dynamics of the Turbulent Suspension
 We describe the turbulent mixing of sediment in the water column as a diffusive process. Such diffusive models for dilute particulate suspensions are widely used even though they are based upon a simple representation of the turbulent motions that sustain the suspended material [Rouse, 1938; Elimelech, 1994; Soulsby, 1997]. Throughout this paper we make the simplifying assumption that volumetric effects on the effective viscosity of the mixture [Richardson and Zaki, 1954] and on the sedimentation and suspension of particulate material may be neglected. In such a dilute suspension, neither hindered settling effects [Hunt, 1969; Batchelor, 1982] nor self-diffusion of sediment driven by particle-particle collisions [Hsu et al., 2004] are considered.
 When variations in the flow and suspension parallel to the bed are neglected, the flow and distribution of particulate material vary only vertically, normal to the bed. Based on the Stokes-Einstein diffusion coefficient, and the settling velocity of a spherical particle of silica sand, diameter 100 μm, in water, the particle Péclet number, a measure of the advective to molecular diffusive transport, is of the order of 109. Thus, molecular diffusion of sand-sized particulate material is assumed to be negligible. Under these conditions, the vertical distribution of a dilute polydisperse suspension within a turbulent flow may be modeled using the Reynolds-averaged mass continuity equation
Here φi(z,t) denotes the volumetric concentration of particulate material of class i in suspension, averaged over an appropriate timescale of the suspending turbulent flow; wsi denotes the settling velocity of the ith particle class; and k(z,t) is an eddy diffusivity function which models turbulent mixing [Fredsøe and Deigaard, 1992]. In this paper particles of class i=0 represent material that cannot be entrained into suspension, whereas particles of classes i=1 to N may be suspended by turbulent diffusion within the flow.
 The free surface of the flow is at z=h, while the surface of the bed is at z=η(t). The vertical axis z is defined such that the bed location at t=0 is z=0, and thus η(0)=0. The depth-averaged concentration Φi of material belonging to class i is given by
and the total depth-averaged concentration, Φ, is thus
Henceforth, the total depth-averaged concentration, Φ, will be referred to as the bulk concentration. The settling velocity for the ith particle class, wsi, is a constant which depends on the particle diameter di. (We will assume that all particles have the same shape and density, although this assumption could readily be relaxed.)
 The eddy diffusivity may be modeled in terms of a time-dependent bed friction velocity and a dimensionless function, , that captures the variation in the eddy diffusivity with height above the bed [Soulsby, 1997; Dorrell and Hogg, 2012],
and where κ is von Kármán's constant. Here is a dimensionless vertical coordinate, defined such that denotes the bed and the free surface (see Figure 1). Following the condition that the concentration of suspended particles is small, φi≪1, it is assumed in this paper that the deviation η of the bed location from its initial location is negligible in comparison to the flow depth, η≪h for all times. Under this assumption, it may be readily seen that to leading order. The bed friction velocity is defined as the square root of the basal shear stress, τb, divided by the fluid density, . Changing flow conditions may be modeled as an increase or decrease in the bed friction velocity over time.
 The advection-diffusion equation (1) for each particle class is solved subject to a specified initial concentration φi(z,0) and to two boundary conditions. At the free surface, a no-flux condition is applied,
At the bed, we specify qi(t), the net upward flux of sediment of class i at the boundary. (The alternative approach of imposing a reference concentration has been shown to produce inappropriate concentration profiles [Armanini and Di Silvio, 1988; Cao and Carling, 2002b; Dorrell and Hogg, 2012].) If qi(t)<0, there is net deposition of these particles from suspension onto the bed, while if qi(t)>0, there is net erosion of these particles from the bed into suspension. At the base of the flow, then, the net flux of material being entrained from the bed [Garcia and Parker, 1993; Cao and Carling, 2002a] is given by
To specify qi(t), we must consider carefully the exchange between the suspension and the bed.
2.2 Exchange Between the Suspension and the Bed
 As noted above, we describe the bed in terms of two regions, as illustrated in Figure 1. Material in the bed is presumed to be at a constant packing concentration φm, independent of grain size distribution [see, for example, Dorrell and Hogg, 2010]. The “deep” part of the bed, occupying the region z<η−δ, consists of immobile sediment, in which the submerged weight of the particles is borne by the granular matrix. The volumetric concentration of particles of class i in the deep bed is written as φdbi(z), with at each depth z. A thin upper region, the active layer (η−δ<z<η), consists of sediment that is “exposed” to the overlying water column and may therefore exchange material with it and with the underlying bed. It is important to note that we present a local description of the exchange, rather than a description that is averaged across the width of a channel. In width-averaged models [Parker, 1991; Parker et al., 2000; Strauss and Glinsky, 2012], the active-layer thickness must accommodate variations in η across the channel, due for example to the presence of bed forms, so δ may be large. In local models, in contrast, δ is comparable with the size of the grains, and so may be treated as small. As well as newly exposed or newly deposited bed material, the population of the active layer may contain particles which inhabit it only briefly before being reentrained. For example, in flows over an inerodible bed, such as bedrock or the base of an experimental flume, the active layer will consist only of particles which are deposited and almost immediately reentrained (see section 4.1.3).
 This concept of an active layer is generally attributed to the original work of Hirano [1971; 1972], but with recent modifications by Armanini , Parker et al. , and Blom and Parker  that account stochastically for processes that control the grain-size-dependent vertical fluxes of sediment within the layer. In the present formulation, we assume that the active layer is vertically uniform in each of the species that it comprises, so it may be characterized by the volumetric concentrations φbi(t), and that the total volume fraction of solids, φm, is constant as in the deep bed (). We comment that our framework could readily be amended to avoid these simplifications and thus include, for example, effects such as vertical mixing, shear-induced dilatancy, variations in layer thickness due to changes in the state of the overlying flow, and changes of the packing fraction due to varying compositions. Here, however, our purpose is to pursue the simplest description.
 The composition of the active layer may be determined by considering mass conservation for each class of sediment. Integrating (1) across the active layer, incorporating possible discontinuities in sediment concentrations at z=η−δ and z=δ, and noting that particle diffusion and sedimentation vanish within the bed, the condition of mass conservation requires that
Here dη/dt and dδ/dt respectively denote the bed and active layer growth rates; φdi≡φdbi(η−δ,t), the concentration of class i evaluated at the lower side of the interface between the active layer and the deep bed; and φwi≡φi(η,t), the concentration of class i in the suspension at the base of the water column (see Figure 1).
 The left-hand side of (7) represents the effect of sediment storage in the active layer. If the rate of change of sediment storage is much smaller than the depositional and erosive fluxes, then these storage terms may be neglected. Formally, we assume both that δ→0 and that dδ/dt→0, so the active layer is vanishingly small and approximately of constant thickness. Thus, changes in sediment storage in the active layer may be neglected, and equation (7) reduces to
 Recall that qi represents the net upward flux of material of class i from the active layer into the turbulent suspension. This consists of a settling flux −wsiφwi, plus an erosive flux Qi. If Ei is the erosive flux from a monodisperse bed consisting entirely of particles of class i, then the erosive flux from a polydisperse bed is modeled, as in Strauss and Glinsky , by weighting this flux by the fraction of the active layer consisting of particles of class i,
In particular, if there are no particles of class i in the active layer (i.e., φbi=0), then this class of material cannot be entrained even if the flow is strong enough to do so (i.e., Qi=0 although Ei>0). Thus, weighting the erosion rate of each particle class by the relative amount of that class in the active layer (9) avoids the need to introduce an empirical “hiding factor” [see, e.g., Einstein and Chien, 1953; Shen and Lu, 1983] to account for particle availability on the bed. However, although for simplicity the model neglects hiding effects associated with the sheltering of particles of one size by those of another, such physics may be readily added to the entrainment function (9). Given equation (9) for Qi, we may write the net entrainment flux qi as
 The erosion rate Ei will, in general, depend on the flow conditions near the bed. A simple model, which we will employ in later sections, expresses it in terms of the Shields number θi as
Here mi is a constant with units of velocity, ρs is the particle density, and θci is the critical Shields number, which must be exceeded for particles of class i to be entrained into suspension. θci is typically a relatively weak function of the particle Reynolds number [Soulsby, 1997]. The exponent may take a wide range of values, depending on the type of material in suspension [Garcia and Parker 1991; 1993]; throughout this paper, a value of is taken for noncohesive sediment [van Rijn, 1984c].
 It remains to consider φdi. Here it is necessary to make a distinction between net-depositional situations, dη/dt>0, and net-erosional situations, dη/dt<0 [Parker, 1991]. (Situations in which dη/dt=0 may be approached as limits of either erosional or depositional behavior: we will consider them below.) For simplicity, below we will use the notation .
 In net-depositional situations, when , the active layer is continually supplying material to the top of the deep bed and is therefore only composed of material in suspension. Consequently, the concentrations at the top of the deep bed must be instantaneously equal to the concentrations in the active layer,
and (8) reduces to
We can substitute (10) into (13) and eliminate to obtain the N independent conditions
(Note that if we include an inerodible class of sediment i=0 as discussed below, with E0=0 and φ0=0, then from above φb0=0, so this class is absent from the active layer.) We may now close the system with the constraint . This implies that the bed growth may also be written as the net deposition rate divided by the change in the total concentration between the bed and deposit [see, e.g., Dorrell and Hogg, 2010],
We can solve the N+1 equations represented by (14) and (15) to obtain the instantaneous values of φbi and of as functions of the other variables representing the state of the flow and suspension.
 In net-erosional situations, when , the composition of the deep bed becomes relevant, because new material is continually being excavated and supplied to the water column by erosion. In this regime the condition (12) no longer applies: the composition of the deep bed influences the composition of the active layer, but the composition of the active layer may differ from that at the top of the deep bed. During net-erosional phases, then, we can substitute (10) into (8) and eliminate to obtain the N independent conditions
again together with the constraint , which leads to the condition (15). This again gives us a set of N+1 equations to solve to find the instantaneous values of φbi and of .
 In general, for a flow which is alternately erosive and depositional, careful budgeting is needed to keep track of φdbi(z). For a flow which is purely erosional, the process is simplified, as we shall see below in section 4.1.2.
 We note that in the limit as , both (14) and (16) reduce simply to
confirming that if there is neither net erosion nor net deposition, then the net flux of each particle class between the active layer and the suspension must vanish.
 Finally, we note that a deep bed that is armored by inerodible particulate material, or indeed entirely inerodible, such as the base of a laboratory flume (section 5.2), may be represented within this framework by introducing an inerodible particle class denoted by i=0, so that φ0(z,t)=0, E0=0, and φd0=φm.
 To illustrate how this represents an inerodible bed, it is helpful to consider a thought experiment. Let us suppose that, initially, the flow is sufficiently vigorous that all material that is initially in suspension can be maintained in suspension: further erosion of the substrate is only prevented by the inerodibility of the class 0 material. At each instant, then, we may assume that and seek solutions to the steady state equations (17) for the bed compositions φbi based on the near-bed concentrations φwi. If these can be consistently solved with , then we may set and continue integrating the system forward in time.
 If at some time it is no longer possible to obtain a consistent solution with because , then the flow has become net-depositional. We must now assume that and determine the bed composition from the depositional equations (14) and (15). Since the inerodible substrate is now being buried, φb0=0 henceforth. We may then continue to assume that the flow is net-depositional as long as we can continue to find consistent solutions for the active-layer composition.
 We will explore some consequences of having an inerodible substrate in sections 3.2 and 5.2 below; in section 4.1.3, we will show in more detail that this behavior emerges naturally as the limit of a model in which one particle class is much less erodible than another.