#### Dynamics of the Turbulent Suspension

[10] 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.

[11] 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 10^{9}. 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

- (1)

Here *φ*_{i}(*z*,*t*) denotes the volumetric concentration of particulate material of class *i* in suspension, averaged over an appropriate timescale of the suspending turbulent flow; *w*_{si} denotes the settling velocity of the *i*th 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.

[12] 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 Φ_{i} of material belonging to class *i* is given by

- (2)

and the total depth-averaged concentration, Φ, is thus

- (3)

Henceforth, the total depth-averaged concentration, Φ, will be referred to as the bulk concentration. The settling velocity for the *i*th particle class, *w*_{si}, is a constant which depends on the particle diameter *d*_{i}. (We will assume that all particles have the same shape and density, although this assumption could readily be relaxed.)

[13] 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],

- (4)

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, *φ*_{i}≪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, *τ*_{b}, divided by the fluid density, . Changing flow conditions may be modeled as an increase or decrease in the bed friction velocity over time.

[14] The advection-diffusion equation (1) for each particle class is solved subject to a specified initial concentration *φ*_{i}(*z*,0) and to two boundary conditions. At the free surface, a no-flux condition is applied,

- (5)

At the bed, we specify *q*_{i}(*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 *q*_{i}(*t*)<0, there is net deposition of these particles from suspension onto the bed, while if *q*_{i}(*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

- (6)

To specify *q*_{i}(*t*), we must consider carefully the exchange between the suspension and the bed.

#### Exchange Between the Suspension and the Bed

[15] 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 *φ*_{m}, 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 *φ*_{dbi}(*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).

[16] This concept of an active layer is generally attributed to the original work of *Hirano* [1971; 1972], but with recent modifications by *Armanini* [1995], *Parker et al.* [2000], and *Blom and Parker* [2004] 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 *φ*_{bi}(*t*), and that the total volume fraction of solids, *φ*_{m}, 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.

[17] 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

- (7)

Here d*η*/d*t* and d*δ*/d*t* respectively denote the bed and active layer growth rates; *φ*_{di}≡*φ*_{dbi}(*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 *φ*_{wi}≡*φ*_{i}(*η*,*t*), the concentration of class *i* in the suspension at the base of the water column (see Figure 1).

[19] Recall that *q*_{i} represents the net upward flux of material of class *i* from the active layer into the turbulent suspension. This consists of a settling flux −*w*_{si}*φ*_{wi}, plus an erosive flux *Q*_{i}. If *E*_{i} 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* [2012], by weighting this flux by the fraction of the active layer consisting of particles of class *i*,

- (9)

In particular, if there are no particles of class *i* in the active layer (i.e., *φ*_{bi}=0), then this class of material cannot be entrained even if the flow is strong enough to do so (i.e., *Q*_{i}=0 although *E*_{i}>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 *Q*_{i}, we may write the net entrainment flux *q*_{i} as

- (10)

[20] The erosion rate *E*_{i} 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 *θ*_{i} as

- (11)

Here *m*_{i} is a constant with units of velocity, *ρ*_{s} is the particle density, and *θ*_{ci} is the critical Shields number, which must be exceeded for particles of class *i* to be entrained into suspension. *θ*_{ci} 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].

[21] It remains to consider *φ*_{di}. Here it is necessary to make a distinction between net-depositional situations, d*η*/d*t*>0, and net-erosional situations, d*η*/d*t*<0 [*Parker*, 1991]. (Situations in which d*η*/d*t*=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 .

[22] 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,

- (12)

and (8) reduces to

- (13)

We can substitute (10) into (13) and eliminate to obtain the *N* independent conditions

- (14)

(Note that if we include an inerodible class of sediment *i*=0 as discussed below, with *E*_{0}=0 and *φ*_{0}=0, then from above *φ*_{b0}=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],

- (15)

We can solve the *N*+1 equations represented by (14) and (15) to obtain the instantaneous values of *φ*_{bi} and of as functions of the other variables representing the state of the flow and suspension.

[24] In general, for a flow which is alternately erosive and depositional, careful budgeting is needed to keep track of *φ*_{dbi}(*z*). For a flow which is purely erosional, the process is simplified, as we shall see below in section 4.1.2.

[25] We note that in the limit as , both (14) and (16) reduce simply to

- (17)

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.

[26] 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 *φ*_{0}(*z*,*t*)=0, *E*_{0}=0, and *φ*_{d0}=*φ*_{m}.

[29] 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.