Vertical sorting and the morphodynamics of bed form–dominated rivers: A modeling framework

Authors


Abstract

[1] Uncertainties and errors in predictions by morphodynamic models of rivers with nonuniform sediment are usually attributed to shortcomings pertaining to the submodel of sediment transport. This mistakenly neglects shortcomings in the submodel of sediment continuity, which describes the vertical sorting and the bed surface composition, whereas the latter is one of the main input parameters for calculating sediment transport and thus morphological changes. Hirano [1971, 1972] was the first to develop a sediment continuity model for nonuniform sediment. Since this commonly used Hirano active layer model and its variants suffer from a number of shortcomings, the authors have developed a new type of sediment continuity model that describes the bed composition and vertical sorting fluxes without distinguishing discrete bed layers. This continuum sorting model is aimed at conditions dominated by bed forms and bed load transport. It is based on (1) the Parker-Paola-Leclair framework for sediment continuity, (2) the Einstein step length formulation, (3) a newly developed lee sorting function, and (4) a newly developed method to account for the variability in bed form trough elevations. The resulting model is deterministic in the computation of the vertical sorting profile and is probabilistic in terms of the riverbed surface due to the presence of dunes.

1. Introduction

[2] In morphological river models the bed material may be characterized by one specific grain size in cases where sediment sorting processes do not play a role. Vertical, lateral, or longitudinal sorting of the bed material may occur when the river bed is characterized by a range in grain sizes as fine grains are picked up and transported more easily than coarse ones. Through affecting the small-scale morphology (e.g., dune dimensions, bed surface composition, and bed roughness), sediment sorting also influences grain size-selective sediment transport, changes in mean bed level, and water levels.

[3] A change in mean bed level results from divergences in the transport rate, which is expressed by the sediment continuity equation or mass balance equation. In cases where the bed material consists of multiple size fractions, divergences in the transport rate of size fractions will result in a change in the composition of the bed surface and/or aggradation or degradation of the river bed. A further complication in the mass balance is the preference of certain size fractions to be deposited at specific bed elevations, i.e., the vertical sorting. A plane bed is often covered with a coarse bed layer (mobile pavement or armor layer), whereas under bed form conditions the coarse size fractions are mainly found in the lower parts of the bed forms (Figure 1).

Figure 1.

Flume experiment A2E conducted by Blom et al. [2003], with an interpretation of the vertical sorting pattern. The flow is from top to bottom.

[4] Blom et al. [2003] describe sediment sorting mechanisms that play a role under bed form-dominated conditions. The above mentioned downward coarsening trend of bed form material results from the avalanching of grains down the bed form lee face [Bagnold, 1941; Allen, 1965]. When conditions are well above the shear stresses for incipient motion of all size fractions in the mixture, the coarse sediment in the lower parts of the bed forms takes part in the sediment transport processes. When conditions of partial transport prevail and a significant amount of coarse material is not or is barely transported by the flow, the coarse sediment gathers below the migrating bed forms, forming an essentially immobile coarse layer. Besides the avalanching process at the bed form lee face and partial transport, the winnowing of fines from the trough surface and subsurface also contributes to the formation of a coarse bed layer underneath migrating bed forms.

[5] The interaction among grain size-selective sediment transport, vertical sorting, and net aggradation or degradation is described in terms of sediment continuity models. Blom [2003] distinguishes four types of sediment continuity models: burial depth models, bed layer models, grain-scale models, and depth-continuous models. Burial depth models describe the time evolution of the transport of (tracer) particles and their burial into the bed [e.g., Crickmore and Lean, 1962]. Strictly speaking, burial depth models are not true sediment continuity models as they do not describe the above interaction. Bed layer models are sediment continuity models in which the bed is divided into a certain number of discrete and homogeneous bed layers [e.g., Hirano, 1971]. Grain-scale models are defined as sediment continuity models that consider sediment entrainment and deposition at the scale of grains [e.g., Tsujimoto and Motohashi, 1990], whereas the other types consider processes at a scale of a large number of bed forms. Depth-continuous models do not distinguish discrete bed layers [Armanini, 1995; Parker et al., 2000]. They include a probability density function of bed surface elevations so as to account for the likelihood of a bed elevation being exposed to the flow.

[6] Hirano [1971, 1972] was the first to develop a sediment continuity model for nonuniform sediment. In his active layer model the bed is divided into a homogeneous top layer, i.e., the active layer, and a nonmoving substrate. The active layer represents the part of the bed that interacts with the flow and that determines the rate and composition of the transported sediment. Sediment fluxes between the active layer and the substrate occur in case of net aggradation or degradation only. Various authors have proposed additions to the Hirano active layer model [also see Blom, 2003]: (1) introduction of a term describing the time evolution of the storage of sediment in a thin bed load layer on top of the active layer [Armanini and Di Silvio, 1988; Parker, 1991; Di Silvio, 1992]; (2) introduction of a formulation for suspended load transport [Armanini and Di Silvio, 1988; Holly and Rahuel, 1990; Di Silvio, 1992]; (3) introduction of a formulation for particle abrasion [Parker, 1991]; (4) modifications to the composition of the depositional flux to the substrate [Parker, 1991; Hoey and Ferguson, 1994; Toro-Escobar et al., 1996]; (5) introduction of an additional layer below the active layer to account for vertical sediment exchange due to occasionally deep bed form troughs [Ribberink, 1987; Di Silvio, 1992]; (6) subdivision of the nonmoving substrate into different layers for bookkeeping purposes [Sloff et al., 2001]; and (7) introduction of a mixing coefficient to account for the effects of biological mixing on tidal flats in estuaries (i.e., bioturbation) [Van Ledden and Wang, 2001]. Yet, the Hirano bed layer model and its variants suffer from a number of shortcomings. They fail to describe vertical sorting fluxes through bed form migration, i.e., through grain size-selective deposition down the bed form lee face and the variability in trough elevations. In most sediment continuity models, vertical sediment fluxes within the bed occur through net aggradation or degradation only, whereas flume experiments have shown that these fluxes also occur in situations without net aggradation or degradation [e.g., Blom et al., 2003]. Another problem of the bed layer models is that in certain situations, their set of equations becomes elliptic in parts of the space-time domain [Ribberink, 1987]. Solving the set of equations then requires future time boundaries, which is physically unrealistic. A final problem is that the definition of the thickness of the bed layers remains rather arbitrary. From a physical point of view, it is not straightforward to distinguish between the range of bed elevations interacting with the flow regularly (i.e., the active layer), the range interacting with the flow only occasionally (i.e., the exchange layer in the Ribberink two-layer model), and the range not interacting with the flow at all (i.e., the substrate). In morphological models the bed layers' thicknesses are therefore usually simply used as calibration parameters.

[7] In order to overcome these shortcomings, the authors have developed a new depth-continuous sediment continuity model, i.e., a continuum sorting model. The present paper describes its components and its derivation. The continuum sorting model is based on the framework for sediment continuity introduced by Parker et al. [2000]. This Parker-Paola-Leclair (PPL) framework offers the possibility of describing the vertical sorting fluxes continuously over bed elevations and relating the sorting fluxes at a certain bed elevation to its likelihood of being exposed to the flow. The PPL model is called a framework as formulations for the grain size-specific and elevation-specific entrainment and deposition fluxes remained to be derived. In the present paper the authors derive formulations for these fluxes for conditions dominated by bed forms and bed load transport.

[8] While this paper describes a new mathematical modeling framework for taking into account the impact of vertical sorting upon the large-scale morphodynamics of bed form-dominated rivers, two follow-up papers (A. Blom et al., manuscripts in preparation, 2004) will consider the simplification of this modeling framework to two reduced sorting models (an equilibrium sorting model and a sorting evolution model, respectively) as well as application of these reduced models to measured data from flume experiments.

2. Parker-Paola-Leclair (PPL) Framework for Sediment Continuity

[9] Sediment conservation of size fraction i at elevation z is given by Parker et al. [2000]:

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where equation imagei(x, z, t) denotes the concentration of size fraction i at elevation z (equation imagei = cbequation imagesequation imagei), equation imageei (x, z, t) denotes the deposition density of size fraction i defined such that equation imageeidxdz is the volume of sediment of size fraction i that is deposited per unit width and time in a bed element with sides dx and dz at elevation z, equation imageei(x, z, t) denotes the entrainment density of size fraction i defined likewise, equation imagei (x, z, t) denotes the volume fraction content of size fraction i at elevation z (Figure 2), equation images(x, z, t) denotes the probability distribution of bed surface elevations indicating the probability that the bed elevation is higher than z, and cb denotes the concentration of sediment in the bed (cb = 1 − λb, where λb denotes the porosity). The bar indicates that the parameter is averaged over some horizontal distance, e.g., a large number of bed forms, x denotes the horizontal coordinate on the scale of series of bed forms, z denotes the vertical coordinate, and t denotes the time coordinate.

Figure 2.

A series of bed forms, for which equation imagei expresses the volume fraction content of size fraction i at elevation z, equation images the probability that the bed surface elevation is higher than z, equation imagee is the probability density that the bed surface elevation is equal to z, and equation imagea is the mean bed level.

[10] The probability density function (PDF) of bed surface elevations, equation imagee(x, z, t), expresses the probability density that the bed surface elevation equals z or the likelihood of elevation z being exposed to the flow (Figure 2). Integration of the PDF of bed surface elevations, equation imagee, over bed elevations −∞ to z yields the probability of exceeding a certain bed elevation, equation images:

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where equation images not only indicates the probability that the bed surface elevation is higher than z but also equals the proportion of the bed at elevation z that is covered with sediment [Crickmore and Lean, 1962].

[11] We now perform the following coordinate transformation: equation image = x, equation image = t, and equation image = zequation imagea, wherein equation image is the deviation from the mean bed level, equation imagea(x, t) (Figure 2). Applying the chain rule yields that the time derivative ∂equation images/∂t in equation (1) can be written as

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since

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where equation images(equation image, equation image, equation image) denotes the probability distribution of bed surface elevations relative to the mean bed level, equation imagea. With equation (3), equation (1) becomes

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Adding up equation (5) over all grain sizes yields

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where equation imagee(x, z, t) denotes the deposition density defined such that equation imageedxdz is the volume of all size fractions deposited in a bed element with sides dx and dz at elevation z per unit width and time (equation imagee = equation imageei, where N denotes the total number of size fractions) and equation imagee(x, z, t) the entrainment density defined likewise (equation imagee = equation imageei). Integration of equation (6) over all bed elevations yields

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where equation image(x, t) denotes the volume of all grain sizes deposited per unit area and time (equation image = equation imageedz) and equation image(x, t) denotes the volume of all grain sizes entrained per unit area and time (equation image = equation imageedz). Appendix A shows that the integral in equation (7) equals zero so that equation (7) reduces to

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in which we recognize the sediment continuity or mass balance equation, where equation imagea(x, t) denotes the bed load transport rate averaged over a series of bed forms.

[12] Equations (5), (6), and (8) compose the fundamental set of equations of the PPL framework for sediment continuity. Now, to complete this set of equations, we will derive formulations for the deposition and entrainment densities, equation imageei and equation imageei, for bed form-dominated conditions. This is done in a few steps. First, we will analyze the migration of a single bed form, in which we successively derive a relation between the grain size-specific deposition and entrainment fluxes over the bed form stoss face (section 3.1), derive a formulation for the deposition rate over the lee face (section 3.2), derive a formulation for the composition of the lee deposit (section 3.3), convert the equations into bed elevation-specific ones (section 3.4), and develop a model describing the grain size-specific deposition down the lee face of a single bed form (section 3.5). Secondly, we will couple the single bed form migration approach to the PPL framework (section 4.1). Then, we will introduce the irregularity of bed forms by incorporating the statistics of bed form trough elevations (section 4.2). Section 5 will briefly describe the reduction of the resulting continuum sorting model to both an equilibrium sorting model and a sorting evolution model, which will be explained in detail in two follow-up papers (A. Blom et al., manuscripts in preparation, 2004). Section 6 presents some results of two submodels of the continuum sorting model.

3. Migration of a Single Bed Form

3.1. Einstein Step Length Formulation

[13] We now divide each bed form into a stoss and a lee face (Figure 3) and assume that on the stoss side, deposition and entrainment occur simultaneously, while on the lee side, only deposition occurs. At the stoss face we apply the step length formulation first introduced by Einstein [1950] in order to relate the grain size-specific deposition rate over the bed form stoss face to the grain size-specific entrainment rate. The Einstein step length, Λ, is defined as the average distance covered by a particle from the moment it is picked up until saltation ceases and a period of rest on the bed, i.e., the rest period, begins. On the basis of experiments [Einstein, 1937], Einstein [1950] proposes that the average step length is a linear function of the grain diameter:

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where d denotes the grain size and α denotes the dimensionless step length. For uniform sediment, Einstein [1937] and Fernandez-Luque and Van Beek [1976] found that α is a constant equal to 100 and 288, respectively. Yalin [1977] and others suggest that α slightly increases with increasing shear stress. For nonuniform sediment, Tsujimoto [1990] found that the dimensionless step length increases from 10 to 50 with increasing shear stress. For simplicity, we assume the dimensionless step length to be independent of shear stress, whence it is not affected by the variation of shear stress over the stoss face. In addition, we assume the step length small compared to the stoss length so that we may neglect the reduction in step length when a particle falls over the crest of the bed form.

Figure 3.

Division of bed form into stoss and lee sides, with accompanying entrainment and deposition fluxes.

[14] Let us first consider the relation among entrainment, deposition, bed load transport, and step length in more detail. For uniform sediment and after [Nakagawa and Tsujimoto, 1980], Parker et al. [2000] give the fundamental relations between the deposition rate D and the entrainment rate E and between the bed load transport rate q and the entrainment rate E:

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where fp(ξ) denotes the probability density that a particle, once it is entrained, travels a distance ξ. Equation (10) expresses that a particle is deposited at x when it has been entrained a certain distance upstream from x multiplied by the probability density that its step length is equal to this distance. Equation (11) expresses that a particle passes a cross section at x when it has been entrained a certain distance upstream from x multiplied by the probability density that its step length is equal to or larger than this distance. Tsujimoto and Motohashi [1990] propose a relation for the PDF of step lengths, fp, for plane bed conditions and uniform sediment. For simplicity, we assume the step length to be a deterministic parameter so that fp reduces to the Dirac delta function and equations (10) and (11) yield

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Equation (12) expresses that the deposition rate at a certain point on the stoss face equals the entrainment rate one step length upstream of this point. Equation (13) expresses that sediment passing the cross section at x has been picked up between x and one step length upstream of x. If the entrainment rate E does not vary significantly within the distance of one step length Λ, equation (13) reduces to

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This relation was first used by Einstein [1950].

[15] We now return to nonuniform sediment. Let the weighted entrainment rate Esi(x) denote the volume of sediment of grain size di locally entrained from the stoss face per unit area and time:

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where x denotes the horizontal coordinate on the scale of a single bed form (Figure 3) and the subscript s indicates the stoss face. The subscript u indicates the case of sediment of only this size fraction. Equation (15) expresses that the entrainment of size fraction i at coordinate x equals the entrainment as if it were uniform sediment of grain size di multiplied by its proportion at the bed surface at x. This is comparable to calculating the transport rate of size fraction i in a sediment mixture using qi = Fiqiu, where qiu, like Esiu, may include hiding exposure effects. Equation (12) tells us that at the stoss face the weighted deposition rate Dsi of size fraction i at x equals the weighted entrainment rate of this size fraction one step length upstream of x:

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where

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Similar to equation (13), the local bed load transport rate of size fraction i at the stoss face is given by

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where qsi(x) denotes the transport rate of size fraction i at coordinate x at the stoss face (qsi = qsFqsi), qs(x) denotes the total transport rate at coordinate x at the stoss face, and Fqsi(x) denotes the volume fraction content of size fraction i in the transported mixture at coordinate x at the stoss face.

[16] The present section has shown how the Einstein step length formulation enables us to relate the grain size-specific deposition rate to the grain size-specific entrainment rate at the stoss face of a bed form.

3.2. Deposition Rate at the Lee Face

[17] In the present section a formulation for the average total deposition rate over the bed form lee face will be derived by applying the sediment continuity equation in equation (8) to the local bed form surface:

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where η(x) denotes the local bed surface elevation. Averaging equation (21) over one bed form yields

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wherein entrainment on the lee face is neglected and where λ denotes the bed form length, λs denotes the horizontal length of the stoss face, λl denotes the horizontal length of the lee face, Dl denotes the average total deposition rate on the lee face, and Esnet denotes the net entrainment rate on the stoss face, which will be discussed in the next paragraph. The bed form-averaged transport rate is given by qa = (1/λ) equation imageq(x) dx, and the bed form-averaged bed level is given by ηa = (1/λ) equation image η(x) dx. Note that the average transport rate, qa, and the mean bed level, ηa, are here defined as averaged over a single bed form. They can still show a spatial variation between bed forms, as suggested by equation (22).

[18] The net entrainment rate on the stoss face, Esnet, equals

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which is found by introducing equation (16). Consistent with equation (18), we neglect the contribution from −Λi to 0 in the second integral so that equation (23) reduces to

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Comparison of equation (24) with equations (19) and (20) shows that the total rate of sediment transport approaching the bed form crest, qtop, is related to the net entrainment rate, Esnet, in the following way:

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in which

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where qtop(x, t) denotes the total bed load transport rate at the bed form crest, qtopi(x, t) denotes the bed load transport rate of size fraction i at the bed form crest, and Ftopi(x, t) denotes the volume fraction content of size fraction i in the sediment transported over the bed form crest. Using equation (25), equation (22) can be written as

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which tells us that the average lee face deposition rate is determined by (1) the net entrainment rate on the stoss face (which equals the amount of sediment transported over the crest divided over the length of the stoss face) and (2) the divergence in the total bed load transport rate.

[19] Thus equation (28) provides another component of the formulations for the vertical sorting fluxes required for the new continuum sorting model: the formulation for the average total deposition rate over the lee face.

3.3. Composition of the Lee Face Deposit

[20] Now, we need to find a formulation for the composition of the lee face deposit. For this purpose, we again apply the sediment continuity equation (8) to the local bed form surface, yet now we consider only a single size fraction:

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where Di, Ei, and qi are related to D, E, and q according to D = equation imageDi, E = equation imageEi and q = equation imageqi. Averaging equation (29) over one bed form yields

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which, using equations (23) and (25), can be reformulated to

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where Fleei denotes the volume fraction content of size fraction i in the deposit at the bed form lee face, qai denotes the bed form-averaged bed load transport rate of size fraction i(qai = Faiqa), and Fai denotes the bed form-averaged volume fraction content of size fraction i in the bed load transport. Equation (31) tells us that the composition of the lee face deposit, Fleei, is determined by (1) the composition of sediment transported over the bed form crest and by (2) the divergence in the bed load transport of size fractions.

[21] In order to solve for the average total deposition rate over the bed form lee face, Dl, in equation (28) and the composition of the lee face deposit, Fleei, in equation (31), we need a formulation for the average bed load transport rate of size fraction i, qai. For that purpose, we first consider the transport rate over the lee face and apply equation (21) to the bed form lee face. We assume the lee face of each bed form to have a uniform slope and apply the simple wave theory to bed form migration [Bagnold, 1941], which yields that Dl is uniform over the lee face (Dl (x) = Dl = −∂ql/∂x). As a boundary condition, we assume that the transport rate is equal to zero at the trough (ql = 0 at x = λ), which results in

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In reality, in the average transport rate of size fraction i over the lee face, qlai, there will be a slight bias toward the coarser particles since, on average, they will be transported farther down the lee face than the finer ones. For simplicity, this has been neglected. The bed form-averaged bed load transport rate of size fraction i, qai, is then obtained by using equation (19) for the stoss face and equation (33) for the lee face:

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Using equations (31), (28), and (35), we are now able to solve for the composition of the total lee face deposit. As a next step, we need to develop a lee sorting function that describes how the size fractions in the lee deposit are distributed over the bed form lee face. First, the present formulations dependent on x will be transformed into formulations dependent on z.

3.4. Elevation-Specific Formulations

[22] Since formulations in the PPL framework are elevation-specific, we need to transform the formulations derived in the previous few sections into formulations dependent on bed elevation z. Appendix B shows in detail how qtopi in equation (26) and qai in equation (34) can be written as equations (B16) and (B17):

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[23] Thus we have converted the formulations previously written as a function of the coordinate x into formulations dependent on bed elevation z so as to make them suitable for being incorporated into the PPL framework.

3.5. Lee Sorting Function

[24] In the present section we will derive a formulation for the grain size-specific deposition down the bed form lee face. To that end, we introduce the quantity Fleeloci(z, t), which denotes the volume fraction content of size fraction i in the sediment deposited at elevation z at the bed form lee face. The amount of sediment of size fraction i deposited at elevation z at the lee face then equals DlFleeloci(z). We define Fleeloci as

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where ωi (z) is the lee sorting function, which determines to what extent a specific size fraction that is transported over the bed form crest is deposited at a certain elevation of the lee face. A first constraint to the lee sorting function is that integration of the composition of the deposited sediment over the lee face, Fleeloci, must result in the composition of the lee face deposit, Fleei:

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where equation imageFleeloci = 1 and 0 ≤ Fleeloci ≤ 1.

[25] Generally, the fine size fractions in a sediment mixture prefer to be deposited at the upper parts of the lee face and the coarse size fractions at its lower parts [Blom et al., 2003; Kleinhans, 2004]. On the basis of these insights, we now assume that the coarse size fraction is distributed over this uniformly sloped lee face with its volume fraction content decreasing linearly with elevation:

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where J (z) is a Heaviside function which equals 1 when considering an elevation covered by the specific bed form (Appendix B). The lee sorting parameter δi is a grain size-specific constant, which will be considered in detail in the next few paragraphs, and equation image* is a dimensionless vertical coordinate relative to the average elevation of the lee face:

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by definition, equation image* = −(1/2) at the lower limit of the lee face (where z* = η*b) and equation image* = (1/2) at the upper limit of the lee face (where z* = η*t). It can be found that equation (38) together with equation (36) indeed meets the constraint in equation (37).

[26] In order to have the δi submodel capture the principal effects of lee sorting, the authors believe that the submodel should at least incorporate the effects of (1) the difference in grain size between size fraction i and the geometric mean grain size of the lee deposit, expressed by ϕmlee − ϕi; (2) the arithmetic standard deviation of the lee deposit, σa; and (3) the dimensionless bed shear stress, τ*b.

[27] It is likely that the larger the difference in grain size between a specific size fraction i and the geometric mean grain size of the lee deposit (ϕmlee − ϕi), the larger the effect is of grain sorting at the lee face and thus the larger δi should be (see Figure 4).

Figure 4.

Lee sorting of three size fractions in one sediment mixture.

[28] To understand the effect of the gradation of the mixture, specified by σa, we consider one specific grain size ϕi in two mixtures: a widely graded mixture and a narrowly graded mixture (Figure 5). The two mixtures are characterized by identical geometric mean grain sizes of the lee deposit, ϕmlee. In this example, size fraction i is finer than the geometric mean grain size of the lee deposit. Namely, the larger the grain size on ϕ scale, the smaller is the grain size di. Figure 5 illustrates why for the widely graded mixture, the sorting of size fraction i at the lee face will be less than in the narrowly graded mixture, despite the difference between ϕmlee and ϕi for the two mixtures being identical. This is caused by size fraction i being relatively closer to the geometric mean grain size for the widely graded mixture, due to its larger gradation. For the widely graded mixture the amount of size fractions finer than size fraction i is larger and these all need to be deposited in the upper part of the lee face.

Figure 5.

Influence of the gradation of a mixture on sorting at the lee face.

[29] Allen [1965] and Kleinhans [2002] have shown that the lee face sorting trend becomes smaller with increasing shear stress.

[30] On the basis of these considerations and as a first step, the authors propose the following formulation for the lee sorting parameter δi:

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where

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where dmlee denotes the geometric mean grain size of the lee deposit in meters, ϕmlee denotes the geometric mean grain size of the lee deposit on ϕ scale, ϕi denotes the grain size of size fraction i on ϕ scale (ϕi = −2log 1000 di with di in meters), and τ*b denotes the dimensionless bed shear stress averaged over the bed form length. The constant γ weights the relative importance of the grain size term on the right-hand side of equation (40), while the value of κ sets the relative importance of the dimensionless bed shear stress term. Section 6 will show some general results of the lee sorting function.

[31] Finally, we consider two bed forms having different bed form heights and assume that the sediment transported over their crests is identical in composition. The formulation for Fleeloci in equations (36) and (38) implies that at the lower (and upper) boundaries of the two bed forms the volume fraction content of each size fraction in the deposit is identical. It implies that the sorting trend becomes stronger with decreasing bed form height. Studying the sorting process of sediment over a delta face (i.e., at a sudden and large transition in water depth), Kleinhans [2002] indeed found that the delta height has little influence on the volume fraction content of size fractions deposited at the upper and lower boundaries of the delta front.

[32] In summary, we propose a new simple trend-based lee sorting function meant for deriving a formulation for the grain size-specific deposition over the bed form lee face. It seems to be the simplest form of a lee face sorting model that has any hope of capturing the effects in question.

4. PPL Framework and Bed Form Migration Approach

4.1. Regular Bed Form Size

[33] In the present section the bed form migration approach will be coupled to the Parker-Paola-Leclair framework for sediment continuity. We first convert the entrainment and deposition rates as used in the bed form migration approach into entrainment and deposition densities as used in this framework. Subsequently, in section 4.2 we incorporate the irregularities of bed forms by taking into account the stochastic nature of the trough elevations.

[34] In section 3 we distinguished an entrainment flux at the stoss face, a deposition flux at the stoss face, and a deposition flux at the lee face so that we rewrite the right-hand terms in the fundamental equation of the PPL framework, equation (5):

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where equation imageeis, equation imageeis, and equation imageeil denote the entrainment density of size fraction i on the stoss face, the deposition density on the stoss face, and the deposition density on the lee face, respectively. For clarity, equation (45) only shows the argument in the z direction and leaves out the arguments x and t.

[35] Note that in the present section we deal with series of regular bed forms so that the parameters averaged over a series of bed forms are simply identical to the corresponding parameters for a single bed form, e.g.,

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Now, let us consider the relation between the entrainment density equation imageeis in equation (45) and the entrainment rate equation imagesiu. The entrainment rate equation imagesiu is defined as the volume of sediment of size fraction i picked up from the bed per unit length, width, and time, in case only size fraction i is present. As explained near equation (15), equation imagesiu may include hiding exposure effects. Thus the volume of sediment of size fraction i picked up from the stoss face from a bed layer with a thickness dzs per unit width and time equals

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where dxs is the horizontal extent over which the bed layer with thickness dzs is exposed to the flow at the stoss face. According to equation (B13), dxs can be written as

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The entrainment density equation imageeis of size fraction i is defined such that equation imageeisdxdz is the volume of size fraction i entrained from a bed element with sides dx and dz at elevation z at the stoss face, per unit width and time. Thus the entrainment density at the stoss face equals

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Then, in accordance with the Einstein step length formulation, the deposition density at the stoss face

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and the deposition density at the lee face

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where equation imageleeloci allows for grain size sorting down the avalanche lee face (section 3.5). The bed load transport rate of size fraction i averaged over the series of bed forms, equation imageai, equals qai in equation (B17). Likewise, the bed load transport rate of size fraction i over the bed form crest and averaged over the series of bed forms, equation imagetopi, equals qtopi in equation (B16).

[36] Thus we have coupled the single bed form migration approach to the PPL framework for sediment continuity, but the resulting set of equations is still only valid for a series of regular dunes. In reality, the variation of bed form geometry can be large, which gives rise to vertical sorting fluxes at bed elevations that are only reached by bed forms with very deep troughs.

4.2. Irregular Bed Form Size

[37] In this section the irregularities in bed form size are taken into account by incorporating the statistics of the trough elevations. We suppose that each individual bed form in a series of irregular bed forms is characterized by the vertical distance from its trough to the mean bed level. The latter is also indicated in terms of the relative trough elevation Δbb = equation imagea − ηb). The PDF of relative trough elevations is given by equation imageηb. Note that all bed forms in the series are assumed to have the same characteristic shape, expressed by the dimensionless PDFs of bed surface elevations for the stoss and the lee face, equation image*se and equation image*le, respectively. As mentioned in section 3.3, each lee face is assumed to have a uniform slope so that equation image*le = J. Note that the term irregular bed forms refers to irregularity in bed form dimensions but not in shape. All bed form-specific features, e.g., bed form height and bed form length, are assumed to be simply related to the relative trough elevation by

display math
display math
display math
display math

where Δa denotes the average bed form height and λa denotes the average bed form length. (Also see Figure 6 for the definition of these parameters.) The crests of the bed forms are assumed to have the same absolute distance to the mean bed level as the troughs, and the steepness of the lee faces is assumed to be equal to the angle of repose (ν). The bed form length is assumed to be proportional to the bed form height, and the ratio of the average bed form length is assumed to be equal to the average bed form height.

Figure 6.

Bed form parameters. For clarity, the bed forms are here assumed to have a triangular shape.

[38] One may rightfully claim that the parameters Δ, λ, and λl may be related to the relative trough elevation in a way different from equations (50)–(52). The purpose of these formulations is to account for the irregularity of bed forms in a relatively simple way so that the method can be incorporated in the new continuum sorting model. Equations (50)–(52) are not supposed to be generally valid, and when applying these formulations, their applicability should be checked against data. Note that the following part of the paper is also valid if equations (50)–(52) were not applicable.

[39] Let us now address the procedure of averaging some general bed form-specific parameter, μ, over a series of irregular bed forms. This requires the parameter to be weighted by the proportion of the total length of the series of bed forms that is related to a specific relative trough elevation. Yet the PDF of relative trough elevations, equation imageηb, only expresses the likelihood that a certain trough elevation occurs and not what proportion of the total length of the series of bed forms is involved. Therefore, in averaging some general bed form-specific parameter, μ, over various bed forms, it should not only be weighted by the probability density that a specific relative trough elevation occurs but also by the bed form length:

display math

where equation image is called the overall parameter μ. The weighted bed form length, equation image, normalizes equation (54), and

display math

For clarity, we now define an adapted PDF of relative trough elevations, equation imageb, which equals the PDF of relative trough elevations, equation imageηb, weighted by the proportion of the total length of the series of bed forms that is related to a specific relative trough elevation:

display math

so that the overall parameter μ in equation (54) can be rephrased to

display math

[40] Likewise, using equations (B6) and (B4), we find that the overall PDF of bed surface elevations, equation imagee, is now determined by the characteristic bed form shape, equation image*se, and by the PDF of relative trough elevations, equation imageb, according to

display math

where J, Δ, λ, λs, and λl are all related to the specific relative trough elevation, Δb. The overall probability distribution of bed surface elevations, equation images, is given by equation (2).

[41] The derivation of the entrainment and deposition densities averaged over a series of irregular bed forms (equation imageeis, equation imageeis, and equation imageeil) is similar to those for regular bed forms, given by equations (47), (48), and (49). Yet we now average the right-hand terms over all trough elevations:

display math
display math
display math

Herein we assume that the vertical sorting profile within an individual bed form, Fi, is identical to the sorting profile averaged over the series of bed forms, equation imagei. We now find the average bed load transport rate of size fraction i, equation imageai, and the average sediment transport over the bed form crest, equation imagetopi, by averaging qai in equation (B17) and qtopi in equation (B16) over all trough elevations:

display math
display math

[42] Thus we have coupled the single bed form migration approach to the PPL framework for sediment continuity for a series of irregularly sized bed forms. The result of section 4 is a new continuum sorting model for bed form-dominated conditions, although the model still lacks a submodel for the grain size-specific entrainment over the stoss face of a bed form.

5. Simplified Continuum Sorting Models

[43] In order to be able to compute the time evolution of the sorting profile, equation imagei, and the mean bed level, equation imagea, we need to make the set of equations complete. A straightforward way to do so would be to acquire a submodel for the grain size-selective entrainment rate over bed forms. As far as is known to the authors, Tsujimoto [1990] was the only one who developed an entrainment model especially for nonuniform sediment. Unfortunately, this entrainment model has never been verified for a wide range of conditions, and it is not known whether it gives a good description of entrainment along bed forms as flow patterns over bed forms are strongly nonuniform and very complex. Another source of uncertainty is the variation of a representative measure for the skin friction over the stoss face, which is not easily determined. Moreover, McLean et al. [1994] experimentally found that the sediment transport rate over a bed form does not directly depend on the variation of the mean bed shear stress over it. They argue that the sediment transport rate should be related to the statistics of the near-bed turbulence as lift and drag forces on grains are directly related to instantaneously fluctuating velocities.

[44] As it is questionable whether an entrainment model based on mean skin friction would give satisfactory results in predicting the size-selective entrainment rate over the stoss face, it was decided to take on another approach, which is explained in detail by Blom [2003] as well as in two follow-up papers (A. Blom et al., manuscripts in preparation, 2004). For equilibrium conditions (∂/∂t = 0) the continuum sorting model is reduced to an equilibrium sorting model. For nonequilibrium conditions the continuum sorting model is reduced to a relaxation-type sorting evolution model. It solves for the time evolution of both the vertical sorting profile and the composition of the bed load transport from the initial sorting profile, the total bed load transport rate, and the PDF of relative trough elevations. In addition, methods are proposed to account for vertical sediment fluxes through both a change in time of the PDF of relative trough elevations and net aggradation or degradation.

6. Results of Submodels

[45] As the continuum sorting model needs to be reduced to either the equilibrium sorting model or the sorting evolution model first, we can not yet show results of the full continuum sorting model. We do show some results of the submodels of bed form size as well as some general results of the submodel for lee-side sorting. In this section we consider the equilibrium phases of experiments B2 and A2 described by Blom et al. [2003]. During these experiments, uniform conditions were maintained (∂/∂x = 0), and the transported sediment was recirculated in order to avoid net aggradation or degradation over the length of the flume (∂equation imagea/∂t = 0). The sediment mixture consisted of three well-sorted size fractions: dfine = 0.68 mm, dmedium = 2.1 mm, and dcoarse = 5.7 mm. The sediment transport consisted solely of bed load transport, and the bed was covered by dunes.

[46] First, we study the validity of equations (51) and (50). Figure 7 (top) shows the relation between measured bed form lengths and bed form heights for individual bed forms, together with equation (51). The relation between the measured bed form lengths and bed form heights shows large scatter, but equation (51) covers the possible trend reasonably well. The large scatter seems partly due to the bed form length (i.e., the distance between two consecutive troughs) also being determined by the adjacent bed forms (see Figure 8). Figure 7 (bottom) shows the relation between measured relative trough elevations, Δb, and bed form heights, Δ, for individual bed forms. In spite of the large scatter, and although equation (50) deviates from the measured trend for the higher dunes in experiment B2E, equation (50) represents the trend reasonably well.

Figure 7.

(top) Relation between (1) measured bed form lengths and bed form heights for individual bed forms (small dots), (2) the average the bed form length, λa, and the average bed form height, Δa (large dot), and (3) the bed form length and bed form height according to equation (51) (line). (bottom) Relation between (1) measured relative trough elevations and bed form heights for individual bed forms (small dots) and (2) the relative trough elevation and bed form height according to equation (50) (line).

Figure 8.

Influence of adjacent bed forms upon the bed form length.

[47] Secondly, we analyze the sensitivity of the lee-side sorting profile, Fleeloci, to the constants γ and κ in the lee sorting function for experiment A2. It is simply assumed that the composition of the lee face deposit, Fleei, equals the measured average volume fraction contents of the three size fractions in the bed load transport, equation imageai, namely 0.38, 0.38, and 0.24. The average bed shear stress was equation imageb = 4.6 Nm−2. First we vary γ over the values [−1, −0.5, 0.5, 1], while κ is set such that the bed shear stress has no effect on the avalanche mechanism at the lee face at all (κ = 0). Figure 9 (left) shows the resulting computed lee-side sorting profiles for an individual bed form. For negative values of γ the coarse material prefers being deposited at the higher elevations of the active bed. This is contradictory to experimental studies on the avalanche mechanism at the lee face, so we may conclude that γ must have a positive value. Furthermore, it shows that the larger γ, the stronger is the sorting trend in the active bed. To study the effect of κ, we vary κ over the values [−0.5, 0, 0.5, 1], while γ = 0.3 (see Figure 9 (right)). It clearly shows that the smaller the value for κ, the smaller is the sorting in the active bed. Also, kappa should be positive as the lee face sorting trend is expected to become smaller with increasing shear stress. In the work of Blom [2003] and in the first follow-up paper (A. Blom et al., manuscript in preparation, 2004), the constants γ and κ are used as calibration coefficients.

Figure 9.

(left) Influence of the constant γ in the lee sorting function on the computed lee-side sorting profile of a single bed form, Fleeloci, while κ = 0. (right) Influence of the constant κ in the lee sorting function on the computed lee-side sorting profile of a single bed form, Fleeloci, while γ = 0.3.

7. Discussion and Conclusions

[48] The authors have developed a new depth-continuous sediment continuity model, which is based on (1) the Parker-Paola-Leclair (PPL) framework; (2) the Einstein step length formulation; (3) a newly developed lee sorting function; and (4) a newly developed method to account for the variability in bed form trough elevations. The present paper presents the derivation of formulations for the grain size-specific and bed elevation-specific entrainment and deposition fluxes as required for the PPL framework. The resulting continuum sorting model incorporates sorting fluxes through (1) the grain size-selective deposition down the avalanche lee face, (2) the variability in bed form trough elevations, and (3) net aggradation or degradation of the river bed.

[49] The present research has focused on the development of a framework for taking into account the effects of vertical sorting in modeling river morphodynamics. The framework contains various submodels (e.g., lee-side sorting; the step length; bed form dimensions; the time evolution of the PDF of trough elevations), which strongly need further development. For instance, the presently proposed lee sorting function still has a very simple form. Also, it would be worthwhile to compare the choice for a deterministic step length to a stochastic one.

[50] In its present form the new continuum sorting model incorporates only bed load transport. Further research is required to also include suspended load transport. Yet a simple way of incorporating suspended load transport is proposed by Blom [2003]. Application of the present model should be limited to bed form-dominated conditions. Further study is required to derive formulations for the grain size-specific and elevation-specific entrainment and deposition fluxes under plane bed conditions.

[51] The present paper has explained the derivation of a fundamental mathematical framework for taking into account the impact of vertical sorting upon the large-scale morphodynamics of bed form-dominated rivers. Two follow-up papers (A. Blom et al., manuscripts in preparation, 2004) will consider the simplification of this modeling framework to an equilibrium sorting model and a sorting evolution model, respectively. These follow-up papers will also discuss the application of the reduced models, wherein computed vertical sorting profiles are compared to the vertical sorting, as measured in the experiments by Blom et al. [2003], Ribberink [1987], and Kleinhans [2002]. The performance of the sorting evolution model and the commonly used Hirano active layer model will be compared.

Appendix A:: Integral of Time Derivative equation images

[52] Near equations (28a) and (28b) in the work of Parker et al. [2000], it is argued that the integral ∂equation images/∂t over z in equation (7) should vanish if equation imagee is symmetric in equation image, i.e., around the mean bed level, equation imagea. However, the authors found that the integral term in equation (7) should vanish regardless of whether or not equation imagee is symmetric in equation image. Let us consider the integral

display math

where r = ∞ and where equation image denotes the bed elevation relative to the mean bed level (equation image = zequation imagea (x, t)). We now substitute

display math

into equation (A1),

display math

and rephrase equation (A3) by integrating in parts according to

display math

where

display math

This yields

display math

wherein the integral term vanishes, as by definition:

display math

In addition, by definition, equation images → 1 as r → −∞ and equation images → 0 as r → ∞ so that ∂equation images/∂t → 0 as r → ±∞. Under the condition that ∂equation images/∂t converges to 0 faster than r converges to ±∞ (a condition that holds for, e.g., a Gaussian distribution, in which the standard variation is allowed to vary in time), it is seen that the integral I → 0 as r → ∞, thus demonstrating that the integral term in equation (7) should vanish, regardless of whether or not equation imagee is symmetric in equation image.

Appendix B:: Elevation-Specific Formulations

[53] Since formulations in the PPL framework are elevation-specific, we need to transform the formulations dependent on x on bed form scale into formulations dependent on bed elevation z. To that end, we introduce the dimensionless coordinates x*s, x*l, and z*:

display math

where x*s and x*l denote the dimensionless horizontal coordinates for the stoss face and lee face, respectively, and z*s denotes the dimensionless vertical coordinate. Δ denotes the bed form height, which is given by Δ = ηt − ηb, where ηb and ηt denote the lower and upper limits of the bed form, respectively. The dimensionless local elevation of the bed surface η is now expressed by

display math

By definition, the dimensionless mean bed level, η*a, equals 0. In case the bed form has a triangular shape, the dimensionless lower limit, η*b, equals −(1/2), and the upper limit, η*t, equals (1/2) (see Figure B1). As we assume each lee face to have a uniform slope, the PDF of active bed elevations at the lee face yields

display math

where J(z) is a Heaviside function that equals 1 when considering an elevation covered by the specific bed form.

Figure B1.

Dimensionless distance of one step length in horizontal and vertical directions for a triangular bed form.

[54] We now define dimensionless PDFs of the bed surface elevations for the stoss side, p*se, and for the lee side, p*le:

display math
display math

where dη* = dη/Δ. The bed form-averaged probability density of bed surface elevations is given by

display math

Like pe = −∂Ps/∂η in equation (4), the probability density of bed surface elevations for the stoss face equals

display math
display math

where Pss(z) denotes the probability that the stoss face elevation is higher than z. Now, comparing equation (B8) with equation (B4) and studying the example of a triangular bed form in Figure B2 yields

display math

For the stoss face we now introduce the function fds, which expresses the dimensionless elevation of the stoss face as a function of the dimensionless horizontal coordinate and its inverse, the function gds:

display math
display math

Comparison of equations (B9) and (B11) yields

display math
Figure B2.

Relation between x*s and the probability distribution of bed surface elevations Pss for a triangular bed form.

[55] In order to transform the equations listed in the previous few sections into equations dependent on bed elevation z, we rewrite the integrals over dx, e.g., in equation (26), into integrals over dz. When considering the stoss face, we replace dx by

display math

In addition, we translate the step length Λi into a dimensionless vertical step length, η*stepi:

display math

where η*stepi = ηstepi/Δ and Λ*i = Λis. This is illustrated in Figure B1. Using equation (B13) to manipulate qtopi in equation (26), we find that

display math

For clarity, we no longer present the arguments in the parameters Esiu, Fi, and pse as x coordinates (x = λsgds((z − ηa/Δ))), but we abbreviate these functional relations such that Esiu(z), Fi(z), and pse(z) are their equivalents, respectively. Then, qtopi in equation (B15) and qai in equation (34) are written as

display math
display math
Notation
( )*

superscript denoting a dimensionless parameter.

equation image

parameter is horizontally averaged over a series of bed forms.

equation image

parameter is relative to the mean bed level.

cb

sediment concentration within the bed (cb = 1 − λb).

equation imagei

concentration of size fraction i at elevation z, averaged over a series of bed forms.

di

grain size of size fraction i, m.

dmlee

geometric mean grain size of the lee deposit, m.

equation image

volume of deposited sediment per unit area and time, summed over all size fractions and averaged over a series of bed forms, m s−1.

equation imagee

deposition density defined like equation imageei but summed over all size fractions, s−1.

equation imageei

deposition density of size fraction i defined such that equation imageeidxdz is the volume of size fraction i deposited in a bed element with sides dx and dz at elevation z, per unit width and time, averaged over a series of bed forms, s−1.

Dl

deposition rate at the lee face, m s−1.

Dsi

volume of size fraction i locally deposited onto the stoss face, per unit area and time, m s−1.

equation image

volume of entrained sediment per unit area and time, summed over all size fractions and averaged over a series of bed forms, m s−1.

equation imagee

entrainment density defined like equation imageei but summed over all size fractions, s−1.

equation imageei

entrainment density of size fraction i, defined such that equation imageeidxdz is the volume of size fraction i entrained from a bed element with sides dx and dz at elevation z, per unit width and time, averaged over a series of bed forms, s−1.

Esnet

net entrained volume of all size fractions on the stoss face, per unit area and time, m s−1.

Esi

volume of size fraction i locally entrained from the stoss face, per unit area and time, m s−1.

Esiu

volume of size fraction i locally entrained from the stoss face, per unit area and time, if only sediment of size fraction i would be present, m s−1.

fds

elevation of the stoss face as a function of its horizontal coordinate, dimensionless.

fp(ξ)

probability density that step length equals ξ.

equation imageai

volume fraction content of size fraction i in the bed load transport, averaged over a series of bed forms.

equation imagei

volume fraction content of size fraction i in the bed at elevation z, averaged over a series of bed forms.

equation imageleei

volume fraction content of size fraction i in the lee deposit, averaged over a series of bed forms.

Fleeloci

volume fraction content of size fraction i in the sediment deposited at elevation z at the lee face.

equation imagetopi

volume fraction content of size fraction i in the bed load transport over the bed form crest, averaged over a series of bed forms.

gds

horizontal coordinate of the stoss face as a function of its elevation, dimensionless.

( )i

subscript indicating the number of the size fraction.

J(z)

Heaviside function, which equals 1 when considering an elevation covered by the bed form.

( )l

subscript indicating the lee face.

N

total number of size fractions.

equation imageηb

probability density function of trough elevations relative to the mean bed level for a series of bed forms, indicating the probability density that the trough elevation equals z, m−1.

equation imageb

adapted probability density function of trough elevations relative to the mean bed level for a series of bed forms, indicating the probability density that the trough elevation equals z, weighted by the horizontal distance involved, m−1.

equation imagee

probability density function of bed surface elevations for a series of bed forms, indicating the probability density that the bed surface elevation equals z, m−1.

equation image*e

probability density function of bed surface elevations for a series of bed forms, dimensionless.

pe

probability density function of bed surface elevations for an individual bed form, m−1.

equation images

probability distribution of bed surface elevations for a series of bed forms, indicating the probability that the bed surface elevation is higher than z.

q

volume of bed load transport per unit width and time (excluding pores), m2 s−1.

equation imagea

volume of bed load transport per unit width and time (excluding pores), averaged over a series of bed forms, m2 s−1.

equation imagetop

volume of bed load transport at the bed form crest per unit width and time (excluding pores), averaged over a series of bed forms, m2 s−1.

( )s

subscript indicating the stoss face.

t

time coordinate, s.

x

horizontal coordinate, m.

x*

horizontal coordinate on the scale of an individual bed form, dimensionless.

z

vertical coordinate, m.

equation image

vertical coordinate relative to the mean bed level equation imagea, m.

z*

vertical coordinate relative to the mean bed level equation imagea, dimensionless.

equation image*

vertical coordinate relative to the mean bed level equation imagea, dimensionless.

α

step length, dimensionless.

γ

constant in lee sorting function.

δi

lee sorting parameter.

Δ

bed form height, m.

Δa

bed form height averaged over a series of bed forms, m.

Δb

trough elevation relative to the mean bed level, i.e., the relative trough elevation, m.

η

local bed surface elevation, m.

ηa

bed surface elevation averaged over a single bed form, m.

equation imagea

bed surface elevation averaged over a series of bed forms (mean bed level), m.

ηb

bed form trough elevation, m.

ηbmax

highest bed form trough elevation, m.

ηbmin

lowest bed form trough elevation, m.

ηt

bed form crest elevation, m.

ηstepi

step length in z direction for size fraction i, m.

κ

constant in the lee sorting function.

λ

bed form length, m.

λa

bed form length averaged over a series of bed forms, m.

λb

porosity.

Λi

step length of size fraction i, m.

ν

angle of repose, deg.

ρ

density of water, kg m−3.

ρs

density of sediment, kg m−3.

σa

arithmetic standard deviation of the composition of the lee deposit; since its unit equals ϕ's unit, which is nonsensical (2log mm), it is left out.

equation imageb

bed shear stress averaged over a series of bed forms, N m−2.

ϕi

grain size of size fraction i on ϕ scale; since ϕ's unit is nonsensical (2log mm), it is left out.

ϕmlee

geometric mean grain size on ϕ scale of the lee deposit.

ωi

lee sorting function, specifying to what extent a specific size fraction that is transported over the bed form crest is deposited at elevation z of the lee face.

Acknowledgments

[56] The first author worked on the project in the pursuit of a doctoral degree at the Department of Civil Engineering of the University of Twente. The study was supported by the Institute for Inland Water Management and Waste Water Treatment (Rijkswaterstaat RIZA) of the Ministry of Transport, Public Works, and Water Management in the Netherlands, by WL Delft Hydraulics, and by the University of Twente. The Netherlands Organization for Scientific Research (NWO) and the Prince Bernhard Cultural Foundation are acknowledged for their financial support for a 3 month stay by the first author at the St. Anthony Falls Laboratory. Jan S. Ribberink and Huib J. de Vriend are gratefully acknowledged for their scientific support during the project.

Ancillary