Rouse revisited: The bottom boundary condition for suspended sediment profiles

Journal Pre-proof Rouse revisited: The bottom boundary condition for suspended sediment profiles

Bernard P. Boudreau, Paul S. Hill PII:

S0025-3227(19)30045-3

DOI:

https://doi.org/10.1016/j.margeo.2019.106066

Reference:

MARGO 106066

To appear in:

Marine Geology

Received date:

17 January 2019

Revised date:

16 October 2019

Accepted date:

22 October 2019

Please cite this article as: B.P. Boudreau and P.S. Hill, Rouse revisited: The bottom boundary condition for suspended sediment profiles, Marine Geology (2019), https://doi.org/10.1016/j.margeo.2019.106066

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© 2019 Published by Elsevier.

Journal Pre-proof

Rouse Revisited: The Bottom Boundary Condition for Suspended Sediment Profiles

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Paul S. Hill1

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Bernard P. Boudreau1,*

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Department of Oceanography, Dalhousie University, Halifax NS B3H4R2, Canada

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*corresponding author

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Keywords: sediment, suspension, diffusion, Rouse profile, bottom boundary condition

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Journal Pre-proof ABSTRACT

A linearly depth-dependent diffusivity is commonly used to model the vertical distribution of dilute, near-bed, suspended sediment in marine fluid environments, i.e., the Rouse model; this simplified form predicts unfortunately infinite concentration at the bed. That singularity causes problems in linking bed models to suspended sediment models. A common remedy to this problem is to assume a reference concentration and height above the bed, but some studies point to the arbitrariness of that procedure and

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suggest, instead, the existence of a residual eddy diffusion near the bed, which

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removes the singularity. In these past studies, the residual diffusion, εo, has been assigned a value based on the Nikuradse roughness height, ks = 2.5ds, where ds is the

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(mean) equivalent sand-grain, diameter. We examine this assumption by fitting a residual diffusion model to 14 time-averaged suspended-sediment profiles available in

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the literature, without assuming the Nikuradse length. This procedure produces εo

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values from 0.6 to 13 cm2 s-1, which are much greater than both the (molecular) kinematic viscosity of water and the expected Brownian diffusion coefficient of the sediment grains of the reported sizes. These residual diffusion coefficients indicate

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mixing scale heights at the bed, εo/??u∗, where ?? is von Kármán’s constant and u∗ is the shear velocity, are 7.4-56.0X greater than the Nikuradse height, ks. Although defined by

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only limited data, we also find that the residual diffusion correlates with grain size and

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the roughness Reynolds number but not with the Rouse parameter, nor with the shear and settling velocities. In addition, the residual diffusion coefficient correlates with van Rijn's roughness scale, again based on the same limited data, suggesting that bedforms influence residual diffusion near the bed.

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Journal Pre-proof INTRODUCTION

Suspended transport is the dominant form of sediment movement in many natural turbulent flows, including those in the marine realm. Prediction of sediment movement in such flows requires models that reliably represent the suspension process and the resulting sediment distribution with depth.

One-dimensional modelling of suspended sediment transport has a long history,

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well covered in various textbooks and review articles, and as such, not repeated or

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surveyed here. One aspect of past work is, nevertheless, of central interest to this paper: modeling of the vertical distribution of dilute suspended sediment in steady flow

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over a sediment bed, with or without net deposition, i.e., the profile. The first model to simulate convincingly this profile, without net deposition, was offered by Rouse (1937).

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He advanced that this profile was the result of a balance between settling, which moved

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sediment particles towards the bed, and turbulence, in the form of diffusion, which acted

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to re-suspend (lift) the particles, creating a so-called equilibrium balance, i.e.,

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(1)

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where z is height above a notional sediment-water interface (cm), C(z) is the concentration of suspended sediment at height z (volume fraction or percent), ws is the settling velocity of the grains (cm s-1), and εs is the (vertical) turbulent diffusion coefficient for suspended sediment (cm2 s-1), which is usually assumed to be height dependent. The interface is notional because real beds are made of irregular grains and have roughness elements. Nevertheless, acoustic and laser methods can now be used to define a reliable average surface with an assigned zero depth if the elements do not dominate the bottom, e.g., Wang and Tang (2009) or Wilson and Hay (2016). The turbulent diffusion coefficient εs is often assumed equal to that for turbulent momentum transport within the flow ε, i.e., a turbulent Prandtl or Schmidt Number (ε/εs)

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Journal Pre-proof of 1, although that has been debated (e.g., Toorman, 2008). This assumption allows observations and theories developed to explain and predict ε to be applied to εs. In particular, at the time of Rouse’s work, measurements of velocity profiles above rigid beds and use of Prandtl mixing-length theory led to a linear depth dependence for εs (Schlichting, 1968), i.e.,

εs = ?? u∗ z

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(2)

where ?? is von Kármán’s constant (0.4) and u∗ is the shear velocity for the flow. The

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constancy and exact value of Von Kármán’s constant has been debated, but that need

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not detain us here.

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When Eq. (1) is applied to the portion of a flow near the bed, so that the

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distribution of the suspended matter near the free-surface or near the top of the boundary layer of the flow is not of immediate concern, then Eq. (1) can be solved with

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Sleath, 1984):

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Eq. (2) to obtain the simplified Rouse profile (Swart, 1976; Smith, 1977; Laursen, 1982;

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(3)

where Ca is the concentration at a reference height za and Ro is the Rouse number (or parameter),

(4)

The reference height and concentration are needed because the original and simplified Rouse profiles, e.g., as given by Eq. (3), demand an infinite concentration at z = 0, i.e., the notional bed. This infinity problem has been recognized from the very 4 of 24

Journal Pre-proof inception of the Rouse model, and it results from the disappearance of εs at z = 0. To overcome this failing, the application of the Rouse equation is limited to heights above a reference height, za. Direct transport communication between the bed and the suspended matter profile does not, therefore, exist in this model, which is a challenge to predicting important related processes, such as erosion rates and tracer/contaminant exchange rates. Nevertheless, Eq. (3) remains popular for interpreting suspended sediment data in marine flows. (Alternatively, Murphy (1985), Nielsen (1992), Boudreau (1997) and Hill and McCave (2001) suggest replacing, at least partially, the diffusion

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model with a non-local transport model, which does not generate infinities, but that

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strategy has not generated great interest.)

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The primary driver of the popularity of Eq. (3) is its mathematical simplicity, regardless of the conspicuous singularity. As well, this equation contains enough

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adjustable parameters (Ca, za, Ro) to fit observed suspended matter profiles; however,

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direct fitting of Eq. (3) to data involves only 2 parameters, as Ca and za are combined in that equation. It is only at the post fitting stage that Ca and za can somehow be separated. There have been determined attempts to set both Ca and za independently

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from any data set and to define them as functions of environmental and sedimentological parameters, e.g., van Rijn (1984), Smith and McLean (1977), and Lee

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et al. (2004), but these efforts have not been able to claim unequivocal success.

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Moreover, the notion of independently calculable Ca and za has a drawback that is seldom discussed: in a statistical sense, it privileges the concentration at za when, in reality, we do not know values there any better than at any other depth.

Of prime interest here is a suggestion, apparently developed independently, by Bogargi (1974) and Taylor and Dyer (1977), and later adopted by Huang et al. (2008) and Kundu and Ghoshal (2014), that the Prandtl mixing length can be no smaller than the Nikuradse (sand) roughness height, ks = 2.5ds, where ds is the mean equivalentsand-diameter of the roughness. Consequently, Eq. (2) was modified to read

εs = ?? u∗ (z + ks)

(7) 5 of 24

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and the modified Rouse profile became

(8)

where Co is either a reference concentration (Bogargi, 1974) or, more correctly, the (apparent) concentration of suspended sediment at the notional seafloor at z = 0 (Taylor

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and Dyer, 1977). Equation (8) removes the singularity at the (notional) sediment-water

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interface. Oddly, none of the proponents of Eq. (8) asked the crucial question: is the Nikuradse roughness height the appropriate scale for Eqs. (7) and (8)? That is the

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question we explore, and the result of that investigation is notable.

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RESIDUAL DIFFUSION

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Instead of assuming that a residual diffusion is created at a scale related to the Nikuradse roughness height, we reverse this logic and assume that there is a residual

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diffusion at the bed, ε0, and ask instead: what is the length scale of the process that

εs = ?? u∗ z + ε0

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creates it? Consequently, we replace Eq. (7) with

(9)

Equation (9) finds observational support in Lee (2008), his Fig. 4. Lee (2008) used detailed vertical distributions from an acoustic backscatter profiling sensor (ABS) to calculate the values of εs, in dimensionless form, i.e., his K+c or K+cw, as a function of height at two marine sand sites (Dounreay, Scotland, and Duck, USA). It is quite clear from most of his plots that dimensionless εs does not extrapolate to zero at the notional seabed. Similar non-zero intercepts in the calculated εs are suggested in the results of

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Journal Pre-proof Rose and Thorne (2001), i.e., their Fig. 7, and Cheng et al. (2013), i.e., their Fig. 3, and maybe even Coleman (1970), i.e., his Fig. 4, if the imposed curve is ignored.

Substitution of Eq. (9) into Eq. (1) leads to

(10)

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where C0 is the (volume fraction) concentration at z = 0.

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Note that, while similar to Eq. (1), Eq. (10) is well behaved at z = 0 and contains again only three parameters, i.e., C0, ε0 and Ro; thus Eq. (10) also removes the problem

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in the original Rouse formulation without increasing the number of parameters or

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drastically changing the form of the solution. Parameters ws and u∗ also appear in both formulations, but these can be independently determined. In fact, we argue that C0 and

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ε0 are superior choices over Ca and za because using the former does not involve an arbitrary choice for the reference height. Equation (10) also re-establishes dynamic

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communication with the bed and eliminates the need for an arbitrary gap layer with infinite mixing employed in some numerical models. Equation (10) is mathematically

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similar to Eq. (8), but it makes no assumption about the relation of ε0 to the Nikuradse roughness height, ks.

It is easy to raise objections to Eq. (10). We even agree with some of these concerns. For example, the region near the bed is an area of (relatively) high suspended matter concentration, even if the remainder of the profile can be considered dilute. The higher concentration may increase the molecular viscosity of the fluid and, thus, dampen the effective eddy viscosity (e.g., Richardson and Zaki, 1954; van Rijn, 1984); this works against a residual diffusion but does not necessarily negate it. As mentioned above, the settling velocity may decrease (Kynch, 1951), instead of εs

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Journal Pre-proof tending to a constant; there is really no experimental data to segregate properly the effects of ws and εs changes near the bed, and we choose to rely on the eddy diffusivity.

Particle-particle interactions may become important near the bed, and that is certainly true in air for sands; however, because of the high inertia of water and the difficulty for particle-particle interactions for fine-grained sediments, we are not convinced that this is necessarily important in marine environments. On that point,

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none of our adopted assumptions precludes that ε0 could not be, in part, a reflection of

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particle-particle interactions.

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Finally, the conditions for the Prandtl mixing-length theory, and the resultant Fickian diffusion, may be violated near the bed; in fact, visualizations of motions in

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boundary layers, e.g., Corino and Brodkey (1969), Kim et al. (1971), Jackson (1976),

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and Cantwell (1981), unequivocally demonstrate that the conditions for the Prandtl mixing-length theory are violated in all turbulent boundary layers. Specifically, the

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motions that create mixing are not only local (or infinitesimal) in nature, but are affected by large scale motions, i.e., bursts, up and down sweeps, rolls, etc., that span over the

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entire boundary layer (Ligrani, 1989). Despite this clear contradiction, eddy diffusion remains almost exclusively the transport mode for suspended sediment modelling. We

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follow that tradition here but see Boudreau (1997) for an example of an alternative.

In addition to the above, the differentiation of suspended and bed loads becomes an issue in the immediate vicinity of the sediment-water interface. Data becomes difficult to obtain with precision and assignment of sediment concentrations to either type of transport somewhat arbitrary. The presence of bed forms (not quantitatively described in any of the quoted studies below), creates the usual problem of positioning these models. These facts make application of Eqs. (3), or (8), or (10) difficult, but we shall nevertheless attempt this.

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RESULTS

To illustrate the use of Eqs. (8) and (10), we apply them to some of the suspended matter profiles published in five previous studies, i.e., Einstein and Chien (1955), Coleman (1986), Lyn (1986), Sumer et al. (1996), as reported in Hsu et al. (2003), and Rose and Thorne (2001), assuming these represent equilibrium situations. These studies vary considerably in the amount of suspended materials, ranging from

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volume fractions of ~0.4 to as low as 0.001, and in their sampling/measurement heights.

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Thorne (2001) studied profiles at marine sites.

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In addition, most of these profiles were obtained in laboratory flumes, while Rose and

Not all profiles reported in these papers were refit; we restricted ourselves to

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those with sufficient data to obtain fits with reasonable statistical errors. These fits are

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illustrated in Fig. 1 (panels A-O). The input parameters are summarized in Table 1 and the results in Tables 2 and 3. All these sediments are in the fine to medium range of

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sands, see column 2 of Table 1. Reported values of ws and u∗ are provided in columns 3 and 4 of Table 1, respectively, and estimates of the kinematic viscosity of the water

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are provided in column 5 of that table. The resulting Ro and roughness Reynolds numbers, Red, are given in columns 6 and 7 of Table 1 (Red = u∗ds/??, where ?? is the

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kinematic viscosity of the solution).

Fits of data in Figure 1 with Eq. (3) are given as Ca×(za)Ro in column 2 and Ro in column 3 of Table 2, with the R2 of the regression given in column 4. The resulting Ro values in column 3 of Table 2 are similar, but not identical to those from the estimates in column 6 of Table 1, and nothing anomalous stands out. Fits with Eq. (3) cannot give separate estimates of Ca and za but only the combination Ca×(za)Ro; external information must be provided to obtain the separate parameters, which in these cases, we do not have.

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Journal Pre-proof Table 2 also contains the best fit values of Co and Ro obtained with Eq. (8) in columns 5 and 6, with corresponding R2 values in column 7 − fits not shown in Figure 1. Most of the Ro values are similar to those obtain with the simplified Rouse equation, Eq. (3), and from independent estimates in Table 1.

The parameter fits with Eq. (10) are displayed in Table 3, and the extracted values of εo and ws are given in columns 5 and 6, while column 7 contains the

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corresponding R2 for these fits. These calculated ws values are similar to those reported in column 3 of Table 1 but not identical, tending to be larger; this latter

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observation reflects that the Ro values in column 3 of Table 3, obtained by fitting Eq.

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(10), tend to be larger than the values in column 5 of Table 1, derived from the reported ws and u∗ values in columns 3 and 4, respectively. Given the scatter in most of this

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data, we are unwilling to read too much into these relatively small differences. In

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particular, the Ro values from the Rose and Thorn (2001) high density data sets are almost identical within statistical error between the models, which suggests that

the bases of the models.

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differences in Ro are related to data scatter and divergent fitting procedures, rather than

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The residual diffusion coefficients, ε0 in column 5 of Table 3, range in magnitude

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from 0.6 to 13 cm2 s-1, with a mean value of 2.7 cm2 s-1 and a median of 0.96 cm2 s-1. Such values are markedly larger that either the (molecular) kinematic viscosity of water (~1.5x10-2 cm2 s-1) or the expected Brownian diffusion coefficient for sand grains of this size range (~2-3x10-12 cm2 s-1), either of which would place lower limits on ε0.

Discussion Estimation of Residual Mixing. Our aim here is not to determine if Eqs. (3) or (8) or (10) provide better fits to the chosen experimental/observational data. All these models have sufficient free parameters to statistically fit most available data sets (see 10 of 24

Journal Pre-proof the R2 values in Tables 2 and 3), and that is not in dispute. It is true that the interfacial concentration Co derived with Eqs. (8) and (10) do not agree, see Table 2, column 5 versus Table 3, column 2. Equation (8) suggests far higher Co values than Eq. (10). Conversely the extracted Ro values are generally lower with Eq. (8) than with Eq. (10), except with the high-resolution data obtained by Rose and Thorne (2001), where the Ro values are similar, while the Co values remain distinctly dissimilar. But this comparison is not the point of our study; the problem here is one of basis

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or conception, not the statistical fitting of data. We developed Eq. (10) to test the suggestion by Bogargi (1974) and Taylor and Dyer (1977), and adopted by Huang et al. (2008)

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and Kundu and Ghoshal (2014), that the implied mixing length for the residual eddy diffusion

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should be set a priori to a minimum determined by the Nikuradse roughness height, i.e., ks in Eq. (8). Equation (10) can be used in this determination because it makes no

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assumption about the "size" of the mixing scale associated with the residual diffusion,

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thus allowing us to calculate the former.

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The calculated ε0 values imply mixing scales equal to ε0/??u∗ - see Table 3, column 4. This calculated scale is compared to the Nikuradse roughness height (ks =

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2.5 ds) in Table 3, column 8, and in Fig. 2 for each case in Table 1. For each and every

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profile, the mixing length scale (ε0/??u∗) is between ~7.4-56X greater than the Nikuradse roughness height for that case (mean ratio of 20.5). This establishes that the Nikuradse roughness height is not the determinant of the residual diffusion at the bed, even if it sets a minimum.

Trend Analysis. Grain size does, nevertheless, play a prominent role in our results. We plotted ε0 against the ds, ws, u∗, and Ro, as well as ε0/?? against the roughness Reynolds number (Red) and found significant linear correlations only with ds and Red, as seen in Fig. 3. These linear relations are, admittedly, strongly defined by only two or three points, which should temper over-interpretation of or over-confidence in these results; nevertheless, the correlations are significant. It can easily be argued 11 of 24

Journal Pre-proof that these correlations are expected as: (a) larger grains are harder to move, so if they are moving (as in these studies), more lift is needed and ε0 must consequently be larger with grain size (Fig. 3A), and (b) larger grains make larger bedforms, producing more turbulence (Fig. 3B).

The linear fit in Fig. 3B contains a surprise. This linear fit was obtained by forcing it through the origin to remove the small but pesky intercept value. The relationship in

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Fig. 3B implies that

(11)

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ε0/?? ≃ 26 Red

Given the small size of our database, we do not claim that Eq. (11) is universal or even

(12)

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established. If Eq. (11) is substituted into Eq. (10), then

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where the roughness scale for the residual diffusion, zr, is

(13)

Equation (12) is our logical equivalent to Eq. (8) of the original residual mixing theory of Bogargi (1974) and Taylor and Dyer (1977). With ?? = 0.4, so that ks = ds/??, our residual mixing length, zr, is theoretically 26 times larger than ks used in Eq. (8); this factor is close to the mean of the individual determinations of the zr/ks ratio given as Column 8 of Table 3. Far more research is needed to validate this result.

Causes of Residual Mixing and Trends. Various factors and processes can create residual mixing and account for the observed trend described above: 12 of 24

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(1) The idea that ε must be zero at a bed is derived from the behavior of a fluid at an immobile, solid and impermeable surface, where both the tangential and normal components of a flow must disappear to properly conserve momentum and mass; yet, a bed of uncemented sediment at the seafloor is usually erodible and mobile. Thus, even under stationary conditions, turbulent bursts can penetrate and resuspend previously deposited sediment, which will then again be returned to the bed by deposition. The bed surface is then dynamic. On average, sediment grains continuously cross the

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notional interface, both upwards and downwards. These upward motions demand that ε

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≠ 0 at the notional interface, and this establishes the necessity of residual mixing.

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(2) The correlation with grain size suggests that a property or process related to ds is a dominant contributor. Such factors would include the nature and size of bed

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forms, which could shed size-related eddies. Larger grained sediments have larger

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sized bedforms and lower cohesiveness, both of which would encourage more effective residual diffusion. This very point was raised by Bogargi (1974), Taylor and Dyer (1977), and Huang et al. (2008), yet they continued to utilize the Nikuradse length, ks, in

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Eq (8), and that is the failure at issue here. Bedforms heights and their effects were not reported in the chosen studies; however, at the suggestion of a reviewer, we employed

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the formulas available in van Rijn (2007) to estimate the roughness height, kvR,

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expected for bedforms under the fluid dynamic conditions in the cited experiments. Table 4 displayed the quantities used to calculate kvR, and plots of this parameter against both ε0 and ε0/(?? u∗) are illustrated in Fig. 4A and 4B, respectively. Again, there is a linear correlation in each case, with a relatively high R2-value. As ε0 correlates with ds (Fig. 3A) and ds is a component in the calculation of kvR, the correlation in Fig. 4A is not unexpected; nevertheless, the relationship illustrated in Fig. 4B between kvR and ε0/(?? u∗) argues that bedform generated roughness and the residual mixing length are correlated and decidedly similar in magnitude. That latter fact may indicate that bed roughness generates the residual diffusion, even if the trend is largely defined by the same two points as in Fig. 3.

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Journal Pre-proof (3) Another potential factor is permeability, which also scales with ds. Higher permeability could allow more penetration of water movement and pressure fluctuations into the bed, increasing the erosion rate and the apparent residual diffusion. At this time, we consider this point highly speculative.

(4) There remains the possibility that the chosen profiles may not be at equilibrium and, thus, might generate anomalous bottom diffusivities. While possible,

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the original studies raised no issues related to disequilibrium.

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Clearly, much more research is suggested by our findings, and this will require highly resolved data near the sediment bed, like the Rose and Thorne (2001) profiles.

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Experiments with finer grained sediments are also indicated, and the ε0 dependence on ds may change radically, even reverse, as cohesiveness increases. If the parameters of

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the residual-diffusion model can be firmly established, then this model may become

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more popular in describing and predicting sediment suspension because it does not generate an infinity at the bed and is thus well-behaved for both theoretical and computational applications. In particular, it allows for direct exchange between the bed

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the suspended sediment, without the use of an artificial reference concentration. This would also be highly desirable in radiotracer and contaminant bed-exchange research

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(Rutgers van der Loeff and Boudreau, 1997).

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Journal Pre-proof SUMMARY

We have re-examined the basis and application of a modified Rouse Model, as suggested by Bogargi (1974), Taylor and Dyer (1977), Huang et al. (2008) and Kundu and Ghoshal (2014), which includes residual diffusion at the sediment water interface and which they equated to the Nikuradse length, ks. We tested that latter assumption by calculating the residual diffusion directly from suspended matter profiles and back-

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calculating the related mixing depth.

Our model is applied to 14 well-known suspended matter profiles from the

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literature. We obtain εo values that range from 0.6 to 13 cm2 s-1. The resulting εo values

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indicate roughness scales for near-bed suspended matter mixing which are on average ~12.8 times larger than the Nikuradse length. The Nikuradse length would appear to be

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a poor representation of the roughness that induces near-bed mixing, contrary to the

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assumption in Bogargi (1974), Taylor and Dyer (1977), Huang et al. (2008) and Kundu and Ghoshal (2014). εo values appear to correlate with ds, Red and also with the van Rijn (2007) roughness parameter, suggesting that bedform roughness generates

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residual mixing (diffusion) at the nominal sediment-water interface.

Acknowledgements. BPB was supported by funds from the Killam Trust and Dalhousie University. PH thanks NSERC for support. We thank our reviewers for demanding evaluations that significantly improved our paper. Data used in this paper can be found in the cited original papers.

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Journal Pre-proof REFERENCES

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Bogargi, J., 1974. Sediment Transport in Alluvial Streams. Budapest, Hungary: Akademiai Kiado, 826 pp. Boudreau, B.P., 1997. A mathematical model for sediment-suspended particle exchange. Journal of Marine Systems 11, 279-303. Cantwell, B.J. (1981). Organized motion in turbulent flows. Ann. Rev. Fluid Mech. 13, 457-515. Cheng, C., Song, Z.-y., Wang, Y.-g., Zhang, J.-s., 2013. Parameterized expressions for an improved Rouse equation. International Journal of Sediment Research 28, 523-534, doi:10.1017/jfm.2014.498. Coleman, N.L., 1970. Flume studies of the sediment transfer coefficient. Water Resour. Res. 6, 801-809. Coleman, N.L., 1986. Effects of suspended sediment on the open-channel velocity distribution. Water Resour. Res. 22, 1377-1384. Corino, E.R., Brodkey, R.S.,1969. A visual investigation of the wall region in turbulent flow. J. Fluid Mech. 37, 1-30. Einstein, H.A., Chien, N., 1955. Effects of heavy sediment concentration near the bed on velocity and sediment distribution. MRD Series 8, Institute of Engineering Research, University of California, Berkeley. Hill, P.S., McCave, I.N., 2001. Suspended particle transport in benthic boundary layers. In: Boudreau, B.P., Jørgensen, B.B., (Eds.), The Benthic Boundary Layer: Transport Processes and Biogeochemistry. Oxford, UK: Oxford University Press. pp. 78-103. Hsu, T.-J., Jenkins, J.T., Liu, P.L.-F., 2003. On two-phase sediment transport: Dilute flow. J. Geophys. Res. 108, 3057, doi:10.1029/2001JC001276. Huang, S.-h., Sun, Z.-l., Xu, D., Xia, S.-s., 2008. Vertical distribution of sediment concentration. J. Zhejiang University Science A 9, 1560-1566, doi:10.1631/jzus.A0720106. Jackson, R.G., 1976. Sedimentological and fluid-dynamic implications of the turbulent bursting phenomenon in geophysical flows. J. Fluid Mech. 77, 531-560. Kim, H.T., Kline, S.J., Reynolds, W.C., 1971. The production of turbulence near a smooth wall in a turbulent boundary layer. J. Fluid Mech. 50, 133-160. Kundu, S., & Ghoshal, K., 2014. Effects of secondary current and stratification on suspension concentration in an open channel flow. Environ. Fluid Mech. 14, 13571380. Kynch, G.J., 1952. A theory of sedimentation. Trans. Faraday Soc. 48, 166–176. Laursen, E.M., 1982. A concentration distribution formula from the revised theory of Prandtl mixing length. In: First International Symposium on River Sedimentation Guanghua Press, Beijing, China, pp. 219-233. 16 of 24

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Lee, G.-h., 2008. Sediment eddy diffusivity and selective suspension under waves and currents on the inner shelf. Geosci. J. 12, 349-359, doi:10.1007/s12303-0080035-4. Lee, G.-h., Dade, W.B., Friedrichs, C.T., Vincent, C.E., 2004. Examination of reference concentration under waves and currents on the inner shelf. J. Geophys. Res. 109, C02021, doi:10.1029/2002JC001707. Ligrani, P.M.,1989. Structure of turbulent boundary layers. In: Cheremisinoff (Ed.), Encyclopedia of Fluid Mechanics, vol. 8, Aerodynamics and Compressible flow. Houston, TX: Gulf Publishing Company, pp. 111-189. Lyn, D.A., 1986. Turbulence and turbulent transport in sediment-laden open-channel flows. Pasadena, CA: PhD Thesis, Cal. Inst. Tech. Murphy, P.J., 1985. Equilibrium boundary condition for suspension. J. Hydraulic Eng. 111, 108-117. Nielsen, P., 1992. Coastal Bottom Boundary Layers and Sediment Transport. Advanced Series on Ocean Engineering, vol. 4. Singapore: World Scientific, 324 pp. . Richardson, J.F., Zaki, W.N., 1954. The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chem. Eng. Sci. 3, 65-73. Rose, C.P., Thorne, P.D., 2001. Measurements of suspended sediment transport parameters in a tidal estuary. Cont. Shelf Res. 21, 1551-1575. Rouse, H., 1937. Modern concepts of the mechanics of fluid turbulence. Trans. Am. Soc. Civil Eng. 102, 463-543. Rutgers van der Loeff, M.M., Boudreau, B.P., 1997. The effect of resuspension on chemical exchanges at the sediment-water interface in the deep sea — A modelling and natural radiotracer approach. J. Mar. Syst. 11, 305-342. Schlichting, H. (1968). Boundary-Layer Theory. New York, NY: McGraw-Hill, 748 pp. Sleath, J.F.A., 1984. Sea Bed Mechanics. New York, USA: John Wiley & Sons, WileyInterscience, 335 pp. Smith, J.D., 1977. Modeling of sediment transport on continental shelves. In: Goldberg, E.D, McCave, I.N., O'Brien, J.J. & Steele, J.H. (Eds.), The Sea, vol. 6. New York, USA: John Wiley & Sons, Wiley-Interscience, pp. 539-577. Smith, J.D., Maclean, S.R., 1977. Spatially averaged flow over a wavy surface. J. Geophys. Res. 82, 1735-1746, doi:10.1016/j.ijheatfluidflow.2011.02.016. Sumer, B.M., Kozakiewicz, A., Fredsøe, J., Deigaard, R., 1996. Velocity and concentration profiles in sheet-flow layer of movable bed. J. Hydraulic Eng. 122, 549-558. Swart, D.H., 1976. Predictive equations regarding coastal transport. In: Johnson, J.W. (Ed.), Proceedings of the 15th Coastal Engineering Conference, Vol. 2. New York, USA: American Society of Civil Engineers, pp. 1113-1132.

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Taylor, P.A., Dyer, K.R., 1977. Theoretical models of flow near the bed and their implications for sediment transport. In: Goldberg, E.D, McCave, I.N., O'Brien, J.J. & Steele, J.H. (Eds.), The Sea, vol. 6. New York, USA: John Wiley & Sons, WileyInterscience, pp. 579-601. Toorman, E.A., 2008. Vertical mixing in the fully developed turbulent layer of sedimentladen open-channel flow. J. Hydraulic Eng. 134, 1225-1235, doi:10.1061/(ASCE)0733-9429(2008)134:9(1225). van Rijn, L.C., 1984. Sediment transport, Part II: Suspended load transport. J. Hydraulic Eng. 110, 1613-1641. van Rijn, L.C., 2007. Unified view of sediment transport by currents and waves. I: Initiation of motion, bed roughness, and bed-load transport. J. Hydraulic Eng. 133, 649-667. Wang, C.-C., Tang, D., 2009. Seafloor roughness by a laser line scanner and a conductivity probe. IEEE J. Oceanic Eng. 34, 459-465. Wilson, G.W., Hay, A.E., 2016. Acoustic observations of near-bed sediment concentration and flux statistics above migrating sand dunes. Geophys. Res. Lett. 43, doi:10.1002/2016GL069579. Winterwerp, J.C., van Kesteren, W.G.M., 2004. Introduction to the Physics of Cohesive

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Sediment in the Marine Environment. Development in Sedimentology 56, Elsevier,

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Amsterdam, 466 pp.

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(3)&(4)

(2),(4)&(5)

5.4 5.4 10.4 7.0 8.6 9.8 36.8 42.1 6.6 5.4 4.3 4.4 7.4 8.2 27.3

ro

0.67 0.67 1.83 0.62 0.51 0.44 0.91 0.79 0.40 0.45 0.61 0.60 1.60 1.45 0.97

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0.0083 0.0083 0.0083 0.009 0.009 0.009 0.009 0.009 0.0124 0.0124 0.0124 0.0124 0.01 0.01 0.01

re

4.1 4.1 4.1 4.81 5.94 6.77 5.52 6.32 6.3 5.6 4.1 4.2 3.9 4.3 10.1

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1.1 1.1 3.0 1.2 1.2 1.2 2.0 2.0 ~1.0 ~1.0 ~1.0 ~1.0 2.5 2.5 3.9

na

0.011 0.011 0.021 0.013 0.013 0.013 0.06 0.06 0.013 0.012 0.013 0.013 0.019 0.019 0.027

ur

Col_2 Col_5 Col_24 H/S_1 H/S_2 H/S_3 H/S_7 H/S_9 R&T_3 R&T_4 R&T_5 R&T_6 1957EQ 1957ST E&C_12

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Table 1: Environmental parameter values for the suspended matter profiles in Figure 1. (1) Reported/Inferred (2) (3) (4) (5) (6) (7) (8) Case* Ro ds ws u∗ ?? Red ks † 2 from (cm) (cm/s) (cm/s) (cm /s) from (cm) 0.0275 0.0275 0.0525 0.0325 0.0325 0.0325 0.15 0.15 0.0325 0.03 0.0325 0.0325 0.0475 0.0475 0.0675

* Col_2, Col_5, and Col_24 are taken from Colman (1986), H/S_1, H/S_2, H/S_3, H/S_7 and

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H/S_9 are taken from Sumer et al. (1996), as reported by Hsu et al. (2003), R&T_3, R&T_4, R&T_5 and R&T_6 are taken from Rose and Thorne (2001), 1957EQ and 1957ST are taken from Lyn (1986), and E&C_12 is taken from Einstein and Chien (1955). † Calculated as ks = 2.5ds.

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* ks = 2.5 ds

0.99 0.94 0.96 0.99 0.70 0.94 0.91 0.91 0.97 0.99 0.99 0.99 0.997 0.99 0.98

1.08 2.80 1.90 22.1 26.6 32.8 84.1 75.2 0.61 0.63 0.52 0.33 62.3 20.9 88.1

ro

0.71 0.62 0.74 1.18 0.59 0.43 0.63 0.63 0.52 0.49 0.48 0.44 2.30 1.67 1.16

Ro

0.72 0.63 0.77 0.53 0.45 0.36 0.72 0.54 0.53 0.50 0.48 0.44 2.43 1.78 1.27

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Co (%)

re

0.079 0.28 0.18 2.61 4.55 8.58 18.8 18.8 0.098 0.12 0.10 0.072 0.034 0.084 2.61

R2

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Col_2 Col_5 Col_24 H/S_1 H/S_2 H/S_3 H/S_7 H/S_9 R&T_3 R&T_4 R&T_5 R&T_6 1957EQ 1957ST E&C_12

Ro

na

(cm)

Ro

ur

Ca(za)Ro

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Case

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Table 2: Values of fitted and derived parameters using the Rouse and related Rouse equations when applied to the profiles in Figure 1. Rouse Eq (3) Related Rouse Eq (8)* (2) (3) (4) (5) (6) (7) (1)

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R2

0.99 0.94 0.97 0.93 0.93 0.94 0.93 0.93 0.975 0.99 0.99 0.99 0.997 0.99 0.99

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(cm /s)

0.49 1.54 0.49 0.29 0.33 0.24 5.88 3.85 1.56 0.49 0.33 0.27 0.61 1.16 0.66

0.80 2.52 0.80 0.57 0.79 0.66 13.0 9.70 3.94 1.0 0.54 0.43 0.96 1.95 2.7

ws (cm/s)

R2

ro

(cm)

2

1.72 2.16 1.70 3.10 3.18 2.22 8.30 5.00 1.70 1.2 0.82 0.77 6.0 7.1 8.9

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1.05 1.32 1.05 1.59 1.34 0.82 3.76 2.01 0.67 0.55 0.50 0.46 3.93 4.22 2.23

εo

re

0.63 0.59 0.63 21.1 24.7 30.2 37.4 41.1 0.13 0.19 0.19 0.14 1.60 1.21 21.4

εo/κu∗

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Ro

na

Col_2 Col_5 Col_24 H/S_1 H/S_2 H/S_3 H/S_7 H/S_9 R&T_3 R&T_4 R&T_5 R&T_6 1957EQ 1957ST E&C_12

Co (%)

ur

Case

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Table 3: Values of fitted and derived parameters from Eq (10) when applied to the suspended matter the profiles in Figure 1. Residual Rouse Eq (10) (2) (3) (4) (5) (6) (7) (8)* (1)

0.97 0.98 0.97 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.998 0.99 0.997

ε0/??u∗ks † 17.8 56.0 9.3 8.9 10.2 7.4 39.2 25.7 48.0 16.3 10.2 8.3 12.8 24.4 9.8

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* Ratio of column (4) of Table 3 and column 8 of Table 1. † Mean: 20.3, Median: 12.8, Std Deviation: 15.3, Std Error: 4.0; Skewn.: 1.24; Kurt.: 0.196

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Table 4: Parameter values used in computing the van Rijn roughness parameter, kvR, from Eq (5) in van Rijn (2007)*. (1) (2) (3) (4) (5) UC ds kvR Case ?? -1

(dimensionless)

(cm)

0.011 0.011 0.021 0.013 0.013 0.013 0.06 0.06 0.013 0.012 0.013 0.013 0.019 0.019 0.027

618 618 609 363 549 678 59 68 544 469 242 414 129 129 1029

0.22 0.22 0.42 0.26 0.26 0.26 8.6 8.2 0.26 0.24 0.26 0.26 1.86 1.867 0.54

re

-p

ro

of

(cm)

~100 ~100 ~100 83 102 114 72 77 102 91 68 89 ~60 ~60 189

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Col_2 Col_5 Col_24 H/S_1 H/S_2 H/S_3 H/S_7 H/S_9 R&T_3 R&T_4 R&T_5 R&T_6 1957EQ 1957ST E&C_12

(cm s )

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* Uc is the mean flow velocity and ?? is a mobility parameter, as given in van Rijn (2007). Other symbols are as defined here.

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FIGURE CAPTIONS

Figure 1. Plot of 14 suspended sediment profiles and the fits provided by the Residual Rouse model, equation (10). Sources as indicated in the panels. Corresponding fitted parameter values are in Table 3. Note that mismatches of the Residual Rouse model in the upper reaches of the flow result from the neglect of the upper-surface that is present

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in the original Rouse model.

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Figure 2. Plot of the observed residual mixing length at the (nominal) sediment-water interface, ε0/??u∗, as calculated from the fits with Eq (10) versus the Nikuradse

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roughness height, ks = 2.5 ds, for each of the cases reported in Tables 1-3. The ε0/??u∗

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values are reported in Table 3, column 4. The Nikuradse roughness heights are reported in Table 1, column 8. The 1:1 line is displayed, and the data would fall on that

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line if the observed residual mixing length was indeed the Nikuradse length. Ratios of

na

these lengths are given in Table 3, column 8 and range from 7.4-56.

ur

Figure 3. Plots of the residual diffusion coefficient, ε0, as a function of: A the reported

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grain diameter (ds) and B the estimated roughness Reynolds Number (Red).

Figure 4. Plots of the van Rijn (2007) roughness parameter, kvR, as a function of: A the

residual diffusion coefficient, ε0, from Table 3 and B the residual mixing length at the sediment-water interface, ε0/(?? u∗). Conflict of Interest

The authors declare no conflicts of interest. Highlights

1 - Tests the validity of a Rouse model with residual sediment diffusivity at the sediment-water interface

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3 - Residual diffusivity from data appears to correlate with both the roughness Reynolds number and the van Rijn roughness parameter

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Figure 1r1

Figure 1r2

Figure 2

Figure 3

Figure 4