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**Earth Surface Dynamics**
An interactive open-access journal of the European Geosciences Union

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**Research article**
03 Jun 2020

**Research article** | 03 Jun 2020

Entrainment and suspension of sand and gravel

^{1}Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA^{2}Ven Te Chow Hydrosystems Laboratory, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA^{3}Department of Geology, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA^{4}Department of Earth, Environmental and Planetary Sciences, Rice University, Houston, TX 77005, USA^{5}Department of Geography, Simon Fraser University, Burnaby, British Columbia, Canada^{6}School of Environmental Science, Simon Fraser University, Burnaby, British Columbia, Canada

^{1}Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA^{2}Ven Te Chow Hydrosystems Laboratory, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA^{3}Department of Geology, University of Illinois at Urbana-Champaign, Champaign, IL 61801, USA^{4}Department of Earth, Environmental and Planetary Sciences, Rice University, Houston, TX 77005, USA^{5}Department of Geography, Simon Fraser University, Burnaby, British Columbia, Canada^{6}School of Environmental Science, Simon Fraser University, Burnaby, British Columbia, Canada

**Correspondence**: Michael P. Lamb (mpl@gps.caltech.edu)

**Correspondence**: Michael P. Lamb (mpl@gps.caltech.edu)

Abstract

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The entrainment and suspension of sand and gravel are
important for the evolution of rivers, deltas, coastal areas, and submarine
fans. The prediction of a vertical profile of suspended sediment
concentration typically consists of assessing (1) the concentration near the
bed using an entrainment relation and (2) the upward vertical distribution of
sediment in the water column. Considerable uncertainty exists in regard to
both of these steps, especially the near-bed concentration. Most
entrainment relations have been tested against limited grain-size-specific
data, and no relations have been evaluated for gravel suspension, which can
be important in bedrock and mountain rivers. To address these issues, we
compiled a database with suspended sediment data from natural rivers and
flume experiments, taking advantage of the increasing availability of
high-resolution grain size measurements. We evaluated 12 dimensionless
parameters that may determine entrainment and suspension relations and
applied multivariate regression analysis. A best-fit two-parameter equation
(*r*^{2}=0.79) shows that near-bed entrainment, evaluated at 10 % of
the flow depth, decreases with the ratio of settling velocity to
skin-friction shear velocity (${w}_{\mathrm{s}i}/{u}_{\ast \mathrm{skin}}$), as in
previous relations, and increases with Froude number (*Fr*), possibly due to its
role in determining bedload-layer concentrations. We used the Rouse equation
to predict concentration upward from the reference level and evaluated the
coefficient *β*_{i}, which accounts for differences in the turbulent
diffusivity of sediment from the parabolic eddy viscosity model used in the
Rouse derivation. The best-fit relation for *β*_{i} (*r*^{2}=0.40) indicates greater relative sediment diffusivities for rivers with
greater flow resistance, possibly due to bedform-induced turbulence, and
larger ${w}_{\mathrm{s}i}/{u}_{\ast \mathrm{skin}}$; the latter dependence is
nonlinear and therefore different from standard Rouse theory. In
addition, we used empirical relations for gravel saltation to show that our
relation for near-bed concentration also provides good predictions for
coarse-grained sediment. The new relations extend the calibrated parameter
space over a wider range in sediment sizes and flow conditions compared
to previous work and result in 95 % of concentration data throughout the
water column predicted within a factor of 9.

How to cite

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How to cite.

de Leeuw, J., Lamb, M. P., Parker, G., Moodie, A. J., Haught, D., Venditti, J. G., and Nittrouer, J. A.: Entrainment and suspension of sand and gravel, Earth Surf. Dynam., 8, 485–504, https://doi.org/10.5194/esurf-8-485-2020, 2020.

1 Introduction

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The suspension of sediment by water plays a critical role in the dynamics of rivers, river deltas, shallow marine environments, and submarine canyons and fans. For example, suspended sediment dominates the load of lowland rivers and builds land in subsiding river deltas and coastal landscapes (Ma et al., 2017; Syvitski et al., 2005; Wright and Parker, 2004a). The transport of sediment on the continental shelf is dominated by the suspension of mud and sand due to wave and current bottom traction (Cacchione et al., 1999; Nittrouer et al., 1986). Suspended sediment provides the negative buoyancy of turbidity currents that move sand and gravel to the deep sea. The suspension of gravel is important in large floods, such as outburst floods (Burr et al., 2009; Larsen and Lamb, 2016), and in steep mountain canyons (Hartshorn et al., 2002), where it can contribute to bedrock erosion (Lamb et al., 2008a). Suspended sediment transport is also important in landscape engineering, such as river restoration (Allison and Meselhe, 2010), fish habitat (Mutsert et al., 2017), and the capacity of dams and reservoirs (Walling, 2006). The balance between entrainment and deposition from suspension determines patterns of deposition and erosion in these environments and therefore controls landform morphology and stratigraphic evolution (Garcia and Parker, 1991; Paola and Voller, 2005). To predict suspended sediment flux across these environments, we need robust theory for the entrainment of sediment from the bed and the vertical distribution of suspended sediment in the water column.

We focus here on understanding the suspended sediment load of cohesionless grains that are entrained from the bed (i.e., suspended bed material) rather than wash load. Most models for sediment suspension are based on the application of Rouse theory (Rouse, 1937; Vanoni, 1946),

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{C}{{C}_{a}}}={\left[{\displaystyle \frac{\frac{H-z}{z}}{\frac{H-a}{a}}}\right]}^{P},\end{array}$$

where *C* (L^{3} L^{−3}) is the volumetric sediment concentration at
elevation (*z*; L) above the bed, *C*_{a} (dimensionless) is the reference
near-bed concentration at *z*=*a* (unit: L), and *H* (L) is the flow depth. *P*
denotes the dimensionless Rouse number,

$$\begin{array}{}\text{(2)}& P={\displaystyle \frac{{w}_{\mathrm{s}}}{\mathit{\beta}\mathit{\kappa}{u}_{\ast}}},\end{array}$$

in which *w*_{s} (L T^{−1}) is the particle settling velocity, *κ* is the
dimensionless von Karman constant of 0.41, *u*_{∗} (L T^{−1}) is the bed
shear velocity, and *β* is a dimensionless factor that accounts for
differences between the turbulent diffusivity of sediment and the parabolic eddy
viscosity model used to derive the Rouse equation
(e.g., Graf and
Cellino, 2002). Although more sophisticated models exist, some of which
abandon the Rouse theory entirely in favor of a more rigorous turbulence
model (Mellor and Yamada, 1982), the Rouse equation remains a
useful and tractable approach for modeling and field application
(Graf
and Cellino, 2002; van Rijn, 1984; Wright and Parker, 2004b). The Rouse
equation was derived assuming an equilibrium suspension whereby the upwards
volumetric flux of sediment per unit area due to turbulence (*F*_{z}; L T^{−1})
is balanced by a downwards gravitational settling flux (*C**w*_{s})
(Fig. 1) (Rouse, 1937). It predicts the shape of the concentration–depth
profile, with a greater gradient in concentration for larger *P* (Fig. 1). By
mass balance, the difference between entrainment, *F*_{za}, and settling,
*C*_{a}*w*_{s}, fluxes per unit area near the bed (at *z*=*a*) controls the
bed deposition rate, d*z*_{b}∕d*t* (L T^{−1}) (Parker, 1978), i.e.,

$$\begin{array}{}\text{(3)}& (\mathrm{1}-\mathit{\lambda}){\displaystyle \frac{\mathrm{d}{z}_{\mathrm{b}}}{\mathrm{d}t}}={F}_{za}-{C}_{a}{w}_{\mathrm{s}}={w}_{\mathrm{s}}\left({E}_{\mathrm{s}}-{C}_{a}\right),\end{array}$$

where *λ* (dimensionless) is the bed porosity, and ${E}_{\mathrm{s}}\equiv {F}_{za}/{w}_{\mathrm{s}}$
is a dimensionless sediment entrainment rate or entrainment parameter
(Garcia and Parker, 1991). At steady state, Eq. (3) reduces to
*C*_{a}=*E*_{s}; thus, the near-bed entrainment rate, *E*_{s}, is necessary
to predict both the vertical distribution of suspended sediment at steady
state (i.e., *C*_{a} in Eq. 1) and the rate of erosion and deposition for
disequilibrium suspensions (Eq. 3).

The application of Eqs. (1)–(3) requires the specification of *β*,
*E*_{s} and *a*, and the approach in previous work has been to identify
important dimensionless variables and fit data from flume experiments and
natural rivers to these variables (e.g., Smith and McLean, 1977; van Rijn,
1984; Garcia and Parker, 1991). Owing to differences in datasets analyzed
and the dimensionless variables explored, there is considerable deviation in
the form of previous models and their predictions (Figs. 2 and 3, Tables 1
and 2). For example, Einstein (1950), Engelund and Fredsøe (1976), van
Rijn (1984), and Smith and McLean (1977) used a relation for *a* that depends
primarily on grain diameter (*D*; L) and secondarily on ${u}_{\ast}/{w}_{\mathrm{s}}$ or Shields number, ${\mathit{\tau}}_{\ast}={\mathit{\tau}}_{\mathrm{b}}/\left({\mathit{\rho}}_{\mathrm{s}}-{\mathit{\rho}}_{\mathrm{w}}\right)gD$ (where *τ*_{b} (ML^{−1}T^{−2}) is bed stress, *ρ*_{s} (M L^{−3}) is sediment density, and *ρ*_{w} (M L^{−3}) is fluid
density). Their rationale was based on the idea that there is a well-mixed
near-bed zone of bedload transport and that suspended sediment is entrained
from this zone (Fig. 1b). Thus, they developed relations for *a* that scale
with the height of bedload saltation. In contrast, Garcia and Parker (1991)
and Wright and Parker (2004b) argued for a simpler approach and used a
reference height that is a small fraction of the flow depth, and they
proposed *a*=0.05*H* as a useful reference height, with no dependence on *D*.

In practice the factor *β* is often neglected (i.e., *β*=1)
under the assumption that the parabolic eddy viscosity assumption used to
derive Eq. (1) represents the turbulent transport of fine suspended
sediment well. *β*<1 in Eq. (2) has been attributed to damping of
turbulence due to sediment-induced stratification, which alters the eddy
viscosity (Einstein,
1955; Graf and Cellino, 2002; Wright and Parker, 2004a), and to
flocculation, which increases the settling velocity of fine-grained sediment (e.g., Bouchez et al., 2011; Droppo and
Ongley, 1994). Some formulas (van Rijn, 1984; Santini et al.,
2019) and datasets (Graf
and Cellino, 2002; Lupker et al., 2011) indicate *β*>1,
which implies enhanced mixing of sediment relative to momentum (Table 1;
Fig. 2). Graf and Cellino (2002) showed that turbulence generated by bedform
roughness results in better sediment mixing and thus in a higher *β*.
Field and flume data (Chien, 1954; Coleman, 1970; van Rijn,
1984) indicate that *β* is greater than unity for the coarse grain size
fraction of the suspended material (Greimann and Holly, 2001).
Nielsen and Teakle (2004) argued that the Fickian
diffusion model in the derivation of the Rouse equation is not valid for
steep concentration gradients, resulting in *β*>1. Most
previous relations show a trend of decreasing *β* with increasing
${u}_{\ast}/{w}_{\mathrm{s}}$ (van Rijn, 1984; Graf and Cellino,
2002; Wright and Parker, 2004b) (Fig. 2a), which could be due to turbulence
damping associated with high concentration suspensions. Wright and Parker (2004b) showed that *β* is also a function of reference concentration
divided by slope (*C*_{a}∕*S*) (Fig. 2b), which they attributed to sediment
stratification. Santini et al. (2019) found that *β* is
a function of ${u}_{\ast}/{w}_{\mathrm{s}}$ and the ratio between flow depth
and bed grain size (*H*∕*D*) (Fig. 2c).

Compared to *β*, even larger uncertainty exists in the dimensionless
entrainment rate, *E*_{s}. In previous work, *E*_{s} was assessed by measuring
the near-bed concentration for equilibrium suspensions, since *E*_{s}=*C*_{a} under those conditions (i.e., $\frac{\mathrm{d}{z}_{\mathrm{b}}}{\mathrm{d}t}=\mathrm{0}$ in Eq. 3).
Previous work found that *C*_{a} was a function of bed stress and grain size
(Fig. 3; Table 2) (Einstein, 1950; Engelund and Fredsoe, 1976; Smith and
McLean, 1977; van Rijn, 1984; Cedik and Rodi, 1984; Akiyama and Fukushima,
1986; Garcia and Parker, 1991; Wright and Parker, 2004b). While holding bed
stress constant, most relations predict smaller *C*_{a} with increasing grain
size across the sand regime, as expected, but some surprisingly show
increasing *C*_{a} for coarse sands and gravel (Akiyama and Fukushima, 1986;
Garcia and Parker, 1991; Wright and Parker, 2004b) (Fig. 3a, c). Due to the
greater weight and settling velocity of larger particles, this behavior is
unrealistic and likely occurs because these coarse particles are outside
the data range used to fit the relations. Despite the importance of gravel
suspension in steep mountain rivers and megafloods, data do not exist to
evaluate the models in the gravel regime. All relations show increasing
*C*_{a} with bed stress, as expected, but there are orders of magnitude
differences in the predictions for *C*_{a} (Fig. 3b). Part of this deviation
is due to differences in the reference level (*z*=*a*) (Fig. 3b), but even
when using a common reference level (*a*=0.05*H*; Fig. 3d), significant
variability still exists. Some of the variance is also due to extrapolating
the models beyond the range in which they were calibrated; for example, van
Rijn (1984) predicts concentrations greater than 100 % at high shear
stresses.

We revisited the problem of sediment entrainment and the suspension of
cohesionless bed sediment by compiling a large database of sediment-size-specific data for bed–sediment mixtures, testing existing relations against
the database, and proposing improved relations for *E*_{s} and *P*. Several
existing formulas for *E*_{s} are applicable only to uniform-sized bed
sediment (Akiyama and Fukushima, 1986; Celik and Rodi, 1984;
Einstein, 1950; Engelund and Fredsøe, 1976; van Rijn, 1984; Smith and
Mclean, 1977) (Table 2). However, given the strong control of grain size on
near-bed concentrations, accurate formulas likely need to make grain-size-specific predictions for sediment mixtures
(Garcia and Parker, 1991;
McLean, 1992; Wright and Parker, 2004b). Our database takes advantage of
high-resolution grain size measurements using laser diffraction
(Lupker
et al., 2011; Gitto et al., 2017; Haught et al., 2017; Santini et al.,
2019), which allows a single concentration profile to be separated into many
grain-size-specific concentration profiles. Our dataset also contains a wide
range grain sizes, extending into the silt regime (median bed sizes range
from 44 to 517 µm), and expands on the number of field
measurements compared to previous efforts. We are unaware of studies on
*C*_{a} for gravel, but data do exist for saltation heights, velocities, and
bedload fluxes for gravel, and calibrated relations for these variables also
exist (Chatanantavet
et al., 2013; Sklar and Dietrich, 2004). To attempt to better constrain
suspension of very coarse sand and gravel, we used existing semiempirical
saltation theory for gravel to check for consistency between sand suspension
data and what might be expected for near-bed gravel concentrations.

2 Methods

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Based on previous theory, we searched for available datasets from rivers and
flume experiments that had suspended sediment profiles (*C*(*z*)), depth-averaged
flow velocity (*U*), flow depth (*H*), channel bed slope (*S*), bed material
grain size (*D*), and the grain size distribution of the bed material and
suspended sediment samples (Tables 3 and S1 in the Supplement). Some of the experimental
studies used a narrow grain size distribution, and, like previous work, we
assumed that the sediment distribution from these studies was uniform. Many
of the older datasets were used in empirical regressions from previous
relations (e.g.,
Garcia and Parker, 1991; van Rijn, 1984; Wright and Parker, 2004b). In
addition, we used a river dataset from the Yellow River (Moodie,
2019), which provides a fine-grained end-member. In total, our database
contains 180 concentration profiles from eight rivers and 62 profiles from six
different experimental studies. We analyzed only the grain fractions coarser
than 62.5 µm (i.e., sand). The mud fraction was present in the bed only
in small amounts, and following previous work, mud was assumed to require a
different approach, potentially due to supply limitation (Garcia,
2008), cohesion, or flocculation. Grain size distributions in older studies
were determined from sieve analysis of bed material and suspended sediment
samples. The more recent studies used laser diffraction techniques, which
have the advantage that a larger number of grain size classes can be
distinguished. We calculated the grain-size-specific suspended sediment
concentration (*C*_{i}) using

$$\begin{array}{}\text{(4)}& {C}_{i}={f}_{i}{C}_{\mathrm{tot}},\end{array}$$

where *f*_{i} (dimensionless) is the mass fraction of grains of the *i*th size
and *C*_{tot} is the total suspended sediment concentration for all sizes. In
addition, we computed the *D*_{50} (median grain size) and *D*_{84} of the bed
material using linear interpolation between the logarithm of *D* and the
cumulative size distribution.

Concentration profiles in the database typically contain three to eight measurements
in the vertical dimension. The Rouse profile was fitted to the profile data
for each grain size class in log-transformed space using linear least
squares to find *P*_{i} (Fig. 4a). Confidence bounds (68 %; 1*σ*)
for the fitted coefficients were obtained using the inverse *R* factor from QR
decomposition of the Jacobian. Data were excluded from further steps in the
analysis if the ratio between the upper and lower bound of the confidence
interval was greater than 10, as these data do not
follow a Rouse relation for unknown reasons (e.g., measurement error) and
would appear as sparse outliers. We also eliminated points with *P*_{i}<0.01, as these profiles are nearly vertical, which poorly constrains settling velocity to be very near zero. Of the data analyzed, 201 points (15 %)
were excluded based on these criteria. Some studies (Bennett et al., 1998; Muste et al., 2005)
have shown that *P* (or *β*) can vary over the flow depth, but this effect
cannot be incorporated into our approach; instead we found one value of
*P*_{i} that best fit the concentration profile for each grain size class.

We used the Rouse equation (Eq. 1) for each grain size class to extrapolate
or interpolate the concentration to a reference level at 10 % of the flow
depth; i.e., we set *a*=0.1*H* and ${C}_{ai}={C}_{i}(z=a$). Extrapolation to
very near the bed can be difficult due to large concentration gradients and
poorly constrained near-bed processes such as interactions with bedforms.
However, a reference level that is too far away from the bed may poorly
capture the exchange of sediment with the bed. Previous researchers used
reference levels that were either at some fraction of the flow depth
(Akiyama and Fukushima, 1986; Celik and
Rodi, 1984; Garcia and Parker, 1991; Wright and Parker, 2004b) or that were
related to the bed roughness height
(Einstein, 1950; Engelund and
Fredsøe, 1976; Garcia and Parker, 1991; Smith and Mclean, 1977) (Table 2). We explored the collapse of the entrainment data for both types of
reference levels. For each reference level, we fit our preferred and
best-fitting two-parameter entrainment relation (as described in the
Results section) to the data with ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ as the first
parameter and the Froude number as the second parameter. We found that a
reference level at a fraction of the flow depth gave a better collapse of
the entrainment data than a reference level related to the saltation layer
height. Furthermore, we also tested different flow depth fractions and found
that the fit improved, in a least-squared sense, as the reference level
moved to a larger fraction of the flow depth (Fig. 4b). However, there is
little change in *r*^{2} once the reference level is higher than
∼10 % of the flow depth. Therefore, we used a reference
level at 10 % of the flow depth for all results shown below.

For sediment mixtures, the grain-size-specific near-bed concentration is
partially controlled by the fraction of each grain size class in the surface
bed material. To account for this effect, Garcia and Parker (1991)
introduced an entrainment rate (*E*_{si}) for each grain size class that is
linearly weighted by the fraction of that material in the bed:

$$\begin{array}{}\text{(5)}& {E}_{\mathrm{s}i}={\displaystyle \frac{{C}_{ai}}{{F}_{\mathrm{b}i}}},\end{array}$$

where *C*_{ai} is the near-bed concentration of that grain size class and
*F*_{bi} is the mass fraction of bed material that falls in that grain size
class. For uniform sediment, the entrainment rate (*E*_{si}) is equivalent to
the near-bed concentration (*C*_{ai}).

Here we review the independent parameters that we evaluated for dependencies
with a dimensionless entrainment rate, *E*_{si}, and for *β*_{i} in the
Rouse number. The primary group of parameters describes the ratio between
bed stress and grain size or grain settling velocity. These parameters
include the ratio between shear velocity and settling velocity (${u}_{\ast}/{w}_{\mathrm{s}}$), whereby we evaluated the total shear
velocity as

$$\begin{array}{}\text{(6)}& {u}_{\ast}=\sqrt{{\mathit{\tau}}_{\mathrm{b}}/{\mathit{\rho}}_{\mathrm{w}}}=\sqrt{gHS},\end{array}$$

assuming steady, uniform unidirectional flow. Others have proposed that
entrainment depends on the skin-friction portion of the total shear stress,
*u*_{∗skin}, rather that the total shear stress. To estimate
*u*_{∗skin} , we used the Manning–Strickler relation,

$$\begin{array}{}\text{(7)}& {\displaystyle \frac{U}{{u}_{\ast \mathrm{skin}}}}=\mathrm{8.1}{\left({\displaystyle \frac{{H}_{\mathrm{sk}}}{{k}_{\mathrm{s}}}}\right)}^{\mathrm{1}/\mathrm{6}},\end{array}$$

where *k*_{s}=3*D*_{84} is the grain roughness on the bed, *H*_{sk} is
the depth due to skin friction, and ${u}_{\ast \mathrm{skin}}=\sqrt{g{H}_{\mathrm{sk}}S}$ (e.g., Wright and Parker, 2004b). To calculate the particle settling
velocity, we followed Ferguson and Church (2004) for each
grain size class,

$$\begin{array}{}\text{(8)}& {w}_{\mathrm{s}i}={\displaystyle \frac{Rg{D}_{i}^{\mathrm{2}}}{{C}_{\mathrm{1}}\mathit{\nu}+{\left(\mathrm{0.75}{C}_{\mathrm{2}}Rg{D}_{i}^{\mathrm{3}}\right)}^{\mathrm{0.5}}}},\end{array}$$

in which $R=\left({\mathit{\rho}}_{\mathrm{s}}-{\mathit{\rho}}_{\mathrm{w}}\right)/{\mathit{\rho}}_{\mathrm{w}}$ is the
submerged specific density of sediment, *ν* (L^{2} T^{−1}) is the kinematic viscosity of the
fluid, *C*_{1}=18 and *C*_{2}=1 are constants set for natural sediment,
and *D*_{i} is the grain diameter within the size class of interest. Another
parameter that relates to the ratio between bed stress and gravity acting on
the grains is the Shields number,

$$\begin{array}{}\text{(9)}& {\mathit{\tau}}_{\ast}={\displaystyle \frac{{\mathit{\tau}}_{\mathrm{b}}}{\left({\mathit{\rho}}_{\mathrm{s}}-\mathit{\rho}\right)g{D}_{i}}},\end{array}$$

and we again assumed steady, uniform flow to find *τ*_{b}=*ρ**g**H**S*.
Similar to the shear velocity, it is also possible to calculate a Shields
number for the skin-friction component of the total shear stress,

$$\begin{array}{}\text{(10)}& {\mathit{\tau}}_{\ast \mathrm{skin}}={\displaystyle \frac{{\mathit{\tau}}_{\mathrm{skin}}}{\left({\mathit{\rho}}_{\mathrm{s}}-\mathit{\rho}\right)g{D}_{i}}},\end{array}$$

where ${\mathit{\tau}}_{\mathrm{skin}}=\mathit{\rho}{u}_{\ast \mathrm{skin}}^{\mathrm{2}}$ by definition. Shields numbers can be rewritten in terms of ${u}_{\ast}/{w}_{\mathrm{s}i}$ through the use of a particle drag coefficient,

$$\begin{array}{}\text{(11)}& {C}_{\mathrm{d}}={\displaystyle \frac{Rg{D}_{i}}{{w}_{\mathrm{s}i}^{\mathrm{2}}}},\end{array}$$

which we also evaluated (i.e., ${\mathit{\tau}}_{\ast}=\frac{{\mathit{\tau}}_{\mathrm{b}}}{\left({\mathit{\rho}}_{\mathrm{s}}-\mathit{\rho}\right)g{D}_{i}}={\left(\frac{{u}_{\ast}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{2}}\frac{\mathrm{1}}{{C}_{\mathrm{d}}}$).

The next group of parameters describes dimensionless particle sizes, including the particle Reynolds number,

$$\begin{array}{}\text{(12)}& {R}_{\mathrm{p}}={\displaystyle \frac{{u}_{\ast}{D}_{i}}{\mathit{\nu}}}.\end{array}$$

Likewise, this parameter can also be calculated with the skin-friction component of the shear velocity,

$$\begin{array}{}\text{(13)}& {R}_{\mathrm{p},\mathrm{skin}}={\displaystyle \frac{{u}_{\ast \mathrm{skin}}{D}_{i}}{\mathit{\nu}}}.\end{array}$$

A particle Reynolds number can be defined without shear velocity as

$$\begin{array}{}\text{(14)}& {\mathit{\text{Re}}}_{\mathrm{p}}={\displaystyle \frac{\sqrt{Rg{D}_{i}}{D}_{i}}{\mathit{\nu}}}.\end{array}$$

For sediment mixtures, the relative particle size might play a role due to
hiding and exposure effects (Garcia and Parker, 1991; Wright and Parker,
2004b); this effect can be captured with $\frac{{D}_{i}}{{D}_{\mathrm{50}}}$.
Sediment-induced density stratification can decrease entrainment by damping
near-bed turbulence, and this effect is thought to be most important in
deep, low-gradient rivers (Wright and Parker, 2004b). Wright and Parker (2004b) proposed that the ratio of near-bed concentration to bed slope is a
good predictor for stratification, $\frac{{C}_{a}}{S}$, for which they used
*C*_{a} at 5 % of the flow depth. Large, low-gradient rivers also have
small Froude numbers and low bed slopes, so we evaluated the Froude number and
slope as additional parameters. The Froude number was calculated as

$$\begin{array}{}\text{(15)}& \mathit{\text{Fr}}={\displaystyle \frac{U}{\sqrt{gH}}}.\end{array}$$

The entrainment rate could also be affected by turbulence or changes to the boundary layer from bed roughness or bedforms, which tend to correlate with a flow resistance friction coefficient (e.g., Engelund and Hansen, 1967),

$$\begin{array}{}\text{(16)}& {C}_{\mathrm{f}}={\displaystyle \frac{{u}_{\ast}^{\mathrm{2}}}{{U}^{\mathrm{2}}}}.\end{array}$$

We also evaluated *H*∕*D*_{50} as a proxy for flow resistance due to grain
roughness.

In order to find relations that explain the variation in our best-fit
*E*_{si} and *P*_{i} values from the vertical concentration profiles, we
regressed the *E*_{si} and *P*_{i} values against the 12 variables described
above (Eqs. 6–16). In some applications, like reconstructing flow
conditions from sedimentary strata, it is useful to have an entrainment
relation that depends on *u*_{∗skin}, while for forward modeling a
relation based on *u*_{∗} is preferred. The two shear velocities are
highly correlated; therefore, we explored two versions of the fit relations
using either *u*_{∗} or *u*_{∗skin}, but not both at the same time.
Because the Rouse parameter, *P*_{i}, by definition depends inversely on
${u}_{\ast}/{w}_{\mathrm{s}i}$ (or ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$) (Eq. 2), we found the best-fit relations for *P*_{i} rather than
*β*_{i} to avoid spurious correlation. We then solved for the
equivalent relation for *β*_{i} using the definition of *P*_{i} (Eq. 2). We started the analysis by testing all models with one explanatory
variable and ranked the models according to the coefficient of determination
from linear least-squares regression (*r*^{2}) evaluated in log–log space.
Next, a second parameter was used in addition to the first best-fitting
parameter, and the resulting two-parameter models were ranked according to
*r*^{2}. The procedure was repeated with additional parameters until the
increase in *r*^{2} was smaller than 0.04. For the fitting of
multiparameter models, we varied the exponents on each parameter in the
model simultaneously to find the combination of exponents that yielded the
best fit. This approach gave a higher *r*^{2} compared to the stepwise
approach used in previous work (e.g., Garcia and Parker, 1991) of first
fitting the dominant variable and then fitting the secondary variables to
the residuals. In addition, we tested fitting with the York method (Table S2), which gives less weight to data with large errors (York,
1968), but found only minor differences, so all results presented use the
simpler linear least-squares method. All parameters were used to evaluate
relations for *E*_{si} and *β*_{i} except for *D*_{i}∕*D*_{50},
which we used only for *E*_{si} since it is relevant for particle–particle
interactions at the bed surface. In the results we report two versions of
the best-fitting one-, two-, and three-parameter models: one version that is
based on the total bed shear stress and one version that is based on the
skin-friction component of the bed shear stress. Model fits using all
possible combinations of the input parameters are given in Table S2.

Although gravel suspension is important in bedrock and steep mountain
rivers, as well as during megafloods, we are not aware of datasets of near-bed
concentration in the gravel range. Following previous work
(McLean, 1992; Lamb et al., 2008a), our
approach was to derive the near-bed concentration in the bedload layer and
then use Rouse theory to predict that concentration at 0.1*H* to compare with
the sand dataset (e.g., Fig. 1b). The near-bed volumetric concentration
within the bedload layer can be calculated by continuity as

$$\begin{array}{}\text{(17)}& {C}_{\mathrm{b}}={\displaystyle \frac{{q}_{\mathrm{b}}}{\left({H}_{\mathrm{b}}{U}_{\mathrm{b}}\right)}},\end{array}$$

where *q*_{b} (L^{2} T^{−1}) is the volumetric bedload flux per unit width,
*H*_{b} (L) is the bedload-layer thickness, and *U*_{b} (L T^{−1}) is the bedload
velocity. Most relations for bedload flux take the form

$$\begin{array}{}\text{(18)}& {\displaystyle \frac{{q}_{\mathrm{b}}}{\sqrt{\left(Rg{D}^{\mathrm{3}}\right)}}}=m{\left({\mathit{\tau}}_{\ast}-{\mathit{\tau}}_{\ast \mathrm{c}}\right)}^{n},\end{array}$$

where *m* and *n* are empirical constants, which we set to *m*=5.7 and *n*=1.5
(Fernandez Luque and van Beek, 1976), and *τ*_{∗c} is
the critical Shields number at initial motion, which we set according to
Lamb et al. (2008b):

$$\begin{array}{}\text{(19)}& {\mathit{\tau}}_{\ast \mathrm{c}}=\mathrm{0.15}{S}^{\mathrm{0.25}}.\end{array}$$

The bedload-layer height and velocity were determined from Chatanantavet et al. (2013). They compiled a large dataset of gravel saltation observations and found a good fit with the following relations:

$$\begin{array}{}\text{(20)}& {\displaystyle \frac{{U}_{\mathrm{b}}}{U}}=\mathrm{0.6},\text{(21)}& {\displaystyle \frac{{H}_{\mathrm{b}}}{H}}=\mathrm{0.6}{\left(\mathit{\text{Fr}}{\left({\displaystyle \frac{{D}_{i}}{H}}\right)}^{\mathrm{2}}\right)}^{\mathrm{0.3}}.\end{array}$$

Equations (17)–(21) were combined with a flow resistance relation (Eq. 7,
assuming no form drag) and *C*_{d}=0.7 for gravel
(Lamb et al., 2017) to calculate *C*_{b} in
the bedload layer using a numerical iterative scheme. Manipulating the
equations revealed that *C*_{b} is a function of only *τ*_{∗} and
*Fr*. To calculate *C*_{a} for gravel, we extrapolated the concentration profile
(Fig. 1b) from the top of the saltation layer to 0.1*H* using the Rouse
equation (Eq. 1). To obtain the Rouse number, we used our best-fit
one-parameter model that uses total shear velocity (*u*_{∗}) (see Table 4). We then used a wide range of input parameters ($\mathrm{0.1}<\mathit{\text{Fr}}<\mathrm{1}$ and $\mathrm{1}<{\mathit{\tau}}_{\ast}<\mathrm{1000}$) relevant to
the suspension of gravel in mountain rivers and large floods to predict a range
of expected values of *C*_{a} for gravel. Although Eqs. (20) and (21) have not
been tested for high Shields numbers in the suspension regime, they have
reasonable limiting values (*U*_{b}=0.6*U*, *H*_{b}=*H*) and provide a
starting place to compare sand entrainment and gravel saltation theories.

3 Results

Back to toptop
Equation (1) was fit to the concentration profile data wherein *P*_{i} was
treated as a fitting parameter. Then *P*_{i} was regressed against the
hypothesized controlling variables (Eqs. 11–16). Figure 5 and Table 4 show
the results for *P*_{i} including our best-fitting one-, two-, and
three-parameter models. The variable with the most explanatory power is
${u}_{\ast}/{w}_{\mathrm{s}i}$ (${P}_{i}=({u}_{\ast}/{w}_{\mathrm{s}i}{)}^{-\mathrm{0.45}}$; *r*^{2}=0.33). The
best-fit two-parameter model that includes *C*_{f} is better than the
predictions of the one-parameter model (*r*^{2}=0.40). Going from a
two-parameter model to a three-parameter model brings a smaller improvement
(*r*^{2}=0.43). Therefore, we recommend the two-parameter model for
combined accuracy and simplicity:

$$\begin{array}{}\text{(22)}& {P}_{i}=\mathrm{0.145}{\left({\displaystyle \frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}}\right)}^{-\mathrm{0.46}}{C}_{\mathrm{f}}^{-\mathrm{0.3}},\end{array}$$

although there are a number of models using different variables that yield
similar *r*^{2} values (Table S2), and some combinations might be
preferred over others depending on the application. Equation (22) indicates that
sediment is better mixed in the water column with larger ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ and with a larger bed roughness
coefficient, *C*_{f}. Equation (22) performs well compared to previous
relations, as is shown by a box plot of measured-to-predicted ratios (Fig. 6). For the best-fit two-parameter model, the measured-to-predicted ratio
falls between 0.74 and 1.29 for 50 % of the data. Using a Rouse number
with a constant *β*=0.94 also provides a reasonably good fit and an
improvement over several more involved relations (Fig. 6).

Equation (22) can be rewritten for *β*_{i} using Eq. (2), and by
assuming that *u*_{∗} in Eq. (2) is actually *u*_{∗skin}, as

$$\begin{array}{}\text{(23)}& {\mathit{\beta}}_{i}=\mathrm{16.82}{\left({\displaystyle \frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}}\right)}^{-\mathrm{0.54}}{C}_{\mathrm{f}}^{\mathrm{0.3}}.\end{array}$$

Because some of the dimensional quantities appear in multiple dimensional
variables and because the dimensional variables are not necessarily
independent from each other, we tested for spurious correlations by
rearranging the two-parameter relation for *β*_{i} to isolate the
dimensional dependencies on grain size and skin-friction shear velocity
(Fig. 7). The data and our model show a decrease in suspended sediment
mixing in the water column with larger grain sizes, as expected (Fig. 7a).
Previous relations show the same trend, but the relations of van Rijn (1984)
and Graf and Cellino (2002) have stronger dependencies in the sand regime.
Suspended sediments are better mixed with increasing skin-friction shear
velocity (Fig. 7b). The data suggest that *P*_{i} varies proportionally to
${u}_{\ast}^{-\mathrm{0.4}}$, whereas standard Rouse theory (Eq. 2) indicates that
*P*_{i} is proportional to ${u}_{\ast}^{-\mathrm{1}}$.

Results for the best-fit dimensionless entrainment rate are shown in Fig. 8
and Table 5, and all possible variable combinations are given in Table S1.
The following one-parameter relation gives the best fit with the data for
*E*_{si} (*r*^{2}=0.61) (Fig. 8a):

$$\begin{array}{}\text{(24)}& {E}_{\mathrm{s}i}=\mathrm{4.23}\times {\mathrm{10}}^{-\mathrm{5}}{\left({\displaystyle \frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}}\right)}^{\mathrm{1.94}}.\end{array}$$

The Froude number was the most significant second parameter:

$$\begin{array}{}\text{(25)}& {E}_{\mathrm{s}i}=\mathrm{4.74}\times {\mathrm{10}}^{-\mathrm{4}}{\left({\displaystyle \frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}}\right)}^{\mathrm{1.77}}{\mathit{\text{Fr}}}^{\mathrm{1.18}},\end{array}$$

which has a significantly better fit (*r*^{2}=0.79) than the best-fitting one-parameter relation (Fig. 8b). The addition of a third variable to
the model gives little further improvement of the fit (Fig. 8c; *r*^{2}=0.80), and many of the variables used as a third parameter give a similar
level of improvement (Table S1). Using Eq. (25), 80 % of the entrainment
data are predicted within a factor of 3 (Figs. 8b, 9). Along with our
proposed new relation (Eq. 25), we also compared the dataset against the
previous relations (Fig. 9). The box plots in Fig. 9 highlight that some
relations systematically underpredict (Wright and Parker, 2004b) or overpredict the entrainment rate (Garcia and Parker, 1991). In addition, previous
relations have a larger spread in measured-to-predicted ratios than Eq. (25). To check for spurious correlation in the dimensionless variables, we
rearranged our two-parameter entrainment relation (Eq. 25) to isolate the
dependencies of entrainment rate on grain size (Fig. 10a) and skin-friction
shear velocity (Fig. 10b). Our relation indicates that entrainment depends
on grain size to the −3.1 power in the sand range, whereas previous
relations suggest a much weaker dependence. Compared to previous work, our
relation also suggests a relatively weak dependence on skin-friction shear
velocity (${E}_{\mathrm{s}i}\propto {u}_{\ast \mathrm{skin}}^{\mathrm{1.77}}$).

Similar to Garcia and Parker (1991) and Wright and Parker (2004b), we modified Eq. (25) such that the predicted entrainment rate is limited at 0.3, as total suspended sediment concentrations greater than 30 % by volume are not physically reasonable for dilute, turbulent flows. In addition, a threshold (0.015), best fit by eye, was added to the entrainment relation because the concentration data fall below the trend of the regression relation at the lower flow strengths (Fig. 8b), suggesting a threshold of significant sediment entrainment at the reference level. The resulting equation has the following form (Fig. 8b):

$$\begin{array}{}\text{(26a)}& {E}_{\mathrm{s}i}={\displaystyle \frac{\mathrm{4.74}\times {\mathrm{10}}^{-\mathrm{4}}{\left({\left(\frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{1.5}}\mathit{\text{Fr}}-\mathrm{0.015}\right)}^{\mathrm{1.18}}}{\mathrm{1}+\mathrm{3}\left(\mathrm{4.74}\times {\mathrm{10}}^{-\mathrm{4}}{\left({\left(\frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{1.5}}\mathit{\text{Fr}}-\mathrm{0.015}\right)}^{\mathrm{1.18}}\right)}}.\end{array}$$

The equivalent formula for the best-fit two-parameter model without a form drag correction (Table 5) is

$$\begin{array}{}\text{(26b)}& {E}_{\mathrm{s}i}={\displaystyle \frac{\mathrm{7.04}\times {\mathrm{10}}^{-\mathrm{4}}{\left({\left(\frac{{u}_{\ast}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{0.945}}\mathit{\text{Fr}}-\mathrm{0.05}\right)}^{\mathrm{1.81}}}{\mathrm{1}+\mathrm{3}\left(\mathrm{7.04}\times {\mathrm{10}}^{-\mathrm{4}}{\left({\left(\frac{{u}_{\ast}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{0.945}}\mathit{\text{Fr}}-\mathrm{0.05}\right)}^{\mathrm{1.81}}\right)}}.\end{array}$$

In Sect. 3.1 and 3.2 we found the best-fit models for the dimensionless
entrainment rate (*E*_{si}) and Rouse number (*P*_{i}). However, ultimately,
we want to predict the sediment concentration throughout the water column, and
the best-fit models for (*E*_{si}) and (*P*_{i}) do not necessarily combine to
yield the best-fit model for sediment concentration owing to nonlinearity
in *E*_{si}, *P*_{i}, and the Rouse equation (Eq. 1). Here we used different
combinations of our preferred one-, two-, and three-parameter models for
entrainment (*E*_{si}) and the Rouse number (*P*_{i}) to predict the grain-size-specific concentrations at each data point in the water column for all our
entries in the database and assessed model performance (Table 6). The
concentration predictions improve as more parameters are added to the
entrainment model, whereas a Rouse model with more than one parameter makes
the predictions worse (Table 6). Our preferred model for concentration
throughout the water depth uses the two-parameter model to predict the
entrainment rate $\left({E}_{\mathrm{s}i}=\mathrm{4.74}\times {\mathrm{10}}^{-\mathrm{4}}{\left(\frac{{u}_{\ast \mathrm{skin}}}{{w}_{\mathrm{s}i}}\right)}^{\mathrm{1.77}}{\mathit{\text{Fr}}}^{\mathrm{1.18}}\right)$ and the one-parameter model for the Rouse number $\left({P}_{i}=({u}_{\ast}/{w}_{\mathrm{s}i}{)}^{-\mathrm{0.45}}\right)$ (Fig. 11). Such a model gives significantly better
predictions than the most basic formulation that uses one-parameter models
for entrainment and the Rouse number (*r*^{2} of 0.87 versus 0.65; Table 6). The
goodness of fit is also fairly constant over the height of the flow; the
upper 1∕3 of the flow has a slightly lower *r*^{2} (0.80) than the
predictions from the lowest 1∕3 of the flow (*r*^{2}=0.89) (Fig. 11);
95 % of the data are predicted within a factor of 9.

The suspended sediment data that we used to calibrate the entrainment
relation cover material in the sand range. To evaluate how our entrainment
relation performs for coarser suspended sediment, we used the empirical
saltation equations for gravel to infer bedload-layer concentrations,
*C*_{b}, and interpolated these to the reference level (0.1*H*) to infer
*C*_{a} for gravel (Sect. 2.3). Importantly, the gravel concentrations at
the reference height can be predicted from the saltation model (Sect. 2.3)
using only the independent parameters of *Fr* and *τ*_{∗},
similar to our best-fitting two-parameter entrainment model. To compare to
gravel, we assumed a uniform sediment size and used the version of our
best-fit two-parameter model that does not include a form drag correction
(Eq. 26b). This was done because gravel saltation studies were typically
performed over a planar bed, whereas form drag likely played a significant
role in the sand datasets owing to dunes. The marked parameter space in
Fig. 12 shows the expected range of *C*_{b} and *C*_{a} for a wide range of
model input parameters for gravel: $\mathrm{0.1}<\mathit{\text{Fr}}<\mathrm{1}$ and $\mathrm{1}<{\mathit{\tau}}_{\ast}<\mathrm{1000}$. Predicted concentrations
in the gravel bedload layer are up to several orders of magnitude higher
than the predictions from our entrainment relation at 10 % of the flow
depth (Fig. 12). However, due to the rapid decrease in sediment
concentration away from the bed predicted by the Rouse profile, the
concentration inferred at 0.1*H* for gravel overlaps the empirical
relation for sand, implying that Eq. (27) might also be a good predictor of
gravel entrainment.

4 Discussion

Back to toptop
The explanatory variables in the models for *P*_{i} and *β*_{i} are
${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ and *C*_{f}. The dependency on these
parameters could reflect processes that result in the turbulent diffusivity
for suspended sediment being different from the parabolic relation used in
the derivation of the Rouse profile (e.g., Bennett et al., 1998). Compared
to standard Rouse theory, in which the Rouse parameter is inversely
dependent on ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ (or ${u}_{\ast}/{w}_{\mathrm{s}i}$), our results indicate a significant
nonlinearity. Our one-parameter model indicates that concentration profiles
are better mixed than standard theory for small ${u}_{\ast}/{w}_{\mathrm{s}i}$
and are more stratified for larger ${u}_{\ast}/{w}_{\mathrm{s}i}$, with a
transition point at about ${u}_{\ast}/{w}_{\mathrm{s}i}=\mathrm{7}$, corresponding to
*β*_{i}=1. Sediment-induced stratification is often cited
(Winterwerp, 2006; Wright and Parker, 2004b, a) as a factor that decreases the mixing of
sediment (i.e., *β*<1). This effect is particularly important
when the absolute concentration is high and may help explain why our best-fit
model is more stratified (i.e., *β*<1) than standard Rouse
theory for large ${u}_{\ast}/{w}_{\mathrm{s}i}$. An alternative hypothesis by
Nielsen and Teakle (2004) is that for steep concentration gradients, the
size of the turbulent eddies is large relative to the mean height of
sediment in the flow. Under these circumstances, large eddies might more
effectively mix sediment that is concentrated close to the bed. This process
may explain the better mixed concentration profiles we observed (i.e.,
*β*_{i}>1) compared to standard theory at small
${u}_{\ast}/{w}_{\mathrm{s}i}$ when near-bed concentration gradients are
large.

Our relation implies that *β*_{i} correlates positively with the flow
friction coefficient *C*_{f}. This dependency could be from bedforms;
bedforms increase *C*_{f} due to form drag (Engelund and Hansen,
1967), and they may also increase the vertical mixing of sediment by
deflecting transport paths up the stoss side of dunes and mixing suspended
sediment in the turbulent wake in the lee of dunes. Similarly, Santini et
al. (2019) found that *β* correlated positively with *H*∕*D* (Fig. 2c), which
is another measure for bed roughness in flows without bedforms. In
agreement, Graf and Cellino (2002) reviewed a number of experimental studies
and found that *β*<1 for experiments without bedforms and
*β*>1 for experiments with bedforms. Flow resistance
(*C*_{f}) can also be smaller in flows with sediment-induced stratification,
which correlates with smaller *β*_{i} (e.g., Wright and Parker,
2004a).

Our new empirical relation for sediment entrainment suggests that only
${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ (or ${u}_{\ast}/{w}_{\mathrm{s}i}$) and *Fr* are
needed to predict entrainment rate, similar to the forward model we
developed for the suspension of gravel (Sect. 2.3). The ratio ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ describes the fluid forces relative to
gravitational settling; similar parameters have appeared in all previous
relations that we reviewed (Table 2). The reason for the increase in
entrainment with the Froude number is less clear. A small Froude number implies
a deep and low-gradient flow, and Froude numbers are typically smaller in
natural rivers compared to flume experiments. Wright and Parker (2004b)
introduced an entrainment relation with a bed slope dependency and argued
that entrainment in large low-sloping rivers is reduced due to
stratification effects. Sediment-induced stratification causing the damping of
turbulence might be the cause of the *Fr* dependency in our relation.
Regardless, we found a better fit with the data using *Fr* than using
$\frac{{C}_{a}}{S}$ or *S* (Table S2), the parameters suggested by Wright and
Parker (2004b). The Froude number might also influence the size and shape of
bedforms (Vanoni, 1974), which can affect boundary layer dynamics
and near-bed turbulence, as well as surface waves. The Froude number is the ratio
between inertial to gravitational forces or, more formally, the ratio of a
unidirectional flow velocity to the celerity of a shallow water wave (i.e.,
$\mathit{\text{Fr}}=U/\sqrt{gH}$). However, we do not think these are the reasons why the
dependence of *E*_{si} on *Fr* emerges in our analysis; rather, our forward model
(Sect. 2.3) suggests that it emerges as a controlling parameter because of
the role of *U* and *H* in determining bedload-layer concentrations, which is
discussed in more detail in Sect. 4.3.

Many of the dimensionless parameters we evaluated correlate with each other
in rivers. While ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ or ${u}_{\ast}/{w}_{\mathrm{s}i}$ was consistently the dominant variable,
several of the possible secondary variables had similar explanatory power as
the ones given in our preferred models. For sediment entrainment, some of
the more highly ranked variables are *C*_{a}∕*S* and *τ*_{∗skin}
in one-parameter models and *H*∕*D*_{50}, *S* and *D*_{i}∕*D*_{50} in two-parameter
models; for the Rouse number they include *R*_{p,skin} and *τ*_{∗} for one-parameter models and *Re*_{p} and *C*_{a}∕*S* in two-parameter models (Table S2). In addition, there is some systematic deviation
between datasets for different rivers (Fig. 8), and different parameters and
exponents might better minimize residuals at specific locations. While our
model has been fit to a large dataset, it has not been validated with
independent data. More data are clearly needed on grain-size-specific
concentration profiles for equilibrium suspensions under a wide range of
${u}_{\ast}/{w}_{\mathrm{s}i}$, *Fr*, and particle sizes, including coarse
sand and gravel.

Combining the predictions for the grain-size-specific reference
concentration and Rouse parameter allows for the calculation of the grain-size-specific sediment concentration throughout the water column (Fig. 11). Our
relation shows that the grain-size-specific sediment concentration at any
given elevation can be predicted to relatively high accuracy (*r*^{2}=0.87) using the preferred combination of a two-parameter entrainment
relation and the one-parameter Rouse number relation (Sect. 3.3). However,
it is unclear how to evaluate the sediment concentration below the reference
level, which could constitute a significant portion of the sediment load.
Bedload fluxes are notoriously difficult to estimate; further, it is unclear
if the region below the reference level is entirely bedload or if it is
bedload and suspended load (e.g., Fig. 1b). Ashley et al. (2020) showed how
bedload sediment fluxes can be estimated from discharge-averaged suspended
sand concentrations, but their method does not predict the concentration
profile. One approach might be to assume that the sediment concentration is
uniform below the reference level at *C*_{i}=*C*_{ai}, as might be the
case in a well-mixed bedload layer (e.g., McLean, 1992) (Fig. 1b). However,
this assumption is inconsistent with our analysis of the saltation
equations, which shows that bedload layers should have greater
concentrations and be less than 10 % of the flow depth (Fig. 12),
especially for sand in deep flows. A second approach is to use the Rouse
profile to extrapolate towards the bed or towards the top of the bedload
layer; however, this approach is also problematic because the Rouse profile
predicts infinitely large concentrations at the bed. Some of our datasets
had concentration measurements below the reference level (Fig. 13). The data
are scattered, but the binned data points suggest a nearly constant
concentration with elevation below the reference level. For lack of a better
method, we propose using the geometric mean of these data as *C*_{i}=1.66*C*_{ai} to approximate the concentration–depth profile below the
reference level (Fig. 13).

The transport of sand and gravel is often modeled using different empirical formulas, which hinders the modeling of systems of mixed gravel–sand transport (Wilcock and Crowe, 2003) and gravel–sand transitions (Paola et al., 1992; Lamb and Venditti, 2016). Gravel can also be in suspension in bedrock and steep mountain rivers, as well as during large floods (Hartshorn et al., 2002; Larsen and Lamb, 2016), and it would be useful to have an entrainment relation to model sediment transport and bedrock erosion in these settings (Lamb et al., 2008a; Scheingross et al., 2014). Unfortunately, sand and gravel are often treated separately, which limits our ability to develop a unified sediment transport theory. Our entrainment relation for sand matches expectations from gravel saltation models (Fig. 12), suggesting that the entrainment relation may be used for sand, gravel, or mixtures of the two. However, we currently lack data on gravel suspension profiles, and developing methods to acquire such data should be the focus of future efforts. It is also unclear how entrainment is complicated by bimodal mixtures (Wilcock and Crowe, 2003) and complex flow hydraulics in steep rivers with large roughness that can significantly affect lift forces and near-bed turbulence (Lamb et al., 2017).

The good fit between the modeled gravel concentrations and the measured sand
data for near-bed sediment concentration suggests that the bedload-layer
equations (Sect. 2.3) might also be used to generate a forward model for
near-bed sediment concentration that works for sand and gravel systems,
similarly to previous efforts (e.g., McLean, 1992). To evaluate this
possibility, we used the saltation equations (Eqs. 17–21) to calculate
sediment concentrations within the bedload layer for conditions
corresponding to our dataset entries for sand-bedded rivers. To extend
Eq. (18) to grain size mixtures, we let *D*=*D*_{i} and ${\mathit{\tau}}_{\ast \mathrm{c}}={\mathit{\tau}}_{\ast \mathrm{c}i}$ and used a hiding function (Parker et
al., 1982) so that

$$\begin{array}{}\text{(27)}& {\mathit{\tau}}_{\ast \mathrm{c}i}={\mathit{\tau}}_{\ast \mathrm{c}\mathrm{50}}{\left({\displaystyle \frac{{D}_{i}}{{D}_{\mathrm{50}}}}\right)}^{-\mathit{\gamma}},\end{array}$$

where *τ*_{∗c50} is the critical Shields number for the median-sized
bed sediment. Equation (28) accounts for the hiding of smaller grains between
larger grains, which renders the smaller grains less mobile, and the
exposure of larger grains into the flow, which renders them more mobile. For
*γ*=1, all grains in a bed mixture move at the same bed stress,
while for *γ*=0, the critical stress for motion is proportional to
grain weight. Gravel-bedded rivers typically have *γ*=0.9
(Parker, 1990). For sand, we evaluated the critical Shields
number following Brownlie (1981),

$$\begin{array}{}\text{(28)}& {\mathit{\tau}}_{\ast \mathrm{c}\mathrm{50}}=\mathrm{0.5}\left[\mathrm{0.22}{\mathit{\text{Re}}}_{\mathrm{p}}^{-\mathrm{0.6}}+\mathrm{0.06}\times {\mathrm{10}}^{\left(-\mathrm{7.7}{\mathit{\text{Re}}}_{\mathrm{p}}^{-\mathrm{0.6}}\right)}\right],\end{array}$$

rather than Eq. (19), which is intended for gravel only. We then calculated the concentrations at 10 % of the flow depth using the Rouse equation (Eq. 1) with the best-fit one-parameter model for the Rouse number. Although the saltation equations in Sect. 2.3 were calibrated for gravel, similar relations have also been used for sand (e.g., Lamb et al., 2008a).

Surprisingly, the near-bed sand concentrations for the entries in our
database (Sect. 2.1) are relatively accurately predicted by the bedload
forward model without any parameter fitting (Fig. 14b). In fact, the
predictions by the bedload-layer forward model are only slightly worse than
the predictions by our preferred two-parameter entrainment model (Fig. 14a)
that was fit to the data (*r*^{2} of 0.68 vs. 0.87). Importantly, the forward
model yields the same controlling parameters as the empirical model, namely
the Shields number, the Froude number, and the bed grain size distribution. The
forward model suggests that the dependence of *E*_{si} on ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ (or ${u}_{\ast}/{w}_{\mathrm{s}i}$) is
predominantly because larger ${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$
correlates with larger *τ*_{∗} and larger bedload-layer
concentrations (Eqs. 17 and 18), as well as more efficient mixing of the
bedload-layer sediment up to the reference height (Eq. 1). Less intuitive is
the dependence of *E*_{si} on the Froude number. In the forward model, this
dependence is because larger *Fr* for the same *τ*_{∗} correlates with
larger *D*_{i}∕*H* (which can be shown by manipulating Eq. 7). In turn, larger
*D*_{i}∕*H* with constant *τ*_{∗} correlates with greater bedload fluxes
(${q}_{\mathrm{b}}\propto {D}_{i}^{\mathrm{3}/\mathrm{2}}$ in Eq. 18), smaller bedload-layer heights (due
to smaller *H* in Eq. 21), and slower bedload-layer velocities (due to smaller
*U* in Eq. 20), all of which increase the sediment concentration in the bedload
layer (Eq. 17). Thus, the forward model indicates that the *Fr* dependency on *E*_{si} emerges because bedload-layer dynamics depend on *U* and *H*, and explanations
for this dependency that rely on stratification, bedforms, or surface waves
are not necessary. The bedload-layer model, while slightly more complicated
to implement, may provide a more robust solution when working outside the
parameter space used to derive the empirical model, since it has a more
physical basis. For example, the forward model can explicitly account for
gravity and other physical properties of the sediment and fluid. In
addition, the bedload-layer model allows for a more mechanistic link between
the bedload and suspended load and avoids uncertainty in how to evaluate
the sediment concentration below the reference level. The model might likely
be improved by accounting for bedform-induced form drag, especially for
sand-bedded rivers. That said, more accurate predictions are still achieved
with the empirical entrainment relation.

5 Conclusions

Back to toptop
We proposed new empirical models for the entrainment of bed material into
suspension and for the shape of the concentration profile governed by the
Rouse parameter. The models were obtained by regression against suspended
sediment data from eight different rivers and six experimental studies. The
data cover a wide range of bed material grain sizes (44–517 µm) and
flow depths (0.06–32 m) and include grain-size-specific data with up to 60
size classes. Our analysis of these data suggests that the near-bed sediment
concentration increases with the ratio between shear velocity or settling
velocity (${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}}$ or ${u}_{\ast}/{w}_{\mathrm{s}i}$) and the Froude number – both parameters also emerge as the
key controlling variables in a forward model based on bedload-layer
concentrations. A parameter such as ${u}_{\ast}/{w}_{\mathrm{s}i}$, which
represents the ratio of fluid force to particle settling, was also present
in previous relations. The Froude number dependence is less clear; it could
be due to stronger sediment-induced stratification in large low-*Fr* rivers,
but our forward model indicates that it emerges because flow velocity and
flow depth impact bedload-layer concentrations. Our preferred Rouse
parameter model for the shape of the concentration profile suggests that
the sediment concentration is better mixed in the water column with larger
${u}_{\ast \mathrm{skin}}/{w}_{\mathrm{s}i}$ and a larger bed friction coefficient,
*C*_{f}. The Rouse number is not inversely proportional to ${u}_{\ast}/{w}_{\mathrm{s}i}$, unlike standard Rouse theory,
indicating that sediment is more stratified than expected with ${u}_{\ast}/{w}_{\mathrm{s}i}>\sim \mathrm{7}$ and better mixed than expected
with ${u}_{\ast}/{w}_{\mathrm{s}i}<\sim \mathrm{7}$, possibly due to the
competing effects of sediment-induced stratification when absolute
concentrations are large (corresponding to large ${u}_{\ast}/{w}_{\mathrm{s}i}$)
and enhanced turbulent mixing when concentration gradients are steep
(corresponding to small ${u}_{\ast}/{w}_{\mathrm{s}i}$). The dependence of
the Rouse number on the bed friction coefficient might result from increased
turbulence close to the bed in rivers with large bed roughness or bedforms.
We also demonstrated that near-bed concentrations can be accurately
predicted with saltation equations that have been tested previously for
gravel, suggesting a unified framework to model sand and gravel transport in
rivers.

Data availability

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Data availability.

All data used in the analysis are provided in Table S1 of the Supplement. Model results for all possible variable combinations are given in Table S2.

Supplement

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Supplement.

Table S1 contains all the suspended sediment data that were used to find the empirical relations. Table S2 contains all entrainment and Rouse number models ranked according to goodness of fit as indicated by *r*^{2}. The supplement related to this article is available online at: https://doi.org/10.5194/esurf-8-485-2020-supplement.

Author contributions

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Author contributions.

MPL and GP conceived the study. JdL compiled data and led data analysis with input from MPL. MPL and JdL wrote the initial paper. AJM, DH, JGV, and JAN supplied suspended sediment data and contributed to the final paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank everyone involved in the suspended sediment data collection in the Yellow River (Brandee Carlson, Hongbo Ma) and the Fraser River (Nicola Rammell, Kate Donkers, Jacqui Brown, Michael Wong, Michelle Linde, and Alex Gitto). We thank the reviewers and editor Eric Lajeunesse for their helpful comments.

Financial support

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Financial support.

This research has been supported by the US National Science Foundation (grant no. EAR 1427262), the Caltech Discovery Fund, and the NSF Graduate Research Fellowship (grant no. EAR 1842494).

Review statement

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Review statement.

This paper was edited by Eric Lajeunesse and reviewed by two anonymous referees.

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