# A co-spectral budget model links turbulent eddies to suspended sediment
concentration in channel flows

###### Abstract

The vertical distribution of suspended sediment concentration (SSC) remains a
subject of active research given its relevance to a plethora of problems in
hydraulics, hydrology, ecology, and water quality control. Much of the
classical theories developed over the course of 90 years represent the effects
of turbulence on suspended sediments (SS) using an effective mixing length or
eddy diffusivity without explicitly accounting for the energetics of turbulent
eddies across scales. To address this gap, the turbulent flux of sediments is
derived using a co-spectral budget (CSB) model that can be imminently used in
SS and other fine particle transport models. The CSB closes the pressure-
redistribution effect using a spectral linear Rotta scheme modified to include
isotropoziation of production and interactions between turbulent eddies and
sediment grains through a modified scale-dependent de-correlation time. The
result is a formulation similar in complexity to the widely used Rouse’s
equation but with all characteristic scales, Reynolds number, and Schmidt
number effects derived from well-established spectral shapes of the vertical
velocity and accepted constants from turbulence models. Finally, the proposed
CSB model can recover Prandtl’s and Rouse’s equations under restricted
conditions.

Water Resources Research

Department of Civil and Environmental Engineering, Duke University, Durham,
NC, USA Nicholas School of the Environment, Duke University, Durham, NC, USA
Shuolin<EMAIL_ADDRESS>
A suspended sediment concentration (SSC) equation for turbulent flows is
proposed and tested.

The equation is derived from a co-spectral budget that accounts for energy
distribution in all eddy sizes.

The effects of Reynolds number and a scale-dependent Schmidt number on SSC are
explicitly described.

##  1 Introduction

In his classic treatise on sediment transport, Hans Albert Einstein (HAE)
presented a definition of suspended sediments (SS) and the role of turbulence
in maintaining suspension as follows Einstein (1950):

”The characteristic definition of a suspended solid particle is that its
weight is supported by the surrounding fluid during its entire motion. While
being moved by the fluid, the solid particle, which is heavier than the fluid,
tends to settle in the surrounding fluid. If the fluid flow has only
horizontal velocities, it is impossible to explain how any sediment particle
can be permanently suspended. Only if the irregular motion of the fluid
particles, called turbulence, is introduced can one show that sediment may be
permanently suspended.”

This operational definition is now standard in textbooks and research articles
alike Dey (2014); Green Coco (2014); Dey Ali (2020). Despite some 80 years of
research, the dominant factors controlling suspended sediment concentration
(SSC) in streams continue to draw interest due to its multiple connections to
ecosystem benefits and water quality degradation issues Muste . (2005); Long
Pavelsky (2013); Nazeer . (2014); Dai . (2016); Huai . (2019, 2020); Tseng
Tinoco (2020). High SSC can intercept photosynthetically active radiation
necessary for sustaining submerged aquatic plants in lakes and rivers. The
presence of high SSC is also related to eutrophication and corollary water
quality issues Yujun . (2008); Kellogg . (2014), clogging of gills of fish and
other aquatic organisms, accelerating the denitrification process Liu .
(2013). In certain cases, sediments provide necessary nutrients to aquatic
plants and are of primary significance to sustaining nearshore ecosystems such
as floodplains and marshes. Their role in element-cycling has been highlighted
in several studies Lupker . (2011); Mohtar . (2020) as well. Another issue is
the connection between SSC and micro/nano-plastics in saline environments.
Recent work has shown that SS can promote polystyrene nano plastics settling
in the presence of saline conditions, prompting further interest in SSC
distribution in natural waters Y. Li . (2019).

Even in the most idealized flow condition with a balance between the
gravitational settling flux and the vertical turbulent sediment flux, the
description of SSC remains a recalcitrant problem. A model for the turbulent
vertical flux is required and is often derived using Reynolds’ analogy Dey
(2014) where eddies are assumed to transport momentum and SS similarly. This
analogy was the cornerstone of the well-celebrated Rouse’s formula Rouse
(1939) that assumes sediment diffusivity is proportional to eddy viscosity.
Since the early work of O’Brien O′Brien (1933), Prandtl and von Kármán von
Karman (1934), these analogies have spawned numerous theories and closure
models for the mixing length Vanoni (1984); Nie . (2017); Bombardelli Jha
(2009); Bombardelli Moreno (2012); Dey (2014). However, these models make no
explicit contact with turbulent eddies and their associated kinetic energy
distribution in the vertical direction. It is precisely the scale-wise
vertical turbulent kinetic energy component that maintains sediments in
suspension Scully Friedrichs (2003); Mazumder Ghoshal (2006); Dey (2014) as
noted by HAE.

The turbulent vertical flux of SS is directly modeled here from the spectrum
of turbulent eddies thereby providing a new perspective on Reynold’s analogy,
the multiple length scales involved in describing SSC, and the emergence of
Reynolds, Rouse, Schmidt, and Stokes numbers when linking eddy viscosity with
eddy diffusivity for SS. The role of the Reynolds number has been introduced
in prior studies as a damping correction to the mixing length Van Driest
(1956); Wallin Johansson (2000); Nezu Azuma (2004) whereas the Rouse number is
operationally used in the classification of sediment load. The proposed
approach uses a co-spectral budget model (CSB) derived from an approximated
Navier-Stokes equation in spectral form for the Reynolds stress and SS
turbulent flux. It uses a spectral Rotta scheme modified to include the
isotropization of the production term for the pressure decorrelation effect
Katul . (2013) and a Schmidt number effect similar in form to van Rijin’s bulk
formulation van Rijn (1984) for linking the fluid and particle velocity
decorrelation time scales, explicitly made here scale-dependent. The newly
proposed formulation and a simplified solution derived from it are tested with
several published experiments that span a wide range of flow conditions and
grain properties (diameter and density). A comparison against the widely-used
Rouse formula is featured and discussed.

##  2 Theory

###  2.1 Definitions and General Considerations

As a starting point to review models for SSC profiles in streams, a prismatic
rectangular channel with constant width $B$ and bed slope $S_{o}$ is
considered. The flow is assumed to be steady and uniform with constant water
depth $H$ and flow rate $Q$. For small slopes, a balance between gravitational
and frictional forces for a length segment $\Delta x$ along the flow direction
$x$ yields

$\rho(BH\Delta x)gS_{o}=2\tau_{s}(H\Delta x)+\tau_{o}(B\Delta x),$ (1)

where $\tau_{s}$ is the side stress, $\tau_{o}$ is the bed stress, $g$ is the
gravitational acceleration, and $\rho$ is the fluid density. This expression
can be re-arranged as

$u_{*}^{2}=\frac{\tau_{o}}{\rho}={gHS_{o}}\left(1+\frac{2H}{B}\frac{\tau_{s}}{\tau_{o}}\right)^{-1},$
(2)

where $u_{*}$ is the friction (or shear) velocity. For the case where
$\tau_{s}=\tau_{o}$, $u_{*}^{2}=gR_{h}S_{o}$ with $R_{h}=H(1+2H/B)^{-1}$ being
the hydraulic radius. However, in many SS laboratory experiments, the channel
bed is covered with sediments whereas the channel sides remain smooth to
permit optical access. This difference in roughness between sides and bed
leads to $\tau_{s}/\tau_{o}\ll 1$. This assumption can be combined with
$H/B\leq 1$ usually selected to minimize secondary circulation to result in
$\tau_{o}/\rho\approx gHS_{o}$. This approximation is adopted throughout.
Fully turbulent flow conditions are also assumed to prevail so that the bulk
Reynolds number $Re_{b}=U_{b}H/\nu>500$, where $\nu$ is the kinematic
viscosity and $U_{b}$ is the bulk or depth-averaged velocity given as

$U_{b}=\frac{Q}{BH}\approx\frac{1}{H}\int_{0}^{H}\overline{u}(z)dz,$ (3)

where $\overline{u}(z)$ is the mean velocity at vertical distance $z$ from the
channel bed (positive upwards), and overline indicates ensemble-averaging
usually determined from time averaging. For such a flow, the Reynolds-averaged
mean continuity equation for SSC in steady and planar homogeneous flow at high
$Re_{b}$ yields Richter Chamecki (2018)

$\frac{\partial\overline{C}(z)}{\partial t}=0=-\frac{\partial}{\partial
z}\left[\overline{w^{\prime}C^{\prime}}-w_{s}\overline{C}-\Phi(z)\right],$ (4)

where $t$ is time, $C=\overline{C}+C^{\prime}$ is the instantaneous volumetric
SSC in the flow, primed quantities are the fluctuating component, $w$ is the
instantaneous vertical velocity component with $\overline{w}=0$ (assuming
water is of constant $\rho$), $\overline{w^{\prime}C^{\prime}}$ is the
turbulent vertical flux that requires a closure model, $w_{s}$ is the terminal
velocity of sediment grains, and $\Phi(z)$ arises from particle inertia. In
the regime where particle inertia is weak, to a leading approximation,
$\Phi(z)$ is given by Ferry Balachandar (2001); Richter Chamecki (2018)

$\Phi(z)=\tau_{p}\overline{\left[C\frac{Dw^{\prime}}{Dt}\right]}=\tau_{p}\overline{C}\frac{\partial\sigma_{w}^{2}}{\partial
z}+\tau_{p}\overline{\left[C^{\prime}\frac{Dw^{\prime}}{Dt}\right]},$ (5)

where $\tau_{p}=w_{s}/g$ is a particle time scale,
$\sigma_{w}^{2}=\overline{w^{\prime}w^{\prime}}$ is the vertical velocity
variance at $z$ and $D(.)/Dt$ is the material derivative (local and advective)
along a fluid particle trajectory. The $\Phi(z)$ is the sum of a turbophoretic
effect that arises due to finite $\partial\sigma_{w}^{2}/\partial z$ in
inhomogeneous flows such as channels Reeks (1983); Sardina . (2012); Johnson .
(2020) and a turbulent concentration-vertical acceleration interaction terms.
In equation 4, the overall significance of $\Phi(z)$ at any $z$ depends on a
local Stokes number $St(z)=\tau_{p}/\tau_{K}(z)$ where
$\tau_{K}(z)=[\nu/\epsilon(z)]^{1/2}$ is the Kolmogorov time scale formed by
the local turbulent kinetic energy dissipation rate $\epsilon(z)$ and $\nu$ as
reviewed elsewhere Bragg . (2021). An associated length scale to $\tau_{K}$ is
$\eta=(\nu^{3}/\epsilon)^{1/4}$, which is the Kolmogorov micro-scale
representing eddy sizes impacted by viscous effects at $z$. Upon defining the
Kolmogorov velocity as $v_{k}=\eta/\tau_{K}$, the Kolmogorov micro-scale
Reynolds number $Re_{k}=v_{k}\eta/\nu=1$, meaning that both turbulence and
viscous effects are equally important at scales commensurate to $\eta$
Tennekes Lumley (1972). In the limit $St\to 0$, the particle vertical velocity
is given by the sum of the local vertical fluid velocity minus $w_{s}$, and
$\Phi(z)$ can be ignored relative to the turbulent flux at $z$, an assumption
routinely invoked in operational models for SSC. To allow for a ’bulk’ Stokes
number $St_{b}$ to be formulated, thereby facilitating comparisons across
experiments, $\tau_{K,b}=(\nu/\epsilon_{b})^{1/2}$ is proposed where
$\epsilon_{b}$ is the over-all bulk dissipation rate in clear water.
Thermodynamic considerations require that the work per unit mass per unit time
to move clear water at $U_{b}$ is $(gS_{o})U_{b}$. For steady-state conditions
(i.e. turbulent kinetic energy is stationary), this mechanical work produces
turbulence that is then dissipated by the action of viscosity leading to an
increase in the internal energy of the fluid. Hence,

$\epsilon_{b}=(gS_{o})U_{b};~{}~{}~{}\tau_{K,b}=\sqrt{\frac{\nu}{\epsilon_{b}}};~{}~{}~{}{\rm
and}~{}~{}~{}St_{b}=\left(\frac{w_{s}}{g}\right)\tau_{K,b}^{-1}.$ (6)

It is assumed that $\Phi$ is small and can be ignored when $St_{b}\ll 1$
(although, more precisely, $\Phi$ can only be ignored when $\max[St_{b},St]\ll
1$). Another estimate of bulk Stokes number is $St_{+}=\tau_{p}(u_{*}/H)$
Greimann . (1999); Greimann Holly Jr (2001), where $(H/u_{*})$ is presumed to
represent an outer-layer eddy turnover time. Noting that $gS_{o}=u_{*}^{2}/H$,
the two bulk Stokes numbers can related using $St_{b}=St_{+}(Re_{b})^{1/2}$. A
critique for using $St_{+}$ as a bulk Stokes number measure have been
discussed elsewhere Greimann . (1999); Richter Chamecki (2018).

With regards to the terminal sediment velocity, a simplified expression for
$w_{s}$ that recovers many prior formulae Tan . (2018); Huai . (2020) is used
here and is given by Cheng (1997)

$w_{s}=\frac{\nu}{d_{s}}\left[\sqrt{25+1.2d_{s}^{2}\left(\frac{\rho_{s}-\rho}{\rho}\frac{g}{\nu^{2}}\right)^{2/3}}-5\right]^{3/2},$
(7)

where $\rho_{s}$ is the sediment grain density (with $\rho_{s}/\rho>1$), and
$d_{s}$ is the sediment grain diameter. This $w_{s}$ is smaller than the
Stokes settling velocity ($w_{st}$)

$w_{st}=\frac{1}{18}\frac{g}{\nu}\left(\frac{\rho_{s}-\rho}{\rho}\right)d_{s}^{2},$
(8)

except when $w_{st}d_{s}/\nu\ll 1$. The comparison between the two settling
velocities is shown in Figure 1 for reference.

Figure 1: Comparison between the empirical sediment settling velocity $w_{s}$
used here and the Stokes settling velocity $w_{st}$ for different sediment to
fluid density ratios. The one-to-one line is shown. The comparison between
$w_{s}$ and $w_{st}$ for the data sets explored here is also featured as
inset.

Since $w_{st}$ only applies to creeping flow past a sphere, equation 7 is used
as it covers a wider range of $w_{s}d_{s}/\nu$.

The mode of sediment transport is operationally related to $w_{s}$ and some
measure of the strength of turbulence based on bulk flow properties. One such
measure is the Rouse number $R$ or ’unit’ Rouse number $R^{*}$ given by

$R^{*}=\frac{1}{\kappa}\frac{w_{s}}{u_{*}};\quad R=\frac{1}{\beta}R^{*};$ (9)

where $\kappa=0.41$ is the von Kármán constant and $\beta=Sc^{-1}$ is an
inverse turbulent Schmidt number ($Sc$). The Rouse number is routinely used
for classifying sediment load: $R>2.5$ for bedload, $0.8<R<2.5$ for SS, and
$R<0.8$ for washload. To solve for $\overline{C}$, models linking
$\overline{w^{\prime}C^{\prime}}$ to $\overline{C}$ as well as estimates for
$Sc$ (and $\Phi$, though this is ignored here) are required in equation 4, and
those models are to be briefly covered.

###  2.2 Conventional Formulations and Revisions

Conventional approaches (including Rouse and O’Brien) for modeling SSC begin
by ignoring $\Phi(z)$ and employing a gradient-diffusion approximation (or
some non-Fickian revision to it) given as

$\overline{w^{\prime}C^{\prime}}=-D_{s}\frac{d\overline{C}}{dz},$ (10)

where $D_{s}$ is the sediment turbulent diffusivity. To estimate $D_{s}(z)$,
existing theories approximate $D_{s}(z)$ by $\nu_{t}/Sc$ or $\beta\nu_{t}$,
where $\nu_{t}$ is the turbulent or eddy viscosity ($\nu_{t}/\nu\gg 1$). When
the mixing length hypothesis is further invoked to model $\nu_{t}$ as a
product of a characteristic length and velocity, it yields

$\nu_{t}=l_{o}\left(l_{o}\,\left|\frac{d\overline{u}}{dz}\right|\right),$ (11)

where $l_{o}$ is a generic mixing length to be externally supplied that can
vary with $z$. Dimensional analysis and similarity theory represent

$\frac{d\overline{u}}{dz}=\frac{\sqrt{-\overline{u^{\prime}w^{\prime}}(z)}}{l_{o}(z)},$
(12)

where $u^{\prime}$ is the longitudinal velocity fluctuation, and
$\overline{u^{\prime}w^{\prime}}$ is the momentum turbulent flux at height $z$
that can be estimated from the mean momentum balance using Dey (2014)

$\frac{-\overline{u^{\prime}w^{\prime}}(z)}{u_{*}^{2}}=\left(1-z_{n}\right),$
(13)

where $z_{n}=z/H$ is the normalized water depth. With this estimate of
$\overline{u^{\prime}w^{\prime}}(z)$, it follows directly that

$\frac{d\overline{u}}{dz}=\frac{u_{*}}{l_{o}}\left(1-z_{n}\right)^{1/2};\nu_{t}=u_{*}{l_{o}}\left(1-z_{n}\right)^{1/2};D_{s}=\beta{u_{*}l_{o}}\left(1-z_{n}\right)^{1/2}.$
(14)

These expressions ensure that as $z_{n}\rightarrow 1$,
${d\overline{u}}/{dz}\rightarrow 0$, $\nu_{t}\rightarrow 0$, and
$D_{s}\rightarrow 0$. For $z_{n}\ll 1$ but $z^{+}>50$ (i.e. above the buffer
layer) where $z^{+}=zu_{*}/\nu$ is a normalized distance in wall units Pope
(2000) that can also be interpreted as a local Reynolds number ($Re_{s}$),
$l_{o}$ is constrained by the channel bottom so that $l_{o}=\kappa z$. In this
case, ${d\overline{u}}/{dz}\approx u_{*}/(\kappa z)$ and $\overline{u}(z)$
varies logarithmically with $z$, $\nu_{t}=\kappa zu_{*}$, and
$D_{s}=\beta\kappa zu_{*}$ (i.e. linear in $z$). As $z_{n}\rightarrow 1$, the
largest eddies are restricted by $H$ so that $l_{o}\propto H$ instead of $z$.
Combining these two arguments using $l_{o}=\kappa z(1-z_{n})^{1/2}$ yields the
quadratic diffusivity profile reported in a number of stream flow studies
Fischer . (2013) and direct numerical simulations (DNS) of stratified
atmospheric flows on inclined planes Giometto . (2017). Assuming
$\beta=Sc^{-1}=1$, the SSC profiles associated with the linear and quadratic
$D_{s}(z)$ are

$\displaystyle\frac{\overline{C}(z_{n})}{\overline{C_{b}}}=\Bigg{\\{}\begin{array}[]{l
l}(\frac{z_{n}}{z_{n,b}})^{-R^{*}},&\textrm{linear
diffusivity},~{}~{}\textrm{Prandtl's power law}\\\
(\frac{z_{n}}{1-z_{n}}\frac{1-z_{n,b}}{z_{n,b}})^{-R^{*}},&\textrm{quadratic
diffusivity},~{}~{}\textrm{Rouse's formula}\end{array},$ (17)

where $\overline{C_{b}}$ is a reference concentration at height
$z_{n,b}=z_{b}/H$ and $R^{*}=R$ when setting $\beta=1$. The $R^{*}$ in
equation 17 is commonly replaced by a fitted $R$ (or $\beta$ is no longer
unity) as discussed elsewhere Muste . (2005); Dey (2014). The analysis using
fitted $R$ is termed here as ’fitted’ Rouse’s formula. Other models for
$l_{o}$ have been introduced but only two are singled out for illustrating
differences in approaches to adjusting conventional formulations (usually for
$\kappa z$): (i) $l_{o}=\kappa zV_{n}(z_{n})$, where $V_{n}=1-\exp(-z^{+}/26)$
(labeled as the van Driest damping function); (ii) $l_{o}=\kappa
z(1-z_{n})^{m_{1}}$, where

$m_{1}=\frac{1}{2}\left[1+a_{e}\left(\frac{\overline{C}}{C_{R}}\right)\right],$
(18)

$C_{R}$ is some reference concentration and $a_{e}$ is an empirical
coefficient Umeyaina (1992); Mazumder Ghoshal (2006); Castro-Orgaz . (2012).
In the second case, the mixing length is assumed to vary with SSC and recovers
$l_{o}=\kappa z(1-z_{n})^{1/2}$ only for clear water. However, in the presence
of sediments, $m_{1}$ varies with $z_{n}$ (and $R$). In the first case,
deviations from a linear mixing length is made to dependent on $z^{+}$
(instead of $H$), which is appropriate in the viscous and buffer regions of
smooth boundary layers. Another revision to equation 10 is to re-cast
turbulent transport in fractional derivatives to emphasize its non-Fickian
aspect Nie . (2017). In this approach, the fractional order becomes a
parameter that must be determined from experiments depending on how SS
trajectories deviate from Brownian trajectories Sun . (2020). In practice, the
order of the fractional derivative is set as a ’free’ parameter and must
implicitly include the $Sc$ effect. This approach is not pursued further here.

###  2.3 Turbulent Stress and SS Flux Budgets

Simplified turbulent stress and SS flux budgets are now considered. For a
stationary and planar homogeneous flow in the absence of subsidence
($\overline{w}=0$), these budgets reduce to

$\displaystyle\frac{\partial{\overline{w^{\prime}u^{\prime}}}}{\partial t}=0$
$\displaystyle=$
$\displaystyle-\overline{w^{\prime}w^{\prime}}\frac{\partial\overline{u}}{\partial
z}-\frac{\partial{\overline{w^{\prime}w^{\prime}u^{\prime}}}}{\partial
z}+\overline{p^{\prime}\frac{\partial u^{\prime}}{\partial
z}}-\epsilon_{wu},~{}~{}$ (19)
$\displaystyle\frac{\partial{\overline{w^{\prime}C^{\prime}}}}{\partial t}=0$
$\displaystyle=$
$\displaystyle-\overline{w^{\prime}w^{\prime}}\frac{\partial\overline{C}}{\partial
z}-\frac{\partial{\overline{w^{\prime}w^{\prime}C^{\prime}}}}{\partial
z}+\overline{p^{\prime}\frac{\partial C^{\prime}}{\partial
z}}-\epsilon_{wc}-w_{s}\overline{\left(w^{\prime}\frac{\partial
C^{\prime}}{\partial z}\right)},$

where $p^{\prime}$ is the turbulent pressure, $\epsilon_{wu}$ and
$\epsilon_{wc}$ are molecular destruction terms assumed to be small when
compared to the pressure-decorrelation terms at high Reynolds numbers Katul .
(2013). The turbulence- particle interaction term requires closure that may be
achieved by commencing with a local decomposition given by,

$\displaystyle w_{s}\overline{\left(w^{\prime}\frac{\partial
C^{\prime}}{\partial
z}\right)}=w_{s}\left[\left(\frac{\partial\overline{w^{\prime}C^{\prime}}}{\partial
z}\right)-\left(\overline{C^{\prime}\frac{\partial w^{\prime}}{\partial
z}}\right)\right].$ (20)

When assuming $\Phi(z)=0$ in equation 4 (i.e. no particle inertia),
$\overline{w^{\prime}C^{\prime}}=w_{s}\overline{C}$ thereby allowing one of
the two terms in the difference shown in equation 20 to be linked to variables
that are explicitly modeled. The other term (i.e.
$\overline{C^{\prime}{\partial w^{\prime}}/{\partial z}}$) still necessitates
a closure. A heuristic model that maintains maximum simplicity is to set

$\overline{C^{\prime}\frac{\partial w^{\prime}}{\partial
z}}=b_{1}\frac{{\partial\overline{w^{\prime}C^{\prime}}}}{{\partial z}},$ (21)

where $b_{1}$ is a positive or a negative constant. Upon setting
$\overline{w^{\prime}C^{\prime}}=w_{s}\overline{C}$, this heuristic closure
model yields Huang . (2014),

$\displaystyle w_{s}\overline{\left(w^{\prime}\frac{\partial
C^{\prime}}{\partial z}\right)}=w_{s}\left[\left(\frac{\partial
w_{s}\overline{C}}{\partial z}\right)-b_{1}\left(\frac{\partial
w_{s}\overline{C}}{\partial
z}\right)\right]=\alpha^{\prime}w_{s}^{2}\frac{\partial\overline{C}}{\partial
z},$ (22)

where $\alpha^{\prime}=1-b_{1}$ is a constant. When $|b_{1}|\ll 1$, then
$\alpha^{\prime}=1$ and

$\displaystyle w_{s}\overline{\left(w^{\prime}\frac{\partial
C^{\prime}}{\partial z}\right)}=w_{s}^{2}\frac{\partial\overline{C}}{\partial
z}.$ (23)

Whether $b_{1}$ or $\alpha^{\prime}$ are strictly closure constants
independent of sediment and/or flow conditions cannot be a priori ascertained.
To do so requires another scaling analysis based on different assumptions and
approximations. In this proposed scaling analysis, $C^{\prime}$ is assumed to
vary with a turbulent quantity such as $\sigma_{c}$, and $w^{\prime}$ to vary
with $\sigma_{w}$. Hence,

$\displaystyle\overline{\left(C^{\prime}\frac{\partial w^{\prime}}{\partial
z}\right)}=A_{F}\left[\sigma_{c}(z)\right]\frac{\partial\sigma_{w}}{\partial
z}=A_{F}\left[\frac{\overline{w^{\prime}C}}{u_{*}}F_{1}\left(z_{n}\right)\right]\frac{\partial\sigma_{w}}{\partial
z},$ (24)

where $A_{F}$ is a flux-variance Albertson . (1995) similarity constant that
can be positive or negative depending on the sign of the correlation
coefficient between $C^{\prime}$ and $\partial w^{\prime}/\partial z$, and
$F_{1}(z_{n})$ is an unknown dimensionless function describing the sediment
concentration variance with $z_{n}$ above and beyond the
$\overline{w^{\prime}C^{\prime}}$ variations with $z_{n}$. Since the goal is
to determine the minimum governing variables impacting $b_{1}$ or
$\alpha^{\prime}$ while assuming $b_{1}$ is independent of $z_{n}$, equations
24 and 21 can be equated to yield

$\displaystyle
A_{F}\left[\frac{\overline{w^{\prime}C^{\prime}}(z_{n})}{u_{*}}F_{1}\left(z_{n}\right)\right]\frac{\partial\sigma_{w}}{\partial
z}=b_{1}\frac{{\partial\overline{w^{\prime}C^{\prime}}(z_{n})}}{{\partial
z}}.$ (25)

Re-arranging to infer $b_{1}$ results in

$\displaystyle
b_{1}=A_{F}\left[\overline{w^{\prime}C^{\prime}}\left(\frac{\partial\overline{w^{\prime}C^{\prime}}}{\partial
z}\right)^{-1}\right]\left[\frac{1}{u_{*}}F_{1}\left(z_{n}\right)\frac{\partial\sigma_{w}}{\partial
z}\right].$ (26)

With the assumption that $b_{1}$ is not dependent on $z_{n}$, additional order
of magnitude arguments must now be invoked to assess the sediment/flow
variables that impact its magnitude: (i) $\partial\sigma_{w}/\partial z\sim-
u_{*}/H$ (likely valid except near the channel bottom), (ii) $F_{1}$ is
roughly a constant, (iii) $\partial\overline{w^{\prime}C^{\prime}}/{\partial
z}=w_{s}\partial\overline{C}/\partial z$, and (iv)
$\overline{w^{\prime}C^{\prime}}/(\partial\overline{C}/\partial z)\sim-
D_{s,avg}$ where $D_{s,avg}=(1/H)\int_{0}^{H}D_{s}(z)dz\sim u_{*}H$. Inserting
these order of magnitude arguments into equation 26 result in

$\displaystyle
b_{1}\sim\mathrm{sgn}(A_{f})\frac{u_{*}H}{w_{s}}\left[\frac{1}{u_{*}}\frac{u_{*}}{H}\right]\sim\mathrm{sgn}(A_{f})\frac{u_{*}}{w_{s}}.$
(27)

Equation 20 is used to suggest a pragmatic closure in equation 22 that applies
to only one of two terms, and this one term itself is only one term in the
overall flux budget. Given the interplay between these multiple terms, the
overall model results for $\overline{C}$ may be robust to uncertainties in
this closure vis-a-vis externally imposing $Sc$ or $\beta^{-1}$ directly on
the eddy diffusivity as common in prior models.

Upon ignoring the flux transport terms (triple moments), and closing the
pressure decorrelation terms using a linear Rotta scheme that accounts for the
isotropization of the production yields

$\displaystyle-(1-C_{I})\sigma_{w}^{2}\frac{\partial\overline{u}}{\partial
z}-A_{R}\frac{\overline{w^{\prime}u^{\prime}}}{\tau}=0,\quad\left[-(1-C_{I})-\alpha^{\prime}\frac{w_{s}^{2}}{\sigma_{w}^{2}}\right]\sigma_{w}^{2}\frac{\partial\overline{C}}{\partial
z}-A_{R}\frac{\overline{w^{\prime}C^{\prime}}}{\tau}=0,$ (28)

where $\tau$ is a turbulent relaxation time scale, $C_{I}=3/5$ is the
isotropization of the production constant determined from rapid distortion
theory Pope (2000), and $A_{R}=1.8$ Katul . (2013); Katul Manes (2014) is the
Rotta constant assumed to be the same for momentum and SS. It directly follows
from these simplified budgets that a model of maximum simplicity for $Sc$ may
be derived as

$Sc^{-1}(z_{n})=\frac{D_{s}}{\nu_{t}}=1+\alpha\left(\frac{w_{s}}{\sigma_{w}}\right)^{2},$
(29)

where $\alpha=\alpha^{\prime}/(1-C_{I})$, though $\alpha^{\prime}$ or $b_{1}$
can vary themselves with $u_{*}/w_{s}$ as noted earlier. It is necessary to
point out that when $\alpha\geq 0$, equation 29 is opposite to what is
predicted by the so-called ’crossing-trajectories’ effect for heavy particles
settling in a turbulent flow. The crossing trajectories arise when particle
trajectories cross trajectories of fluid elements under the influence of
gravity. This effect invariably forces particles to move from a region of
highly correlated flow to another less correlated region Wells Stock (1983).
In this manner, particles lose velocity correlation more rapidly than the
corresponding fluid points and thus must disperse less. Thus, the crossing
trajectories effect requires $Sc>1$ Csanady (1963); Duman . (2016).

###  2.4 The Co-spectral Budget Model

The models so far make no explicit contact with the phenomenon they perpetrate
to represent: turbulent eddies and their energy distribution. The proposed
approach here uses a co-spectral budget model (CSB) to achieve such a link.
The CSB is derived from an approximated Navier-Stokes equation in a spectral
form that links turbulent eddies of different sizes to
$\overline{w^{\prime}C^{\prime}}$. The CSB derivation commences by noting that
$\overline{w^{\prime}C^{\prime}}$ and $\overline{u^{\prime}w^{\prime}}$ both
satisfy the normalizing properties,

$-\overline{w^{\prime}C^{\prime}}=\int_{0}^{\infty}\phi_{wc}(k)dk,~{}~{}-\overline{u^{\prime}w^{\prime}}=\int_{0}^{\infty}\phi_{wu}(k)dk,$
(30)

where $\phi_{wc}(k)$ and $\phi_{wu}(k)$ are the co-spectral density functions
of the turbulent vertical velocity-turbulent sediment concentration and
turbulent vertical-longitudinal velocities, respectively, and $k$ is the
wavenumber or inverse eddy size. The co-spectral budgets associated with
equation 2.3 have been derived elsewhere and simplify to Bos . (2004); Cava
Katul (2012); Katul . (2013); Katul Manes (2014),

$\displaystyle\frac{\partial}{\partial
t}\phi_{wu}(k)=0=P_{wu}(k)+T_{wu}(k)+\pi_{wu}(k)-2\nu k^{2}\phi_{wu}(k),$ (31)
$\displaystyle\frac{\partial}{\partial
t}\phi_{wc}(k)=0=P_{wc}(k)+T_{wc}(k)+\pi_{wc}(k)-\nu(1+Sc_{m}^{-1})k^{2}\phi_{wc}(k),$
(32)

where $P_{wu}(k)=({d\overline{u}}/{dz})E_{ww}(k)$ and
$P_{wc}(k)=({d\overline{C}}/{dz})E_{ww}(k)$ are the stress and flux production
terms at $k$, $E_{ww}(k)$ is the vertical velocity spectrum satisfying the
normalizing relation $\sigma_{w}^{2}=\int_{0}^{\infty}E_{ww}(k)dk$,
$T_{wu}(k)$ and $T_{wc}(k)$ are turbulent transfer terms, $\pi_{wu}(k)$ and
$\pi_{wc}(k)$ are pressure-velocity and pressure-scalar decorrelation terms,
and $Sc_{m}$ is the molecular Schmidt number (not related to $Sc$). Invoking a
spectral-based Rotta model that includes the isotropization of the production
as before, the pressure-scalar co-variance in k-space can be modeled as

$\displaystyle\pi_{wu}(k)=-A_{R}\frac{1}{t_{ww}(k)}\phi_{wu}(k)-C_{I}P_{wu}(k),\quad\pi_{wc}(k)=-A_{R}\frac{1}{t_{r}(k)}\phi_{wc}(k)-C_{I}P_{wc}(k),$
(33)

where $A_{R}\approx 1.8$ and $C_{I}=3/5$ are as before, $t_{ww}(k)$ and
$t_{r}(k)$ are the decorrelation time-scale of the turbulent stress and
particle concentration. A model of maximum simplicity is to assume that these
two wavenumber dependent time scales are related using a wavenumber-dependent
$Sc(k)$ given by,

$t_{r}(k)={t_{ww}(k)}{Sc^{-1}(k)},~{}~{}\textrm{with}~{}{Sc^{-1}(k)}={1+\alpha{(w_{s}\,k\,t_{wc})^{2}}},$
(34)

where $t_{wc}=\min(t_{ww},f_{o}~{}t_{K,b})$ with $f_{o}$ being a constant (a
plausibility argument to such $t_{wc}(k)$ representation is discussed later),
$Sc$ is modeled in analogy to equation 29 albeit in a spectral form e.g. the
local characteristic turbulent velocity is estimated by $(kt_{wc})^{-1}$ using
a one-way coupling approach Elghobashi (1994),
$t_{ww}(k)\propto\epsilon^{-1/3}k^{-2/3}$ is interpreted as a characteristic
time scale derived from dimensional analysis assuming $\epsilon$ is the
conserved quantity across the energy cascade of $E_{ww}(k)$, and $\epsilon$ is
the turbulent kinetic energy dissipation rate. One plausible choice for the
proportionality constant is $C_{o}^{-1/2}$ so as to recover a Kolmogorov time
scale in the inertial subrange, where $C_{o}=0.65$ is the Kolmogorov constant
for the vertical velocity component.

For scalewise integration, it is also necessary to maintain a bounded
$t_{r}(k)$ as $k\rightarrow 0$ for any $z_{n}$. We set $t_{r}(k)=t_{r}(k_{c})$
when $k<k_{c}$, where $k_{c}$ is the smallest inverse length scale where
$E_{ww}(k)$ increases with increasing $k$. The viscous-destruction terms are
negligible when compared to the Rotta terms for $k\eta\ll 1$. Since
$T_{wu}(k)$ and $T_{wc}(k)$ do not contribute to the net production or
destruction of $\phi_{wu}(k)$ and $\phi_{wc}(k)$ but only redistribute them
across scales (i.e. $\int_{0}^{\infty}T_{wu}(k)dk=0$, and
$\int_{0}^{\infty}T_{wc}(k)dk=0$), they are ignored for simplicity Bonetti .
(2017). Adopting these simplifications,

$\displaystyle\phi_{uw}(k)=\left(\frac{1-C_{I}}{A_{R}}\right)\frac{d\overline{u}}{dz}\left[E_{ww}(k)t_{ww}(k)\right],\quad\phi_{wc}(k)=\left(\frac{1-C_{I}}{A_{R}}\right)\frac{d\overline{C}}{dz}\left[E_{ww}(k)t_{r}(k)\right].$
(35)

To integrate these equations across $k$ and derive turbulent shear stress and
sediment flux at any height $z_{n}$, an expression for $E_{ww}(k)$ is
required. A model for $E_{ww}(k)$ that captures known spectral features at an
arbitrary $z_{n}$ is shown in Figure 2.

Figure 2: Left: A typical $E_{ww}(k)$ at $z_{n}$ from the channel bottom. The
very low wavenumber range are assumed to follow the Saffman spectrum
($E_{ww}(k)$ $\propto$ $k^{2}$) until $k_{c}=1/H$. The Saffman spectrum is
then connected using a flat transition (i.e. wall effects introduce energy
splashing) to the inertial subrange at $k_{o}$ $\propto$ $1/z$ where
$E_{ww}(k)$ $\propto$ $k^{-5/3}$. The black curves are extracted from
measurements Nikora Goring (2002) with different flow conditions using ADV and
do not resolve the viscous dissipation range in the vicinity of $k_{e}=1/\eta$
or the presumed Saffman spectrum. Right: The $\sigma_{w}^{2}/u_{*}^{2}$
profile modeled from scale-wise integration of $E_{ww}(k)$ and its simplified
form (i.e. ignoring the Saffman contribution and extending the inertial
subrange indefinitely to fine scales). The measured $\sigma_{w}^{2}/u_{*}^{2}$
profiles are from experiments described elsewhere Raupach (1981); Nikora
Goring (2002); Heisel . (2020). They include field experiments and wind-tunnel
experiments over a wide range of roughness types and Reynolds number
conditions. The direct numerical simulations (DNS) for a smooth channel (red)
are also included for comparisons Heisel . (2020).

The $E_{ww}(k)$ is now piece-wise approximated as

$E_{ww}(k)=\left\\{\begin{array}[]{l
l}E_{kol}(k_{o})k_{c}^{-2}k^{2},&\mathrm{if}~{}0\leq k\leq k_{c}\\\
E_{kol}(k_{o}),&\mathrm{if}~{}k_{c}\leq k\leq k_{o}\\\
E_{kol}(k),&\mathrm{if}~{}k_{o}\leq k\leq k_{e}\end{array}\right.,$ (36)

where $k_{c}=H^{-1}$, $k_{o}=(\kappa z)^{-1}$ and $k_{e}=\eta^{-1}$ are three
characteristic wavenumbers that mark the key transitions in $E_{ww}(k)$
between $H$ and the characteristic eddy scales bounding the inertial subrange
Bonetti . (2017); Katul . (2013); S. Li Katul (2019); Ayet Katul (2020), and
$E_{kol}(k)=C_{o}\epsilon(z)^{2/3}k^{-5/3}$ is the Kolmogorov spectrum. In the
case of $E_{kol}(k)$, the transfer of energy across scales shapes the energy
cascade and is necessary for obtaining the $k^{-5/3}$ scaling. The transfer of
stress across scales, as given by $T_{wu}(k)$, was ignored in the CSB model
here. The inclusion of the transfer term in the energy cascade (indirectly
specified by $E_{ww}(k)$) but not in the CSB may appear paradoxical. This is
not so as the role and significance of the transfer terms are quite different
when analyzing scale-wise energy and stress budgets Bos . (2004). In the
inertial subrange where $E_{kol}(k)\sim k^{-5/3}$, a $\phi_{uw}(k)\sim
k^{-7/3}$ has also been reported and confirmed in numerous boundary layer
experiments and simulations of wall-bounded flows Pope (2000). A balance
between production and dissipation terms in the CSB model leads to a
$\phi_{uw}(k)\sim(d\overline{u}/dz)E_{ww}(k)t_{ww}(k)$, which recovers the
$[(d\overline{u}/dz)\epsilon^{1/3}]k^{-7/3}$ scaling in the inertial subrange.
Inclusion of $T_{wu}(k)$ necessarily leads to $\phi_{uw}(k)$ that must deviate
from a $k^{-7/3}$ scaling in the inertial subrange as discussed elsewhere D.
Li . (2015). Moreover, the constants emerging from a production balancing
dissipation in the scale-wise CSB model for the inertial subrange,
$[(1-C_{I})/A_{R}]C_{o}^{1/2}=0.18$, does recover the accepted co-spectral
similarity constant whose numerical value was determined at $0.15-0.16$ from
wind tunnel studies, atmospheric surface layer studies, and DNS Katul .
(2013). For these reasons (i.e. $T_{wu}(k)$ ignored within the inertial
subrange) and because $\int_{0}^{\infty}T_{wu}(k)dk=0$, $T_{wu}(k)$ is ignored
at all $k$. This assumption is also compatible with ignoring the triple
moments in equations 2.3.

The only remaining term needed to describe the magnitude of $E_{ww}(k)$ at all
$k$ is $\epsilon(z)$. A model of maximum simplicity is to relate $\epsilon(z)$
to the mechanical production $P_{wu}(z)$ of the turbulent kinetic energy
budget using Pope (2000)

$\epsilon(z)=\frac{P_{wu}(z)}{\phi(z_{n})}=\phi^{-1}(z_{n})\left(-\overline{u^{\prime}w^{\prime}}\frac{d\overline{u}}{dz}\right)=\phi^{-1}(z_{n})u_{*}^{2}\left(1-\frac{z}{H}\right)\frac{d\overline{u}}{dz},$
(37)

where $\phi(z_{n})$ is a modification function to account for the imbalance
between the local mechanical production and local dissipation terms in the
turbulent kinetic energy budget. For stationary and planar-homogeneous flow
conditions without any mean vertical advection and in the absence of any
transport terms, $\epsilon(z)\approx P_{wu}(z)$ and $\phi(z_{n})\approx 1$.
While this estimate may be acceptable in the log-region describing
$\overline{u}(z)$, deviations near the channel bottom ($\phi(z_{n})>1$) and
near the water surface ($\phi(z_{n})<1$) are expected. Hence, $\phi(z_{n})$
must be viewed as a depth-dependent function Kim . (1987); Pope (2000) though
its variation from unity is not considered here to maintain maximum
simplicity. A plausibility argument for ignoring its variation from unity is
that $\overline{w^{\prime}C^{\prime}}\propto\left[\phi(z_{n})\right]^{-1/3}$
(shown later), which makes the SSC calculations less sensitive to
$\phi(z_{n})$ deviations from unity. This point is considered later in the
context of modeling $\sigma_{w}^{2}(z_{n})$ based on the assumed $E_{ww}(k)$
shape.

Returning to the choice of $t_{wc}=\min(t_{ww},f_{o}~{}t_{K,b})$ and the
choice $f_{o}$, as $z_{n}\rightarrow 1$,
$\overline{w^{\prime}u^{\prime}}\rightarrow 0$, $P_{wu}(z_{n})\rightarrow 0$,
and thus $\epsilon\rightarrow$ (i.e. no turbulence) near the free water
surface. With $\epsilon\rightarrow$, $t_{ww}(k)\rightarrow\infty$ (along with
$\tau_{k}\rightarrow\infty$ and $\eta_{k}\rightarrow\infty$). That
$t_{ww}(k)\rightarrow\infty$ is not problematic for the closure scheme of
$\pi_{wu}(k)$ and $\pi_{wc}(k)$ as those terms are expected to decay near the
free water surface and this decay remains compatible with
$t_{ww}(k)\rightarrow\infty$. The problem of $\epsilon\rightarrow 0$ arises in
maintaining a finite $Sc^{-1}(k)$ dominated by turbulent processes thereby
necessitating a finite $\epsilon$ in the calculation of $Sc^{-1}(k)$ that
cannot be readily inferred from $P_{wu}(z_{n})$. To ensure that the particle
interaction time scale $t_{wc}$ remains bounded in $Sc^{-1}(k)$, an adhoc
minimal value of $\epsilon$, set to be $0.1\%~{}\epsilon_{b}$, is proposed.
This choice of minimal $\epsilon_{b}$ prevents $\epsilon\rightarrow 0$ as
$z_{n}\rightarrow 1$ in the $Sc(k)$ formulation only. This minimal threshold
set to ensure a finite $\epsilon$ in $Sc(k)$ (mainly near the free water
surface) leads to $f_{o}=\sqrt{1000}\approx 31$.

##  3 Results and Discussion

###  3.1 Co-spectral Budget Model

By scale-wise integrating $\phi_{uw}(k)$ and using
$u_{*}^{2}(1-z_{n})=\int_{o}^{k_{e}}\phi_{uw}(k)$ dk, the velocity gradient
$d\overline{u}/dz$ at $z$ is obtained as

$\displaystyle\frac{d\overline{u}}{dz}=A_{\pi}^{-3/4}\phi^{1/4}(z_{n})\left(1-\frac{z}{H}\right)^{1/2}\left[\frac{15}{4}-\frac{8}{3}\left(\frac{k_{c}}{k_{o}}\right)^{1/3}-\frac{3}{4}\left(\frac{k_{o}}{k_{e}}\right)^{4/3}\right]^{-3/4}\left(k_{o}u_{*}\right),$
(38)

where $A_{\pi}=({1-C_{I}})\sqrt{C_{o}}/{A_{R}}\approx 0.18$, and the vertical
velocity variance can be derived by scale-wise integrating $E_{ww}(k)$ as,

$\displaystyle\frac{\sigma_{w}^{2}}{u_{*}^{2}}=\frac{5}{2}C_{o}A_{\pi}^{-1/2}\phi^{-1/2}(z_{n})\left[1-\frac{4}{15}\frac{k_{c}}{k_{o}}-\frac{3}{5}\left(\frac{k_{e}}{k_{o}}\right)^{-2/3}\right]\left[\frac{15}{4}-\frac{8}{3}\left(\frac{k_{c}}{k_{o}}\right)^{1/3}-\frac{3}{4}\left(\frac{k_{e}}{k_{o}}\right)^{-4/3}\right]^{-1/2}(1-z_{n}).$
(39)

Likewise, the SSC turbulent flux is solved as

$\displaystyle-\overline{w^{\prime}C^{\prime}}=A_{\pi}\phi^{-1/3}(z_{n})\Omega(z)u_{*}^{2/3}\left[\left(1-\frac{z}{H}\right)\frac{d\overline{u}}{dz}\right]^{1/3}\frac{d\overline{C}}{dz}=-w_{s}\overline{C},$
(40)

with $\Omega(z_{n})$ given by

$\displaystyle\Omega(z_{n})=\int_{0}^{k_{c}}Sc^{-1}(k_{c})k_{c}^{-8/3}k_{o}^{-5/3}k^{2}dk+\int_{k_{c}}^{k_{o}}k_{o}^{-5/3}Sc^{-1}(k)k^{-2/3}dk+\int_{k_{o}}^{k_{e}}Sc^{-1}(k)k^{-7/3}dk.$
(41)

Therefore, the turbulent Schmidt number $Sc(z_{n})$ can be determined from the
CSB model as

$\displaystyle
Sc(z_{n})=\frac{\nu_{t}}{D_{s}}=\Omega^{-1}(z_{n})\left[\frac{15}{4}-\frac{8}{3}\left(\frac{k_{c}}{k_{o}}\right)^{1/3}-\frac{3}{4}\left(\frac{k_{o}}{k_{e}}\right)^{4/3}\right]k_{o}^{-4/3}.$
(42)

Because the determination of $k_{e}=1/\eta$ (where
$\eta=(\nu^{3}/\epsilon)^{1/4}$) requires an estimate of
$\epsilon(z_{n})=P_{uw}(z_{n})$ and thus an estimate of $d\overline{u}/dz$, an
iterative scheme is needed to determine $d\overline{u}/dz$ and $k_{e}$ at
every $z_{n}$ from equation 38. Once determined, the $E_{ww}(k)$, $Sc(z_{n})$,
$\overline{w^{\prime}C^{\prime}}$ and the subsequent SSC profile can be
computed at each $z_{n}$ by solving equations 39, 40, and 42 for
$\sigma_{w}^{2}$, $\overline{w^{\prime}C^{\prime}}$ and $Sc$. Since there is
no analytical solution to this system, a numerical integration using a 3rd-
order Adams–Bashforth method is employed.

Before proceeding to the analysis of SSC, an assessment of the assumed shape
of $E_{ww}(k)$, its transition wavenumbers, as well as the consequence of the
assumption of $\phi(z_{n})\approx 1$ is conducted in Figure 2. The predicted
$\sigma_{w}^{2}/u_{*}^{2}$ and its simplified version using $E^{1}_{ww}(k)$
without the Saffman spectrum and assuming $k_{e}\rightarrow\infty$ are
compared against two sets of experiments: (i) wind tunnel experiments
conducted over a wide range of surface roughness types Raupach (1981) and (ii)
field experiments Nikora Goring (2002) of the sediment flow in the Balmoral
Irrigation Canal (New Zealand). The wind-tunnel experiments used a hot-wire
probe whereas the field experiments used acoustic Doppler velocity (ADV)
measurements that do not resolve the viscous dissipation regime. As expected,
the predicted $\sigma_{w}^{2}/u_{*}^{2}$ here exceeds the measurements because
the spectral shapes assumed in $E_{ww}(k)$ account for a much broader range of
eddy sizes than the experiments interrogate. Specifically, the Saffman and
dissipation ranges are not resolved by the flume experiments whereas the wind
tunnel experiments resolve a limited dissipation range but are not conducted
over a sufficiently long enough sampling period to cover the Saffman spectrum.
Nonetheless, the model recovers key features of the $(\sigma_{w}/u_{*})^{2}$
profile: a rapid increase with $z_{n}$ near the surface, a peak at
$(\sigma_{w}/u_{*})^{2}=1.9$, and a quasi-linear decline as $z_{n}\rightarrow
1$. The peak $(\sigma_{w}/u_{*})^{2}=1.9$ is compatible with near-neutral
atmospheric surface layer measurements ($=1.8$) where lateral confinements of
the flow are absent (unlike flumes and wind tunnels) and where $H/\eta$ far
exceeds those obtained in laboratory studies.

Now the comparisons of the CSB results with $\alpha$ temporarily set as a
’free’ parameter with (i) Prandtl’s power law solution and (ii) Rouse’s
formula are shown in Figure 3.

Figure 3: The predicted SSC, $Sc$, and $D_{s}$ profiles based on the CSB model
when setting $d_{s}$=1 mm, $\rho_{s}$=1.2 g cm-3, $u_{*}$=3 cm s-1, and
$U_{b}/u_{*}$=$10$. The reference position is at $z_{n,b}$=$0.01$. Different
$\alpha$ values and Reynolds numbers ($Re_{*}$=$u_{*}H/\nu$) are featured to
illustrate overall sensitivity of the normalized SSC profile to these
parameters. The Prandtl and Rouse model predictions of SSC are shown for
reference in the top-left panel. The Reynolds number is varied by alerting
$\nu$.

The computed SSC and $Sc$ profiles are also presented when the flow conditions
and sediment properties are externally supplied. For Prandtl’s power-law and
Rouse’s formula, the bulk Schmidt number was set to unity. However, the CSB
model allows for a depth-dependent $Sc(z_{n})$, which is set by $\alpha$. When
$\alpha=0$, $Sc(z_{n})=1$ in the entire channel, consistent with equation 29.
When $\alpha>0$, $Sc(z_{n})$ varies with depth and is generally greater in the
near-bed region and becomes smaller with increasing $z_{n}$. However, because
of the imposition of a finite $\epsilon$ near the water surface
($=0.001\epsilon_{b}$), $Sc(z_{n})$ increases back to near unity when
$z_{n}\rightarrow 1$. Rouse’s equation and CSB models exhibit different
behavior near the water surface. Rouse’s equation yields a zero-concentration
at $z_{n}=1$ whereas the CSB model does not. One advantage to the CSB approach
is its ability to resolve the dependence of $\overline{C}/\overline{C_{b}}$ on
Reynolds number. Using different $\nu$, variations in $Re_{*}=u_{*}H/\nu$ can
be generated and their effects on CSB model predictions tracked. Recall that
$H/\eta_{b}$ (modeled in the CSB) scales as $Re_{*}^{3/4}$, and the effects of
this scale separation on the shape of the vertical velocity spectrum, sediment
flux co-spectrum, and the resulting $\overline{C}/\overline{C_{b}}$ profiles
are explicitly determined. The effects of $\alpha$ are much more significant
than the effects of $Re_{*}$, which is heuristically supportive for using
Direct Numerical Simulation runs (lower $Re_{*}$) to further explore the CSB
approach. As earlier noted, the implications of setting
$t_{wc}=\min(t_{ww},f_{o}~{}t_{K,b})$ with $f_{o}=\sqrt{1000}$ are most
visible on the $Sc(z_{n})$ profile near the free water interface. Altering
$f_{o}$ primarily modifies the thickness of the region near the water
interface impacted by the imposed finite $t_{wc}$ (or finite $\epsilon$ in the
$Sc(k)$ determination). However, the CSB model itself is not expected to be
valid in this zone as the assumed shape of $E_{ww}(k)$ is not realistic, the
flux transport terms can be finite, and turbo-phoretic effects may also be
large in this vicinity. In sum, predictions from the CSB model near the free
water surface must be treated with skepticism and caution.

###  3.2 Recovery of the Rouse and Prandtl equations

Whether a Rouse equation can be recovered from the CSB model under certain
simplifications is now examined. Any explicit model must include $Sc$ and
approximations to equation 42. Assuming $k_{c}/k_{o}\rightarrow 0$ and
$k_{o}/k_{e}\rightarrow 0$ in $\Omega(z_{n})$ only (i.e. setting the area
under the Saffman spectrum to zero that is then partially compensated for by
extending the inertial subrange to $k_{e}\rightarrow\infty$), the Schmidt
number derived from equation 42 can be approximated as

$Sc^{-1}\approx
1+B_{\pi}\left(\frac{w_{s}}{u_{*}}\right)^{2},~{}\textrm{with}~{}B_{\pi}=\frac{\sqrt{15A_{\pi}}}{3C_{o}}\alpha\approx
0.84\alpha,$ (43)

which directly recovers the quadratic model for $Sc^{-1}$ reported elsewhere
van Rijn (1984); Bombardelli Moreno (2012) as expected. With $R\approx 0$,
equation 43 indicate $\beta=Sc^{-1}\rightarrow 1$ thereby recovering Rouse’s
original assumption (i.e. SS resemble passive scalars in this case). This
estimate of $\beta$ also allows for the determination of the model coefficient
$\alpha$ using a separate data set and model runs shown in Figure 4.

Figure 4: The model coefficient $\beta$ based on different formulae,
experiments, and model runs. The experiments and model runs presented here are
described elsewhere Jha Bombardelli (2009).

Figure 4 shows different predictions of $\beta$, including
$\beta=1+2(w_{s}/u_{*})^{2}$ van Rijn (1984) and
$\beta=1.3+3(w_{s}/u_{*})^{2}$ Jha Bombardelli (2009) for model results that
explicitly consider particle-fluid interactions. Moreover, with $Sc$ provided
in equation 43, the SS diffusivity is derived as,

$\frac{D_{s}(z)}{\kappa
zu_{*}}=\frac{1}{Sc}\left(1-z_{n}\right)=\left[1+B_{\pi}\left(\frac{w_{s}}{u_{*}}\right)^{2}\right]\left(1-z_{n}\right)$
(44)

where $\kappa zu_{*}$ is the eddy viscosity in the log-region of
$\overline{u}(z)$. Depending on choices made for $\alpha$ or $B_{\pi}$, a
number of empirical relations can be recovered including the widely used
Rouse’s equation and variants on it Hunt (1954). For a given $\alpha$, an
analytical solution for the SSC can be derived and compared with published
experiments. The SSC solution for an arbitrary $\alpha$ is given as

$\frac{\overline{C}(z_{n})}{\overline{C_{b}}}=\left(\frac{z_{n}}{1-z_{n}}\frac{1-z_{n,b}}{z_{n,b}}\right)^{-R_{+}}.$
(45)

where the power exponent $R_{+}$ is defined as

$R_{+}=\frac{1}{1+B_{\pi}\left({w_{s}}/{u_{*}}\right)^{2}}{\frac{w_{s}}{\kappa
u_{*}}}.$ (46)

When $\alpha=0$ (or $B_{\pi}=0$), a quadratic diffusivity profile O′Brien
(1933) as well as Rouse’s formula Rouse (1939, 1937) for SSC given in equation
17 are recovered. Furthermore, in the limit of ($z_{n}\ll 1$) a linear
diffusivity profile von Karman (1934) along with the classic power law
solution are also recovered from equation 45. The consequences on
$\sigma_{w}^{2}$ of setting the Saffman spectrum to zero and extending the
inertial subrange to $k\rightarrow\infty$ on $\sigma_{w}^{2}$ are briefly
discussed using Figure 2. As expected, these approximation over-estimate
$(\sigma_{w}/u_{*})^{2}$ in the near-wall region and underestimate
$(\sigma_{w}/u_{*})^{2}$ in the outer layer when compared to a $E_{ww}(k)$
that accommodates the Saffman spectrum (i.e. large scale effects) but
truncates the inertial subrange at $1/k_{e}$. These effects cannot be readily
ignored and may influence the choices made about $\alpha$.

###  3.3 Comparison with Experiments

The CSB model given by equations 40 and 38 and its simplified version featured
in equation 45 are compared with published experiments Greimann Holly Jr
(2001); Vanoni (1984); Tseng Tinoco (2020) summarized in Table 1. We assume
$\Phi(z)=0$ thereby neglecting inertial effects for compatibility with
operational models (e.g. the Rouse model). The comparisons are shown in Figure
5. For these experiments, all the reported parameters including measured
$u_{*}$, $d_{s}$, and $\rho_{s}$ and the fitted $\beta$ (needed for assessing
the fitted Rouse equation) and $\alpha$ (needed for evaluating the numerical
CSB model) are presented in Table 1.

Table 1: Summary of published experiments and parameters used in model-data comparisons. When setting $Sc$ =$1$, not all runs are classified as SS (or $0.8$ $\leq$ ${R^{*}}$ $\leq$ $2.5$) even though sediments were reported as suspended. While $St_{b}$ is not very small for (b) and (c), $St_{+}$ $\ll$ $1$. Calculated densimetric and critical Froude numbers ($Fr_{d}$ and $Fr_{dc}$) are presented along with the roughness Reynolds number $Re_{pa}$. All experiments lie in the fully-rough ($Re_{pa}>100$) or transitional ($3<Re_{pa}<100$) regimes in fully-developed turbulence ($Re_{b}\geq 500$). Run | (a) | (b) | (c) | (d) | (e) | (f)
---|---|---|---|---|---|---
| Flow Properties
$H$ (m) | 0.10 | 0.52 | 0.50 | 0.10 | 0.10 | 0.10
$B$ (m) | $-$ | 0.84 | 0.84 | 0.15 | 0.15 | 0.15
$U_{b}$ (measured, m s-1) | 1.98 | 3.95 | 3.63 | 0.31 | 0.22 | 0.17
$Re_{b}=U_{b}H/\nu\times 10^{-4}$ | 18.7 | 193 | 170 | 2.9 | 2.1 | 1.6
$Re_{pa}=u_{*}d_{s}/\nu$ | 103 | 169 | 166 | 16 | 13 | 7
$u_{*}$ (cm s-1) | 7.67 | 20.0 | 20.0 | 1.7 | 1.4 | 0.8
$z_{n,b}\times 10^{2}$ (measured) | 6.3 | 2.6 | 3.0 | 5.0 | 3.8 | 5.6
$z^{+}_{n,b}(=u_{*}z_{b}/\nu)$ | 456 | 2747 | 2988 | 85 | 53 | 45
$U_{b}/u_{*}$ (measured) | 25.8 | 19.4 | 18.2 | 18.2 | 15.7 | 21.3
| Sediment Properties
$\rho_{s}/\rho$ | 1.05 | 2.65 | 2.65 | 1.20 | 1.20 | 1.20
$d_{s}$ (mm) | 1.42 | 0.88 | 0.88 | 1.00 | 1.00 | 1.00
$w_{s}$ (cm s-1) | 1.7 | 10 | 10 | 2.9 | 2.9 | 2.9
| Dimensionless Model Parameters
$R*=w_{s}/(\kappa u_{*})$ | 0.5 | 1.2 | 1.2 | 4.3 | 5.2 | 9.1
$\alpha$ | 27.7 | 14.5 | 16.3 | 0.6 | 0.4 | 0.2
$\beta$ (Rouse) | 1.3 | 1.6 | 1.8 | 2.2 | 2.2 | 2.8
$R=w_{s}/(\beta\kappa u_{*})$ | 0.4 | 0.8 | 0.7 | 1.9 | 2.4 | 3.2
$\beta$ (Prandtl) | 0.9 | 1.1 | 1.2 | 1.4 | 1.5 | 1.9
$R=w_{s}/(\beta\kappa u_{*})$ | 0.6 | 1.1 | 1.0 | 3.0 | 3.5 | 4.8
$St_{b}\left(={w_{s}}/{g}\sqrt{{gS_{o}U_{b}}/{\nu}}\right)$ | 0.57 | 5.63 | 5.4 | 0.09 | 0.06 | 0.03
$St_{+}\left(=\tau_{p}u_{*}/H\right)\times 10^{3}$ | 1.3 | 4.0 | 4.1 | 0.5 | 0.4 | 0.2
$Fr_{d}\left(=U_{b}/\sqrt{(\rho_{s}/\rho-1)gd}\right)$ | 75 | 33 | 30 | 7 | 5 | 4
$Fr_{dc}$ | 3 | 4 | 4 | 3 | 3 | 3
$U_{b}/u_{*}$ (CSB rough bed) | 12.4 | 15.2 | 15.6 | 13.0 | 14.0 | 13.5
$U_{b}/u_{*}$ (CSB smooth bed) | 27.2 | 33.1 | 33.5 | 23.5 | 23.7 | 22.4

In the experiments, the sediments covered the bed and were assumed to have
reached an equilibrium state where equation 4 applies Tseng Tinoco (2020). The
densimetric Froude number $Fr_{d}$ and the critical densimetric Froude number
$Fr_{dc}$ whose formulation is described elsewhere Ali Dey (2017, 2018); S. Li
Katul (2019) are also presented in Table 1. In all cases, the $U_{b}$,
$\rho_{s}/\rho$, and $d/H$ result in $Fr_{d}>Fr_{dc}$ meaning that sediments
can be released from the bed and must be balanced by sediments depositing onto
the bed. Thus, the experiments do not strictly abide by HAE’s definition of SS
as sediments here are not remain permanently suspended. Across the
experiments, the flow variables $U_{b}$ and $u_{*}$ varied from 10 cm s-1 to
40 cm s-1 and 0.8 to 8 cm s-1, respectively. However,
$U_{b}/u_{*}=(8/f_{dw})^{1/2}$, related to the Darcy-Weisbach friction factor
$f_{dw}$, varied much less (15-25) as may be anticipated in fully rough flow
over a channel bed covered by grains of similar $d_{s}$. The particle
properties $\rho_{s}/\rho$ and $d_{s}$ varied from 1.05 to 2.65 and 0.88 to
1.4 mm, respectively. The consequence of these variations is that the
empirically derived settling velocity $w_{s}$ is much smaller than the Stokes
settling velocity as shown in the inset of Figure 1. Collectively, these
experiments span wide-ranging particle sizes (in the SS range) and flow
properties from different sources. The lowest measured sediment concentration
near the channel bottom is close to the surface ($z_{n,b}\in[0.026,0.063]$)
but remains above the buffer region $z^{+}=u_{*}z_{b}/\nu>30$ as shown in
Table 1. For some runs, the $z^{+}<100$ and wall-blockage effects (not
considered here) can impact $E_{ww}(k)$ and $d\overline{u}(z)/dz$ McColl .
(2016), which introduce obvious uncertainties. As shown in Table 1,
experiments (a)-(c) are characterized by $St_{b}>0.5$, which may be indicative
that $\Phi(z)$ is not small. Experiments (d)-(f) are characterized by a small
$St_{b}$ as assumed by the CSB and Rouse’s formula.

Figure 5: The predicted SSC profiles normalized by $C_{b}$ selected at the
measurement height with the highest reported concentration. The panel labeling
follows Table 1 with the top panel showing the comparisons from earlier
measurement i.e. (a) Wang Qian (1992); Greimann Holly Jr (2001) and (b)-(c)
Vanoni (1984), and the bottom panel showing the comparisons using recent
experiments (d)-(f) Tseng Tinoco (2020). For experiments in panels (a)-(c),
the $St_{b}$ is not small ($>0.5$).

Figure 5 confirms that the fitted Rouse formula and fitted Prandtl formula
(i.e. the $R$ model, allowing $\beta$ to be fitted) offer good agreements with
some measurements (for (a)-(b) and (d)-(f) respectively) at all depths. Given
that the simplified CSB model is identical to Rouse’s formula, an agreement
between the fitted Rouses’s formula and the measurements can also be
juxtaposed to the simplified CSB model. However, the numerical CSB model
provides reasonable agreements for all the runs when allowing $\alpha$ to
vary. Allowing $\alpha$ to be a free parameter has several advantages when
compared to $\beta$ in the fitted Rouse equation. Setting $\beta$ as constant
implies $Sc$ is constant at all $z_{n}$ while setting $\alpha$ as constant
incorporates some of the local variations in $Sc$ with $z_{n}$ (albeit near
the free water surface, maintaining a finite $\epsilon$ can be problematic
without adjustments). The impact of minor variations in particle sizes is
shown in the shaded area: the particle sizes are increased/decreased by 20% to
illustrate model sensitivity to $d_{s}$. Uncertainty in sediment composition
(and thus $d_{s}$ and $w_{s}$) can be a factor in determining SSC uncertainty
but not in all cases (runs d,e,f). While the SSC model does not require
$\overline{u}$ (only $d\overline{u}/dz$), the predicted $U_{b}$ from the CSB
turbulent stress budget can be compared against measured $U_{b}$ for a
plausibility check. The modeled $U_{b}$ requires $u_{*}$ along with a boundary
condition specified here as $\overline{u}(z_{n,b})/u_{*}$ at $z_{n,b}$. A
number of choices can be made about this boundary condition. Given that
$z_{n,b}$ is sufficiently distant from the wall, the most direct of those
choices is the log-law for two end-member cases: (i) fully rough with an
externally imposed surface roughness and (ii) hydrodynamically smooth. In both
cases, the mean velocity at $z_{n,b}$ is approximated as

$\frac{\overline{u}(z_{b})}{u_{*}}=\frac{1}{\kappa}\log\left(\frac{z_{b}}{z_{o}}\right);\quad\quad\frac{\overline{u}(z_{b})}{u_{*}}=\frac{1}{\kappa}\ln\left(z^{+}_{n,b}\right)+5,$
(47)

where $z_{o}$ is the momentum roughness length. The $z_{o}$ can be related to
$d_{s}$ by $z_{o}\approx d_{s}/30$ where the grain diameter is assumed
constant. In all cases, the roughness Reynolds number
$Re_{pa}=u_{*}d_{s}/\nu>3$ but in some cases, the flow is not fully rough
(i.e. transitional with $3<Re_{pa}<100$). For this reason, the CSB model
forced by both rough and smooth surface boundary conditions at $z_{n,b}$ are
featured in Table 1. The agreement between measured and the range of CSB
modeled $U_{b}/u_{*}$ for these two end-member cases appears reasonable. Runs
(a) and (f) are closer to a smooth-wall case whereas runs (b), (c), and (e)
are better approximated by a rough-wall boundary condition. Run (d) falls in-
between these two end-member cases. While Run (f) had the smallest $Re_{pa}=8$
and a near-smooth wall approximation may be justifiable, run (a) had an
$Re_{pa}>100$. We do not have a clear explanation as to why $U_{b}$ in run (a)
is better approximated by a smooth wall boundary condition.

An investigation of the relation between fitted $\alpha$ (and $\beta$) and
$w_{s}/u_{*}$ is undertaken and shown in Figure 6. A near-linear relation
between $\alpha^{-1}$ and $w_{s}/u_{*}$ indirectly supports the heuristic
closure adopted for $\overline{C^{\prime}{\partial w^{\prime}}/{\partial z}}$
with some caveats.

Figure 6: The dependence of fitted $\alpha$ and $\beta$ on the $w_{s}/u_{*}$.
The red, blue and black dashed lines show the fitted trend-lines of
$\alpha^{-1}$ and $\beta$ from Rouse and Prandtl equations respectively. The
cyan dashed line is $\beta$=$1+$ $2(w_{s}/u_{*})^{2}$ van Rijn (1984)
extrapolated for large $w_{s}/u_{*}$.

In the regime $w_{s}/u_{*}\gg 1$, the closure model with
$b_{1}\sim\mathrm{sgn}(A_{f})u_{*}/w_{s}$ leads to an
$\alpha^{-1}\sim-\mathrm{sgn}(A_{f})(1-C_{I})(w_{s}/u_{*})$ and
$\beta\sim-\mathrm{sgn}(A_{f})/(1-C_{I})(w_{s}/u_{*})$, both of which are
negative unless $\mathrm{sgn}(A_{f})$ is negative. The relation in Figure 6
indicates a positive slope between fitted $\alpha^{-1}$ and $w_{s}/u_{*}$,
suggesting that the coefficient $A_{f}$ in the flux-variance similarity
closure (i.e. equation 23) is negative. More broadly, to what extend this
closure is general and how robust are its results in the context of SSC
profile predictions cannot be unpacked from the experiments here and is better
kept for a future research topic.

##  4 Model Limitations

The treatment of suspended sediments as a dilute mixture is an obvious model
limitation. This assumption requires particles to settle independently and
that the solid volume can be ignored relative to the water volume. For the
experiments considered here, this assumption is reasonable. Another
restrictive assumption is setting $\Phi=0$ Kind (1992); Chamecki . (2007). A
$\Phi=0$ also leads to
$\overline{C}=\overline{w^{\prime}C^{\prime}}/w_{s}\rightarrow 0$ at
$z_{n}\rightarrow 1$, which may not be general. Given the large vertical
gradients in $\sigma_{w}^{2}$ near the channel bottom and near the free water
surface, turbophoretic effects can be significant in these two regions
Caporaloni . (1975); Guha (1997); Marchioli Soldati (2002); Zhao Wu (2006);
Katul . (2010); Chamecki . (2007). The turbophoretic effect act to increase
the SS concentration near the water surface; however, the measurements here
(runs a-c) suggest that for the $St_{b}>1$ cases, the SS concentrations near
the water surface experience a decline as $z_{n}\rightarrow 1$ instead of an
increase. This finding can be used to suggest that $\Phi=0$ may be plausible
as the turbophoretic term was shown to dominate $Phi$ near the water surface
Richter Chamecki (2018); Bragg . (2021). The CSB budget formulation here (i.e.
equation 32) ignored the flux transfer term and their vertical variation. In
the case of the turbulent stress, ignoring the flux transfer term (and its
vertical gradients) altogether guarantees that the co-spectrum between
$w^{\prime}$ and $u^{\prime}$ in the inertial subrange maintains a $k^{-7/3}$
scaling. This $k^{-7/3}$ scaling has been observed in numerous boundary layer
studies reporting co-spectra thereby offering indirect justification for this
assumption. The flux transport terms (i.e. the vertical gradients of triple
moments in the Reynolds averaged equations) have also been ignored. These
terms have been studied less for stress and sediment flux turbulent budgets
compared to their turbulent kinetic energy budget counterparts. The work here
highlights the need for an assessment of these terms relative to their
mechanical production terms. The CSB model also assumes that the linear Rotta
scheme (slow component) with an isotropization of production (rapid component)
applies equally to SS and momentum fluxes without adjustments in constants
(i.e. $A_{R}=1.8$ and $C_{I}=3/5$). Hence, any departure from these
established constants must be absorbed by $t_{ww}(k)/t_{r}(k)$, which
manifests itself as a Schmidt number effect (or $\alpha$ variations).

The assumed shape of $E_{ww}(k)$ is also over-simplified and certainly not
reflective of what is known about the energetics near the surface
($z^{+}<100$) such as wall-blockage. Moving away from the wall region itself,
other ’shape issues’ arise. For example, near the spectral transition from
inertial to viscous regimes, usually occurring at around $k\eta\approx 0.1$,
$E_{ww}(k)$ experiences a bottleneck that is absent here Saddoughi Veeravalli
(1994); Katul . (2015). Likewise, as $k\eta>0.1$ and increases further into
the viscous regime, $E_{ww}(k)$ decays exponentially Pope (2000). Hence,
extending the inertial subrange to $k\eta=1$ is not intended to capture all
such mechanisms impacting the vertical velocity spectrum. Instead, it allows
for some compensation of loss in energy due to censoring $E_{ww}(k)$ at
$k\eta=1$ while introducing extra energy due to an expected overestimation of
the extrapolated inertial subrange spectrum in this vicinity. On a more
positive note, while the full details of the turbulent kinetic energy cascade
across scales are not explicitly considered, their effects remain implicitly
contained in the assumed shape of $E_{ww}(k)$. As such, some of these effects
can be accommodated (e.g. the bottleneck, viscous cutoff, etc…) by various
revisions to $E_{ww}(k)$ (e.g. including a bump around $k\eta=0.1$, resolving
the viscous cutoff region using the Pao spectral shape or variants Pope (2000)
on it, etc…).

It is to be noted that the co-spectral budget is integrated scale-wise, which
means that the precise shape of $E_{ww}(k)$ in the vicinity of $k\eta\approx
1$ is less crucial. Moving beyond the shape issues of $E_{ww}(k)$ and focusing
on its primary input variable $\epsilon(z_{n})$, the approach assumes
turbulent kinetic energy production is balanced by its dissipation at every
$z_{n}$ (i.e. $\phi(z_{n})=1$), which is certainly not realistic for all
$z_{n}$. However, as previously mentioned, deviations from unity in
$\phi(z_{n})$ may be ameliorated by the sub-unity exponent ($-1/3$) dependence
in the SSC budget. An exception to this statement is the particle time scale
$t_{wc}(k)$ in $Sc(k)$. A $\phi(z_{n})=1$ as $z_{n}\rightarrow 1$ leads to an
unbounded $Sc^{-1}(k)$ and thus an uncertain $D_{s}$ shape in the vicinity of
the free surface. A plausible adjustment to the $Sc^{-1}(k)$ calculations
based on maintaining a minimal $\epsilon$ ($=0.001\epsilon_{b}$) was
introduced here though this correction remains adhoc. Last, the turbulent SS
flux from the CSB model(s) follows the same form as gradient-diffusion closure
upon ignoring both - turbulent flux transport and scale-wise transfer terms.
However, a key advantage here is that the effective diffusion coefficient
$D_{s}$ from the CSB model contains contributions from turbulent eddies and
Schmidt numbers at all scales. The proposed Schmidt number (or $\alpha$) is
consistent with bulk Schmidt number formulations such as those by van Rijin’s
and other one-way coupling schemes (i.e. particle transport does not impact
the flow) when $Sc<1$ Bombardelli Moreno (2012). For dense mixture or other
aeolian particles in the atmosphere, the particle Schmidt number can be larger
than unity Csanady (1963) implying other particle-fluid interaction models are
required. When using the CSB model, the $\alpha$ used for the determination of
the Schmidt number is treated as a single fitted parameter. Hence, the CSB
model offers the same number of free parameters as the fitted Rouse equation.
What was found here is that $\alpha^{-1}$ varies linearly with $w_{s}/u_{*}$
when combining all the experiments. A plausibility argument as to why $\alpha$
depends on $w_{s}/u_{*}$ was also offered. In some instances, the addition of
a single fitted parameter may be desirable in hydraulic models as discussed
elsewhere Papke Battiato (2013); Battiato Rubol (2014); Rubol . (2018); S. Li
. (2019), but an increasing number of free model parameters does not
necessarily lead to a better physical understanding. The sediment settling
velocity estimated in equation 7 is commonly based on a mass-median-diameter
from particle size distribution measurements, which however may not be an
optimized characteristic size as shown by some in-situ measurements Williams .
(2007). Large variations in $d_{s}$ can have a substantial impact on SSC
profiles, which may be more significant than models for $\alpha$.

##  5 Conclusion

Operational modeling of SSC in turbulent flows continues to be a formidable
challenge in hydraulics, hydrology, ecology, and water quality control. The
work here establishes a new link between the spectrum of vertical velocity and
SS turbulent flux, which was then used to arrive at expressions for the SSC
profile. The spectrum of vertical velocity is characterized by multiple
scaling regimes that include the Saffman spectrum ($E_{ww}(k)\sim k^{+2}$),
the ’energy splashing’ effect due to the presence of a wall ($E_{ww}(k)\sim
k^{0}$), and the much-studied inertial subrange regime ($E_{ww}(k)\sim
k^{-5/3}$). Finite Reynolds effects are accommodated through a scale
separation between $z$ and the Kolmogorov microscale $\eta$ terminating the
scale-wise extent of the inertial subrange (as a first approximation). This
dependence can be noted when considering the scaling argument
$k_{e}/k_{o}=z/\eta\sim(zu_{*}/\nu)^{3/4}$ Tennekes Lumley (1972). Hence,
increasing $Re_{s}=(zu_{*}/\nu)$ by either increasing $z$ or $u_{*}$ leads to
a widening of the scale-wise extent of the inertial subrange, which then
impacts all subsequent expressions such as $\Omega(z_{n})$ and
$d\overline{u}/dz$. As such, the proposed model is responsive to finite
Reynolds number, Schmidt number, and Rouse number effects. Prior ad-hoc
efforts such as correcting $l_{o}$ by $V_{n}$ (i.e. the van Driest damping
function) can now be interpreted from this new spectral perspective (i.e.
$Re_{s}$ effects become large for small $z$ or $u_{*}$). A simplified solution
to the CSB model in which the Saffman spectrum is truncated but the inertial
subrange is now extended to infinite wave-numbers (i.e.
$Re_{s}\rightarrow\infty$) was shown to recover earlier theories (e.g. Rouse’s
formula). The fitted Rouse’s equation (and by extension the simplified CSB
solution) also describes the measured SSC profiles in all the experiments
considered here provided $\alpha$ (or $\beta$) is allowed to vary with
$w_{s}/u_{*}$. Thus, one of the main novelties here is to provide a spectral
link between the energy distribution in eddies and the SSC shape. Interactions
between turbulent eddies and suspended sediment grains at various heights were
also proposed, resulting in a scale-dependent $Sc$ captured by a single
parameter $\alpha$ that varies with $w_{s}/u_{*}$. Such $Sc$ variations were
formulated in spectral space but recover expected bulk relations between $R$
and $Sc$ identified by other models, experiments, and simulation studies. When
all these findings are taken together, future extension of this work must
focus on upgrading the particle-turbulence interaction scheme and its
signature in a scale-dependent Schmidt number. Such extension will benefit
from targeted DNS runs where all the terms in the particle co-spectrum as well
as $E_{ww}(k)$ can be computed or determined. Likewise, an exploration of
where the sediment flux transport term is significant relative to the
mechanical production term and how to incorporate its effects can be
undertaken from the aforementioned DNS runs.

### Data Availability and Acknowledgements

All the data used were digitized from the published literature Greimann Holly
Jr (2001); Vanoni (1984); Jha Bombardelli (2009); Tseng Tinoco (2020). SL was
supported by a fellowship from the Nicholas School of the Environment at Duke
University. GK and ADB acknowledge support from the U.S. National Science
Foundation (NSF-AGS-1644382, NSF-AGS-2028633, NSF-IOS-1754893, and NSF-
CBET-2042346).

## References

  * Albertson . (1995) albertson1995sensibleAlbertson, JD., Parlange, MB., Katul, GG., Chu, CR., Stricker, H. Tyler, S.  1995\.  Sensible heat flux from arid regions: A simple flux-variance method Sensible heat flux from arid regions: A simple flux-variance method.  Water Resources Research314969–973.
  * Ali Dey (2017) ali2017originAli, SZ. Dey, S.  2017\.  Origin of the scaling laws of sediment transport Origin of the scaling laws of sediment transport.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences473219720160785.
  * Ali Dey (2018) ali2018impactAli, SZ. Dey, S.  2018\.  Impact of phenomenological theory of turbulence on pragmatic approach to fluvial hydraulics Impact of phenomenological theory of turbulence on pragmatic approach to fluvial hydraulics.  Physics of Fluids304045105.
  * Ayet Katul (2020) ayet2020scalingAyet, A. Katul, G.  2020\.  Scaling Laws for the Length Scale of Energy-Containing Eddies in a sheared and thermally Stratified Atmospheric Surface Layer Scaling laws for the length scale of energy-containing eddies in a sheared and thermally stratified atmospheric surface layer.  Geophysical Research Letters4723e2020GL089997.
  * Battiato Rubol (2014) battiato2014singleBattiato, I. Rubol, S.  2014\.  Single-parameter model of vegetated aquatic flows Single-parameter model of vegetated aquatic flows.  Water Resources Research5086358–6369.
  * Bombardelli Jha (2009) bombardelli2009hierarchicalBombardelli, F. Jha, S.  2009\.  Hierarchical modeling of the dilute transport of suspended sediment in open channels Hierarchical modeling of the dilute transport of suspended sediment in open channels.  Environmental Fluid Mechanics92207.
  * Bombardelli Moreno (2012) bombardelli2012exchangeBombardelli, F. Moreno, P.  2012\.  Exchange at the bed sediments-water column interface Exchange at the bed sediments-water column interface.  Fluid Mechanics of Environmental Interfaces, 2nd ed.; Gualtieri, C., Mihailovic, D.T, Eds Fluid Mechanics of Environmental Interfaces, 2nd ed.; Gualtieri, C., Mihailovic, D.T, Eds ( 221–253).
  * Bonetti . (2017) bonetti2017manningBonetti, S., Manoli, G., Manes, C., Porporato, A. Katul, G.  2017\.  Manning’s formula and Strickler’s scaling explained by a co-spectral budget model Manning’s formula and Strickler’s scaling explained by a co-spectral budget model.  Journal of Fluid Mechanics8121189–1212.
  * Bos . (2004) bos2004behaviorBos, W., Touil, H., Shao, L. Bertoglio, JP.  2004\.  On the behavior of the velocity-scalar cross correlation spectrum in the inertial range On the behavior of the velocity-scalar cross correlation spectrum in the inertial range.  Physics of Fluids16103818–3823.
  * Bragg . (2021) bragg2021mechanismsBragg, AD., Richter, DH. Wang, G.  2021\.  Mechanisms governing the settling velocities and spatial distributions of inertial particles in wall-bounded turbulence Mechanisms governing the settling velocities and spatial distributions of inertial particles in wall-bounded turbulence.  Physical Review Fluids66064302.
  * Caporaloni . (1975) caporaloni1975transferCaporaloni, M., Tampieri, F., Trombetti, F. Vittori, O.  1975\.  Transfer of particles in nonisotropic air turbulence Transfer of particles in nonisotropic air turbulence.  Journal of Atmospheric Sciences323565–568.
  * Castro-Orgaz . (2012) castro2012karmanCastro-Orgaz, O., Giráldez, J., Mateos, L. Dey, S.  2012\.  Is the von Kármán constant affected by sediment suspension? Is the von kármán constant affected by sediment suspension?  Journal of Geophysical Research: Earth Surface117F4.
  * Cava Katul (2012) cava2012scalingCava, D. Katul, G.  2012\.  On the scaling laws of the velocity-scalar cospectra in the canopy sublayer above tall forests On the scaling laws of the velocity-scalar cospectra in the canopy sublayer above tall forests.  Boundary-layer Meteorology1452351–367.
  * Chamecki . (2007) chamecki2007concentrationChamecki, M., Van Hout, R., Meneveau, C. Parlange, MB.  2007\.  Concentration profiles of particles settling in the neutral and stratified atmospheric boundary layer Concentration profiles of particles settling in the neutral and stratified atmospheric boundary layer.  Boundary-layer Meteorology125125–38.
  * Cheng (1997) cheng1997simplifiedCheng, NS.  1997\.  Simplified settling velocity formula for sediment particle Simplified settling velocity formula for sediment particle.  Journal of Hydraulic Engineering1232149–152.
  * Csanady (1963) csanady1963turbulentCsanady, G.  1963\.  Turbulent diffusion of heavy particles in the atmosphere Turbulent diffusion of heavy particles in the atmosphere.  Journal of Atmospheric Sciences203201–208.
  * Dai . (2016) dai2016declineDai, Z., Fagherazzi, S., Mei, X. Gao, J.  2016\.  Decline in suspended sediment concentration delivered by the Changjiang (Yangtze) River into the East China Sea between 1956 and 2013 Decline in suspended sediment concentration delivered by the Changjiang (Yangtze) River into the East China Sea between 1956 and 2013\.  Geomorphology268123–132.
  * Dey (2014) dey2014fluvialDey, S.  2014\.  Fluvial Hydrodynamics Fluvial hydrodynamics.  Springer.
  * Dey Ali (2020) dey2020fluvialDey, S. Ali, SZ.  2020\.  Fluvial instabilities Fluvial instabilities.  Physics of Fluids326061301.
  * Duman . (2016) duman2016dissipationDuman, T., Trakhtenbrot, A., Poggi, D., Cassiani, M. Katul, GG.  2016\.  Dissipation intermittency increases long-distance dispersal of heavy particles in the canopy sublayer Dissipation intermittency increases long-distance dispersal of heavy particles in the canopy sublayer.  Boundary-Layer Meteorology159141–68.
  * Einstein (1950) einstein1950bedEinstein, H.  1950\.  The bed-load function for sediment transportation in open channel flows The bed-load function for sediment transportation in open channel flows ( 1026).  Technical Report, US Department of Agriculture, Soil Conservation Service, Washington DC.
  * Elghobashi (1994) elghobashi1994predictingElghobashi, S.  1994\.  On predicting particle-laden turbulent flows On predicting particle-laden turbulent flows.  Applied Scientific Research524309–329.
  * Ferry Balachandar (2001) ferry2001fastFerry, J. Balachandar, S.  2001\.  A fast Eulerian method for disperse two-phase flow A fast Eulerian method for disperse two-phase flow.  International Journal of Multiphase Flow2771199–1226.
  * Fischer . (2013) fischer2013mixingFischer, HB., List, JE., Koh, CR., Imberger, J. Brooks, NH.  2013\.  Mixing in Inland and Coastal Waters Mixing in Inland and Coastal Waters.  Elsevier.
  * Giometto . (2017) giometto2017directGiometto, M., Katul, G., Fang, J. Parlange, M.  2017\.  Direct numerical simulation of turbulent slope flows up to Grashof number ${G}r=2.1\times 10^{11}$ Direct numerical simulation of turbulent slope flows up to Grashof number ${G}r=2.1\times 10^{11}$.  Journal of Fluid Mechanics829589.
  * Green Coco (2014) green2014reviewGreen, M. Coco, G.  2014\.  Review of wave-driven sediment resuspension and transport in estuaries Review of wave-driven sediment resuspension and transport in estuaries.  Reviews of Geophysics52177–117.
  * Greimann Holly Jr (2001) greimann2001twoGreimann, B. Holly Jr, F.  2001\.  Two-phase flow analysis of concentration profiles Two-phase flow analysis of concentration profiles.  Journal of Hydraulic Engineering1279753–762.
  * Greimann . (1999) greimann1999twoGreimann, B., Muste, M. Holly Jr, F.  1999\.  Two-phase formulation of suspended sediment transport Two-phase formulation of suspended sediment transport.  Journal of Hydraulic Research374479–500.
  * Guha (1997) guha1997unifiedGuha, A.  1997\.  A unified Eulerian theory of turbulent deposition to smooth and rough surfaces A unified Eulerian theory of turbulent deposition to smooth and rough surfaces.  Journal of Aerosol Science2881517–1537.
  * Heisel . (2020) heisel2020velocityHeisel, M., Katul, G., Chamecki, M. Guala, M.  2020\.  Velocity asymmetry and turbulent transport closure in smooth-and rough-wall boundary layers Velocity asymmetry and turbulent transport closure in smooth-and rough-wall boundary layers.  Physical Review Fluids510104605.
  * Huai . (2020) huai2020analyticalHuai, W., Yang, L. Guo, Y.  2020\.  Analytical solution of suspended sediment concentration profile: Relevance of dispersive flow term in vegetated channels Analytical solution of suspended sediment concentration profile: Relevance of dispersive flow term in vegetated channels.  Water Resources Research567e2019WR027012.
  * Huai . (2019) huai2019predictingHuai, W., Yang, L., Wang, WJ., Guo, Y., Wang, T. Cheng, Yg.  2019\.  Predicting the vertical low suspended sediment concentration in vegetated flow using a random displacement model Predicting the vertical low suspended sediment concentration in vegetated flow using a random displacement model.  Journal of Hydrology578124101.
  * Huang . (2014) huang2014particleHuang, CW., Launiainen, S., Grönholm, T. Katul, GG.  2014\.  Particle deposition to forests: An alternative to K-theory Particle deposition to forests: An alternative to K-theory.  Atmospheric Environment94593–605.
  * Hunt (1954) hunt1954turbulentHunt, J.  1954\.  The turbulent transport of suspended sediment in open channels The turbulent transport of suspended sediment in open channels.  Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences2241158322–335.
  * Jha Bombardelli (2009) jha2009twoJha, SK. Bombardelli, FA.  2009\.  Two-phase modeling of turbulence in dilute sediment-laden, open-channel flows Two-phase modeling of turbulence in dilute sediment-laden, open-channel flows.  Environmental Fluid Mechanics92237.
  * Johnson . (2020) johnson20Johnson, PL., Bassenne, M. Moin, P.  2020\.  Turbophoresis of small inertial particles: theoretical considerations and application to wall-modelled large-eddy simulations Turbophoresis of small inertial particles: theoretical considerations and application to wall-modelled large-eddy simulations.  J. Fluid Mech.883A27.  10.1017/jfm.2019.865
  * Katul . (2010) katul2010predictingKatul, GG., Grönholm, T., Launiainen, S. Vesala, T.  2010\.  Predicting the dry deposition of aerosol-sized particles using layer-resolved canopy and pipe flow analogy models: Role of turbophoresis Predicting the dry deposition of aerosol-sized particles using layer-resolved canopy and pipe flow analogy models: Role of turbophoresis.  Journal of Geophysical Research: Atmospheres115D12.
  * Katul Manes (2014) katul2014cospectralKatul, GG. Manes, C.  2014\.  Cospectral budget of turbulence explains the bulk properties of smooth pipe flow Cospectral budget of turbulence explains the bulk properties of smooth pipe flow.  Physical Review E906063008.
  * Katul . (2015) katul2015bottlenecksKatul, GG., Manes, C., Porporato, A., Bou-Zeid, E. Chamecki, M.  2015\.  Bottlenecks in turbulent kinetic energy spectra predicted from structure function inflections using the von Kármán-Howarth equation Bottlenecks in turbulent kinetic energy spectra predicted from structure function inflections using the von kármán-howarth equation.  Physical Review E923033009.
  * Katul . (2013) katul2013coKatul, GG., Porporato, A., Manes, C. Meneveau, C.  2013\.  Co-spectrum and mean velocity in turbulent boundary layers Co-spectrum and mean velocity in turbulent boundary layers.  Physics of Fluids259091702.
  * Kellogg . (2014) kellogg2014useKellogg, ML., Smyth, AR., Luckenbach, MW., Carmichael, RH., Brown, BL., Cornwell, JC.Higgins, CB.  2014\.  Use of oysters to mitigate eutrophication in coastal waters Use of oysters to mitigate eutrophication in coastal waters.  Estuarine, Coastal and Shelf Science151156–168.
  * Kim . (1987) kim1987turbulenceKim, J., Moin, P. Moser, R.  1987\.  Turbulence statistics in fully developed channel flow at low Reynolds number Turbulence statistics in fully developed channel flow at low Reynolds number.  Journal of Fluid Mechanics177133–166.
  * Kind (1992) kind1992oneKind, R.  1992\.  One-dimensional aeolian suspension above beds of loose particles—A new concentration-profile equation One-dimensional aeolian suspension above beds of loose particles—a new concentration-profile equation.  Atmospheric Environment. Part A. General Topics265927–931.
  * D. Li . (2015) li2015revisitingLi, D., Katul, GG. Zilitinkevich, SS.  2015\.  Revisiting the turbulent Prandtl number in an idealized atmospheric surface layer Revisiting the turbulent prandtl number in an idealized atmospheric surface layer.  Journal of the Atmospheric Sciences7262394–2410.
  * S. Li Katul (2019) li2019cospectralLi, S. Katul, G.  2019\.  Cospectral budget model describes incipient sediment motion in turbulent flows Cospectral budget model describes incipient sediment motion in turbulent flows.  Physical Review Fluids49093801.
  * S. Li . (2019) li2019meanLi, S., Katul, G. Huai, W.  2019\.  Mean velocity and shear stress distribution in floating treatment wetlands: An analytical study Mean velocity and shear stress distribution in floating treatment wetlands: An analytical study.  Water Resources Research5586436–6449.
  * Y. Li . (2019) li2019interactionsLi, Y., Wang, X., Fu, X., W.and Xia, Liu, C., Min, J., Zhang, W. Crittenden, JC.  2019\.  Interactions between nano/micro plastics and suspended sediment in water: Implications on aggregation and settling Interactions between nano/micro plastics and suspended sediment in water: Implications on aggregation and settling.  Water Research161486–495.
  * Liu . (2013) liu2013accelerationLiu, T., Xia, X., Liu, S., Mou, X. Qiu, Y.  2013\.  Acceleration of denitrification in turbid rivers due to denitrification occurring on suspended sediment in oxic waters Acceleration of denitrification in turbid rivers due to denitrification occurring on suspended sediment in oxic waters.  Environmental Science and Technology4794053–4061.
  * Long Pavelsky (2013) long2013remoteLong, CM. Pavelsky, TM.  2013\.  Remote sensing of suspended sediment concentration and hydrologic connectivity in a complex wetland environment Remote sensing of suspended sediment concentration and hydrologic connectivity in a complex wetland environment.  Remote Sensing of Environment129197–209.
  * Lupker . (2011) lupker2011rouseLupker, M., France-Lanord, C., Lavé, J., Bouchez, J., Galy, V., Métivier, F.Mugnier, J.  2011\.  A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin A Rouse-based method to integrate the chemical composition of river sediments: Application to the Ganga basin.  Journal of Geophysical Research: Earth Surface116F4.
  * Marchioli Soldati (2002) marchioli2002mechanismsMarchioli, C. Soldati, A.  2002\.  Mechanisms for particle transfer and segregation in a turbulent boundary layer Mechanisms for particle transfer and segregation in a turbulent boundary layer.  Journal of Fluid Mechanics468283.
  * Mazumder Ghoshal (2006) mazumder2006velocityMazumder, B. Ghoshal, K.  2006\.  Velocity and concentration profiles in uniform sediment-laden flow Velocity and concentration profiles in uniform sediment-laden flow.  Applied Mathematical Modelling302164–176.
  * McColl . (2016) MccollEA16McColl, K., Katul, G., Gentine, P. Entekhabi, D.  2016\.  Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage Mean-velocity profile of smooth channel flow explained by a cospectral budget model with wall-blockage.  Physics of Fluids283035107.
  * Mohtar . (2020) mohtar2020incipientMohtar, WHMW., Lee, JW., Azha, NIM. Cheng, NS.  2020\.  Incipient sediment motion based on turbulent fluctuations Incipient sediment motion based on turbulent fluctuations.  International Journal of Sediment Research352125–133.
  * Muste . (2005) muste2005twoMuste, M., Yu, K., Fujita, I. Ettema, R.  2005\.  Two-phase versus mixed-flow perspective on suspended sediment transport in turbulent channel flows Two-phase versus mixed-flow perspective on suspended sediment transport in turbulent channel flows.  Water Resources Research4110.
  * Nazeer . (2014) nazeer2014heavyNazeer, S., Hashmi, MZ. Malik, RN.  2014\.  Heavy metals distribution, risk assessment and water quality characterization by water quality index of the River Soan, Pakistan Heavy metals distribution, risk assessment and water quality characterization by water quality index of the river soan, pakistan.  Ecological Indicators43262–270.
  * Nezu Azuma (2004) nezu2004turbulenceNezu, I. Azuma, R.  2004\.  Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows Turbulence characteristics and interaction between particles and fluid in particle-laden open channel flows.  Journal of Hydraulic Engineering13010988–1001.
  * Nie . (2017) nie2017verticalNie, S., Sun, H., Zhang, Y., Chen, D., Chen, W., Chen, L. Schaefer, S.  2017\.  Vertical distribution of suspended sediment under steady flow: existing theories and fractional derivative model Vertical distribution of suspended sediment under steady flow: existing theories and fractional derivative model.  Discrete Dynamics in Nature and Society2017.
  * Nikora Goring (2002) nikora2002fluctuationsNikora, VI. Goring, DG.  2002\.  Fluctuations of suspended sediment concentration and turbulent sediment fluxes in an open-channel flow Fluctuations of suspended sediment concentration and turbulent sediment fluxes in an open-channel flow.  Journal of Hydraulic Engineering1282214–224.
  * O′Brien (1933) obrien1933reviewO′Brien, M.  1933\.  Review of the theory of turbulent flow and its relation to sediment-transportation Review of the theory of turbulent flow and its relation to sediment-transportation.  EOS Transactions American Geophysical Union141487–491.
  * Papke Battiato (2013) papke2013reducedPapke, A. Battiato, I.  2013\.  A reduced complexity model for dynamic similarity in obstructed shear flows A reduced complexity model for dynamic similarity in obstructed shear flows.  Geophysical Research Letters40153888–3892.
  * Pope (2000) pope2001turbulentPope, S.  2000\.  Turbulent Flows Turbulent flows.  Cambridge University Press, Cambridge, U.K.
  * Raupach (1981) Raupach81Raupach, M.  1981\.  Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers.  Journal of Fluid Mechanics108363–382.
  * Reeks (1983) reeks83Reeks, M.  1983\.  The transport of discrete particles in inhomogeneous turbulence The transport of discrete particles in inhomogeneous turbulence.  J. Aerosol Sci.146729 - 739.  https://doi.org/10.1016/0021-8502(83)90055-1
  * Richter Chamecki (2018) richter2018inertialRichter, D. Chamecki, M.  2018\.  Inertial effects on the vertical transport of suspended particles in a turbulent boundary layer Inertial effects on the vertical transport of suspended particles in a turbulent boundary layer.  Boundary-Layer Meteorology1672235–256.
  * Rouse (1937) rouse1937modernRouse, H.  1937\.  Modern conceptions of the mechanics of fluid turbulence Modern conceptions of the mechanics of fluid turbulence.  Transactions of the American Society of Civil Engineers1021463–505.
  * Rouse (1939) rouse1939analysisRouse, H.  1939\.  An analysis of sediment transportation in the light of fluid turbulence An analysis of sediment transportation in the light of fluid turbulence.  SCS-TP-251–40.
  * Rubol . (2018) rubol2018universalRubol, S., Ling, B. Battiato, I.  2018\.  Universal scaling-law for flow resistance over canopies with complex morphology Universal scaling-law for flow resistance over canopies with complex morphology.  Scientific Reports814430.
  * Saddoughi Veeravalli (1994) saddoughi1994localSaddoughi, SG. Veeravalli, SV.  1994\.  Local isotropy in turbulent boundary layers at high Reynolds number Local isotropy in turbulent boundary layers at high reynolds number.  Journal of Fluid Mechanics268333–372.
  * Sardina . (2012) sardina2012wallSardina, G., Schlatter, P., Brandt, L., Picano, F. Casciola, CM.  2012\.  Wall accumulation and spatial localization in particle-laden wall flows Wall accumulation and spatial localization in particle-laden wall flows.  J. Fluid Mech.69950–78.
  * Scully Friedrichs (2003) scully2003influenceScully, ME. Friedrichs, CT.  2003\.  The influence of asymmetries in overlying stratification on near-bed turbulence and sediment suspension in a partially mixed estuary The influence of asymmetries in overlying stratification on near-bed turbulence and sediment suspension in a partially mixed estuary.  Ocean Dynamics533208–219.
  * Sun . (2020) sun2020reviewSun, L., Qiu, H., Wu, C., Niu, J. Hu, BX.  2020\.  A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water A review of applications of fractional advection–dispersion equations for anomalous solute transport in surface and subsurface water.  Wiley Interdisciplinary Reviews: Water74e1448.
  * Tan . (2018) tan2018ruiTan, G., Fang, H., Dey, S. Wu, W.  2018\.  Rui-Jin Zhang’s research on sediment transport Rui-Jin Zhang’s research on sediment transport.  Journal of Hydraulic Engineering144602518002.
  * Tennekes Lumley (1972) tennekes2018firstTennekes, H. Lumley, J.  1972\.  A first course in turbulence A first course in turbulence.  MIT press, Boston Massachusetts, USA.
  * Tseng Tinoco (2020) tsengtwoTseng, C. Tinoco, R.  2020\.  A two-layer turbulence-based model to predict suspended sediment concentration in flows with aquatic vegetation A two-layer turbulence-based model to predict suspended sediment concentration in flows with aquatic vegetation.  Geophysical Research Letterse2020GL091255.
  * Umeyaina (1992) umeyaina1992verticalUmeyaina, M.  1992\.  Vertical distribution of suspended sediment in uniform open-channel flow Vertical distribution of suspended sediment in uniform open-channel flow.  Journal of Hydraulic Engineering1186936–941.
  * Van Driest (1956) van1956turbulentVan Driest, E.  1956\.  On turbulent flow near a wall On turbulent flow near a wall.  Journal of the Aeronautical Sciences23111007–1011.
  * Vanoni (1984) vanoni1984fiftyVanoni, VA.  1984\.  Fifty years of sedimentation Fifty years of sedimentation.  Journal of Hydraulic Engineering11081021–1057.
  * van Rijn (1984) rijn1984sedimentvan Rijn, LC.  1984\.  Sediment transport, part II: suspended load transport Sediment transport, part ii: suspended load transport.  Journal of Hydraulic Engineering110111613–1641.
  * von Karman (1934) vonKarman1934von Karman, T.  1934\.  Some Aspects of the Turbulence Problem Some aspects of the turbulence problem.  Proceedings of the Fourth International Congress of Applied Mechanics, Cambridge, England Proceedings of the Fourth International Congress of Applied Mechanics, Cambridge, England ( 54–91).
  * Wallin Johansson (2000) wallin2000explicitWallin, S. Johansson, A.  2000\.  An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows.  Journal of Fluid Mechanics40389–132.
  * Wang Qian (1992) xingkui1992velocityWang, X. Qian, N.  1992\.  Velocity profiles of sediment-laden flow Velocity profiles of sediment-laden flow.  International Journal of Sediment Research7127–-58.
  * Wells Stock (1983) wells1983effectsWells, M. Stock, D.  1983\.  The effects of crossing trajectories on the dispersion of particles in a turbulent flow The effects of crossing trajectories on the dispersion of particles in a turbulent flow.  Journal of Fluid Mechanics13631–62.
  * Williams . (2007) williams2007highWilliams, N., Walling, D. Leeks, G.  2007\.  High temporal resolution in situ measurement of the effective particle size characteristics of fluvial suspended sediment High temporal resolution in situ measurement of the effective particle size characteristics of fluvial suspended sediment.  Water Research4151081–1093.
  * Yujun . (2008) yujun2008sedimentYujun, Y., Zhaoyin, W., Zhang, K., Guoan, Y. Xuehua, D.  2008\.  Sediment pollution and its effect on fish through food chain in the Yangtze River Sediment pollution and its effect on fish through food chain in the yangtze river.  International Journal of Sediment Research234338–347.
  * Zhao Wu (2006) zhao2006modelingZhao, B. Wu, J.  2006\.  Modeling particle deposition from fully developed turbulent flow in ventilation duct Modeling particle deposition from fully developed turbulent flow in ventilation duct.  Atmospheric Environment403457–466.