Because increasing climatic variability and anthropic
pressures have affected the sediment dynamics of large tropical rivers,
long-term sediment concentration series have become crucial for
understanding the related socioeconomic and environmental impacts. For
operational and cost rationalization purposes, index concentrations are
often sampled in the flow and used as a surrogate of the cross-sectional
average concentration. However, in large rivers where suspended sands are
responsible for vertical concentration gradients, this index method can
induce large uncertainties in the matter fluxes.
Assuming that physical laws describing the suspension of grains in turbulent
flow are valid for large rivers, a simple formulation is derived to model the
ratio (α) between the depth-averaged and index concentrations. The model
is validated using an exceptional dataset (1330 water samples, 249
concentration profiles, 88 particle size distributions and 494 discharge
measurements) that was collected between 2010 and 2017 in the Amazonian
foreland. The α prediction requires the estimation of the Rouse
number (P), which summarizes the balance between the suspended particle
settling and the turbulent lift, weighted by the ratio of sediment to eddy
diffusivity (β). Two particle size groups, fine sediments and sand,
were considered to evaluate P. Discrepancies were observed between the
evaluated and measured P, which were attributed to biases related to the
settling and shear velocities estimations, but also to diffusivity ratios
β≠1. An empirical expression taking these biases into account was
then formulated to predict accurate estimates of β, then P
(ΔP=±0.03) and finally α.
The proposed model is a powerful tool for optimizing the concentration
sampling. It allows for detailed uncertainty analysis on the average
concentration derived from an index method. Finally, this model could
likely be coupled with remote sensing and hydrological modeling to serve as
a step toward the development of an integrated approach for assessing
sediment fluxes in poorly monitored basins.
Introduction
In recent decades, the Amazon Basin has experienced an intensification in
climatic variability (e.g., Gloor et al., 2013; Marengo and Espinoza, 2015),
specifically in extreme events (drought and flood), as well as increasing
anthropic pressure. In the Peruvian foreland, the advance of the pioneer
fronts causes serious changes in land use, which are enhanced by the
proliferation of roads that provide access to the natural resources hosted
by this region. The number of hydropower projects is also rapidly increasing
(e.g., Finer and Jenkins, 2012; Latrubesse et al., 2017; Forsberg et al.,
2017). These global and local changes might increase the erosion rates in
the basin as well as the suspended load interannual variability (e.g.,
Walling and Fang, 2003; Martinez et al., 2009). The sediment transfer
dynamics might also be affected (e.g., Walling, 2006), generating large
ecological impacts on the mega-diverse Amazonian biome and having
socioeconomic consequences on the riverine populations. In such a context,
long-term and reliable sediment series are crucial for detecting, monitoring
and understanding the related socioeconomic and environmental impacts (e.g.,
Walling, 1983, 2006; Horowitz, 2003; Syvitski et al., 2005; Horowitz et al.,
2015). However, there is a lack of consistent data available for this
region, and this lack of data has prompted an increased interest in
developing better spatiotemporal monitoring of sediment transport.
In the large tropical rivers of Peru, the measurement of cross-sectional
average concentrations 〈C〉 (mg L-1) remains a costly and time-consuming task. First, gauging
stations can only be reached after several days of traveling on hard dirt
roads or by the river. Second, there is no infrastructure on the rivers,
and all operations are conducted using small boats under all flow
conditions. Third, the gauging sections have depths that range from the
metric to the decametric scale and widths that range from the hectometer to
the kilometer scale. Such large sections experience pronounced sediment
concentration gradients and grain size sorting, in both the
vertical and the transverse directions (e.g., Curtis et al., 1979; Vanoni,
1979, 1980; Horowitz and Elrick, 1987; Filizola and Guyot, 2004; Filizola et
al., 2010; Bouchez et al., 2011; Lupker et al., 2011; Armijos et al., 2013,
2016; Vauchel et al., 2017). The balance between the local hydrodynamic
conditions and the sediment characteristics (e.g., grain size, density and
shape) drive this spatial heterogeneity. Thus, the sand suspension is
characterized by a high vertical gradient as well as a significant lateral
variability, and the concentration varies by several orders of magnitude; in
contrast, the fine sediments (e.g., clays and silts) are transported
homogeneously throughout the entire river section.
As a consequence, the entire cross section must be explored to provide a
representative estimate of the mean concentration of coarse particles. Thus,
it is necessary to identify a trade-off between the need to sample an
adequate number of verticals and points throughout the cross section and the
need for time-integrated or repeated measurements to ensure the temporal
representativeness of each water sample (Gitto et al., 2017). The
second-order moments of the Navier–Stokes equations induce this temporal
concentration variability, as do the larger turbulent structures (typically
those induced by the bedforms) and the changes in flow conditions (e.g.,
backwaters, floods and flow pulses). Sands and coarse silts are much more
sensitive to velocity fluctuations than clay particles are (i.e., settling
laws are highly sensitive to the diameter of the particles) and are the most
difficult to accurately measure.
Depth-integrated or point-integrated sampling procedures are traditionally
used to determine the mean concentration of suspended sediment in rivers.
However, deploying these methods from a boat is rarely feasible due to the
velocity and depth ranges that are encountered in large Amazonian rivers.
For a point-integrated bottle sampling method, maintaining a position for a
duration long enough to capture a representative water sample (Gitto et al.,
2017) requires anchoring the boat and using a heavy ballast. This type of
operation is very risky without good infrastructure and well-trained staff,
especially when collecting measurements near the river's bottom. Moreover,
this method decreases the number of samples that can be collected in 1 d.
For a depth-integrated sampling method within a deep river, the bottle may
fill up before reaching the water surface if its transit speed is too slow.
Moreover, if the ballast weight is not sufficient to hold the sampler nose
in a horizontal position, the filling conditions are not isokinetic, and,
therefore, the sample will be nonrepresentative.
Indirect surrogate technologies (e.g., laser diffraction technology or
high-frequency acoustic instruments with multi-transducers) may also be
used. These instruments provide access to the temporal variability in
concentration or grain size; however, they have limited ranges,
post-processing complexity (Gray and Gartner, 2010; Armijos et al., 2016)
and higher maintenance costs due the fragility of the instruments.
Thus, sampling methods with instantaneous capture or short-term integration
(<30 s) are preferred. These methods follow a relevant grid of
sample points (Xiaoqing, 2003; Filizola and Guyot, 2004; Bouchez et al.,
2011; Armijos et al., 2013; Vauchel et al., 2017). The mean concentration
〈C〉 (mg L-1) is
determined by combining all samples into a single representative
discharge-weighted concentration value, which is depth-integrated and
cross-sectionally representative (Xiaoqing, 2003; Horowitz et al., 2015;
Vauchel et al., 2017). In the present study, the spatial distribution of the
concentration within the cross section is summarized into a single
concentration profile that is assumed to be representative of the suspension
regime along the river reach. The water depth h (m) becomes the mean
cross-section depth, which is close to the hydraulic radius for large rivers.
Therefore, 〈C〉 will be
hereafter defined as the depth-integration of this concentration profile
C(z), z (m) being the height above the bed, from a reference height
z0 (m) just above the riverbed (z0≪h) to the free
surface, and weighted by the depth-averaged velocity 〈u〉=1h∫z0hu(z)dz (m s-1) (a term list can be found in the appendices):
〈C〉=∫z0hC(z)×u(z)dz∫z0hu(z)dz.
However, to dampen the random uncertainties mainly related to the coarse
sediments, this procedure requires taking a statistically significant number
of samples throughout the cross section, which is also time- and
labor-intensive.
All of these limitations preclude the application of such complete sampling
procedures at a relevant time-step necessary to build up a detailed
concentration series. By analogy with the index velocity method for discharge
computation (Levesque and Oberg, 2012), derived surrogate procedures, called
index sampling methods (Xiaoqing, 2003), are thus preferred. One or a few
“index samples” are taken as proxies of 〈C〉, usually at the water
surface (e.g., Filizola and Guyot, 2004; Bouchez et al., 2011; Vauchel et al., 2017).
The index concentration monitoring frequency is then scheduled to suit the
river's hydrological behavior and minimize the random uncertainties on the
measured index concentration C(zχ) (mg L-1) (Duvert et al.,
2011; Horowitz et al., 2015).
The index concentration method first requires a robust site-specific
calibration between the two concentrations of interest,
〈C〉 and C(zχ), i.e.,
for all hydrological conditions, which cannot always be achieved under field
conditions. Such relations are usually expressed with the following linear
form (e.g., Filizola, 2003; Guyot et al., 2005, 2007; Espinoza-Villar et al.,
2012; Vauchel et al., 2017):
〈C〉=αC(zχ)+ξ,
where the regression slope α and the intercept ξ are the fitted
parameters of the empirical model. In this study, the intercept will be
assumed to be zero (ξ=0). The dispersion and the extrapolation of the α=〈C〉/C(zχ) may
induce substantial uncertainties in the matter fluxes (Vauchel et al., 2017).
Most of this uncertainty is attributable to C(zχ) (Gitto et al.,
2017), particularly when only a single index sample is taken or when a unique
sample position is considered. Indeed, the relation may change around this
position based on the flow conditions.
The index sample representativeness becomes crucial as high-resolution
imagery is increasingly used to link remote-sensing reflectance data with
the suspended sediment concentration (e.g., Mertes et al., 1993; Martinez et al., 2009, 2015; Espinoza-Villar et al., 2012, 2013, 2017; Park and Latrubesse,
2014; Dos Santos et al., 2017). These advanced techniques finally provide a
spatially averaged C(zχ) value for the finest grain sizes at the
water surface of a reach (Pinet et al., 2017), which must be correlated with
the total mean concentration transported in the reach of interest (i.e.,
including the sand fraction when possible) to be a quantitative measurement
(Horowitz et al., 2015). Hence, to improve our knowledge of the sediment
delivery problem (Walling, 1983), these empirical relations deserve
hydraulic-based understanding.
In this study, the ratios α=〈C〉/C(zχ) observed at
eight gauging stations in the Amazonian foreland were analyzed to identify the
main parameters that controlled their variability. Assuming that the shape of
the concentration profiles measured in large Amazonian rivers can be well
described using a physically based model for sediment suspension, the
possibility of deriving a simple formulation for the ratio α using
this model was investigated. This assumption is supported by previous studies
that specifically showed that the Rouse model (Rouse, 1937) can describe
the suspension of sediments in large tropical rivers well (Vanoni, 1979, 1980; Bouchez et al., 2011; Lupker et al., 2011; Armijos et al., 2016). However, the Rouse model predicts a concentration of zero at the water surface, which
is where the index concentration is often sampled. To find an alternative,
other formulations (Zagustin, 1968; Van Rijn, 1984; Camenen and Larson, 2008)
are compared to the data.
Then, the relevance of the derived model in terms of developing a detailed
and reliable sediment flux series with an index method is discussed;
specifically, the ability to accurately estimate the model parameters is
evaluated. Finally, recommendations for the optimized collection of index
samples from large Amazonian rivers are inferred from the proposed model.
Materials and methodsHydrological data acquisition
The hydrological data presented here were collected within the international
framework of the critical zone observatory HYBAM (HYdrogéochimie du Bassin AMazonien – Geodynamical, hydrological and biogeochemical control of erosion, alteration and material transport in the Amazon Basin), which is a long-term monitoring program. A Franco-Peruvian team,
from the IRD (Institut de Recherche pour le Développement) and the
SENAMHI (SErvicio NAcional de Meteologia e HIdrologia), operates the eight
gauging stations of the HYBAM hydrological network in Peru; of these, four
stations control the Andean piedmont fluxes, and four stations control the
lowlands (Fig. 1). The three major Peruvian tributaries of the Amazon
(Solimões) River, i.e., the Ucayali River, the Marañon River and the
Napo River, are monitored. The studied sites cover drainage areas ranging
from approximately 22 000 to 720 000 km2
and have mean discharges ranging from 2100 to 30 300 m3 s-1 (Table 1). These large tropical rivers have flows with
gradually varied conditions, unimodal and diffusive flood waves (except for
the Napo River), and subcritical conditions, which enable backwater effects
(Dunne et al., 1998; Trigg et al., 2009).
Location of the sampling sites in the Amazon Basin. Blue
squares represent piedmont gauging stations, yellow dots represent lowland gauging stations and cyan represents flooded areas.
Hydrologic and sample dataset for the eight sampling stations.
The Amazonian foreland in Peru has a humid tropical regime (Guyot et al.,
2007; Armijos et al., 2013), and large amounts of runoff are produced during
the austral summer. During the austral winter, the maximum continental
rainfall is located to the north of the Equator, in line with the
intertropical convergence zone (Garreaud et al., 2009). Thus, the numerous
water supplies from the Ecuadorian subbasins smooth the seasonality of the
Marañon River flow regime. Located further to the south, the Ucayali
Basin experiences a pronounced dry season (Ronchail and Gallaire, 2006;
Garreaud et al., 2009; Lavado et al., 2011; Santini et al., 2014).
The El Niño–Southern Oscillation (ENSO) might alter these dynamics, as
there are severe low-flow events in El Niño years and heavy rainfall
events in La Niña years (Aceituno, 1988; Ronchail et al., 2002; Garreaud
et al., 2009). These events seriously affect the sediment routing processes
(e.g., Aalto et al., 2003), as do other extreme events unrelated to the ENSO
(e.g., Molina-Carpio et al., 2017).
Sampling strategy
For the reasons outlined in Sect. 1, local observers monitor
surface index concentrations at each station following a hydrology-based
scheme. The sampling depth is typically 20–50 cm below the water surface.
The samples are taken in the mainstream and at a fixed position.
Additionally, HYBAM routinely uses MODIS images to
determine surface concentrations, and these values are calibrated with in
situ radiometric measurements (Espinoza-Villar et al., 2012; Santini et al.,
2014; Martinez et al., 2015).
For calibration purposes (i.e., water level vs. discharge and concentration
index vs. mean concentration), 44 campaigns were conducted during the
2010–2017 period. These campaigns included the collection of 494 discharge
measurements, 249 sediment concentration profiles and 1330 water samples.
The dataset covers contrasted regimes, including periods of extreme droughts
(e.g., 2010) and periods of extreme floods (e.g., 2012 and 2015) (Espinoza et
al., 2012, 2013; Marengo and Espinoza, 2015). Thus, the sampled concentrations
spanned a wide range (Table 1), which represented the river
hydrological variability well. A 600 kHz Teledyne RDI Workhorse acoustic Doppler
current profiler (ADCP) was used and coupled with a 5 Hz GPS sensor to
correct for the movable bed error (e.g., Callède et al., 2000; Vauchel et
al., 2017).
A point sampling method was preferred to estimate
〈C〉 (Filizola, 2003; Guyot et
al., 2005; Vauchel et al., 2017) to capture the vertical concentration
distribution. The sampling for concentration determination was usually
performed at the following height (h) from the bed: ∼0.98h, 0.75h, 0.5h, 0.25h, sometimes at ∼0.15h and finally
at ∼0.1h, at three verticals that divided the cross section
according to the river width or the flow rate. Each vertical was assumed to
be representative of the flow in the corresponding subsection. Sampling was
performed from a boat drifting on a streamline immediately after the ADCP
measurements were collected. The sampler capacity was 650 mL, with a filling
time of ∼10 s, which allowed for a short time
integration along the streamline passing by the sample point. Considering the
waves at the free surface, the boat's pitch and roll and the bedforms, the
accuracy of the vertical position of the sampler may be evaluated as ±0.5 m. This variability leads to substantial uncertainty in the zones with
high concentration gradients. The operation time was approximately 2–5 h,
depending on the river sites. Steady conditions were observed during the
sampling operation.
Finally, samples for the characterization of the bed material PSD were
collected at four sites: BEL, REQ, REG and TAM. The bed material was dragged
on the riverbed.
Analytical methods
The concentrations Cϕ for two main grain size fractions ϕ were
further determined: the sand fraction (ϕ=s) was separated from the
silt/clay fraction (ϕ=f) using a 63 µm sieve (cf. Standart
Methods ASTM D3977), according to the Wentworth (1922) grain size
classification for noncohesive particles. The water samples were filtered
using 0.45 µm cellulose acetate filters (Millipore) that were then
dried at 50 ∘C for 24 h.
Particle size analysis was performed with a Horiba LA-920-V2 laser
diffraction sizer. The entire sampled volume was analyzed, with several
repetitions demonstrating excellent analytical reproducibility. For each
size group ϕ, the arithmetic mean diameter dϕ (m) was
calculated:
dϕ=∑idiXi∑iXi,
where Xi is the relative content in the PSD for the class of diameter
dϕ. The settling velocities wϕ corresponding to the
diameters dϕ derived from the PSD were computed using the Soulsby (1997) law, which assumed a particle density of 2.65 g cm-3.
Theory for modeling vertical concentration profiles
Schmidt (1925) and O'Brien (1933) proposed a diffusion–convection equation to
model the time-averaged vertical concentration distribution Cϕ(z) of
grains settling with a velocity wϕ (m s-1). The grain size,
shape and density are considered to be uniform. The equation is expressed as
follows:
εϕ∂Cϕ∂z=-wϕCϕ,
where the term on the left side is the rate of upward concentration
diffusion caused by turbulent mixing, balanced by the settling mass flux in
the right term. εϕ (m2 s-1) is the
sediment diffusivity coefficient that characterizes the particle exchange
capacity for two eddies positioned on both sides of a horizontal fictitious
plane. εϕ is assumed to be proportional to the
momentum exchange coefficient εm (m2 s-1) (Rouse, 1937):
εϕεm=βϕ,
where the βϕ parameter is similar to the inverse of a
turbulent Schmidt number (Graf and Cellino, 2002; Camenen and Larson, 2008).
It may be depth-averaged (Van Rijn, 1984) or considered to be independent of
the height above the bed (Rose and Thorne, 2001).
The main issue of the Schmidt–O'Brien formulation (Eq. 4) is the
expression of the vertical distribution of the sediment mass diffusivity
εϕ. Once this term is modeled (models are given
in the following), Eq. (4) is depth-integrated from the reference height
z0 to the free surface to obtain the expression of the concentration
distribution along the water column. The concentration Cϕz0 is then required to determine the magnitude of the profile and can
be evaluated using a bed-load transport equation (e.g., Van Rijn, 1984; Camenen
and Larson, 2008) or measured directly. However, in the case of a sampling
operation, the large concentration gradient observed near the riverbed would
force the operator to sample water at the reference height z0 with a
very high precision to minimize uncertainties; however, achieving such a high
level of precision is rarely possible. Hence, it is preferable to choose a
more reliable reference concentration in the interval z=[z0,h]. Thus, the following formulae resulting from Eq. (4) are written
using Cϕ(zχ) instead of Cϕz0 as a
reference.
Building on Prandtl's concept of mixing length distribution, O'Brien (1933)
and Rouse (1937) expressed the sediment diffusion profile using the following
parabolic form:
εϕz=βϕκu∗z1-zh,
where κ is the Von Kármán constant, and u∗ (m s-1) is the shear velocity. This expression leads to the
classic Rouse equation (Rouse, 1937) for suspended concentration
profiles. For zχ∈z0,h:
CϕzCϕ(zχ)=zχz×h-zh-zχPϕ,
where Pϕ=wϕ/βϕκu∗ is the Rouse suspension parameter, i.e., the ratio
between the upward turbulence forces and the downward gravity forces.
Pϕ is the shape factor for the concentration profile. The Rouse
formulation is widely used in open channels and suits the observed
profiles in the Amazon River well (Vanoni, 1979, 1980; Bouchez et al.,
2011; Armijos et al., 2016). However, the Rouse formulation predicts a
concentration of zero at the water surface. Three other simple models, for which Cϕ(h)≠0, have been selected in this work to overcome this problem, i.e., the Zagustin (1968), Van Rijn (1984) and Camenen and Larson (2008) models.
Zagustin (1968) proposed a formulation for the eddy diffusivity distribution
based on experimental measurements and a defect law for the velocity
distribution. The following variable changes were introduced: Z=(h-z)/z, and the sediment
diffusivity formulation proposed by Zagustin (1968) is
εϕZ=βϕκ3u∗hZ1-Z23.
This leads to an expression with a finite value at the water surface:
Cϕ(z)Cϕ(zχ)=expPϕΦzχ-Φ(z)Φ=12lnZ3+1Z-13Z3-1Z+13+3arctan3ZZ2-1.
As the proposed diffusivity profile is slightly different from the parabolic
form, this expression leads to Pϕ values that are approximately
7 % lower than those obtained with the Rouse theory (Zagustin, 1968).
Van Rijn (1984) proposed a parabolic-constant distribution for sediment
diffusivity, i.e., a parabolic profile in the lower half of the flow depth
(Eq. 6) and a constant value in the upper half of the flow depth (Eq. 10), which corresponds to the maximum diffusivity predicted by the
Prandtl–Von Kármán theories. Indeed, some authors have reported
measurements with constant sediment diffusivity in the upper layers
(Coleman, 1970; Rose and Thorne, 2001).
εϕz≥0.5h=βϕ4κu∗h.
Therefore, for z≥0.5h, the concentration profile is exponential,
with a finite value at the free surface:
Cϕ(z)Cϕ(zχ)=zχh-zχPϕexp-4Pzh-12.
In addition, Van Rijn (1984) introduced a coefficient to account for the
dampening of the fluid turbulence by the sediment particles. This
coefficient value is equal to the unity if the sediment diffusion
εϕ distribution is concentration-independent, which was
an assumption used in the present work because of the range of concentrations
measured in the Amazonian lowland rivers (Table 1), and discussed further in
Sect. 4.1.
Camenen and Larson (2008) showed that the depth-averaged sediment diffusivity
εϕ=βϕ6κu∗h is
a reasonable approximation of the Prandtl–Von Kármán parabolic form
(Eq. 10) that does not significantly affect the prediction of the
concentration profiles in large rivers (Pϕ<1), except near the
boundaries. This simple expression for εϕ lead to an
exponential sediment concentration profile:
Cϕ(z)Cϕ(zχ)=exp6Pϕhzχ-z.
This profile has practical interest: there is no need to define the
reference level z0 accurately or estimate the corresponding
concentration C0 (Camenen and Larson, 2008).
A general expression for the ratio αAssumptions and formalism
Cϕ(z) can be expressed by each of the models presented in this
section (Eqs. 7, 9, 12) and substituted into Eq. (1) to calculate 〈Cϕ〉.
Then, the development of the expression αϕ=〈Cϕ〉/Cϕ(zχ)
would lead to the following equation, which is similar to Eq. (2),
where the parameters driving αϕ are identified:
〈Cϕ〉Cϕ(zχ)=αϕz0h,zχh,Pϕ,u.
However, the PSDs observed in large rivers are rather broad (e.g., Bouchez et
al., 2011; Lupker et al., 2011; Armijos et al., 2016) and may be binned in a
range of n grain size fractions ϕ, as modeling the concentration
profiles requires the diameter of sediment in suspension dϕ to be
almost constant throughout the water depth if there is not a narrow PSD.
Assuming that the interaction between sediment classes ϕ is
negligible, it is possible to apply Eq. (7) and use a multiclass
configuration to describe the PSD:
α=∑ϕ=1nαϕXϕ,
where Xϕ is the mass fraction of each grain size fraction
measured for the index sample with the concentration Cϕ(zχ) (∑i=1nXϕ=1). Moreover, it can be shown that the weight of the
velocity distribution on the depth-averaged concentration may be neglected in
Eq. (1) when the suspension occurs throughout the water column, i.e., when
Pϕ<0.6 . Thus, if Pϕ<0.6, it is possible to express
αϕz0/h,zχ/h,Pϕ.
A key issue is then to provide a proper model of the PSD using a limited
number of sediment classes. In this study, the available dataset provides
concentrations for fine particles (0.45<df<63µm) and sand particles (ds≥63µm). Then, the ratio
α may be formalized as follows:
α=Xfαf+Xsαs.
Thus, if at height zχ the mass fraction of each group is
accurately known after sieving, α may be calculated for the whole
PSD.
Model proposed for the ratio αϕ prediction in Amazonian large rivers
The depth-integration of the Camenen and Larson formulation (Eq. 12) is considered to be a reasonable approximation of the measured
〈C〉 in large rivers
(Camenen and Larson, 2008), with a simple expression that is independent of
the z0 term, which differs from the other theories presented above.
Moreover, in the next section, the fit of the suspension models to the
measured concentration profiles will show that the Zagustin model provides
the best fit to the observations, particularly in the upper layer of the
flow. Thus, in this work, Cϕ(zχ) will be expressed using the
Zagustin model (Eq. 9), and
〈C〉 will be expressed using
the Camenen and Larson model.
Because the Zagustin model causes the Rouse number (Pϕ′) to be
slightly smaller than that calculated with the Rouse model (Pϕ′≈0.93Pϕ, according to Zagustin, 1968), we obtain the
following expression for predicting the ratio αϕ:
αϕ(zχ,Pϕ)=exp6Pϕzrh1-exp-6Pϕ6Pϕexp0.93PϕΦzr-Φzχ,
where zr is a reference height required for expressing Cϕ(zχ) with the Zagustin model (zr replaces zχ in
Eq. 9). Taking zr=0.5h, the previous expression is
simplified:
αϕ(zχ,Pϕ)=exp3Pϕ1-exp-6Pϕ6Pϕexp0.93PϕΦ(h2)-Φzχ.
Nevertheless, other formulations might be inferred from the suspension
models. For instance, the Camenen and Larson formulation could be
alternatively used to model Cϕ(zχ) in the central region of
the flow 0.2h,0.8h, which leads to a simpler
expression:
αϕ(zχ,Pϕ)=16Pϕexp6Pϕzχh1-exp-6Pϕ.
Model fitting strategy
To obtain a reach-scale profile, the fit to the concentrations averaged at
each normalized depth z/h was assessed. It was assumed that the energy
gradient, the mean bed roughness factor and the mean diameter did not
significantly change from one subsection to another, even if the
point-to-point variability was high (Yen, 2002). Thus, the depth becomes the
main factor influencing the Pϕ in the transverse direction. In the
cross sections studied here, the variation in depth from one vertical to the
next was not sufficient to significantly influence the Rouse number. Cϕ(zχ) is then the average of several representative samples taken across the river width at the same relative height (zχ/h).
After the first data cleaning of the sampled points, a robust and iteratively
re-weighted least squares regression technique was used to minimize the
influence of the outlier values. The weight values (W) between z0 and
h were assigned with the following parabolic function, similar to the eddy
diffusivity expression (Eq. 6): Wz=z(1-z).
Thus, the half-depth point, where the mixing term is the highest, has the
largest influence.
Based on the ADCP velocity profile measurements, the parameter z0 was
fixed at z0=10-3h. Indeed, when z0<10-2h,
〈C〉 is no longer sensitive to z0 (Eq. 1), even if the Rouse number is
not accurately known (Van Rijn, 1984). Hence,
(h-z0)/(z-z0)≈z/h can be assumed when considering
reach-scale flow conditions.
Shear velocity estimation from ADCP transects
The velocity transects measured with an acoustic Doppler current profiler
(ADCP) were used to estimate the shear velocities u∗ from the
vertical velocity gradient through the fit of the logarithmic inner law
(e.g., Sime et al., 2007; Gualtieri et al., 2018). An average of 30 ADCP
“ensembles” (i.e., measurement verticals of velocity), corresponding to
about 40 to 70 m in the cross-sectional direction, were required to obtain
robust u∗ values. This was consistent with the methodology applied
by Armijos et al. (2016) (50–60 ensembles, corresponding to 10 % of the
total width of the section) or Lupker et al. (2011) (30 ensembles, 40 to 70 m). Following these findings, the velocity profiles were averaged over a
spanwise length of about 60 m around each concentration profile position.
Then, an average of the fitted shear velocities was calculated for each ADCP
measurement. The flow over the first 30 m from the riverbanks has been
neglected, given the low velocities and the depths in this small area of the
cross section.
Furthermore, the imprecise knowledge of the exact bed elevation, the side
lobe interferences, the beam angle (which induces a large measurement area),
and the instrument's pitch and roll all cause the ADCP velocity data to be
inaccurate in the inner flow region (i.e., the region of the flow under bed
influence ∼z0,0.2h). However, a fit over the
entire height of the measured velocity (∼0.06h to the ADCP
“blanking depth” plus the transducer depth) leads to more robust shear
velocity values. For that reason, the shear velocities were assessed in the
zone between 0.1h and 0.85h.
ResultsData analysisIndex concentration relations calibrated for surface index samples
The observed α ratios for total concentration (i.e., concentration
including fine particles and sands) were calculated for a surface index using
each field measurement carried out (Fig. 2). Empirical relationships for
estimating the total mean concentration from the surface index samples (Eq. 2) were calibrated using these data . The α ratios observed
at the three stations monitoring the Ucayali Basin fluxes (i.e., LAG, PIN,
and REQ) are similar (1.3<α<1.5) (Fig. 2). At BEL (Napo River), the
α values observed are higher (α≅1.7). Conversely,
different trends with larger scatter are observed in the Marañon Basin.
The α ratios observed at REG (α≅2.3) are higher than
those at BOR (α≅1.5) and CHA (α≅1.4), which are
similar to those at LAG, PIN and REQ (Ucayali Basin). However, the observed
α values at BOR fluctuate between two main trends, which are
represented by the CHA–LAG–PIN–REQ group and the BEL–REG group. At TAM,
similarly, the α values rather follow the REG trend at low
concentrations before evolving between the REQ and REG trends.
Observed ratios α=〈C〉/C(h) of the total mean concentration to the total surface
concentration, stacked by river basin, with trend lines. For the Amazon
River basin at TAM (a), the REG and REQ trend lines were
reported. Dashed lines denote the first bisector.
This variability suggests that the α ratio is site-dependent and
potentially variable with the flow conditions. It could reflect differences
in the basin characteristics (e.g., lithology and climate spatial
distributions), then in sediment sources (e.g., mineralogy and PSD) and could
relate to the sediment routing in the lowland. The first group
(CHA–LAG–PIN–REQ) could be representative of a same source of sediments (the
Central Andes), as few lateral inputs come swelling these rivers discharges
in the lowland (Guyot et al., 2007; Armijos et al., 2013; Santini et al.,
2014). Conversely, in the Marañon lowland, the Ecuadorian tributaries
supply almost 55 % of the water discharge and could significantly
contribute to the river sediment load. The Napo River example (Laraque et
al., 2009; Armijos et al., 2013) shows that the lowland part of the basin
can be the main sediment source for these Ecuadorian tributaries. The river
incision of this secondary source, and/or the Ecuadorian Andes, could
provide coarser elements than the central Andean source does and explain why
the ratios α are higher at BEL and REG than at the other sites.
The concentration dataset highlights the control of the sand mass fraction
Xs on the ratio α (Fig. 3): α increases with Xs.
If finest particles of the PSD are dominant in the index concentrations
Cχ sampled in the upper layers of the flow, this wash load is
supply-limited, depending on the matter availability, rainfall upon the
sources and sediment entrainment processes occurring on the weathered
hillslopes. Wash load is then routed through the foreland without important
mass fluctuations (e.g., Yuill and Gasparini, 2011). Significant exchanges
between the floodplain and main channel lead to some dilution but also to
some remobilization of the huge floodplain sediment stocks of the coarser
elements that were previously deposited (e.g., clay aggregates, silts and
fine sands). Conversely, the sand transport regime is capacity-limited,
depending only on the available energy to route the sediments. As the
flow energy significantly decreases with the decreasing bed slope, the sand
suspended load is gradually decoupled from the wash load in the floodplain,
and the wash-load concentration is no longer a good proxy of the coarse
particle concentration. The floodplain incision mechanically increases the
sand mass fraction Xs in the suspended load. Implicitly, the PSD mean
diameter shifts with Xs, but it does not mean that there is any
change in the physical properties (e.g., diameter, density and shape) of the
sand fraction. This shift directly affects α, as the vertical
concentration gradient depends on the balance between the turbulence
strength and the settling velocity (Eq. 4). This result highlights the key
challenge of providing a proper model of the PSD using a limited number of
sediment classes, and validates the discrete approach proposed to model α.
Observed ratios α=〈C〉/C(h) of the total mean
concentration to the total index concentration sampled at the water surface
vs. the sand mass fraction Xs.
Particle size profiles
The measured particle size distributions (PSDs) show a multimodal pattern
(Fig. 4a). This example of a global PSD that includes the entire particle
size range was deconvoluted, assuming a mixture of lognormal
subdistributions (e.g., Masson et al., 2018). On the left side of the PSD, a
weak lognormal mode was detected in the clay range, but it was negligible in
comparison to the silt volume. A fairly uniform fraction of fine sands
(ds≅80µm) that were transported in suspension throughout
the water column with a nearly constant mode over depth was identified. This
fraction approximately corresponds to the diameters less than the 10th
percentile of the riverbed PSD. A second sand class (ds≅200µm) is transported as graded suspension with a strong vertical
gradient limited to the lower part of the water column (z/h<0.2). The Rouse
number Pϕ varies from one to six for this class of sediments, suggesting
that bed load may be non-negligible. However, the concentration dataset does
not contain any bed-load sample close enough to the riverbed and taken with a
relevant integration time to assess this argument.
(a) Multimodal modeling of a typical PSD vertical profile. Gray
lines represent the PSD measured at the Requena gauge station (16 March 2015) on the Ucayali
River. Sampling depths are mentioned on top of each panel. Green, blue, pink and
yellow roughly correspond to the following particle size groups: clays,
silts and flocculi, very fine sands–fine sands, and bed material,
respectively. The dashed gray line is the sum of the subdistributions. (b) Particle diameters dϕ measured at the eight sampling stations for the fine and sand fractions.
Concerning the whole dataset of fine sediment mean diameters df (Fig. 4b), no vertical gradient was observed for fine sediments, indicating that there
was homogeneous mixing throughout the water column, except near the
air–water interface, where the calculated df tended to decrease. In contrast, a gradient was observed for the sand fraction. Indeed, ds
varied from approximately 300 to 500 µm near the bottom to 80 to 100 µm near the surface. The increased sand diameters ds in the bottom
0.2h of the water column may be explained by bed material inputs (see
the yellow distribution in Fig. 4a).
Nevertheless, modeling the PSD with two size groups, which were
characterized by a diameter dϕ that was almost constant throughout
the water column, was reasonably suitable for the observed PSD, although two
more classes (i.e., clay and bed material) could be considered to improve
this model. Thus, an average of the diameters derived from the PSD was
calculated to summarize the PSD data into one single mean diameter dϕ (m) per site for each size group ϕ.
Suspension model suitability with the measured profiles
The suspension models (Eqs. 7, 9, 11, 12) were fitted to the concentration
data to evaluate their suitability to the observed profiles. The dataset
confirmed that the Zagustin model causes the Rouse number (Pϕ′)
to be slightly smaller than that calculated using the Rouse model: Pϕ′≈0.93Pϕ. The fitted Pϕ values showed
low variability and were summarized by single average values per site, as
shown in Table 2. This low variability indicates that there is a dynamic
equilibrium between the settling velocity wϕdϕ and the shear velocity u∗ under nominal flow
conditions, although some extreme values (Pϕ>0.5) were measured
during severe drought events at the lowland stations.
Summary of suspended sediment transport parameters for each site
and size group.
Mean results for the fine particle fraction Mean results for the sand fraction StationRh (m)u∗ (m s-1)Xs(h)df (µm)wf (m s-1)Pfβfαfαfds (µm)ws (m s-1)Psβsαsαscodezχ=hzχ=0.5hzχ=hzχ=0.5hLAG6.80.1015 %16±3 %2.5×10-40.03±16 %0.211.11.0148±4 %1.8×10-20.27±9 %1.73.51.1PIN7.20.1420 %14±3 %1.9×10-40.01±14 %0.241.01.0124±2 %1.4×10-20.23±12 %1.12.31.0REQ12.50.097 %16±4 %2.5×10-40.05±8 %0.141.21.0114±7 %1.2×10-20.39±7 %0.86.41.2BOR7.90.1710 %16±4 %2.5×10-40.02±17 %0.171.01.0118±10 %1.3×10-20.28±16 %0.64.31.1CHA7.40.2016 %––0.01±19 %–1.11.0––0.24±7 %–2.61.0REG16.70.1414 %15±3 %2.2×10-40.07±10 %0.061.41.0132±7 %1.4×10-20.44±5 %0.67.51.2TAM18.60.126 %17±4 %2.8×10-40.08±14 %0.071.51.0141±10 %1.7×10-20.44±5 %0.88.01.2BEL10.10.1017 %19±2 %2.5×10-40.04±16 %0.151.21.0192±19 %2.8×10-20.32±8 %2.14.31.2Mean10.90.1313 %162.4×10-40.040.161.21.01381.6×10-20.331.14.81.1MethodMeasuredFitted on u(z)MeasuredDerived fromSoulsbyFitted on Czwf/(Pfκu∗)FittedFittedDerived fromSoulsbyFitted on Czws/(Psκu∗)FittedFitted(log-law)measured PSDlaw(Rouse model)on obs.on obs.measured PSDlaw(Rouse model)on obs.on obs.
The Rouse numbers obtained for the fine fraction reflect a suspension regime
that is close to the ideal wash load (Pf<0.1, 1≤αf(h,Pf)≤1.5). Additionally, regarding the Rouse numbers corresponding to
the sand fraction, they reflect a well-developed suspension in the entire
water column for the piedmont station group (0.2<Ps<0.3) and for the
lowland station group (0.35<Ps<0.45), with a significant concentration
gradient (2.3≤αs(h,Ps)≤7.5).
Due to the availability, for a given site, of a single mean value of wϕdϕ per size group, only the corresponding mean values of the diffusivity ratio βϕ=wϕ/Pϕκu∗ were calculated, considering the mean shear velocities (Table 2).
Sediment diffusivity profiles
The diffusivity profiles εϕ(z) were derived from the
measured concentration profiles using the discrete form of Eq. (4). In
order to capture the small variations in εϕ, accurate
sampling is key: the calculation of εϕ requires precise
concentration and sampling height values, particularly for the fine fraction,
which experiences low vertical concentration gradients.
Nevertheless, the overall shapes of the derived εϕz profiles were in good agreement with the Rouse and Zagustin
theories and were slightly closer to the latter (Fig. 5). Given the high
scatter of the diffusivity values, Camenen and Larson's expression of
depth-averaged diffusivity is a reasonable approximation, except near the
bottom and top edges of the diffusivity profiles, where the data departs
gradually from this model. However, the constant diffusivity value suggested
by Van Rijn (1984) for the upper half of the water column clearly
overestimates the diffusivity for z>0.75h. The diffusivity around
z=0.75h is, however, overestimated by all the models. The low
concentrations near the water surface could result in an underestimation of
the εϕ(z≈0.75h) values calculated from the
difference ΔCϕ=Cϕz≈0.5h-Cϕz≈h (Eq. 4). Thus,
detailed measurements are required in the upper layer of the flow to confirm
the shapes of the εϕz profiles in this zone
where the air–water interface and the secondary currents can influence the
turbulent mixing profiles.
Dimensionless sediment diffusivity coefficient derived from the
measured concentration profiles.
Concentration profile suitability
Overall, the suspension models (Eqs. 7, 9, 11, 12) fit well with the
observed profiles (Fig. 6): for 92 % of the profiles fitted, the
coefficients of correlation (r) were superior to 0.9 % and 100 % of the
r were superior to 0.7, except near the edges where the highest discrepancies
between the two exponential expressions (Van Rijn, 1984; Camenen and Larson,
2008) and the Rouse and Zagustin models appear. The concentrations sampled
at the bottom edge confirmed the general shape of the Zagustin and Rouse
models, despite the uncertainties in the concentrations measured in this
zone. Near the water surface, the nonzero values predicted by the Zagustin
model were often the closest to the observed concentrations.
Typical examples of measured concentration profiles Cϕ(z/h), fitted with the Rouse, Van Rijn,
Zagustin, and Camenen and Larson models.
The use of the Camenen and Larson model to calculate the mean concentration
〈Cϕ〉 seems to be a reasonable
approximation. Indeed, for the range of nominal Rouse numbers considered here
(Pϕ<0.6), the bottom concentration gradient has little influence on
〈Cϕ〉 because the velocity
decreases rapidly with depth in this region of the flow. Moreover, the
top-layer concentrations are too low to weight significantly on 〈Cϕ〉.
(a) Predicted and observed αϕi ratios as a function of the Rouse number. Filled circles
and squares are the mean αϕ values observed per site for the sand and fine mass fractions,
respectively. Unfilled circles denote observed αs values. The red to pink rainbow set of solid
lines correspond to the general model prediction (Eq. 17), and each
represents 10 % of the water height. Dashed lines are for the simplified
model (Eq. 18). (b) Predicted vs. observed mean α ratios
per site (i.e., total concentration).
The comparison between the predicted and observed mean αϕ
values per site and size group (Fig. 7a) allows for the validation of the
general model proposed in this work (Eq. 17). To show the model's
ability to predict how αϕ changes with flow conditions at one
specific site, this model was also compared with all of the αϕ
values observed at the water surface and at mid-depth (Fig. 7a). The
observations follow the model trend well, despite the high scatter of the
αs(h,Ps) values, which is caused by the low diffusivity and
concentration in coarse material near the water surface and by the
uncertainty of the exact z position of the samples. At mid-depth, the
αs(0.5h,Ps) values have lower scatter.
Nevertheless, the αs sensitivity to the Rouse number remains
moderate for most of the hydraulic conditions encountered, except for the
extremely low flow rates, i.e., when Ps>0.5. The αf
sensitivity to changes in flow conditions is very small. Then, considering
the small contribution of the low waters to the sediment budget and the
small Rouse number variations for the nominal hydraulic conditions at a
specific site (Table 2), the use of the mean αϕ coefficients per site seems to be reasonable for assessing reliable
sediment budgets. Regarding the simplified model (Eq. 18), a reasonable
approximation is expected to be found in the central region of the flow, but
the values gradually depart from the observations near the water surface and
the riverbed.
Finally, the mean ratios α(h) per site were computed (Eq. 15) using the predicted mean αf(h,Pf) and αs(h,Ps) (Eq. 17) and the mean mass fractions Xf and
Xs measured at the water surface (Table 2). The observed vs. predicted
α ratios are in excellent agreement (r2=0.97)
(Fig. 7b) and validate the prediction ability of the model when the Rouse
numbers are accurately known.
Discussion on the model applicability
The equations proposed in this work (Eqs. 17, 18) for modeling the ratios
αϕ(zχ,Pϕ) become very sensitive when both the
index sample is taken near the river surface (zχ≈h) and the
Rouse number is rather large (Pϕ>0.4) (Fig. 7a). This is a first
limitation for the model applicability, if a monitoring of the index
concentration at a deeper level on the water column is not technically
feasible. In addition, for rivers with Rouse numbers greater than 0.6 (i.e.,
when the suspension does not occur in the entire water column because the
particle are too coarse, in comparison to the strength of the flow, to be
uplifted at the water surface), the weight of the velocity distribution in
the model can no longer be neglected as it was in this work (see the
assumption, Sect. 2.3.1). Furthermore, the higher the Rouse number, the
more difficult the concentration measurement is to perform. Then, the
accuracy of the model also depends on the concentration measurement procedure
chosen and related uncertainties. These uncertainties depend on the
point-sampling integration-time which must be long enough to be
representative (Gitto et al., 2017), on the volume of water collected and on
the sampling position(s) defined in the cross section.
Therefore, estimating Pϕ with a low uncertainty is a key issue to
predict accurate αϕratios during sediment
concentration monitoring. This estimation can be achieved (1) via the
estimation of the hydraulic parameteru∗, wϕ and βϕ, or (2) empirically using detailed point concentration
measurements. Then, if the Rouse number variability is significant during
the hydrological cycle, the empirical relationship between u∗ or
h and the Pϕ fitted on measured concentration profiles may be
calibrated.
Estimation of the diffusivity ratio βϕ
For many decades, studies based on flume experiments or measurements in
natural rivers have shown that βϕ usually departs from the
unity. The sediment diffusivity increases (βϕ>1) with
bedforms or movable bed configurations (Graf and Cellino, 2002; Gualtieri et
al., 2017); specifically, the boundary layer thickness tends to be thin just
before the bedforms crest, and then it peels off at the leeward side
(Engelund and Hansen, 1967; Bartholdy et al., 2010). This trend implies there
are anisotropic macroturbulent structures, with eddies that convect large
amounts of sediments to the upper layers and settle further after eddy
dissipation. Thus, bedforms locally modify the ratio between the laminar and
turbulent stresses, inducing different lifting profile shapes in the inner
region (e.g., Kazemi et al., 2017) and causing the mixing length theory to
fail in the overlap region. Centrifugal forces driven by turbulent motion and
applied on the grains could also enhance the particle exchange rate between
eddies (Van Rijn, 1984). Conversely, the suspension is dampened (βϕ<1) when the large suspended particles do not fully respond to all velocity fluctuations, such as passive scalars.
(a) Ratio of sediment to eddy diffusivity βϕ as a function of the ratio wϕ/u∗, with points shaded
according to the water level h. (b) Idem, after correction of
the ratio (wϕ/u∗). Circles and squares are the mean values of βs and βf calculated per site,
respectively.
Van Rijn (1984), Rose and Thorne (2001) and Camenen and Larson (2008)
attempted to model βϕ as a function of the ratio wϕ/u∗ for sand and silt particles. However, the measured
βϕ encompasses poorly understood physicochemical processes as well
as uncertainties and bias of the wϕ and u∗ estimations,
which might partly explain the shifts along the wϕ/u∗axis
between the three abovementioned laws and the βϕ inferred in this
study from measured profiles of concentration, particle diameter and
velocity (Fig. 8a).
With regard to wϕ, a major difficulty comes from the need to
divide the PSD into various size groups and to summarize each
subdistribution with a single characteristic diameter (e.g., mode, median,
mean), and different values of wϕ(dϕ) are calculated
according to the choices made.The aggregation process is a supplementary
complicating factor (Bouchez et al., 2011) but is probably not the main issue
in these white rivers with little organic matter (Moquet et al., 2011;
Martinez et al., 2015). Indeed, the results of Bouchez et al. (2011) are
probably biased because the authors used a single diameter to summarize the
entire PSD, which is highly sensitive to the flow conditions. However, this
bias would not concern the sand group because the shear modulus experienced
in large Amazonian rivers would prevent the formation of large aggregates.
The choice of a settling law (e.g., Stokes; Zanke, 1977; Cheng, 1997; Soulsby,
1997; Ahrens, 2000; Jiménez and Madsen, 2003; Camenen, 2007) may also
induce bias on wϕ. In these laws, the sediment density is a key
parameter that is often neglected, as natural rivers comprise a diversity of
minerals with contrasting density ranges.
Conversely, the shear velocity estimation also suffers from
uncertainties in terms of the velocity measurements and biases that are
induced by the method used (Sime et al., 2007). For instance, the departures
from logarithmic velocity profiles increase with the distance to the bed
(e.g., Guo et Julien, 2008) in sediment-laden flows (e.g., Castro-Orgaz et
al., 2012), which could be relevant to deep Amazonian rivers. Indeed, the
mixing length expansion could reach a maximum before the water surface, as
the energetic eddy size cannot expand ad infinitum far from the flow zone under the
influence of bed roughness because of the increasing entropy. The log-law
assumptions (i.e., constant shear velocity throughout the water column and
mixing length approximation) would no longer be valid, and the velocity
profiles would follow a defect law in the outer region. This raises the need
to find a suitable model for the velocity distribution in large rivers,
leading to an unbiased estimate of the shear velocity.
Thus, it is not surprising to find discrepancies between the empirical laws
and the observations based on the experimental conditions. Here, the Rose
and Thorne (2001) empirical law is the closest to the observed βϕ (Fig. 8a), with departures that seem to be a function of the water level.
We assume that a global correction of the different bias on the wϕ/u∗ term would depend on the flow depth as well as on the skin
roughness, which partly influences the formation and expansion of the
turbulence structures and thus influences the velocity distribution (Gaudio
et al., 2010). Here, ds is considered instead of the skin roughness
height, as few riverbed PSDs are available and because it is a key parameter
for the settling law. Thus, the following modification of the Rose and
Thorne (2001) law is proposed:
βϕ=3.1exp-0.19×10-3u∗wϕhds0.6+0.16,
where the coefficient 3.1 comes from the Rose and Thorne (2001) law. Other
numerical values in Eq. (19) were fitted to obtain the best agreement
with the βϕ inferred from the measured concentration profiles
(Table 2). In a similar way to Camenen and Larson (2008), a minimum βϕ-value was found for very small values of
wϕ/u∗. This nondimensional law, which extends below the range of wϕ/u∗ usually considered in previous studies, allows for an enhanced
prediction of βϕ(±0.03) (Fig. 8b).
Furthermore, the dataset did not show any relationship between concentration
and the diffusivity ratio βϕ (not shown here). The
uncertainties in the dataset collected under field conditions do not allow for the further investigation of the influence of second-order factors on the diffusivity
ratio, such as the particle characteristics (shape, grain size,
density, and so on), the aggregation phenomenon or the level of
stratification of the flow (e.g., Van Rijn, 1984; Graf and Cellino, 2002; Pal
and Ghoshal, 2016; Gualtieri et al., 2017).
Fitted Rouse numbers against (a) predicted
Ps (the gray square in the bottom-left
represents the range of variation of Pf), (b) shear velocities and (c) water levels.
Applying Eq. (19) to predict the mean βϕ values per site, the
predicted and fitted Pϕ are in good agreement (Fig. 9a), with
little scatter when considering the uncertainties in the measured
concentrations and therefore on the fitted Pϕ. This result shows
that the shear velocity mainly controls the Rouse number variability at a
given site (Fig. 9b). Therefore, the variations in particle size are a second-order factor. The shear velocity is itself driven by the high amplitude of the water depth in Amazonian rivers (Fig. 9c), and it has hysteresis effects
at the gauging stations located in the floodplain, which are attributed to
the backwater slope variability in these subcritical flood wave contexts
(Trigg et al., 2009). Hence, the accurate monitoring of the water level and
knowledge of the river surface slope, even if limited or biased, would allow
for an acceptable prediction of the Rouse numbers, which could be used to
establish a single βϕ value per site.
Predicting ds from the
riverbed PSD
For fine particles, df can be accurately measured in the water column
because the fine particles are well mixed in the flow.
Regarding the sand particles, such measurements induce uncertainties due to
the particle fluctuations in the current and because the eddy structure
development in the bottom layers of the flow swiftly causes strong grain
size sorting (Fig. 4). The suspended sediment particles are thus
considerably smaller than the bed load or riverbed particles (Van Rijn,
1984).
The diameter of the suspended sand can be assessed by taking a
representative percentile of the riverbed PSD (e.g., Rose and Thorne, 2001).
Alternatively, an empirical expression that considers the flow conditions
was proposed by Van Rijn (1984).
(a) Prediction of the mean diameter
ds at the TAM gauging station for
u∗=0.12 m s-1 (mean flow conditions). (b) Predicted vs. measured ds at BEL, REQ, REG and TAM. The dashed line denotes the first bisector, and the solid line represents the best fit.
Here, the Camenen and Larson (2005) formulae for the estimation of the reference concentration Cϕ(z0) was applied in a multiclass way to the
riverbed PSD, and it was assumed that the size fractions did not influence
each other and there was a uniform sediment density for all grain sizes (2.65 g cm-3). In this formulation, Cϕ(z0) is a function of the
dimensionless grain size d∗, the local Shields parameter
θϕ and of the critical Shields parameter θcr for the
inception of transport (Camenen et al., 2014):
Cϕ(z0)=0.0015θϕexp0.2d∗+4.5θcrθϕ.
This first PSD predicted at the transition level z0 is further diffused
vertically with the Zagustin model (Eq. 19), considering the Soulsby (1997)
settling law in the Pϕ calculations (Fig. 10a). The model
underestimates the measured ds by approximately 10 % (Fig. 10b).
This slight discrepancy might be explained by stochastic and ephemeral
inputs of coarse bed material in the water column, which are not addressed
by the suspension theory.
Sensitivity analysis and recommendations for optimized sampling
procedures
The approximation error ΔPϕ can be evaluated at ±0.03
(from Eq. 19, Fig. 9 or Table 2) and propagated to the
corresponding αϕ(zχ,Pϕ±ΔPϕ) (Fig. 11a). The error on zχ is not considered here but would
increase the αϕ sensitivity in the zones with a high
concentration gradient. Overall, the relative error on αϕ remains moderate for all of the flow conditions experienced by the rivers
studied here (i.e., below ±10 % in the central zone of the flow and
below ±20 % at the water surface), except near the riverbed (Fig. 11a). Nevertheless, for operational applications, this result must be
weighted by the relative error profile of the measured index
concentrations ΔCϕ(zχ)/Cϕ(zχ).
(a) Relative error of the predicted αϕ according to the relative height
zχ/h of the index sampling, for various
Rouse numbers Pϕ. (b) Relative error of the
concentration sampled, inferred from the Zagustin model and
assuming ΔCϕCϕ(zχ=05h)=±10 %. (c) Relative error on
〈Cϕ〉 as a function of zχ/h.
By substituting Cϕ(zχ) with the Zagustin model (Eq. 9) and assuming ΔCϕ(0.5h)/Cϕ0.5h=±10 %, it is possible to model this profile of concentration
uncertainty (Fig. 11b) and to derive the relative uncertainty of
〈Cϕ〉 according to the sampling
height under various flow conditions (Fig. 11c). Here, the considered
uncertainty is a simple function of the concentration. However, coarse
particles are more sensitive to current fluctuations than are fine sediments.
Thus, the sand concentration uncertainty is underestimated, at least in the
region of the flow under bed influence ∼z0,0.2h, where stochastic uplifts of bed sediments impose high variability
on the concentration. Furthermore, the sampling frequency as well as the
number of index samples taken and their positions are important parameters to
consider. The section geometry, the velocity distribution and the transversal
movable bed velocity pattern are important guidelines in the selection of a
sampling position(s). The integration of the lateral variability of the
concentration is not discussed here. Nevertheless, when considering these
assumptions, optimized sampling heights may be defined as follows:
For fine sediments (Pf<0.1), the most accurate
〈Cf〉 is obtained when sampling
the water column at approximately 0.5h. The sampling can also be
achieved at the water surface with a good estimation of 〈Cf〉 (±15 %).
For the sand fraction at the piedmont stations (Ps<0.3), sampling in the
0.2h,0.8h region is recommended to keep the errors
of 〈Cs〉 below ±20 %. Sampling at the water surface is still possible, but there will be
uncertainties between ±20 % and 40 % for 〈Cs〉.
For enhanced monitoring of the sand concentration at the lowland stations
(Ps>0.3), the 0.2h,0.8h zone is preferred over
the water surface, where the αϕ prediction would
require very accurate estimations of Ps and Cs(h).
The proposed αϕ models (Eqs. 17, 18) allow for a
routine protocol with sampling in the central zone of the flow to be achieved: the αϕ can be predicted at each sampling time-step, when the section
geometry is known and when the flow is sufficiently stable to estimate zχ/h. For instance, a single fixed sampling depth could be used.
Alternatively, the Rouse number can be estimated at each sampling time-step
from Eq. (7) by sampling two heights zχ1 and zχ2 of
the water column at each measurement time-step:
Pϕ=lnCϕzχ1Cϕzχ2lnzχ2zχ1h-zχ1h-zχ2.
For instance, the concentration at zχ1=0.7h and at zχ2=0.3h results in
Pϕ=0.59lnCϕ0.3h/Cϕ0.7h.
When considering nominal flow conditions (Pϕ<0.6), the sampling
height above the riverbed h/e≈0.37h (e being the
Euler number) appears to be pertinent for simplified operations, as the
ratios of αϕ(h/e,Pϕ) remain interestingly close
to unity (±10 %) (Fig. 7a). Thus, βf,βs≈0.16,1 could be simply assumed
without inducing large errors in the αϕ(h/e,Pϕ)
estimations. The particles are usually present in a significant amount, and
the turbulent mixing is intense (Eq. 6), while the concentration
gradients are moderate, which also causes for more uncertainty regarding zχ. Interestingly, when considering the depth-averaged velocity 〈u〉≈u(h/e) for velocity
profiles that are logarithmic in nature, the sediment discharge on a vertical
qsϕ (g s-1 m-2) may be expressed as follows:
qsϕ=αϕ×Cϕhe×uhe≅Cϕhe×〈u〉±10%.
Finally, if sampling in the central zone of the flow is not technically
feasible during concentration monitoring, the mean concentration of fine
particles may be estimated with surface index sampling or remote sensing
(Martinez et al., 2015; Pinet et al., 2017). Then, the sand concentration
could be assessed with a sediment transport model that is suitable for large
rivers (e.g., Molinas and Wu, 2001; Camenen and Larson, 2008). To
parameterize such models, improved spaceborne altimeters (e.g., the SWOT – Surface Water Ocean Topography – mission) and hydrological models are already serious alternatives to in situ
discharge, water level and slope measurements (e.g., De Paiva et al., 2013;
Paris et al., 2016).
Conclusion and perspectives
The use of measured concentration profiles with physically based models
describing the suspension of grains in turbulent flow has shown the
possibility to derive a simple model for the prediction of αϕ, i.e., for a given particle size group ϕ. Proper modeling of the
PSD using two hydraulically consistent size groups (i.e., fine particles and
sand) is first required to obtain a characteristic diameter that is mostly
constant for each size group during the hydrological cycle.
The Zagustin profile, with finite values at the water surface, demonstrated
the best suitability in relation to the observed data. Nevertheless, the
Camenen and Larson model was in good agreement with the observations in the
central zone of the flow and was a reasonable approximation of the
depth-averaged concentration.
The Rouse number is the main parameter for αϕ modeling. Variations in Pϕ during the hydrological cycle may be
monitored from a few point concentration measurements or through the
calibration of a relation between the u∗ or h and the
measured Pϕ. Alternatively, a function of wϕ/u∗ and h/ds was proposed to
compute βϕ and predict Pϕ±0.03.
The sensitivity of the αϕ model decreases from the boundaries
to a zone between 0.2h,0.5h which is based on the
flow conditions. At the water surface, the model becomes inaccurate when Ps>0.3, i.e., for flow conditions corresponding to sand suspension in
the lowland. In such a context, sampling in the central zone of the flow is
preferable for sand concentration monitoring. A pertinent sampling height
for optimized concentration monitoring appears to be zχ=0.37h.
This insight into the hydraulic theory leads to enhanced sediment monitoring
practices, with a more accurate estimate of the sediment load, especially in
regions with limited data availability, such as the Amazon Basin. Indeed,
the proposed model is a tool that can be used to predict the αϕ and α ratios and can also be used to select a proper sampling
height for optimized monitoring. Extensively, the model allows for detailed
uncertainty analysis on the
〈C〉 derived from an index method.
Finally, where the cross-section geometry is well known and where no in situ
concentration data exist, the model could allow for an accurate estimation
of the mean concentration in fine sediments
〈Cf〉, with remote-sensing monitoring of the index concentration in fine
sediments on the water surface. Coupling this monitoring with a sand
transport model suitable for large rivers could ensure a better
understanding of the sediment dynamics in the Amazon Basin.
Data availability
The data that support the findings of this study are available from the
following data repository (Santini et al., 2018): 10.6096/DV/CBUWTR. Extra data (water levels, suspended
concentration time series etc.) are also available from the
corresponding author upon request and on the CZO HYBAM website: http://www.so-hybam.org (last access: 1 March 2019).
List of notations
.fFine sediment particles group (0.45 µm <df<63µm).sSand sediment group (ds>63µm)CTime-averaged concentration (mg L-1)dArithmetic mean diameter (m)d∗dgρρw-1υ213 is the dimensionless grain sizegGravitational force (m s-2)hMean water depth (m)ksNikuradse equivalent roughness height (m)PRouse number (–)qsTime-averaged sediment discharge on a vertical (g s-1 m-2)uTime-averaged velocity (m s-1)u∗Shear velocity (m s-1)wSuspended sediment particle settling velocity (m s-1)XMass fraction (–)zHeight above the bed (m)αRatio between mean concentration and index concentration (–)βRatio of sediment to eddy diffusivity (–)εSediment diffusivity coefficient (m2 s-1)εmMomentum exchange coefficient (m2 s-1)κVon Kármán constant (–)υKinematic viscosity (m2 s-1)ρwWater density (kg m-3)ρSediment density (kg m-3)θShield's dimensionless shear stress parameter (–)θcrCritical dimensionless shear stress threshold (–)〈〉Depth-integrated value
Inner law (“law of the wall”):
uz=u∗κln30zks.
Zagustin (1968) defect law:
uz=Umax-2×u∗κarctanhh-zh32.
Author contributions
WS, BC, JLC and PV conceived the study, analyzed the data, and undertook the investigation, methodology and code
development. WS, BC, JLC, JMM and JLG prepared the paper and discussed the data. JMM, JLG, WS, WL and MAP were responsible for funding acquisition as well as project administration and supervision. WS, PV, JC, JJPA, REV and JLG were responsible for the hydrologic data acquisition. NA, FJ and WS undertook the laboratory analysis.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The authors would especially like to acknowledge their colleagues from the
National Agrarian University of La Molina (UNALM) and the Functional
Ecology and Environment laboratory (EcoLab), who contributed to the analysis
of the data used in this study.
Financial support
This research has been supported by the French National Research Institute for
Sustainable Development (IRD), the National Center for Scientific Research
(CNRS – INSU) and the Peruvian Hydrologic and Meteorology Service (SENAMHI).
Review statement
This paper was edited by Robert Hilton and reviewed by Kathryn Clark and two anonymous referees.
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