An Index Concentration Method for Suspended Load Monitoring in Large Rivers of the Amazonian Foreland

Earth Surf. Dynam., 7, 515–536, 2019 https://doi.org/10.5194/esurf-7-515-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License.

An index concentration method for suspended load monitoring in large rivers of the Amazonian foreland

William Santini1,2, Benoît Camenen3, Jérôme Le Coz3, Philippe Vauchel1,2, Jean-Loup Guyot1,2, Waldo Lavado4, Jorge Carranza4, Marco A. Paredes5, Jhonatan J. Pérez Arévalo5, Nore Arévalo6, Raul Espinoza Villar6,7, Frédéric Julien8, and Jean-Michel Martinez1,2 1IRD, Toulouse, 31400, France 2Laboratoire GET, CNRS, IRD, UPS, OMP, Toulouse, 31400, France 3Irstea, UR RiverLy, Lyon-Villeurbanne, 69625 Villeurbanne, France 4SENAMHI, Lima, Lima 11, Peru 5SENAMHI, Iquitos, Peru 6Facultad de Ingeniería Agrícola, UNALM, La Molina, Lima 12, Peru 7IGP, Ate, Lima 15012, Peru 8Laboratoire ECOLAB, CNRS, INPT, UPS, Toulouse, 31400, France Correspondence: William Santini ([email protected])

Received: 15 December 2018 – Discussion started: 21 January 2019 Revised: 30 March 2019 – Accepted: 21 May 2019 – Published: 4 June 2019

Abstract. Because increasing climatic variability and anthropic pressures have affected the sediment dynam- ics 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 con- centrations 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 6= 1. An empirical expression taking these biases into account was then formulated to predict accurate estimates of β, then P (1P = ±0.03) and finally α. The proposed model is a powerful tool for optimizing the concentration sampling. It allows for detailed un- certainty 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.

Published by Copernicus Publications on behalf of the European Geosciences Union. 516 W. Santini et al.: An index concentration method for suspended load monitoring

1 Introduction the need for time-integrated or repeated measurements to ensure the temporal representativeness of each water sam- In recent decades, the Amazon Basin has experienced an in- ple (Gitto et al., 2017). The second-order moments of the tensification in climatic variability (e.g., Gloor et al., 2013; Navier–Stokes equations induce this temporal concentration Marengo and Espinoza, 2015), specifically in extreme events variability, as do the larger turbulent structures (typically (drought and flood), as well as increasing anthropic pressure. those induced by the bedforms) and the changes in flow con- In the Peruvian foreland, the advance of the pioneer fronts ditions (e.g., backwaters, floods and flow pulses). Sands and causes serious changes in land use, which are enhanced by coarse silts are much more sensitive to velocity fluctuations the proliferation of roads that provide access to the natural than clay particles are (i.e., settling laws are highly sensitive resources hosted by this region. The number of hydropower to the diameter of the particles) and are the most difficult to projects is also rapidly increasing (e.g., Finer and Jenkins, accurately measure. 2012; Latrubesse et al., 2017; Forsberg et al., 2017). These Depth-integrated or point-integrated sampling procedures global and local changes might increase the erosion rates are traditionally used to determine the mean concentration in the basin as well as the suspended load interannual vari- of suspended sediment in rivers. However, deploying these ability (e.g., Walling and Fang, 2003; Martinez et al., 2009). methods from a boat is rarely feasible due to the velocity and The sediment transfer dynamics might also be affected (e.g., depth ranges that are encountered in large Amazonian rivers. Walling, 2006), generating large ecological impacts on the For a point-integrated bottle sampling method, maintaining a mega-diverse Amazonian biome and having socioeconomic position for a duration long enough to capture a representa- consequences on the riverine populations. In such a context, tive water sample (Gitto et al., 2017) requires anchoring the long-term and reliable sediment series are crucial for de- boat and using a heavy ballast. This type of operation is very tecting, monitoring and understanding the related socioeco- risky without good infrastructure and well-trained staff, espe- nomic and environmental impacts (e.g., Walling, 1983, 2006; cially when collecting measurements near the river’s bottom. Horowitz, 2003; Syvitski et al., 2005; Horowitz et al., 2015). Moreover, this method decreases the number of samples that However, there is a lack of consistent data available for this can be collected in 1 d. region, and this lack of data has prompted an increased in- For a depth-integrated sampling method within a deep terest in developing better spatiotemporal monitoring of sed- river, the bottle may fill up before reaching the water surface iment transport. if its transit speed is too slow. Moreover, if the ballast weight In the large tropical rivers of Peru, the measurement is not sufficient to hold the sampler nose in a horizontal posi- of cross-sectional average concentrations hCi (mg L−1) re- tion, the filling conditions are not isokinetic, and, therefore, mains a costly and time-consuming task. First, gauging sta- the sample will be nonrepresentative. tions can only be reached after several days of traveling on Indirect surrogate technologies (e.g., laser diffraction tech- hard dirt roads or by the river. Second, there is no infras- nology or high-frequency acoustic instruments with multi- tructure on the rivers, and all operations are conducted us- transducers) may also be used. These instruments provide ac- ing small boats under all flow conditions. Third, the gauging cess to the temporal variability in concentration or grain size; sections have depths that range from the metric to the deca- however, they have limited ranges, post-processing complex- metric scale and widths that range from the hectometer to the ity (Gray and Gartner, 2010; Armijos et al., 2016) and higher kilometer scale. Such large sections experience pronounced maintenance costs due the fragility of the instruments. sediment concentration gradients and grain size sorting, in Thus, sampling methods with instantaneous capture or both the vertical and the transverse directions (e.g., Curtis et short-term integration (< 30 s) are preferred. These meth- al., 1979; Vanoni, 1979, 1980; Horowitz and Elrick, 1987; ods follow a relevant grid of sample points (Xiaoqing, 2003; Filizola and Guyot, 2004; Filizola et al., 2010; Bouchez et Filizola and Guyot, 2004; Bouchez et al., 2011; Armijos et al., 2011; Lupker et al., 2011; Armijos et al., 2013, 2016; al., 2013; Vauchel et al., 2017). The mean concentration hCi Vauchel et al., 2017). The balance between the local hydro- (mg L−1) is determined by combining all samples into a sin- dynamic conditions and the sediment characteristics (e.g., gle representative discharge-weighted concentration value, grain size, density and shape) drive this spatial heterogeneity. which is depth-integrated and cross-sectionally representa- Thus, the sand suspension is characterized by a high vertical tive (Xiaoqing, 2003; Horowitz et al., 2015; Vauchel et al., gradient as well as a significant lateral variability, and the 2017). In the present study, the spatial distribution of the con- concentration varies by several orders of magnitude; in con- centration within the cross section is summarized into a sin- trast, the fine sediments (e.g., clays and silts) are transported gle concentration profile that is assumed to be representative homogeneously throughout the entire river section. of the suspension regime along the river reach. The water As a consequence, the entire cross section must be ex- depth h (m) becomes the mean cross-section depth, which plored to provide a representative estimate of the mean con- is close to the hydraulic radius for large rivers. Therefore, centration of coarse particles. Thus, it is necessary to iden- hCi will be hereafter defined as the depth-integration of this tify a trade-off between the need to sample an adequate num- concentration profile C(z), z (m) being the height above the ber of verticals and points throughout the cross section and bed, from a reference height z0 (m) just above the riverbed

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(z0  h) to the free surface, and weighted by the depth- our knowledge of the sediment delivery problem (Walling, averaged velocity hui = 1 R h u(z)dz (m s−1) (a term list can 1983), these empirical relations deserve hydraulic-based un- h z0 be found in the appendices): derstanding. In this study, the ratios α = hCi/C(zχ ) observed at eight R h × z C(z) u(z)dz gauging stations in the Amazonian foreland were analyzed to hCi = 0 . (1) R h identify the main parameters that controlled their variability. z u(z)dz 0 Assuming that the shape of the concentration profiles mea- However, to dampen the random uncertainties mainly related sured in large Amazonian rivers can be well described us- to the coarse sediments, this procedure requires taking a sta- ing a physically based model for sediment suspension, the tistically significant number of samples throughout the cross possibility of deriving a simple formulation for the ratio α section, which is also time- and labor-intensive. using this model was investigated. This assumption is sup- All of these limitations preclude the application of such ported by previous studies that specifically showed that the complete sampling procedures at a relevant time-step neces- Rouse model (Rouse, 1937) can describe the suspension of sary to build up a detailed concentration series. By analogy sediments in large tropical rivers well (Vanoni, 1979, 1980; with the index velocity method for discharge computation Bouchez et al., 2011; Lupker et al., 2011; Armijos et al., (Levesque and Oberg, 2012), derived surrogate procedures, 2016). However, the Rouse model predicts a concentration of called index sampling methods (Xiaoqing, 2003), are thus zero at the water surface, which is where the index concentra- preferred. One or a few “index samples” are taken as proxies tion is often sampled. To find an alternative, other formula- of hCi, usually at the water surface (e.g., Filizola and Guyot, tions (Zagustin, 1968; Van Rijn, 1984; Camenen and Larson, 2004; Bouchez et al., 2011; Vauchel et al., 2017). The in- 2008) are compared to the data. dex concentration monitoring frequency is then scheduled Then, the relevance of the derived model in terms of de- to suit the river’s hydrological behavior and minimize the veloping a detailed and reliable sediment flux series with an random uncertainties on the measured index concentration index method is discussed; specifically, the ability to accu- −1 C(zχ ) (mg L ) (Duvert et al., 2011; Horowitz et al., 2015). rately estimate the model parameters is evaluated. Finally, The index concentration method first requires a robust site- recommendations for the optimized collection of index sam- specific calibration between the two concentrations of in- ples from large Amazonian rivers are inferred from the pro- terest, hCi and C(zχ ), i.e., for all hydrological conditions, posed model. which cannot always be achieved under field conditions. Such relations are usually expressed with the following lin- ear form (e.g., Filizola, 2003; Guyot et al., 2005, 2007; 2 Materials and methods Espinoza-Villar et al., 2012; Vauchel et al., 2017): 2.1 Hydrological data acquisition hCi = αC(z ) + ξ, (2) χ The hydrological data presented here were collected within where the regression slope α and the intercept ξ are the fitted the international framework of the critical zone observa- parameters of the empirical model. In this study, the inter- tory HYBAM (HYdrogéochimie du Bassin AMazonien – cept will be assumed to be zero (ξ = 0). The dispersion and Geodynamical, hydrological and biogeochemical control of the extrapolation of the α = hCi/C(zχ ) may induce substan- erosion, alteration and material transport in the Amazon tial uncertainties in the matter fluxes (Vauchel et al., 2017). Basin), which is a long-term monitoring program. A Franco- Most of this uncertainty is attributable to C(zχ ) (Gitto et al., Peruvian team, from the IRD (Institut de Recherche pour le 2017), particularly when only a single index sample is taken Développement) and the SENAMHI (SErvicio NAcional de or when a unique sample position is considered. Indeed, the Meteologia e HIdrologia), operates the eight gauging stations relation may change around this position based on the flow of the HYBAM hydrological network in Peru; of these, four conditions. stations control the Andean piedmont fluxes, and four sta- The index sample representativeness becomes crucial as tions control the lowlands (Fig. 1). The three major Peruvian high-resolution imagery is increasingly used to link remote- tributaries of the Amazon (Solimões) River, i.e., the Ucay- sensing reflectance data with the suspended sediment con- ali River, the Marañon River and the Napo River, are mon- centration (e.g., Mertes et al., 1993; Martinez et al., 2009, itored. The studied sites cover drainage areas ranging from 2015; Espinoza-Villar et al., 2012, 2013, 2017; Park and La- approximately 22 000 to 720 000 km2 and have mean dis- trubesse, 2014; Dos Santos et al., 2017). These advanced charges ranging from 2100 to 30 300 m3 s−1 (Table 1). These techniques finally provide a spatially averaged C(zχ ) value large tropical rivers have flows with gradually varied con- for the finest grain sizes at the water surface of a reach (Pinet ditions, unimodal and diffusive flood waves (except for the et al., 2017), which must be correlated with the total mean Napo River), and subcritical conditions, which enable back- concentration transported in the reach of interest (i.e., in- water effects (Dunne et al., 1998; Trigg et al., 2009). cluding the sand fraction when possible) to be a quantita- The Amazonian foreland in Peru has a humid tropical tive measurement (Horowitz et al., 2015). Hence, to improve regime (Guyot et al., 2007; Armijos et al., 2013), and large www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 518 W. Santini et al.: An index concentration method for suspended load monitoring

Figure 1. 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.

Table 1. Hydrologic and sample dataset for the eight sampling stations.

Suspended sediment concentration ranges Particle size distribution Station Site River Basin Mean Period Number Number of Clay and silt Sands Number Number code area discharge considered of concentration (mg L−1) (mg L−1) of PSD of PSD (km2) (m3 s−1) samples profiles profiles samples LAG Lagarto Ucayali 191 000 6700 2011–2017 142 27 9–1340 1–3700 2 10 PIN Puerto Inca Pachitea 22 000 2100 2012–2015 105 27 180–1600 6–2800 0 1 REQ Requena Ucayali 347 000 12 100 2010–2015 213 36 110–1600 5–2300 4 25 BOR Borja Marañon 115 000 5200 2010–2015 130 27 40–1250 2–3400 2 8 CHA Chazuta Huallaga 69 000 3200 2010–2015 141 27 60–1450 5–2330 0 0 REG San Regis Marañon 362 000 18 000 2010–2015 226 39 60–600 5–1600 3 16 TAM Tamshiyacu Amazon 720 000 30 300 2010–2015 223 39 60–960 1–1600 2 21 BEL Bellavista Napo 100 000 7400 2010–2015 150 27 40–340 4–830 1 7 Table Eight sites Amazon 820 000 37 700 2010–2017 1330 249 9–1600 1–3700 14 88 summary amounts of runoff are produced during the austral summer. The El Niño–Southern Oscillation (ENSO) might alter During the austral winter, the maximum continental rainfall these dynamics, as there are severe low-flow events in El is located to the north of the Equator, in line with the in- Niño years and heavy rainfall events in La Niña years tertropical convergence zone (Garreaud et al., 2009). Thus, (Aceituno, 1988; Ronchail et al., 2002; Garreaud et al., the numerous water supplies from the Ecuadorian subbasins 2009). These events seriously affect the sediment routing smooth the seasonality of the Marañon River flow regime. processes (e.g., Aalto et al., 2003), as do other extreme events Located further to the south, the Ucayali Basin experiences unrelated to the ENSO (e.g., Molina-Carpio et al., 2017). a pronounced dry season (Ronchail and Gallaire, 2006; Gar- reaud et al., 2009; Lavado et al., 2011; Santini et al., 2014).

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2.1.1 Sampling strategy rated from the silt/clay fraction (φ = f ) using a 63 µm sieve (cf. Standart Methods ASTM D3977), according to the Went- For the reasons outlined in Sect. 1, local observers mon- worth (1922) grain size classification for noncohesive parti- itor surface index concentrations at each station following cles. The water samples were filtered using 0.45 µm cellulose a hydrology-based scheme. The sampling depth is typically acetate filters (Millipore) that were then dried at 50 ◦C for 20–50 cm below the water surface. The samples are taken 24 h. in the mainstream and at a fixed position. Additionally, HY- Particle size analysis was performed with a Horiba LA- BAM routinely uses MODIS images to determine surface 920-V2 laser diffraction sizer. The entire sampled volume concentrations, and these values are calibrated with in situ was analyzed, with several repetitions demonstrating excel- radiometric measurements (Espinoza-Villar et al., 2012; San- lent analytical reproducibility. For each size group φ, the tini et al., 2014; Martinez et al., 2015). arithmetic mean diameter dφ (m) was calculated: For calibration purposes (i.e., water level vs. discharge and P concentration index vs. mean concentration), 44 campaigns idiXi were conducted during the 2010–2017 period. These cam- dφ = P , (3) Xi paigns included the collection of 494 discharge measure- i ments, 249 sediment concentration profiles and 1330 wa- where Xi is the relative content in the PSD for the class of ter samples. The dataset covers contrasted regimes, includ- diameter dφ. The settling velocities wφ corresponding to the ing periods of extreme droughts (e.g., 2010) and periods diameters dφ derived from the PSD were computed using of extreme floods (e.g., 2012 and 2015) (Espinoza et al., the Soulsby (1997) law, which assumed a particle density of 2012, 2013; Marengo and Espinoza, 2015). Thus, the sam- 2.65 g cm−3. pled concentrations spanned a wide range (Table 1), which represented the river hydrological variability well. A 600 kHz 2.2 Theory for modeling vertical concentration profiles Teledyne RDI Workhorse acoustic Doppler current profiler (ADCP) was used and coupled with a 5 Hz GPS sensor to Schmidt (1925) and O’Brien (1933) proposed a diffusion– correct for the movable bed error (e.g., Callède et al., 2000; convection equation to model the time-averaged vertical con- Vauchel et al., 2017). centration distribution Cφ(z) of grains settling with a velocity −1 A point sampling method was preferred to estimate hCi wφ (m s ). The grain size, shape and density are considered (Filizola, 2003; Guyot et al., 2005; Vauchel et al., 2017) to to be uniform. The equation is expressed as follows: capture the vertical concentration distribution. The sampling ∂Cφ for concentration determination was usually performed at the εφ = −wφCφ, (4) following height (h) from the bed: ∼ 0.98h, 0.75h, 0.5h, ∂z 0.25h, sometimes at ∼ 0.15h and finally at ∼ 0.1h, at three where the term on the left side is the rate of upward con- verticals that divided the cross section according to the river centration diffusion caused by turbulent mixing, balanced by 2 −1 width or the flow rate. Each vertical was assumed to be rep- the settling mass flux in the right term. εφ (m s ) is the resentative of the flow in the corresponding subsection. Sam- sediment diffusivity coefficient that characterizes the particle pling was performed from a boat drifting on a streamline im- exchange capacity for two eddies positioned on both sides of mediately after the ADCP measurements were collected. The a horizontal fictitious plane. εφ is assumed to be proportional ∼ 2 −1 sampler capacity was 650 mL, with a filling time of 10 s, to the momentum exchange coefficient εm (m s ) (Rouse, which allowed for a short time integration along the stream- 1937): line passing by the sample point. Considering the waves at εφ the free surface, the boat’s pitch and roll and the bedforms, = βφ, (5) the accuracy of the vertical position of the sampler may be εm evaluated as ±0.5 m. This variability leads to substantial un- where the βφ parameter is similar to the inverse of a turbu- certainty in the zones with high concentration gradients. The lent Schmidt number (Graf and Cellino, 2002; Camenen and operation time was approximately 2–5 h, depending on the Larson, 2008). It may be depth-averaged (Van Rijn, 1984) river sites. Steady conditions were observed during the sam- or considered to be independent of the height above the bed pling operation. (Rose and Thorne, 2001). Finally, samples for the characterization of the bed mate- The main issue of the Schmidt–O’Brien formulation rial PSD were collected at four sites: BEL, REQ, REG and (Eq. 4) is the expression of the vertical distribution of the sed- TAM. The bed material was dragged on the riverbed. iment mass diffusivity εφ. Once this term is modeled (mod- els are given in the following), Eq. (4) is depth-integrated 2.1.2 Analytical methods from the reference height z0 to the free surface to obtain the expression of the concentration distribution along the water The concentrations Cφ for two main grain size fractions φ column. The concentration Cφ (z0) is then required to deter- were further determined: the sand fraction (φ = s) was sepa- mine the magnitude of the profile and can be evaluated using www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 520 W. Santini et al.: An index concentration method for suspended load monitoring a bed-load transport equation (e.g., Van Rijn, 1984; Came- approximately 7 % lower than those obtained with the Rouse nen and Larson, 2008) or measured directly. However, in the theory (Zagustin, 1968). case of a sampling operation, the large concentration gradi- Van Rijn (1984) proposed a parabolic-constant distribution ent observed near the riverbed would force the operator to for sediment diffusivity, i.e., a parabolic profile in the lower sample water at the reference height z0 with a very high pre- half of the flow depth (Eq. 6) and a constant value in the up- cision to minimize uncertainties; however, achieving such a per half of the flow depth (Eq. 10), which corresponds to the high level of precision is rarely possible. Hence, it is prefer- maximum diffusivity predicted by the Prandtl–Von Kármán able to choose a more reliable reference concentration in the theories. Indeed, some authors have reported measurements interval z = [z0,h]. Thus, the following formulae resulting with constant sediment diffusivity in the upper layers (Cole- from Eq. (4) are written using Cφ(zχ ) instead of Cφ (z0) as a man, 1970; Rose and Thorne, 2001). reference. βφ Building on Prandtl’s concept of mixing length distribu- εφ (z ≥ 0.5h) = κu∗h. (10) tion, O’Brien (1933) and Rouse (1937) expressed the sedi- 4 ment diffusion profile using the following parabolic form: Therefore, for z ≥ 0.5h, the concentration profile is exponen- tial, with a finite value at the free surface:  z  ε z = β κu z − , φ ( ) φ ∗ 1 (6)  Pφ    h C (z) zχ z 1 φ = exp −4P − . (11) −1 Cφ(zχ ) h − zχ h 2 where κ is the Von Kármán constant, and u∗ (m s ) is the shear velocity. This expression leads to the classic Rouse In addition, Van Rijn (1984) introduced a coefficient to ac- equation (Rouse, 1937) for suspended concentration profiles. count for the dampening of the fluid turbulence by the sed- For zχ ∈ [z0,h[: iment particles. This coefficient value is equal to the unity if the sediment diffusion εφ distribution is concentration-  Pφ Cφ (z) zχ h − z independent, which was an assumption used in the present = × , (7) Cφ(zχ ) z h − zχ work because of the range of concentrations measured in the Amazonian lowland rivers (Table 1), and discussed further in where Pφ = wφ/βφκu∗ is the Rouse suspension parameter, Sect. 4.1. i.e., the ratio between the upward turbulence forces and the Camenen and Larson (2008) showed that the depth- downward gravity forces. Pφ is the shape factor for the con- βφ averaged sediment diffusivity ε = κu∗h is a reasonable centration profile. The Rouse formulation is widely used in φ 6 approximation of the Prandtl–Von Kármán parabolic form open channels and suits the observed profiles in the Amazon (Eq. 10) that does not significantly affect the prediction of the River well (Vanoni, 1979, 1980; Bouchez et al., 2011; Armi- concentration profiles in large rivers (P < 1), except near jos et al., 2016). However, the Rouse formulation predicts a φ the boundaries. This simple expression for ε lead to an ex- concentration of zero at the water surface. Three other sim- φ ponential sediment concentration profile: ple models, for which Cφ(h) 6= 0, have been selected in this   work to overcome this problem, i.e., the Zagustin (1968), Van Cφ(z) 6Pφ
= exp zχ − z . (12) Rijn (1984) and Camenen and Larson (2008) models. Cφ(zχ ) h Zagustin (1968) proposed a formulation for the eddy diffu- sivity distribution based on experimental measurements and This profile has practical interest: there is no need to define the reference level z accurately or estimate the correspond- a defect law for the velocity distribution.√ The following vari- 0 able changes were introduced: Z = (h − z)/z, and the sed- ing concentration C0 (Camenen and Larson, 2008). iment diffusivity formulation proposed by Zagustin (1968) is 2.3 A general expression for the ratio α 3 κ  2 2.3.1 Assumptions and formalism εφ (Z) = βφ u∗hZ 1 − Z . (8) 3 Cφ(z) can be expressed by each of the models presented This leads to an expression with a finite value at the water in this section (Eqs. 7, 9, 12) and substituted into Eq. (1) surface: to calculate hCφi. Then, the development of the expression  αφ = hCφi/Cφ(zχ ) would lead to the following equation, Cφ(z)


 = exp Pφ 8 zχ − 8(z) which is similar to Eq. (2), where the parameters driving αφ  Cφ(zχ )      √ are identified: Z3 + 1 (Z − 1)3 √ ! . (9)  1 3Z h i  8 = ln  + 3arctan Cφ z0 zχ   2 Z3 − 1
(Z + 1)3 Z2 − 1 = αφ , ,Pφ,u . (13) Cφ(zχ ) h h As the proposed diffusivity profile is slightly different from However, the PSDs observed in large rivers are rather broad the parabolic form, this expression leads to Pφ values that are (e.g., Bouchez et al., 2011; Lupker et al., 2011; Armijos et

Earth Surf. Dynam., 7, 515–536, 2019 www.earth-surf-dynam.net/7/515/2019/ W. Santini et al.: An index concentration method for suspended load monitoring 521 al., 2016) and may be binned in a range of n grain size frac- zr = 0.5h, the previous expression is simplified: tions φ, as modeling the concentration profiles requires the


exp 3Pφ 1 − exp −6Pφ diameter of sediment in suspension dφ to be almost constant α (z ,P ) = . (17) φ χ φ h −


throughout the water depth if there is not a narrow PSD. As- 6Pφ exp 0.93Pφ 8( 2 ) 8 zχ suming that the interaction between sediment classes φ is negligible, it is possible to apply Eq. (7) and use a multiclass Nevertheless, other formulations might be inferred from the configuration to describe the PSD: suspension models. For instance, the Camenen and Larson formulation could be alternatively used to model Cφ(zχ ) in n X the central region of the flow [0.2h,0.8h], which leads to a α = α X , φ φ (14) simpler expression: φ=1 1  zχ 

where Xφ is the mass fraction of each grain size fraction αφ(zχ ,Pφ) = exp 6Pφ 1 − exp −6Pφ . (18) 6Pφ h measured for the index sample with the concentration Cφ(zχ ) n P ( Xφ = 1). Moreover, it can be shown that the weight of 2.4 Model fitting strategy i=1 the velocity distribution on the depth-averaged concentra- To obtain a reach-scale profile, the fit to the concentrations tion may be neglected in Eq. (1) when the suspension occurs averaged at each normalized depth z/h was assessed. It was throughout the water column, i.e., when Pφ < 0.6 . Thus, if
assumed that the energy gradient, the mean bed roughness Pφ < 0.6, it is possible to express αφ z0/h,zχ /h,Pφ . factor and the mean diameter did not significantly change A key issue is then to provide a proper model of the PSD from one subsection to another, even if the point-to-point using a limited number of sediment classes. In this study, the variability was high (Yen, 2002). Thus, the depth becomes available dataset provides concentrations for fine particles the main factor influencing the Pφ in the transverse direction. (0.45 < df < 63 µm) and sand particles (ds ≥ 63 µm). Then, In the cross sections studied here, the variation in depth from the ratio α may be formalized as follows: one vertical to the next was not sufficient to significantly in- fluence the Rouse number. C (z ) is then the average of sev- α = X α + X α . (15) φ χ f f s s eral representative samples taken across the river width at the same relative height (z /h). Thus, if at height zχ the mass fraction of each group is accu- χ rately known after sieving, α may be calculated for the whole After the first data cleaning of the sampled points, a ro- PSD. bust and iteratively re-weighted least squares regression tech- nique was used to minimize the influence of the outlier val- ues. The weight values (W) between z and h were assigned 2.3.2 Model proposed for the ratio α prediction in 0 φ with the following parabolic function, similar to the eddy dif- Amazonian large rivers fusivity expression (Eq. 6): W (z) = z(1 − z). Thus, the half- The depth-integration of the Camenen and Larson formula- depth point, where the mixing term is the highest, has the tion (Eq. 12) is considered to be a reasonable approximation largest influence. of the measured hCi in large rivers (Camenen and Larson, Based on the ADCP velocity profile measurements, the −3 2008), with a simple expression that is independent of the z0 parameter z0 was fixed at z0 = 10 h. Indeed, when z0 < −2 term, which differs from the other theories presented above. 10 h, hCi is no longer sensitive to z0 (Eq. 1), even if the Moreover, in the next section, the fit of the suspension mod- Rouse number is not accurately known (Van Rijn, 1984). els to the measured concentration profiles will show that the Hence, (h − z0)/(z − z0) ≈ z/h can be assumed when con- Zagustin model provides the best fit to the observations, par- sidering reach-scale flow conditions. ticularly in the upper layer of the flow. Thus, in this work, Cφ(zχ ) will be expressed using the Zagustin model (Eq. 9), 2.5 Shear velocity estimation from ADCP transects and hCi will be expressed using the Camenen and Larson model. The velocity transects measured with an acoustic Doppler Because the Zagustin model causes the Rouse number current profiler (ADCP) were used to estimate the shear ve- 0 locities u∗ from the vertical velocity gradient through the fit (Pφ) to be slightly smaller than that calculated with the Rouse 0 ≈ of the logarithmic inner law (e.g., Sime et al., 2007; Gualtieri model (Pφ 0.93Pφ, according to Zagustin, 1968), we ob- tain the following expression for predicting the ratio αφ: et al., 2018). An average of 30 ADCP “ensembles” (i.e., mea- surement verticals of velocity), corresponding to about 40 to zr


exp 6Pφ 1 − exp −6Pφ 70 m in the cross-sectional direction, were required to obtain α (z ,P ) = h , (16) φ χ φ


robust u∗ values. This was consistent with the methodology 6Pφ exp 0.93Pφ 8(zr) − 8 zχ applied by Armijos et al. (2016) (50–60 ensembles, corre- where zr is a reference height required for expressing Cφ(zχ ) sponding to 10 % of the total width of the section) or Lup- with the Zagustin model (zr replaces zχ in Eq. 9). Taking ker et al. (2011) (30 ensembles, 40 to 70 m). Following these www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 522 W. Santini et al.: An index concentration method for suspended load monitoring

findings, the velocity profiles were averaged over a spanwise length of about 60 m around each concentration profile posi- tion. Then, an average of the fitted shear velocities was cal- culated 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 sec- tion. Furthermore, the imprecise knowledge of the exact bed el- evation, 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 inac- curate in the inner flow region (i.e., the region of the flow under bed influence ∼ [z0,0.2h]). However, a fit over the en- tire 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 veloc- ities were assessed in the zone between 0.1h and 0.85h.

Figure 2. Observed ratios α = hCi/C(h) of the total mean con- 3 Results centration to the total surface concentration, stacked by river basin, with trend lines. For the Amazon River basin at TAM (a), the REG 3.1 Data analysis and REQ trend lines were reported. Dashed lines denote the first bisector. 3.1.1 Index concentration relations calibrated for surface index samples The observed α ratios for total concentration (i.e., concen- most 55 % of the water discharge and could significantly con- tration including fine particles and sands) were calculated tribute to the river sediment load. The Napo River example for a surface index using each field measurement carried out (Laraque et al., 2009; Armijos et al., 2013) shows that the (Fig. 2). Empirical relationships for estimating the total mean lowland part of the basin can be the main sediment source for concentration from the surface index samples (Eq. 2) were these Ecuadorian tributaries. The river incision of this sec- calibrated using these data . The α ratios observed at the three ondary source, and/or the Ecuadorian Andes, could provide stations monitoring the Ucayali Basin fluxes (i.e., LAG, PIN, coarser elements than the central Andean source does and ex- and REQ) are similar (1.3 < α < 1.5) (Fig. 2). At BEL (Napo plain why the ratios α are higher at BEL and REG than at the River), the α values observed are higher (α =∼ 1.7). Con- other sites. versely, different trends with larger scatter are observed in The concentration dataset highlights the control of the ∼ the Marañon Basin. The α ratios observed at REG (α = 2.3) sand mass fraction Xs on the ratio α (Fig. 3): α increases ∼ ∼ are higher than those at BOR (α = 1.5) and CHA (α = 1.4), with Xs. If finest particles of the PSD are dominant in the which are similar to those at LAG, PIN and REQ (Ucayali index concentrations Cχ sampled in the upper layers of the Basin). However, the observed α values at BOR fluctuate be- flow, this wash load is supply-limited, depending on the mat- tween two main trends, which are represented by the CHA– ter availability, rainfall upon the sources and sediment en- LAG–PIN–REQ group and the BEL–REG group. At TAM, trainment processes occurring on the weathered hillslopes. similarly, the α values rather follow the REG trend at low Wash load is then routed through the foreland without im- concentrations before evolving between the REQ and REG portant mass fluctuations (e.g., Yuill and Gasparini, 2011). trends. Significant exchanges between the floodplain and main chan- This variability suggests that the α ratio is site-dependent nel lead to some dilution but also to some remobilization and potentially variable with the flow conditions. It could re- of the huge floodplain sediment stocks of the coarser ele- flect differences in the basin characteristics (e.g., lithology ments that were previously deposited (e.g., clay aggregates, and climate spatial distributions), then in sediment sources silts and fine sands). Conversely, the sand transport regime (e.g., mineralogy and PSD) and could relate to the sediment is capacity-limited, depending only on the available energy routing in the lowland. The first group (CHA–LAG–PIN– to route the sediments. As the flow energy significantly de- REQ) could be representative of a same source of sediments creases with the decreasing bed slope, the sand suspended (the Central Andes), as few lateral inputs come swelling load is gradually decoupled from the wash load in the flood- these rivers discharges in the lowland (Guyot et al., 2007; plain, and the wash-load concentration is no longer a good Armijos et al., 2013; Santini et al., 2014). Conversely, in proxy of the coarse particle concentration. The floodplain in- the Marañon lowland, the Ecuadorian tributaries supply al- cision mechanically increases the sand mass fraction Xs in

Earth Surf. Dynam., 7, 515–536, 2019 www.earth-surf-dynam.net/7/515/2019/ W. Santini et al.: An index concentration method for suspended load monitoring 523

ing throughout the water column, except near the air–water interface, where the calculated df tended to decrease. In con- trast, 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 suit- able 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 sin- gle mean diameter dφ (m) per site for each size group φ.

Figure 3. Observed ratios α = hCi/C(h) of the total mean concen- 3.2 Suspension model suitability with the measured tration to the total index concentration sampled at the water surface profiles vs. the sand mass fraction Xs. The suspension models (Eqs. 7, 9, 11, 12) were fitted to the concentration data to evaluate their suitability to the the suspended load. Implicitly, the PSD mean diameter shifts observed profiles. The dataset confirmed that the Zagustin 0 with Xs, but it does not mean that there is any change in model causes the Rouse number (Pφ) to be slightly smaller 0 ≈ the physical properties (e.g., diameter, density and shape) of than that calculated using the Rouse model: Pφ 0.93Pφ. the sand fraction. This shift directly affects α, as the vertical The fitted Pφ values showed low variability and were sum- concentration gradient depends on the balance between the marized by single average values per site, as shown in Ta- turbulence strength and the settling velocity (Eq. 4). This re- ble 2. This low variability indicates that there is a dynamic
sult highlights the key challenge of providing a proper model equilibrium between the settling velocity wφ dφ and the of the PSD using a limited number of sediment classes, and shear velocity u∗ under nominal flow conditions, although validates the discrete approach proposed to model α. some extreme values (Pφ > 0.5) were measured during se- vere drought events at the lowland stations. The Rouse numbers obtained for the fine fraction reflect 3.1.2 Particle size profiles a suspension regime that is close to the ideal wash load The measured particle size distributions (PSDs) show a mul- (Pf < 0.1, 1 ≤ αf(h,Pf) ≤ 1.5). Additionally, regarding the timodal pattern (Fig. 4a). This example of a global PSD that Rouse numbers corresponding to the sand fraction, they re- includes the entire particle size range was deconvoluted, as- flect a well-developed suspension in the entire water column suming a mixture of lognormal subdistributions (e.g., Mas- for the piedmont station group (0.2 < Ps < 0.3) and for the son et al., 2018). On the left side of the PSD, a weak lognor- lowland station group (0.35 < Ps < 0.45), with a significant mal mode was detected in the clay range, but it was negligible concentration gradient (2.3 ≤ αs(h,Ps) ≤ 7.5). in comparison to the silt volume. A fairly uniform fraction of Due to the availability, for a given site, of a single mean ∼
fine sands (ds = 80 µm) that were transported in suspension value of wφ dφ per size group, only the corresponding throughout the water column with a nearly constant mode mean values of the diffusivity ratio βφ = wφ/Pφκu∗ were over depth was identified. This fraction approximately corre- calculated, considering the mean shear velocities (Table 2). sponds to the diameters less than the 10th percentile of the d =∼ riverbed PSD. A second sand class ( s 200 µm) is trans- 3.2.1 Sediment diffusivity profiles ported as graded suspension with a strong vertical gradient limited to the lower part of the water column (z/h < 0.2). The diffusivity profiles εφ(z) were derived from the mea- The Rouse number Pφ varies from one to six for this class of sured concentration profiles using the discrete form of sediments, suggesting that bed load may be non-negligible. Eq. (4). In order to capture the small variations in εφ, ac- However, the concentration dataset does not contain any bed- curate sampling is key: the calculation of εφ requires precise load sample close enough to the riverbed and taken with a concentration and sampling height values, particularly for the relevant integration time to assess this argument. fine fraction, which experiences low vertical concentration Concerning the whole dataset of fine sediment mean di- gradients. ameters df (Fig. 4b), no vertical gradient was observed for Nevertheless, the overall shapes of the derived εφ (z) pro- fine sediments, indicating that there was homogeneous mix- files were in good agreement with the Rouse and Zagustin www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 524 W. Santini et al.: An index concentration method for suspended load monitoring

Figure 4. (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. theories and were slightly closer to the latter (Fig. 5). Given the secondary currents can influence the turbulent mixing the high scatter of the diffusivity values, Camenen and Lar- profiles. son’s expression of depth-averaged diffusivity is a reason- able approximation, except near the bottom and top edges of the diffusivity profiles, where the data departs gradually from 3.2.2 Concentration profile suitability this model. However, the constant diffusivity value suggested Overall, the suspension models (Eqs. 7, 9, 11, 12) fit well by Van Rijn (1984) for the upper half of the water column with the observed profiles (Fig. 6): for 92 % of the profiles fit- clearly overestimates the diffusivity for z > 0.75h. The dif- ted, the coefficients of correlation (r) were superior to 0.9 % = fusivity around z 0.75h is, however, overestimated by all and 100 % of the r were superior to 0.7, except near the edges the models. The low concentrations near the water surface where the highest discrepancies between the two exponential ≈ could result in an underestimation of the εφ(z 0.75h) val- expressions (Van Rijn, 1984; Camenen and Larson, 2008) = ≈ − ues calculated from the difference 1Cφ Cφ (z 0.5h) and the Rouse and Zagustin models appear. The concentra- ≈ Cφ (z h) (Eq. 4). Thus, detailed measurements are required tions sampled at the bottom edge confirmed the general shape in the upper layer of the flow to confirm the shapes of the of the Zagustin and Rouse models, despite the uncertainties εφ (z) profiles in this zone where the air–water interface and in the concentrations measured in this zone. Near the water

Earth Surf. Dynam., 7, 515–536, 2019 www.earth-surf-dynam.net/7/515/2019/ W. Santini et al.: An index concentration method for suspended load monitoring 525 h 5 . 0 = s χ α h z = s χ z α ) Fitted Fitted ∗ κu s P ( / s s w β ) z ( C 7 % – 2.6 1.0 9 %12 %7 % 1.7 1.116 % 0.85 0.6 %5 % 3.5 2.38 % 0.6 6.4 0.8 1.1 4.3 1.0 2.1 1.2 1.1 7.5 8.0 4.3 1.2 1.2 1.2 ± ± ± ± ± ± ± ± 24 27 23 39 28 44 44 32 s ...... P 0 0 0 0 0 0 0 0.33 1.1 4.8 1.1 ) 2 2 2 2 2 2 2 2 1 Mean results for the sand fraction − − − − − − − − − 10 10 10 10 10 10 10 10 × × × × × × × × (m s 8 4 2 3 4 7 8 6 s ...... w

4 %2 %7 % 1 10 % 1 1 7 % 1 10 %19 % 1 1 2 Figure 5. Dimensionless sediment diffusivity coefficient derived ± ± ± ± ± ± ±

(µm) from the measured concentration profiles. s d 148 124 114 118 –132 141 192 138 –Derived frommeasured PSD Soulsby law 1 Fitted on 0 (Rouse model) on obs. on obs. h 5 . 0 = f χ surface, the nonzero values predicted by the Zagustin model α were often the closest to the observed concentrations. h z

= The use of the Camenen and Larson model to calculate the f χ α z mean concentration hCφi seems to be a reasonable approxi- ) Fitted Fitted

∗ mation. Indeed, for the range of nominal Rouse numbers con- κu f sidered here (Pφ < 0.6), the bottom concentration gradient P ( / f f has little influence on hCφi because the velocity decreases w β rapidly with depth in this region of the flow. Moreover, the ) z ( top-layer concentrations are too low to weight significantly C 16 % 0.2110 % 1.1 0.06 1.0 1.4 1.0 14 %8 % 0.2417 % 0.14 0.1714 % 1.016 % 1.2 0.07 1.0 1.0 0.15 1.0 1.0 1.5 1.2 1.0 1.0 19 % – 1.1 1.0 on hCφi. ± ± ± ± ± ± ± ± The comparison between the predicted and observed mean 03 07 01 05 02 08 04 01 f ...... 0 0 P 0 0 0 0 0 0.04 0.16 1.2α 1.0 φ values per site and size group (Fig. 7a) allows for the val- ) 4 4 4 4 4 4 4 4 1 − − − − − − − − idation of the general model proposed in this work (Eq. 17). − 10 10 10 10 10 10 10 10 Mean results for the fine particle fraction To show the model’s ability to predict how α changes with × × × × × × × × φ (m s 5 2 9 5 5 8 5 4 f ......

w flow conditions at one specific site, this model was also com- pared with all of the αφ values observed at the water sur- face and at mid-depth (Fig. 7a). The observations follow the 3 % 2 3 % 2 3 %4 %4 % 1 2 2 4 %2 % 2 2 model trend well, despite the high scatter of the αs(h,Ps) val- ± ± ± ± ± ± ± (µm)

f ues, which is caused by the low diffusivity and concentration d in coarse material near the water surface and by the uncer- tainty of the exact z position of the samples. At mid-depth, ) h ( s the αs(0.5h,Ps) values have lower scatter. X Nevertheless, the αs sensitivity to the Rouse number re- ) Measured Derived from Soulsby Fitted on z ( )

u mains moderate for most of the hydraulic conditions encoun- 1 − tered, except for the extremely low flow rates, i.e., when (m s

∗ Ps > 0.5. The αf sensitivity to changes in flow conditions is u (log-law) measured PSDvery law small. (Rouse model) Then, considering on obs. on obs. the small contribution of the low waters to the sediment budget and the small Rouse num- (m)

h ber variations for the nominal hydraulic conditions at a spe- R Summary of suspended sediment transport parameters for each site and size group. cific site (Table 2), the use of the mean αφ coefficients per site seems to be reasonable for assessing reliable sediment bud- REG 16.7 0.14 14 % 15 Station LAG 6.8 0.10 15 %Method Measured Fitted 16 on code PINREQBOR 7.2 12.5CHA 7.9 7.4TAM 0.14 0.09BEL 18.6 0.17Mean 0.20 10.1 10.9 20 % 0.12 7 % 0.10 10 % 0.13 16 14 % 16 6 % 16 – 17 % 13 % 17 gets. 19 16 Regarding – 2 0 the simplified model (Eq. 18), a reasonable Table 2. www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 526 W. Santini et al.: An index concentration method for suspended load monitoring

Figure 6. Typical examples of measured concentration profiles Cφ(z/h), fitted with the Rouse, Van Rijn, Zagustin, and Camenen and Larson models. approximation is expected to be found in the central region a first limitation for the model applicability, if a monitoring of the flow, but the values gradually depart from the observa- of the index concentration at a deeper level on the water col- tions near the water surface and the riverbed. umn is not technically feasible. In addition, for rivers with Finally, the mean ratios α(h) per site were computed Rouse numbers greater than 0.6 (i.e., when the suspension (Eq. 15) using the predicted mean αf(h,Pf) and αs(h,Ps) does not occur in the entire water column because the parti- (Eq. 17) and the mean mass fractions Xf and Xs measured cle are too coarse, in comparison to the strength of the flow, at the water surface (Table 2). The observed vs. predicted α to be uplifted at the water surface), the weight of the velocity ratios are in excellent agreement (r2 = 0.97) (Fig. 7b) and distribution in the model can no longer be neglected as it was validate the prediction ability of the model when the Rouse in this work (see the assumption, Sect. 2.3.1). Furthermore, numbers are accurately known. the higher the Rouse number, the more difficult the concen- tration measurement is to perform. Then, the accuracy of the model also depends on the concentration measurement pro- 4 Discussion on the model applicability cedure chosen and related uncertainties. These uncertainties depend on the point-sampling integration-time which must The equations proposed in this work (Eqs. 17, 18) for mod- be long enough to be representative (Gitto et al., 2017), on eling the ratios αφ(zχ ,Pφ) become very sensitive when both the volume of water collected and on the sampling position(s) ≈ the index sample is taken near the river surface (zχ h) and defined in the cross section. the Rouse number is rather large (Pφ > 0.4) (Fig. 7a). This is

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φ Figure 7. (a) Predicted and observed α i 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).

Therefore, estimating Pφ with a low uncertainty is a key the ratio wφ/u∗ for sand and silt particles. However, the mea- issue to predict accurate αφratios during sediment concen- sured βφ encompasses poorly understood physicochemical tration monitoring. This estimation can be achieved (1) via processes as well as uncertainties and bias of the wφ and u∗ the estimation of the hydraulic parameteru∗, wφ and βφ, or estimations, which might partly explain the shifts along the (2) empirically using detailed point concentration measure- wφ/u∗axis between the three abovementioned laws and the ments. Then, if the Rouse number variability is significant βφ inferred in this study from measured profiles of concen- during the hydrological cycle, the empirical relationship be- tration, particle diameter and velocity (Fig. 8a). tween u∗ or h and the Pφ fitted on measured concentration With regard to wφ, a major difficulty comes from the profiles may be calibrated. need to divide the PSD into various size groups and to sum- marize each subdistribution with a single characteristic di- ameter (e.g., mode, median, mean), and different values of 4.1 Estimation of the diffusivity ratio βφ wφ(dφ) are calculated according to the choices made.The For many decades, studies based on flume experiments or aggregation process is a supplementary complicating factor measurements in natural rivers have shown that βφ usually (Bouchez et al., 2011) but is probably not the main issue in departs from the unity. The sediment diffusivity increases these white rivers with little organic matter (Moquet et al., (βφ > 1) with bedforms or movable bed configurations (Graf 2011; Martinez et al., 2015). Indeed, the results of Bouchez and Cellino, 2002; Gualtieri et al., 2017); specifically, the et al. (2011) are probably biased because the authors used a boundary layer thickness tends to be thin just before the bed- single diameter to summarize the entire PSD, which is highly forms crest, and then it peels off at the leeward side (En- sensitive to the flow conditions. However, this bias would not gelund and Hansen, 1967; Bartholdy et al., 2010). This trend concern the sand group because the shear modulus experi- implies there are anisotropic macroturbulent structures, with enced in large Amazonian rivers would prevent the forma- eddies that convect large amounts of sediments to the upper tion of large aggregates. The choice of a settling law (e.g., layers and settle further after eddy dissipation. Thus, bed- Stokes; Zanke, 1977; Cheng, 1997; Soulsby, 1997; Ahrens, forms locally modify the ratio between the laminar and tur- 2000; Jiménez and Madsen, 2003; Camenen, 2007) may also bulent stresses, inducing different lifting profile shapes in the induce bias on wφ. In these laws, the sediment density is a inner region (e.g., Kazemi et al., 2017) and causing the mix- key parameter that is often neglected, as natural rivers com- ing length theory to fail in the overlap region. Centrifugal prise a diversity of minerals with contrasting density ranges. forces driven by turbulent motion and applied on the grains Conversely, the shear velocity estimation also suffers from could also enhance the particle exchange rate between eddies uncertainties in terms of the velocity measurements and bi- (Van Rijn, 1984). Conversely, the suspension is dampened ases that are induced by the method used (Sime et al., 2007). (βφ < 1) when the large suspended particles do not fully re- For instance, the departures from logarithmic velocity pro- spond to all velocity fluctuations, such as passive scalars. files increase with the distance to the bed (e.g., Guo et Julien, Van Rijn (1984), Rose and Thorne (2001) and Camenen 2008) in sediment-laden flows (e.g., Castro-Orgaz et al., and Larson (2008) attempted to model βφ as a function of 2012), which could be relevant to deep Amazonian rivers. In- www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 528 W. Santini et al.: An index concentration method for suspended load monitoring

Figure 8. (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.

deed, the mixing length expansion could reach a maximum sional law, which extends below the range of wφ/u∗ usually before the water surface, as the energetic eddy size cannot considered in previous studies, allows for an enhanced pre- expand ad infinitum far from the flow zone under the influ- diction of βφ(±0.03) (Fig. 8b). ence of bed roughness because of the increasing entropy. The Furthermore, the dataset did not show any relationship be- log-law assumptions (i.e., constant shear velocity throughout tween concentration and the diffusivity ratio βφ (not shown the water column and mixing length approximation) would here). The uncertainties in the dataset collected under field no longer be valid, and the velocity profiles would follow a conditions do not allow for the further investigation of the in- defect law in the outer region. This raises the need to find fluence of second-order factors on the diffusivity ratio, such a suitable model for the velocity distribution in large rivers, as the particle characteristics (shape, grain size, density, and leading to an unbiased estimate of the shear velocity. so on), the aggregation phenomenon or the level of stratifi- Thus, it is not surprising to find discrepancies between the cation of the flow (e.g., Van Rijn, 1984; Graf and Cellino, empirical laws and the observations based on the experimen- 2002; Pal and Ghoshal, 2016; Gualtieri et al., 2017). tal conditions. Here, the Rose and Thorne (2001) empirical Applying Eq. (19) to predict the mean βφ values per site, law is the closest to the observed βφ (Fig. 8a), with depar- the predicted and fitted Pφ are in good agreement (Fig. 9a), tures that seem to be a function of the water level. We assume with little scatter when considering the uncertainties in the that a global correction of the different bias on the wφ/u∗ measured concentrations and therefore on the fitted Pφ. This term would depend on the flow depth as well as on the skin result shows that the shear velocity mainly controls the Rouse roughness, which partly influences the formation and expan- number variability at a given site (Fig. 9b). Therefore, the sion of the turbulence structures and thus influences the ve- variations in particle size are a second-order factor. The shear locity distribution (Gaudio et al., 2010). Here, ds is consid- velocity is itself driven by the high amplitude of the water ered instead of the skin roughness height, as few riverbed depth in Amazonian rivers (Fig. 9c), and it has hysteresis ef- PSDs are available and because it is a key parameter for the fects at the gauging stations located in the floodplain, which settling law. Thus, the following modification of the Rose and are attributed to the backwater slope variability in these sub- Thorne (2001) law is proposed: critical flood wave contexts (Trigg et al., 2009). Hence, the accurate monitoring of the water level and knowledge of the "  0.6# −3 u∗ h river surface slope, even if limited or biased, would allow for βφ = 3.1exp −0.19 × 10 + 0.16, (19) wφ ds an acceptable prediction of the Rouse numbers, which could be used to establish a single βφ value per site. 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 4.2 Predicting ds from the riverbed PSD the measured concentration profiles (Table 2). In a similar way to Camenen and Larson (2008), a minimum βφ-value For fine particles, df can be accurately measured in the water was found for very small values of wφ/u∗. This nondimen- column because the fine particles are well mixed in the flow.

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Figure 9. 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.

Regarding the sand particles, such measurements induce zones with a high concentration gradient. Overall, the relative uncertainties due to the particle fluctuations in the current error on αφ remains moderate for all of the flow conditions and because the eddy structure development in the bottom experienced by the rivers studied here (i.e., below ±10 % in layers of the flow swiftly causes strong grain size sorting the central zone of the flow and below ±20 % at the water (Fig. 4). The suspended sediment particles are thus consid- surface), except near the riverbed (Fig. 11a). Nevertheless, erably smaller than the bed load or riverbed particles (Van for operational applications, this result must be weighted by Rijn, 1984). the relative error profile of the measured index concentrations The diameter of the suspended sand can be assessed by 1Cφ(zχ )/Cφ(zχ ). taking a representative percentile of the riverbed PSD (e.g., By substituting Cφ(zχ ) with the Zagustin model (Eq. 9) Rose and Thorne, 2001). Alternatively, an empirical expres- and assuming 1Cφ(0.5h)/Cφ (0.5h) = ±10 %, it is possible sion that considers the flow conditions was proposed by Van to model this profile of concentration uncertainty (Fig. 11b) Rijn (1984). and to derive the relative uncertainty of hCφi according to Here, the Camenen and Larson (2005) formulae for the es- the sampling height under various flow conditions (Fig. 11c). timation of the reference concentration Cφ(z0) was applied in Here, the considered uncertainty is a simple function of the a multiclass way to the riverbed PSD, and it was assumed that concentration. However, coarse particles are more sensitive the size fractions did not influence each other and there was to current fluctuations than are fine sediments. Thus, the sand a uniform sediment density for all grain sizes (2.65 g cm−3). concentration uncertainty is underestimated, at least in the In this formulation, Cφ(z0) is a function of the dimension- region of the flow under bed influence ∼ [z0,0.2h], where less grain size d∗, the local Shields parameter θφ and of the stochastic uplifts of bed sediments impose high variability critical Shields parameter θcr for the inception of transport on the concentration. Furthermore, the sampling frequency (Camenen et al., 2014): as well as the number of index samples taken and their po- sitions are important parameters to consider. The section ge- 0.0015θφ Cφ(z0) = . (20) ometry, the velocity distribution and the transversal movable  θ  exp 0.2d∗ + 4.5 cr bed velocity pattern are important guidelines in the selec- θφ tion of a sampling position(s). The integration of the lateral This first PSD predicted at the transition level z0 is further variability of the concentration is not discussed here. Never- diffused vertically with the Zagustin model (Eq. 19), consid- theless, when considering these assumptions, optimized sam- ering the Soulsby (1997) settling law in the Pφ calculations pling heights may be defined as follows: (Fig. 10a). The model underestimates the measured d by ap- s h i proximately 10 % (Fig. 10b). This slight discrepancy might – For fine sediments (Pf < 0.1), the most accurate Cf is be explained by stochastic and ephemeral inputs of coarse obtained when sampling the water column at approxi- bed material in the water column, which are not addressed mately 0.5h. The sampling can also be achieved at the h i ± by the suspension theory. water surface with a good estimation of Cf ( 15 %).

– For the sand fraction at the piedmont stations (Ps < 4.3 Sensitivity analysis and recommendations for 0.3), sampling in the [0.2h,0.8h] region is recom- optimized sampling procedures mended to keep the errors of hCsi below ±20 %. Sam- pling at the water surface is still possible, but there will The approximation error 1P can be evaluated at ±0.03 φ be uncertainties between ±20 % and 40 % for hC i. (from Eq. 19, Fig. 9 or Table 2) and propagated to the corre- s sponding αφ(zχ ,Pφ ±1Pφ) (Fig. 11a). The error on zχ is not – For enhanced monitoring of the sand concentration at considered here but would increase the αφ sensitivity in the the lowland stations (Ps > 0.3), the [0.2h,0.8h] zone is www.earth-surf-dynam.net/7/515/2019/ Earth Surf. Dynam., 7, 515–536, 2019 530 W. Santini et al.: An index concentration method for suspended load monitoring

−1 Figure 10. (a) Prediction of the mean diameter ds at the TAM gauging station for u∗ = 0.12 m s (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.

Figure 11. (a) Relative error of the predicted αφ according to the relative height zχ /h of the index sampling, for various Rouse numbers 1C P . (b) Relative error of the concentration sampled, inferred from the Zagustin model and assuming φ (z = 05h) = ±10 %. (c) Relative φ Cφ χ error on hCφi as a function of zχ /h.

preferred over the water surface, where the αφ predic- When considering nominal flow conditions (Pφ < 0.6), tion would require very accurate estimations of Ps and the sampling height above the riverbed h/e ≈ 0.37h (e be- Cs(h). ing the Euler number) appears to be pertinent for simpli- fied operations, as the ratios of α (h/e,P ) remain inter- The proposed α models (Eqs. 17, 18) allow for a routine φ φ φ estingly close to unity (±10 %) (Fig. 7a). Thus, {β ,β } ≈ protocol with sampling in the central zone of the flow to be f s {0.16,1} could be simply assumed without inducing large er- achieved: the α can be predicted at each sampling time-step, φ rors in the α (h/e,P ) estimations. The particles are usually when the section geometry is known and when the flow is φ φ present in a significant amount, and the turbulent mixing is sufficiently stable to estimate z /h. For instance, a single χ intense (Eq. 6), while the concentration gradients are mod- fixed sampling depth could be used. Alternatively, the Rouse erate, which also causes for more uncertainty regarding z . number can be estimated at each sampling time-step from χ Interestingly, when considering the depth-averaged velocity Eq. (7) by sampling two heights z and z of the water χ1 χ2 hui ≈ u(h/e) for velocity profiles that are logarithmic in na- column at each measurement time-step: −1 −2 ture, the sediment discharge on a vertical qsφ (g s m )  C z
 may be expressed as follows: ln φ χ1 C z
φ χ2       Pφ = . (21) h h ∼ h  z h−z
 qsφ = αφ ×Cφ ×u = Cφ ×hui±10%. (22) ln χ2 χ1 e e e zχ −
1 h zχ2 Finally, if sampling in the central zone of the flow is not tech- For instance, the concentration at zχ = 0.7h and at zχ = 1
2 nically feasible during concentration monitoring, the mean 0.3h results in Pφ = 0.59ln Cφ (0.3h)/Cφ (0.7h) . concentration of fine particles may be estimated with sur-

Earth Surf. Dynam., 7, 515–536, 2019 www.earth-surf-dynam.net/7/515/2019/ W. Santini et al.: An index concentration method for suspended load monitoring 531 face index sampling or remote sensing (Martinez et al., 2015; This insight into the hydraulic theory leads to enhanced Pinet et al., 2017). Then, the sand concentration could be as- sediment monitoring practices, with a more accurate estimate sessed with a sediment transport model that is suitable for of the sediment load, especially in regions with limited data large rivers (e.g., Molinas and Wu, 2001; Camenen and Lar- availability, such as the Amazon Basin. Indeed, the proposed son, 2008). To parameterize such models, improved space- model is a tool that can be used to predict the αφ and α ratios borne altimeters (e.g., the SWOT – Surface Water Ocean To- and can also be used to select a proper sampling height for pography – mission) and hydrological models are already se- optimized monitoring. Extensively, the model allows for de- rious alternatives to in situ discharge, water level and slope tailed uncertainty analysis on the hCi derived from an index measurements (e.g., De Paiva et al., 2013; Paris et al., 2016). method. Finally, where the cross-section geometry is well known 5 Conclusion and perspectives and where no in situ concentration data exist, the model could allow for an accurate estimation of the mean concentra- The use of measured concentration profiles with physically tion in fine sediments hCfi, with remote-sensing monitoring based models describing the suspension of grains in turbu- of the index concentration in fine sediments on the water sur- lent flow has shown the possibility to derive a simple model face. Coupling this monitoring with a sand transport model for the prediction of αφ, i.e., for a given particle size group φ. suitable for large rivers could ensure a better understanding Proper modeling of the PSD using two hydraulically consis- of the sediment dynamics in the Amazon Basin. tent 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. Data availability. The data that support the findings of this study The Zagustin profile, with finite values at the water sur- are available from the following data repository (Santini et al., face, demonstrated the best suitability in relation to the ob- 2018): https://doi.org/10.6096/DV/CBUWTR. Extra data (water levels, suspended concentration time series etc.) are also avail- served data. Nevertheless, the Camenen and Larson model able from the corresponding author upon request and on the CZO was in good agreement with the observations in the central HYBAM website: http://www.so-hybam.org (last access: 1 March zone of the flow and was a reasonable approximation of the 2019). depth-averaged concentration. The Rouse number is the main parameter for αφ model- ing. 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 bound- aries to a zone between [0.2h,0.5h] which is based on the flow conditions. At the water surface, the model becomes in- accurate when Ps > 0.3, i.e., for flow conditions correspond- ing 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.

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Appendix A

A1 List of notations

.f Fine sediment particles group (0.45 µm < df < 63 µm) .s Sand sediment group (ds > 63 µm) C Time-averaged concentration (mg L−1) d Arithmetic mean diameter (m)   ! 1 g ρ −1 3 ρw d∗ d is the dimensionless grain size υ2 g Gravitational force (m s−2) h Mean water depth (m) ks Nikuradse equivalent roughness height (m) P Rouse number (–) −1 −2 qs Time-averaged sediment discharge on a vertical (g s m ) u Time-averaged velocity (m s−1) −1 u∗ Shear velocity (m s ) w Suspended sediment particle settling velocity (m s−1) X Mass fraction (–) z Height above the bed (m) α Ratio between mean concentration and index concentration (–) β Ratio of sediment to eddy diffusivity (–) ε Sediment diffusivity coefficient (m2 s−1) 2 −1 εm Momentum exchange coefficient (m s ) κ Von Kármán constant (–) υ Kinematic viscosity (m2 s−1) −3 ρw Water density (kg m ) ρ Sediment density (kg m−3) θ Shield’s dimensionless shear stress parameter (–) θcr Critical dimensionless shear stress threshold (–) hi Depth-integrated value

Main subscripts

.χ Index height value .0 Bottom reference height value .r Reference height value .φ Particle size group φ

A2 Soulsby (1997) settling law (terminal velocity) ν q  w = 10.362 + 1.049d3 − 10.36 . d ∗

A3 Velocity laws Inner law (“law of the wall”):

u∗ 30z u(z) = ln . κ ks Zagustin (1968) defect law:

3 u∗ h − z 2 u(z) = U − 2 × arctanh . max κ h

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Author contributions. WS, BC, JLC and PV conceived the study, Bouchez, J., Métivier, F., Lupker, M., Maurice-Bourgoin, L., Perez, analyzed the data, and undertook the investigation, methodology M., Gaillardet, J., and France-Lanord, C.: Prediction of depth- and code development. WS, BC, JLC, JMM and JLG prepared the integrated fluxes of suspended sediment in the Amazon River: paper and discussed the data. JMM, JLG, WS, WL and MAP were particle aggregation as a complicating factor, Hydrol. Process., responsible for funding acquisition as well as project administration 25, 778–794, https://doi.org/10.1002/hyp.7868, 2011. and supervision. WS, PV, JC, JJPA, REV and JLG were responsible Callède, J., Kosuth, P., Guyot, J. L., and Guimaraes, V. S.: Discharge for the hydrologic data acquisition. NA, FJ and WS undertook the determination by acoustic Doppler current profilers (ADCP): A laboratory analysis. moving bottom error correction method and its application on the River Amazon at Obidos, Hydrolog. Sci. J., 45, 911–924, https://doi.org/10.1080/02626660009492392, 2000. Competing interests. The authors declare that they have no con- Camenen, B.: Simple and general formula for the settling flict of interest. velocity of particles, J. Hydraul. Eng., 133, 229–233, https://doi.org/10.1061/(ASCE)0733-9429(2007)133:2(229), 2007. Acknowledgements. The authors would especially like to ac- Camenen, B. and Larson, M.: A general formula for non-cohesive knowledge their colleagues from the National Agrarian University bed load sediment transport, Estuar. Coast. Shelf S., 63, 249–260, of La Molina (UNALM) and the Functional Ecology and Environ- https://doi.org/10.1016/j.ecss.2004.10.019, 2005. ment laboratory (EcoLab), who contributed to the analysis of the Camenen, B. and Larson, M.: A General Formula for Noncohesive data used in this study. Suspended Sediment Transport, J. Coast. Res., 243, 615–627, https://doi.org/10.2112/06-0694.1, 2008. Camenen, B., Le Coz, J., Dramais, G., Peteuil, C., Fretaud, T., Fal- gon, A., Dussouillez, P., and Moore, S. A.: A simple physically- Financial support. This research has been supported by the based model for predicting sand transport dynamics in the Lower French National Research Institute for Sustainable Development Mekong River, River Flow, Proc. River Flow conference, 3– (IRD), the National Center for Scientific Research (CNRS – 5 September 2014, Lausanne, Switzerland, 2189–2197, 2014. INSU) and the Peruvian Hydrologic and Meteorology Service Castro-Orgaz, O., Girldez, J. 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