RESEARCH ARTICLE
10.1002/2017WR020591
River Mixing in the Amazon as a Driver of Concentration-
Discharge Relationships
Julien Bouchez1 , Jean-Sebastien Moquet1,2, Jhan Carlo Espinoza3 , Jean-Michel Martinez2,
Jean-Loup Guyot2 , Christelle Lagane2, Naziano Filizola4 , Luis Noriega5, Liz Hidalgo Sanchez6,
and Rodrigo Pombosa7
1Institut de Physique du Globe de Paris - Centre National de la Recherche Scientifique - Sorbonne Paris Cite, Paris, France,
2Geosciences Environnement Toulouse, GET/OMP, CNRS/IRD/Universite Paul Sabatier, Toulouse, France, 3Instituto
Geofısico del Peru, Lima, Peru, 4LAPA (Laboratorio de Potamologia da Amazo^nia – Universidade Federal do Amazonas),
Av. General Rodrigo Octavio Jord~ao Ramos 3000, Campus Universitario, Bloco Arthur Reis, Coroado, Manaus, Brazil,
5SENAMHI Calle Reyes Ortiz no. 41 2do Piso, La Paz, Bolivia, 6Universite Pierre et Marie Curie, Paris, France, 7INAMHI
I~naquito N36-14 y Corea, Codigo 16-310, Quito, Ecuador
Abstract Large hydrological systems aggregate compositionally different waters derived from a vari-
ety of pathways. In the case of continental-scale rivers, such aggregation occurs noticeably at confluences
between tributaries. Here we explore how such aggregation can affect solute concentration-discharge (C-
Q) relationships and thus obscure the message carried by these relationships in terms of weathering
properties of the Critical Zone. We build up a simple model for tributary mixing to predict the behavior of
C-Q relationships during aggregation. We test a set of predictions made in the context of the largest
world’s river, the Amazon. In particular, we predict that the C-Q relationships of the rivers draining hetero-
geneous catchments should be the most ‘‘dilutional’’ and should display the widest hysteresis loops. To
check these predictions, we compute 10 day-periodicity time series of Q and major solute (Si, Ca21,
Mg21, K1, Na1, Cl-, SO224 ) C and fluxes (F) for 13 gauging stations located throughout the Amazon basin.
In agreement with the model predictions, C-Q relationships of most solutes shift from a fairly ‘‘chemo-
static’’ behavior (nearly constant C) at the Andean mountain front and in pure lowland areas, to more
‘‘dilutional’’ patterns (negative C-Q relationship) toward the system mouth. More prominent C-Q
hysteresis loops are also observed at the most downstream stations. Altogether, this study suggests that
mixing of water and solutes between different flowpaths exerts a strong control on C-Q relationships of
large-scale hydrological systems.
1. Introduction
The release of solutes to rivers by chemical weathering results from water-rock interactions. Weathering
models account for these interactions by coupling laws for mineral precipitation and dissolution with
water transport (Godderis et al., 2006; Steefel et al., 2005) or by assuming a direct dependency of weath-
ering fluxes on water fluxes (linear as in Berner et al., 1983; or nonlinear as in Bluth and Kump, 1994) or
on water residence time in the system (Maher, 2010). Conversely, empirical observations on the relation-
ship between solute export and water fluxes, as a function of time or across different catchments, may
provide insight into weathering mechanisms and coupling between hydrology and water-rock interac-
tions in the Critical Zone (Calmels et al., 2011; Clow & Mast, 2010; Eiriksdottir et al., 2013; Godsey et al.,
2009; Ibarra et al., 2016; Maher & Chamberlain; 2014; Maher, 2011; Torres et al., 2015; Von Blanckenburg
et al., 2015).
The instantaneous solute export (F, e.g., in mol.s21) from a watershed is equal to discharge (hereafter
termed Q, e.g., in L.s21) times solute concentration (hereafter referred to as C, e.g., in mol.L21):
F5C:Q (1)
As these two metrics can be readily measured, and even possibly measured at a relatively high frequency
(Floury et al., 2017), relationships between Q and C through time (‘‘C-Q relationships’’) have long been a
focus of characterization in hydrology (Anderson et al., 1997; Meybeck, 1976; Saunders & Lewis, 1989) - one
Special Section:
Concentration-discharge
Relations in the Critical Zone
Key Points:
 Modeling predicts that tributary
mixing should affect river
concentration-discharge
relationships, especially in large,
heterogeneous basins
 In the Amazon basin river
concentration-discharge relationships
of heterogeneous basins are the least
‘‘chemostatic’’
 Concentration-discharge
relationships in large systems are
strongly affected by mixing between
waters following different pathways
Supporting Information:
 Supporting Information S1
 Figure S1
 Table S1
 Table S2
Correspondence to:
J. Bouchez,
bouchez@ipgp.fr
Citation:
Bouchez, J., Moquet, J.-S.,
Espinoza, J. C., Martinez, J.-M.,
Guyot, J.-L., Lagane, C., . . . Pombosa, R.
(2017). River mixing in the Amazon as
a driver of concentration-discharge
relationships. Water Resources
Research, 53, 8660–8685. https://doi.
org/10.1002/2017WR020591
Received 16 FEB 2017
Accepted 1 SEP 2017
Accepted article online 12 SEP 2017
Published online 2 NOV 2017
VC 2017. American Geophysical Union.
All Rights Reserved.
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8660
Water Resources Research
PUBLICATIONS
practical interest of this approach being to establish rating curves to estimate C based on Q time series,
which benefit usually from a much higher measurement frequency.
From a theoretical perspective, two extreme scenarios can a priori be distinguished for a given C-Q relation-
ship: one where the release of solute from the watershed, F, is constant, and concentrations C vary following
a dilution law, i.e., as Q21; and one where C is constant, and where F scales as Q. The interest in C-Q relation-
ships has been recently revived by Godsey et al.’s (2009) analysis of data from small basins of the US Hydrol-
ogy Benchmark Network (HBN), suggesting that these end members are best expressed through the
functional fit of C-Q relationships by a power law:
C5a:Qb (2)
where a and b are the fit parameters. Godsey et al. (2009) coined the term ‘‘chemostatic’’ to describe that
for most solutes the power law exponent b is close to 0, i.e., that many small rivers display a limited variabil-
ity in C compared to that in Q. This observation has two main implications: (1) through time, F depends pri-
marily on Q; (2) a mechanism is at play in the Critical Zone, ‘‘buffering’’ variations in C against dilution. To
examine the possible mechanism that could account for their observation of the chemostatic behavior of
river catchments, Godsey et al. (2009) reviewed a set of previous models (Johnson et al., 1969; Langbein &
Dawdy, 1964) and suggested a new ‘‘porosity-permeability-aperture model’’ (‘‘PPA model’’), deemed plausi-
ble and featuring only measurable field parameters, and therefore testable.
The ‘‘initial chemostatic observation’’ of Godsey et al. (2009) has since been refined, in particular by Maher
(2011) who included the trend toward dilution usually observed at high Q (Bluth & Kump, 1994). Maher
(2010, 2011) indeed introduced a 1D-reactive transport model based on a 1st-order dependency of solute
production on C (i.e., F scaling with C - Ceq, where Ceq is the solute concentration at chemical equilibrium),
yielding:
C5C0e
2Dw=q1Ceq 12e
2Dw=q
 
(3)
where Dw is the Damk€ohler coefficient, that depends on the rate of solute release from minerals, flowpath
length, and Ceq (Maher & Chamberlain, 2014). This model has been used recently by Ibarra et al. (2016) to
retrieve the Dw values of monolithological, relatively small basaltic and granitic catchments throughout the
world based on their individual C-Q relationships, and infer the difference in weatherability between these
two rock types.
Other recent studies at a larger scale have shown that power law fits of C-Q relationships could yield b-val-
ues significantly lower than 0 (Moon et al., 2014; Moquet et al., 2016; Torres et al., 2015). In particular, using
a series of four nested catchments in the upper Madre de Dios basin (southwestern Amazon basin), Torres
et al. (2015) observed a systematic decrease of the b-value with basin altitude for most solutes, with a more
chemostatic behavior in mountainous basins and a larger effect of dilution at lowland locations. Based on
the framework introduced by Maher (2011), this was chiefly interpreted as a change in the weathering effi-
ciency from mountainous regions, with shorter surface water residence time and more altered regolith lead-
ing to decreased weathering efficiency in lowland regions compared to mountains (Torres et al., 2015).
River catchments are complex and heterogeneous systems, be it regarding water (rainfall) and element
inputs (e.g., atmospheric wet and dry inputs, weathering, or anthropogenic contamination), or in terms of
other parameters that might influence the motion of water and elements (e.g., soil depth and type, or vege-
tation cover). Although the recent body of work on C-Q relationships has yielded significant insight into the
links between water flow and solute export in weathering systems, interpretation of C-Q relationship has
until now heavily relied on a strong assumption: namely that the whole area upstream the sampling point
is homogenous in terms of properties that are relevant to the interpretation (with the notable exception of
the inclusion of water travel time distributions by Maher, 2011). The issue of environmental heterogeneity
and aggregation of different flowpaths has recently been addressed by Kirchner (2016a, 2016b), who
showed in particular how such heterogeneity strongly affects any inference that can be made on water
travel time distribution from C and Q time series.
One feature that has been overlooked in the most recent analyses of C-Q relationships is the presence of
hysteresis loops, although its existence has been long recognized (e.g., Evans & Davies, 1998; Hornberger
Water Resources Research 10.1002/2017WR020591
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8661
et al., 2001; House & Warwick, 1998; Rose, 2003). Besides a handful of studies (e.g., Duffy & Cusumano,
1998), C-Q loops have often been interpreted as an offset between the discharge of different hydrological
reservoirs within a given catchment (e.g., surface water, soil water, groundwater; e.g., Calmels et al., 2011;
Edokpa et al., 2015) or, when observed downstream from a confluence, between tributaries (Ollivier et al.,
2006; Moquet et al., 2016). Therefore, there seems to be a general agreement that environmental heteroge-
neity should generate hysteresis in C-Q relationships. However, we still lack systematic observations and
theoretical frameworks about C-Q hysteresis loops, as well as potential links between these loops and other
features of C-Q relationships, especially in large, heterogeneous river systems.
In the present paper we examine how catchment heterogeneity affects C-Q relationships, through the time-
varying contribution of different subcatchments. In large watersheds, the process of aggregation of water
pathways is easy to pinpoint as it occurs mostly at confluences of tributaries. In particular, we explore the
hypothesis that previously observed changes in b-values with catchment size catchment could stem solely
from aggregation of heterogeneous subcatchments, as large drainage areas are more likely to collect water
and solutes fluxes that drastically differ in timing and amplitude of temporal variations. To that effect, we
build up a simple model of water and solute mixing from two compositionally different, chemostatic
tributaries, which predicts the trend (b-value) and shape (hysteresis loops) of C-Q relationships in the mixed
water flow, depending on the differences between the two tributaries in terms of solute concentration, vari-
ation of discharge, and timing of peak discharge. We then turn toward the Amazon basin to check these
predictions. The Amazon, which is the largest river basin in the world, is well suited for this approach as its
tributaries drain areas that greatly differ in climate, lithology, or geomorphic characteristics. Altogether, our
study questions the inferences that can be made on C-Q relationships about the properties of weathering
systems, beyond a certain spatial scales and a certain degree of aggregation of water pathways.
2. Study Area
The Amazon basin is the largest fluvial basin in the world with a drainage area of about 5.9 million km2 (Call-
e`de et al., 2010), located in the tropical region between 5816’N and 20828’S and between 79836’W and
50815’W, from the Andes to the tropical Atlantic Ocean. It contributes to 16–18% of the continental dis-
charge (Dai & Trenberth, 2002; Calle`de et al., 2010) and to around 7% of the flux of dissolved species to the
ocean (Moquet et al., 2016).
The Amazon has 5 main tributaries (Figure 1). The Negro River drains the Brazilian Shield in the northern
Amazon; the Solim~oes River drains the Northern and Central Andes and a large part of the Lowlands; the
Madeira River drains the Southern Andes, the Southern Foreland basins and part of the Brazilian shield; and
the Tapajos and Xingu drain the remaining area of the Brazilian shield.
The basin receives around 2,200 mm.yr21 of precipitation (Espinoza et al., 2009a) resulting in a mean dis-
charge of 206,000 m3.s21 (Calle`de et al., 2010). The climate is monsoonal and is mainly controlled by the
seasonal variation of the Southern American Monsoon System (SAMS) throughout the north-south oscilla-
tion of the Intertropical Convergence Zone (ITCZ) (Marengo et al., 2012; Vera et al., 2006). While the north-
western part of the basin receives a high amount of rainfall around the year, generally up to 3,000 mm.yr21,
the drier part of the basin is located in the southern Andes, where semiarid conditions can be met, particu-
larly during the austral winter (dry season). However, over the Andean region, interactions between the
topography and the atmospheric circulation produce a strong rainfall gradient that can vary from 6,000 to
300 mm.yr21 over a few kilometers (Espinoza et al., 2015). The North-South annual rainfall gradient is linked
to a gradient in the seasonality of rainfall. In the northern wetter region of the basin, the seasonality is low
and, the Seasonal Variability Coefficient (SVC) is generally above 0.3. Conversely, in the southern, drier
region of the basin, the seasonality is strongly marked with a SVC of up to 0.8 (Espinoza et al., 2009a). This
seasonal variability of rainfall distribution implies that the discharge of the rivers is variably marked by sea-
sonality. An offset of the maximum of discharge is also observed between the main Amazon tributaries
according to the evolution of the position of the ITCZ and the SAMS intensity along the year (Espinoza
et al., 2009b). Therefore, along the year, the maximum of discharge is first recorded in the southeastern and
southern basins (Tapajos and Madeira rivers), then in the central and western basins (Solim~oes River) and
finally in the northern basins (Negro River). This hydroclimatic gradient is therefore a first cause for hetero-
geneity in water and solute fluxes across the Amazon basin (Figure 1b).
Water Resources Research 10.1002/2017WR020591
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8662
IL
.g
m(
 sdil
os
 d
evl
ossid
 l
at
oT
1
-
)
Mixed riversLowland riversAndean rivers
B.
BOR
ATA
FOR
RUR
TAB
MAN
PVE
FAZ
OBI
LAB
SER
CAI
ITA
-16
-14
-12
-10
-8
-6
-4
-2
0
2
4
1 2 3 4 5 6 7 8
Latitude of the sam
pled station (decimal degree)
Month of the higher discharge
Andean rivers 
Lowland rivers
Mixed rivers
C.
A.
Figure 1. (a) Topographic map of the Amazon basin, gauging stations of the HYBAM network, and main geomorphic domains. The location of the Andean front
(defined as the contour line of the 500 m-a.s.l.) is shown as a thick black line. (b) Time of peak discharge in the year versus latitudinal coordinate for the studied
gauging stations. (c) Box-plot of the distribution of total dissolved solids (sum of major species concentration, including HCO23 ) for the 13 gauging stations of our
database, ranked following their geomorphic context. This plot shows the major difference between solute-rich, Andean rivers and dilute Lowland waters. The red
line is the median of the distribution, the box limits are the upper and lower quartile, the whiskers delimit the data range and crosses are outliers (as defined by
the Python matplotlib boxplot function). In panels A and B, stations corresponding to Andean rivers, pure lowland rivers, and mixed rivers are indicated as white,
black, and gray circles, respectively.
Water Resources Research 10.1002/2017WR020591
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8663
This diversity of geologic and climatic characteristic throughout the Amazon basin generates a strong gradi-
ent in erosion and weathering regimes, which in turn affects the quality of the water transported by rivers
(Figure 1c). Rivers draining the Andes (among which the main tributaries, from south to north: the Mamore,
Beni and Madre de Dios - draining to the Madeira River -, Ucayali, Huallaga, Mara~non and Napo - draining to
the Solim~oes River) are typical of the ‘‘white waters’’ (Gibbs, 1967; Rıos-Villamizar et al., 2014; Sioli, 1964), i.e.,
characterized by high dissolved concentrations and fluxes. Solute fluxes are high (172.106 t.yr21) in the
Andes (Gaillardet et al., 1997, Moquet et al., 2011, 2014a, 2016; Sanchez et al., 2015). This area contributes to
nearly 100% and to around 64% of the Amazon solid and solute annual exports, respectively. In particular,
this region supplies most of Na1, Ca21, Mg21, Cl-, SO224 to the system (Moquet et al., 2016). By contrast, in
the Shields and the Lowlands, solute fluxes are relatively low (Moquet et al., 2016; Wittmann et al., 2011).
The Shields are drained by ‘‘black water’’-type rivers (such as the Negro River), with high organic matter con-
tent reflecting intense weathering processes in strongly cation-depleted regolith, such as podzolization; or
‘‘clear water’’-type rivers (such as the Tapajos, Trombetas or Branco rivers), with high primary productivity
(Gaillardet, 1997; Gibbs, 1967; Sioli, 1964). With solute fluxes of less than 30.106 t.yr21, respectively (Moquet
et al., 2016), these rivers contribute to less than 10% of the Amazon annual fluxes. Finally, rivers draining
the low-standing areas of the Lowlands regions are of the clear-water or black-water type (Rıos-Villamizar
et al., 2014) with low solute loads (Stallard & Edmond, 1983). With solute fluxes of about 70.106 t.yr21, Low-
lands contribute to a minor yet significant part of the solute fluxes of the Amazon (Moquet et al., 2016).
Therefore, hydrochemistry varies significantly between the different geomorphic domains across the Ama-
zon Basin, in particular between the Shield and the Lowlands (Figure 1c). Altogether, the hydroclimatic and
hydrochemical heterogeneity of the Amazon Basin make it a natural laboratory where the behaviour of C-Q
relationships at downstream stations should be significantly influenced by mixing of waters from different
tributaries.
The anthropogenic influence across the Amazon is limited, although large dams are being built on the
Madeira River (Latrubesse et al., 2017), and mining activity exists in the Andes and in the Shield areas. We
also note that a significant release of Na1 and Cl- to the rivers through the exhumation of brines during oil
exploration has been observed between 2007 and 2009 on the Mara~non River, affecting even the Amazon
at its mouth (Moquet et al., 2014b).
In the context of the Amazon Basin, a variety of hydrological, inorganic and biological processes have been
invoked to explain observed C-Q relationships. With increasing Q, the dilution of Cl- delivered by evaporite
dissolution in the Andes (Moquet et al., 2016) or by inputs of deep water through oil extraction in the Peru-
vian foreland (Moquet et al., 2014b), has been attributed to a constant Cl- release flux superimposed on vari-
able water discharge. Conversely, nearly constant concentrations along the hydrological cycle have been
explained by sorption processes, for example, for PO324 in the Solimoes basin (Devol et al., 1995), or by ther-
modynamic equilibrium between soils or suspended matter and water for K and Si for most of the Amazon
subbasins (Moquet et al., 2016). Finally, for some elements concentrations increase with discharge, as is the
case, for example, for PO324 or NO
2
3 in a small watershed in Ecuador (Boy et al., 2008) which was attributed
to the mobilization of matter from the soil surface by runoff during rain events. This behavior is also
observed for major elements in rivers draining Shield areas (Markewitz et al., 2001) or in some basins of the
Andean headwater of the Madeira Basin (Guyot, 1993; Roche & Fernandez, 1988). In both cases, the authors
suggested that solutes are released by weathering of surficial soil layers within the catchment during high
runoff periods. In some rivers, C-Q relationships shift significantly over the hydrological period: for example,
in the Beni basin Guyot (1993) and Moquet et al. (2016) reported a dilutional behavior of major elements
during the dry season and an increase of solute concentration with Q during the wet season, which was
explained by the contrasted hydrological regimes between the dry and wet seasons in this semiarid context
(Roche & Fernandez, 1988).
3. Data Source and Processing
3.1. Database for Concentration and Discharge
We used the SNO-HYBAM (Geodynamical, hydrological, and biogeochemical controls on erosion/weather-
ing and material transport in the Amazon, Orinoco, and Congo basins) observatory discharge and chemical
data recorded at 13 hydrological stations between 2003 and 2008. These stations are located on the four
Water Resources Research 10.1002/2017WR020591
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8664
main Amazon tributaries (Negro, Solim~oes, Madeira and Tapajos rivers), corresponding to drainage areas
between 12 and 4,671.103 km2 (Table 1) and categorized in three main river types: Andean, Lowland, and
mixed rivers (typically downstream stations) (Figure 1). Hydrological data have been produced using con-
ventional hydrological methods. Water levels were recorded daily or twice daily, and the daily discharge
series was calculated from rating curves (discharge versus water level) based on 3-months frequency gaug-
ing using a 600 kHz (Peru, Bolivia and Brazil) or 1,200 kHz (Ecuador) Acoustic Doppler Current Profiler
(ADCP). For hydrochemical data, only major elements (Cl-, SO224 , Na
1, Ca21, Mg21, K1) have been considered
in the present study. Hydrogenocarbonate ion (HCO23 ), although a major dissolved component of river
waters, has not been considered in the present study as its dynamics can be largely controlled by gaseous
exchange (e.g., Abril et al., 2014). A sample of 650 mL was collected monthly at each gauging station. The
samples were filtered on site using a 0.2-mm porosity filter. Concentrations were determined at the GET lab-
oratory (Geoscience Environnement Toulouse, France) for Ecuadorian, Peruvian and Bolivian samples and at
the ‘‘Instituto de Geociencias’’ laboratory of the Universidade de Brasilia (UnB) for Brazilian samples. The
samples were analyzed according to the procedure described in Cochonneau et al. (2006). Cl- and SO224 con-
centrations were measured by ion chromatography, and Na1, Ca21, Mg21, K1 and Si were analyzed by ICP-
AES (Inductively Coupled Plasma-Atomic Emission Spectroscopy). Based on the analyses of geostandards,
the analytical error is less than 10%.
3.2. Computation of Time Series
Daily discharge and monthly hydrochemical data were downloaded from the SNO-HYBAM website (http://
www.ore-hybam.org). Using the HYDRACESS software (Vauchel, 2005), a 10 day periodicity time series was
produced for each station and each parameter, for Q by averaging 10 daily data points and for C by linear
interpolation of the monthly original time series. This allows for having synchronous C and Q databases.
The validity of a linear interpolation of C from the monthly to the 10 day periodicity is justified by the rela-
tively smooth temporal variations of C, and the weak contribution of high-frequency variation in the power
density spectra (see supporting information section S1).
In this paper, in addition to using Q and C times series, we rely on temporal variations of solute fluxes F. In
the scope of the present work, F has two interesting properties:
1. Like Q and unlike C, F can - under certain conditions - be additive during aggregation of water pathways
and therefore fullfill simple mass balance requirements. For example, the sum of the input fluxes of a
Table 1
Description of the Gauging Stations of the HYBAM Network Used in This Study
Geomorphological
domain Hydrosystem River
Station
name Full name
Upstream
stations
inputs
Geographic
coordinates
Annual
discharge
(103 m3.s-1)
Drainage
area
(103 km2)
% basin
in the Andes
(>500m)
Water
sampling
periodX (8) Y(8)
Andes Solim~oes Napo FOR Francisco de
Orellana
276.98 20.474 1.2 12 69% 2000–2008
Solim~oes Upper
Maranon
BOR Borja 277.55 24.47 5 114 91% 2003–2008
Solim~oes Ucayali ATA Atalaya-Lagarto 273.82 210.68 7 191 87% 2003–2008
Madeira Beni RUR Rurrenabaque 267.53 214.44 2 70 92% 2003–2008
Lowland Solim~oes Purus LAB Labrea 264.8 27.252 5.5 226 0% 2004–2008
Negro Upper Negro SER Serrinha 264.83 20.497 17 290 0% 2003–2008
Negro Branco CAI Caracara€ı 261.12 1.819 3.6 125 0% 2003–2008
Tapajos Tapajos ITA Itaituba 255.98 24.283 12 458 0% 2004–2008
Mixed Solim~oes Amazonas/
Solimoes
TAB Tabatinga FOR1 BOR
1ATA
269.96 24.242 35 877 46% 2003–2008
Solim~oes Solimoes MAN Manacapuru TAB1 LAB 260.55 23.345 102 2203 19% 2003–2008
Madeira Madeira PVE Porto Velho RUR 263.92 28.788 18 978 21% 2003–2008
Madeira Madeira FAZ Fazenda Vista
Alegre
PVE 260.03 24.897 26 1314 15% 2003–2008
Amazon Amazon OBI Obidos MAN1 FAZ1
SER1CAI
255.51 21.947 177 4671 13% 2003–2008
Water Resources Research 10.1002/2017WR020591
BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8665
given conservative solute by two tributaries at a confluence is equal to the flux of the same element in
the mainstream after the confluence, whereas this is not true for C. We emphasize that having additive F
requires that the solute is conservative during mixing (Guinoiseau et al., 2016) and that the downstream
F is calculated taking into account potential lateral heterogeneity in the channel (Bouchez et al., 2010).
Spectral analysis tools such as Fourier transform can then also be applied to an additive parameter such
as F, in order to retrieve the properties of water transport systems (see section 5.1).
2. As C variations are not as strong as Q variations in the Amazon (Moquet et al., 2016, and this study), F is
strongly controlled by Q, such that F variations are as easy to characterize empirically as Q variations,
which might not be the case for C variations (see e.g., the case of sine-wave fits in supporting informa-
tion section S2).
Therefore, 10 days frequency F records were computed by multiplication of the 10 days frequency Q and C
records (equation (2)). Mean Q, C, and F values, along with relative standard deviations, are provided in
Tables 2 and 3. In this scope we note that the inferred standard deviations in Q depend on the frequency of
the Q record used (calculations not shown here, but performed for periods between 1 day and 1 month
from the HYBAM database).
Table 2
Mean Discharge (Q) and Solute Flux (F) and Corresponding Relative Standard Deviation (s.d.) for Each Station and Solute
Discharge Si SO224 Cl
- Na1 K1 Mg21 Ca21
Mean
(m3.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
Mean
(kg.s-1)
s.d.
(%)
FOR 1314 36 9.3 38 3.5 38 0.7 52 3.3 39 1.9 38 1.5 37 10 37
BOR 5045 34 27 51 48 41 17 37 21 36 5.2 50 11 47 97 56
ATA 5770 59 29 59 151 50 54 51 50 55 8.8 64 22 52 153 61
RUR 1998 80 7.8 92 41 77 1.6 49 6.1 70 2.3 93 6.8 74 19 78
LAB 5805 69 38 89 15 80 2.8 90 9.9 70 6.6 68 6.4 64 31 63
SER 17245 38 30 39 8.8 122 4.2 80 14 165 7.3 103 2.7 57 8.4 54
CAI 3496 80 21 70 2.7 173 2.7 87 8.1 78 5.3 86 2.8 80 5.8 79
ITA 12619 61 50 59 3.5 76 6.8 71 9.4 63 11 69 6.2 51 18 72
TAB 35844 30 214 52 291 44 282 47 298 37 46 36 84 33 732 35
MAN 105504 28 447 26 412 32 390 26 364 31 98 31 138 28 1008 31
PVE 17850 66 85 76 169 57 15 129 43 62 28 74 42 60 133 66
FAZ 27011 67 114 70 169 60 21 76 61 66 36 74 45 61 126 60
OBI 186330 29 782 30 474 42 345 20 384 17 163 27 185 28 1038 26
Table 3
Mean Solute Concentration (C) and Corresponding Relative Standard Deviation (s.d.) for Each Station and Solute
Discharge Si SO224 Cl
- Na1 K1 Mg21 Ca21
Mean
(m3.s-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
Mean
(mg.L-1)
s.d.
(%)
FOR 1314 36 7.1 14 2.7 22 0.6 41 2.6 25 1.5 16 1.2 23 7.7 18
BOR 5045 34 5.3 29 9.4 23 3.5 39 4.3 32 1 25 2.1 29 18.8 37
ATA 5770 59 5.3 15 31.7 39 12.5 57 10.7 50 1.6 18 4.5 30 28.9 24
RUR 1998 80 4.1 18 25.6 36 1.2 58 3.8 35 1.2 36 4 27 10.7 24
LAB 5805 69 6.6 35 3 59 0.6 65 2.1 42 1.2 28 1.4 42 6.9 43
SER 17245 38 1.8 20 0.7 180 0.2 52 0.8 129 0.4 82 0.2 31 0.5 32
CAI 3496 80 6.4 16 0.8 121 0.8 34 2.5 34 1.5 36 0.8 17 1.6 21
ITA 12619 61 4.2 13 0.3 47 0.7 70 0.8 29 0.9 16 0.6 21 1.4 45
TAB 35844 30 6 34 8.3 40 8.1 41 8.9 40 1.3 15 2.4 20 20.7 18
MAN 105504 28 4.4 11 4.3 39 4.1 41 3.8 33 1 24 1.4 20 10 22
PVE 17850 66 4.7 19 11.1 36 1 60 2.8 37 1.5 15 2.7 30 8.5 34
FAZ 27011 67 4.4 16 7.5 29 1 60 2.9 48 1.4 20 2 28 5.7 31
OBI 186330 29 4.3 27 2.8 50 2.1 42 2.3 36 0.9 28 1 31 5.9 28
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8666
3.3. Main Features of the Discharge, Concentration, and Solute Fluxes Records
The water discharge and chemistry of the dissolved load of the Amazon have been reported and inter-
preted extensively elsewhere (e.g., Gaillardet, 1997), and parts of the present database have been published
and discussed in detail in Moquet et al. (2011, 2016). Therefore, in this paper we do not comment exten-
sively on the values taken by Q and C and the contribution of different solutes to the total load. However,
we provide here a summary on their variation through time.
Discharge generally increases downstream, with the lowest mean Q of the database recorded for the BOR
station (1,314 m3/s) and the largest for Obidos (1,86,330 m3/s; Figure 2 and supporting information Figures
S14–S26). The Andean stations have the lowest mean Q of the database (Table 2). Across the record period,
Q varies from by 29% (relative standard deviation calculated over all 10 day periodicity data points avail-
able) for OBI and by up to 80% for RUR and CAI stations. Some northern Andean stations exhibit a relatively
low variability in Q (e.g., around 35% for FOR or BOR), whereas at some of the southern downstream sta-
tions, such as PVE of FAZ, Q is much more variable (relative standard deviation> 65%) reflecting the differ-
ence in precipitation variability between the two regions (Espinoza et al., 2009b; Molina-Carpio et al., 2017).
At all stations, the C of all considered solutes varies significantly across the period of study (Figure 3 and
supporting information Figures S27–S39). In terms of mean concentration values, the species contributing
most to the anion mass load is HCO23 (not considered in the C-Q analyses in this paper) at all stations,
whereas that contributing the most to the cation mass load is Ca21 for most stations (except CAI and SER
where it is Na1). For all solutes with the exception of Si, the highest mean C is measured on the Ucayali
River at the ATA station. For Si, the highest mean C is obtained at the Andean station of FOR. The relative
variation in mean C across the set of gauging stations is about 30% for all solutes except for Si and K1,
where it is around 20%. Over the period of record, the highest temporal variability in C is generally observed
for Cl- (relative standard deviation: 34–70% depending on the station) and SO224 (22–180%), and the lowest
for Si (11–35%; Table 3). Nevertheless, we emphasize that the station SER exhibits fairly low average concen-
tration values for all solutes, over which analytical issues (e.g., contamination during sampling, or precision
and accuracy of analyses) will have the strongest imprint.
Averaged over all the stations, the highest mean solute flux is recorded for Ca21 (range: 6-1,038 kg.s21) and
the lowest for Cl- (0.7–782.0 kg.s21), Mg21 (1.5–185.4 kg.s21), and K1 (1.9–162.8 kg.s21; Table 2; Figure 2
and supporting information Figures S14–S26). The stations which record the lowest F are the Andean sta-
tions FOR (0.7–9.4 kg.s21 depending on the solute) and RUR (1.6–41.3 kg.s21), due to their relatively small
drainage area and thus low Q; and the stations located on the pure lowland rivers SER (4.2–29.8 kg.s21) and
CAI (2.7–21.2 kg.s21), due to their dilute nature and thus low C. Depending on the station, the temporal var-
iability in F ranges from 20–30% (relative standard deviation) to 80% (Mg21 at OBI) to 173% (SO224 at CAI).
There is no systematic difference in variability between different families of rivers: for Andean stations, the
variability in F ranges from 37% (Ca21 at FOR) to 93% (K1 at RUR), whereas for lowland stations, it is
between 17% (Na1 at OBI) and 173% (SO224 at CAI).
Therefore, our database encompasses a set of gauging stations located throughout the Amazon Basin,
among which three groups can be distinguished: Andean-derived rivers featuring high solute concentra-
tions (FOR, BOR, ATA, RUR), rivers draining only low-relief areas and carrying dilute waters (LAB, SER, CAI,
ITA); and rivers draining both Andean and Lowland regions (TAB, MAN, PVE, FAZ, and OBI) (Table 3 and Fig-
ure 1). The study of concentration-discharge-fluxes relationships across the Amazon basin thus allows us to
address the role of environmental heterogeneity on C-Q relationships, through the aggregation of different
types of water.
4. A Model for Nonreactive Mixing of Water and Solutes
4.1. Model Setup and Numerical Experiment
We consider two water flows carried by two theoretical tributaries 1 and 2, the mixing of which produces a
downstream flow (mixture m; Figure 4). Discharge (q1 and q2) is assumed to vary as a sine wave in each trib-
utary at the same frequency (angular frequency x), but differs in average (v1 and v2), amplitude (w1 and w2)
and phase (u1 and u2). We then consider a conservative solute carried by the two tributaries, which
are both assumed to be chemostatic (their b-exponent is therefore 0). Therefore, the solute concentration
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8667
BOR (Andean station) SER (’pure lowland’ station) MAN (downstream station)
Figure 2. Discharge (Q) and solute fluxes (F) time series for a selection of gauging stations and solutes. The 10 day period data is presented as a thin blue curve, the sine
wave best fit as a thick blue line (see supporting information section S2). The parameters of the best fit (amplitude, mean, and phase), as well as its quality as indexed by
the normalized root mean square deviation (RMSD) are indicated. The left panels correspond to the Andean station BOR (Mara~non); the central panels to the station SER
(Negro) that drain only the Amazon Lowlands; and the right plots to the downstream station MAN (Solim~oes) that represent a mixture of these different types of inputs.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8668
(C1 and C2) in each tributary is constant with time t. This is thought to reflect the near-chemostatic behavior
of small, homogeneous tributaries (Godsey et al., 2009):
q15v11w1sin xt1u1ð Þ (4a)
q25v21w2sin xt1u2ð Þ (4b)
f15c1 v11w1sin xt1u1ð Þð Þ (4c)
f25c2 v21w2sin xt1u2ð Þð Þ (4d)
with f1 and f2 the solute fluxes of the two tributaries. In order to reduce the number of model parameters
that will need to be explored, we adopt the following normalization: T5xt1u1; /5u22u1; Q15
q1
v1
; Q25
q2
v1
;
F15
f1
v1 c1
; F25
f2
v1 c1
; W15
w1
v1
; W25
w2
v2
; rQ5
v2
v1
; and rC5
c2
c1
. This yields:
Figure 3. Concentration (C) and discharge (Q) time series for the station PVE (located on the Madeira River) for the 7
major solutes considered in this study. The 10 day period data is presented as a thin blue curve for C and a thin grey curve
for Q, the sine wave best fit as a thick blue line for C and a dotted black line for Q (see supporting information section S2).
The parameters of the best fits of F time series (amplitude, mean, and phase), as well as its quality as indexed by the nor-
malized root mean square deviation (RMSD) are indicated.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8669
a b c
d e f
g h i
j k l
m n o
Figure 4. Synthetic time series of Q and F as inputs for the tributary mixing model. In this diagram concentration and discharge are normalized to the average of
those of tributary 1. The blue curve corresponds to the Q of and F of tributary 1 (the normalization scheme adopted here collapses these two time series on the
same curve). The continuous red line corresponds to the Q of tributary 2, and the dotted red curve to the F of tributary 2. Each panel corresponds to a different
combination of phase offset (/ in the text) between the two tributaries inputs, and to a different ratio of average Q between these two tributaries (rQ in the text).
The ratio of C between the two tributaries is maintained constant here (rC 5 1.2 see text).
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8670
Q1511W1 sin T (5a)
Q25rQ 11W2 sin T1/ð Þð Þ (5b)
F1511W1 sin T (5c)
F25rcrQ 11W2 sin T1/ð Þð Þ (5d)
Here, Q and F are normalized to the time-averaged discharge and solute flux, respectively, of tributary 1,
and / is the phase offset between the tributaries. W1 and W2 (positive and< 1) are the relative amplitude
of variations of F and Q in each tributary. T is nondimensional and covers a whole period of variation
between 0 and 2p.
We can now predict the evolution of C, Q and F in the mixture (Cm, Qm, and Fm). Importantly, only Q and F
(because we consider a conservative solute) can be summed, whereas the nonadditive property Cm has
to be calculated by dividing Fm by Qm:
Qm5Q11Q25 11 rQ1W1 sin T1rQW2 sin T1/ð Þ (6a)
Fm5F11F25 11rcrQ1W1sin T1 rcrQW2 sin T1/ð Þ (6b)
Cm5
Fm
Qm
5
11W1 sin Tð Þ1rcrQ 11W2 sin T1/ð Þ½ 
11W1 sin Tð Þ1rQ 11W2 sin T1/ð Þ½  (6c)
From equations (6b) and (6c), Cm-Qm relationships can be numerically predicted by varying T between 0
and 2p (a whole cycle of variation, corresponding, for example, to an annual cycle), for a range of values
taken by the model parameters W1, W2, rq, rc , and /. Different Cm-Qm relationships are shown in Figure 5 for
various values of the above-listed parameters. The corresponding inputs Q1, Q2, F1, and F2 are shown in Fig-
ure 4 and the resulting Q and F in the mixture (Qm and Fm) are shown in supporting information Figure S40.
Without phase offset between the tributaries (/5 0), no loop is observed (Figure 5, top), as the extrema in
Q and F are reached simultaneously by the two tributaries. Depending on the values of rc and rq (and for a
given value of W2W1), the Cm-Qm trend can be positive or negative (see also supporting informationsection S3
for an algebraic proof of this observation). Conversely, phase offset between the two tributaries generates
hysteresis in the Cm-Qm relationship. Increasing the concentration ratio rc between the two tributaries
makes the loop ‘‘deeper’’ (i.e., more extended in the vertical direction), while increasing the ratio of average
discharge, w2w1, ‘‘widens’’ this loop (i.e., makes it larger in the horizontal direction). This is because the range of
Q (respectively C) that can be reached by the mixture is higher if the two tributaries have drastically differ-
ent Q (respectively C) - as long as they are not in phase opposition. Altogether, this numerical experiment
shows that a range of very diverse C-Q relationships, both in trend and shape, can be obtained simply by
mixing a conservative solute from two tributaries displaying sine-wave variations in this solute’s flux.
4.2. First-Order Predictions in the Context of the Amazon Basin
Regarding the trend (b-value) of the C-Q relationship in the modeled mixture, we can predict, for different
sets of parameters, the sign taken by the derivative of Cm with respect to Qm as a function of T, which in
turn indicates the slope in the Cm-Qm diagram. In the supporting information section S3, we provide this
derivation, and show that the parameters that will drive the sign of dFmdQm are rc , rq, /, and
W2
W1
or w2w1. The first
parameter rc is related to the difference between the two tributaries in terms of chemical composition,
while the others are related to the hydrological pattern of Q time series. Specifically, a change in the b-expo-
nent after mixing is predicted to happen by our model as long as the tributaries differ in C (rc 6¼ 1), as
shown in Figure 5. In particular, a decrease in the b-value will occur through mixing if the tributary with the
highest C is the one exhibiting the smallest variations in Q (see supporting information section S3). In the
context of the Amazon Basin, Tables 2 and 3 show that in general, northern Andean stations have higher
mean C than their ‘‘pure Lowland’’ counterparts (e.g., FOR-BOR-ATA versus LAB-SER-CAI-ITA). Although the
case of the southern region of the Amazon Basin cannot be examined directly with the data presented in
Table 2, we can predict that downstream stations, where solute-rich, Andean-derived waters are admixed to
more dilute lowland waters, should display lower b-values and therefore more dilutional patterns.
This first-order prediction on the downstream evolution of b-values can be refined for each solute, consider-
ing their main pathways of production in the Amazon basin, for which a wealth of background information
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8671
a b c
d e f
g h i
j k l
m n o
Figure 5. Result of the tributary mixing model in terms of C-Q relationships. In this diagram C and Q are normalized to the average of those of tributary 1. In each panel,
each curve represents a different value of the ratio of C between the two tributaries (defined as rC in the text; black line: 0.2; horizontal grey line: 1.0, thin grey line: 1.2). The
trends of the ‘‘chemostatic’’ and dilutional behaviors are indicated for comparison (the exact vertical position of these trends are arbitrary). Each panel then corresponds to
a different combination of phase offset (/ in the text) between the two tributaries inputs, and to a different ratio of average Q (rQ in the text) between the two tributaries.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8672
exists. The strongest decrease of the b-exponent should be observed for solutes exhibiting the largest dif-
ference in C between the Andean-derived waters and the waters derived from low-relief areas. This is the
case for Na1, Cl- and SO224 , which are dilute in low-relief rivers (Moquet et al., 2016). The main source of
these species in the low-relief area of the Amazon is the atmosphere, because silicate weathering (which
could release Na1) is slow there, and because evaporite and sulfide dissolution (which could release Na1,
SO224 , or Cl
-) are virtually absent (Moquet et al., 2011; Torres et al., 2015). On the other hand, for solutes
which are significantly present in ‘‘pure low-relief’’ waters (rc  1), we predict a smaller change of the b-
value of C-Q relationships in the mixture, such that the C-Q behavior of these solutes should remain closer
to being ‘‘chemostatic’’ throughout the basin. This should be the case for Si and K1, as indicated by the C
data (Table 3) and by pluri-annual budgets showing the significant inputs of these two solutes in lowland
regions (Moquet et al., 2016). It is worth noting that these two elements are important nutrients, and that
therefore their behavior might be influenced by vegetation and/or plankton development in the highly pro-
ductive lowland regions (Fraysse et al., 2006; Konhauser et al., 1992; Lucas, 2001; Moulton et al., 2000). More-
over, thermodynamical equilibrium between solution and secondary silicate minerals could also control the
fluxes of these two particular elements and set their concentration to nearly constant values (e.g., Clymans
et al., 2013; Hughes et al., 2013; Neal et al., 2005).
None of the models recently developed to examine C-Q relationships (e.g., the ‘‘PPA model’’ of Godsey et al.,
2009, or the 1D-reactive transport model of Maher, 2011) can account for C-Q hysteresis loops. However, Moquet
et al. (2016) showed that for the Amazon at Obidos such hysteresis loops could be reproduced by summing the
monthly inputs of the different tributaries. Hysteresis is thus most likely an important feature of C-Q relationships
when tributary mixing needs to be evaluated. Therefore, and aside from the downstream change in the b-expo-
nent of C-Q relationships, following the numerical experiments of section 4.1, C-Q hysteresis loops occur in the
mixture only if there is a phase shift (/ 6¼ 0) between the tributaries (Figure 5). This condition is clearly met in the
Amazon as the precipitation and discharge patterns vary greatly between (1) Andean and Lowland regions due
to orographic effects (Espinoza et al., 2015); (2) different and large subbasins (e.g., Negro, Solim~oes, and Madeira
rivers) due to the migration of the SAMS spatial and variability in intensity during the year (Espinoza et al., 2009b;
Guimberteau et al., 2012; Vera et al., 2006) (Figure 1). Given this hydroclimatic spatial variability across the Amazon
Basin in terms of timing of peak discharges in the year (Figure 1c) and Q variability (Table 2), we can expect that
the largest subbasins, integrating over vast areas of the Amazon Basin, will display wider hysteresis loops.
To summarize, our model leads us to expect that, in the context of the Amazon Basin, C-Q relationships
should become more ‘‘dilutional’’ downstream through mixing of compositionally different tributaries, espe-
cially through the admixture of relatively dilute, ‘‘pure’’ Lowland waters to more solute-rich, Andean-derived
waters. This effect should be more prominent for solutes for which the Andean region is the main source,
i.e., Na1, Cl- and SO224 , whereas for solutes significantly produced in the Lowlands such as K
1 and Si, the C-
Q relationships should remain relatively ‘‘chemostatic.’’ Additionally, the C-Q relationships should also
exhibit wider hysteresis loops downstream. In the following section we test these predictions against obser-
vations summarized in our database.
5. Comparison With Field Data
5.1. Validity of the Assumptions Underlying the Model
Before checking our model predictions against real-world data, it is necessary to evaluate the validity of the
assumptions underlying the model in the context of the Amazon Basin, the natural laboratory chosen for
the present study. Basically, our mixing model represents what we can call from a conceptual point of view
a ‘‘case of aggregation.’’ More physically, this corresponds to a confluence or a set of confluences, plus the
river reaches (including associated floodplains) between the different upstream and downstream stations
of observation. At the time resolution used in this study, two main assumptions are thus underlying our
mixing model. First, the solutes under consideration need to be conservative during water mixing (such
that at all times F11 F25 Fm). Second, any signal smearing and phase shifting, by the effect of transport
alone (Kirchner, 2016a), should be negligible. In a more operational phrasing, this latter condition is equiva-
lent to saying that the upstream and downstream stations of measurements are sufficiently close to one
another such that any effect of transport between these stations (in the river reach and associated flood-
plains) can be neglected.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8673
We first note that the requirement of solute conservativity is most likely fulfilled by the major solutes
under consideration here, as shown by the systematic studies of the ‘‘Encontro das Aguas’’ mixing zone
at the confluence between the Solim~oes and Negro rivers (e.g., Aucour et al., 2003; Guinoiseau et al.,
2016; Laraque et al., 2009). The validity of the second assumption can be evaluated by comparing the
properties (mean, amplitude of variation, phase) of signals (discharge and solute fluxes) upstream and
downstream from real-world cases of aggregation. However, this requires in turn that all the input fluxes
upstream of the case of aggregation are known, in order to obtain a reliable estimate of the summed
input flux. We emphasize that such approach is possible only for additive properties such as discharge
and solute fluxes - provided, again, that these solutes are conservative during mixing - and not for solute
concentration, hence the interest of considering Q and F in the present analysis. In the supporting infor-
mation section S4 and Table S2, the effect of transport on the signals of discharge and solute fluxes is
evaluated in detail for two cases of aggregation in the Amazon for which all input fluxes are known,
namely PVE ! FAZ and MAN1 SER1CAI1 FAZ ! OBI. Our analysis shows that even over the relatively
long river reaches characterizing these two cases of aggregation, and despite the presence of significant
water and solute routing through floodplains, the signals are not significantly affected by transport, at
least at the time resolution (10 day period) of the present study.
To summarize, the two main assumptions underlying our tributary mixing model presented in section 4 are
shown to be broadly valid in the case of the Amazon Basin.
5.2. Concentration-Discharge Relationships in the Amazon
5.2.1. Trends: b-Values
In the Amazon, major solute C-Q relationships are very diverse in shape (Figure 6 and supporting information
Figures S47-S59). We fitted the two parameters of log C5a1b:log Q to the 10 day period C and Q time series
using the Python Numpy polyfit function, to obtain the parameters of equation (3). Most C-Q relationships can
be reasonably fitted with by a power law with a 0-to-negative b exponent, as observed by Godsey et al.
(2009) for small basins, Moon et al. (2014) for large rivers, or Torres et al. (2015) and Moquet et al. (2016) for
rivers of the Amazon system. Concentration-discharge relationships in the Amazon thus cover the full range
of behavior between ‘‘pure dilution’’ (best represented by Cl- at OBI with a b-value of 21.066 0.04; Figures 6
and 7) and ‘‘chemostatic’’ (with b-values undistinguishable from 0: Si at BOR, LAB, PVE and FAZ; Na1 at SER
and ITA; K1 at ATA, SER, and TAB; Mg21 at BOR and CAI; Ca21 at BOR, SER, and CAI; and SO224 at BOR and ITA).
For some stations/solutes C even increases with Q (significantly positive b-values): Si at TAB; K1 at BOR, RUR,
ITA, and PVE; Mg21 at SER; Ca21 at ITA; Cl- at SER. The most consistent trends are usually obtained for solutes
showing a strong dilution effect (such as Cl- at OBI; Figure 6), while a large relative dispersion exists for ele-
ments with a flatter pattern in the C-Q diagrams (such as K1 at ATA; Figure 6).
The largest b-values (in absolute value; Table 4 and Figure 7) are obtained for Cl- (mean of 20.40 [from
21.06 to 0.22]) and Na1 (-0.31 [-0.82 to 0.02]) while Si and K1 display the values closest to 0 (-0.07 [-0.33 to
0.07] and 20.04 [-0.49 to 0.10], respectively). Andean stations usually exhibit b-values closer to 0 (e.g., from
20.04 to 20.32 depending on the solute for FOR) compared to downstream stations (-1.06 to 20.33 for
OBI). Therefore, in general lower b-values are observed for mixed rivers compared to rivers draining only
the Andes or the Lowlands (Figure 7). The effect is stronger for Na1 and SO224 than for Ca
21 and Mg21.
Chlorine displays the lowest b-values of all elements for most basins, but has higher b-values in basins
draining exclusively the low-relief areas (stations LAB, SER, CAI, ITA). Finally, there is almost no difference in
the b-value of Si and K1 across the different river types, with values mostly around 0 (‘‘chemostatic’’), except
for the largest basins (stations MAN and OBI).
These observations, confirming and refining those of Torres et al. (2015) and Moquet et al. (2016) on the
downstream evolution of b-exponents across geomorphic gradients, are therefore entirely compatible with
the predictions based on our tributary mixing model developed in section 4.
5.2.2. Shape: C-Q Hysteresis Loops Quantified by Phase Offset Between Discharge and Solute Fluxes
Concentration-discharge relationships are parametric curves where time is the implicit parameter, and
where two signals (C and Q) are plotted against each other. In such a diagram, loops will appear if there is a
difference in time between the peaks of the two signals. If the two signals vary with no offset between their
peaks, a positive trend, without any loop is obtained (panels a to c in Figures 4 and 5). Similarly, if the two
signals vary in phase opposition, a negative trend, without any loop is obtained a situation close to that
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8674
depicted in (panels m to o in Figures 4 and 5). On the contrary, any offset between the peaks will make the
two signals attain their extrema at different times, thereby generating hysteresis in the parametric curve
(panels d to l in Figures 4 and 5). In the present study we use the phase offset between the signals of dis-
charge and solute fluxes of a given river as a metric for the significance of C-Q hysteresis loops. The validity
of this metric is demonstrated in the supporting information section S5.
The phase offset between F and Q, uF2uQ; is provided in Table 5 and displayed in Figure 8. For most sol-
utes, F peaks lag behind Q peaks (significantly positive values of uF2uQ) at the Andean station BOR and at
. .
.
.
..
BOR ATA
SER
PVE
OBIOBI
ITA
PVE
Figure 6. Concentration-discharge (C-Q) relationships for a selection of gauging stations and solutes. The power law best
fit is indicated as a dotted curve, along with the best fit equation (where concentrations in bracket are in mg.L21 and Q is
in m3.s21). The exponent of this equation is the b-exponent of the C-Q relationship. In such diagrams, a strictly chemo-
static behavior corresponds to a horizontal line. The data points are color-coded following the instantaneous relative Q
change, calculated as the derivative of Q with respect to time, divided by the value of Q, and expressed in %. Negative val-
ues (red dots) correspond consequently to decreasing discharge while positive values (blue dots) correspond to increas-
ing discharge. In order to optimize the use of the colorbar, the colorbar limits were set to1 and - the standard deviation
of the relative Q change recorded at the station. Here the exact values along the colorbar refer only to the Obidos station
(OBI).
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8675
Figure 7. b-exponent power law best fits of C-Q relationships (e.g., Figure 6) for the different domains of the Amazon
Basin, as defined in Figure 1b. A value of 0 corresponds to a chemostatic behavior (meaning that C is constant despite Q
variations); a value of 21 indicates a pure dilutional behavior. Data points are color-coded following the proportion of the
corresponding drainage area that is located in the Andes. The b-values are reported in Table 4.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8676
the downstream stations MAN, PVE, FAZ, and OBI. For the lowland station ITA, most F peaks occur earlier in
the year than the Q peak. For most solutes, uncertainties preclude the measurement of any phase offset
between F and Q peaks for stations FOR, SER, and CAI, In terms of absolute value, the phase offset juF2uQj
of most solutes is in the range 0 (meaning that Q and F vary at pace) to 23 days for Si at TAB and to 89 days
for SO224 at OBI. For a given solute, there are significant differences in phase offset between stations (Table
5). Although phase offset remains in a relatively limited range across the set of stations for Cl-, Na1, and Si,
for the other solutes uF2uQ is higher in mixed rivers than in rivers draining only the Andes or the Lowlands
(Figure 8), indicating that hysteresis loops are more prominent at downstream stations than at upstream
ones. The fact that hysteresis loops are wider at downstream stations is expected from our modeling results,
as these stations collect water from a range of areas spanning different hydroclimatic regimes, and there-
fore are the most likely to have tributaries which differ in timing of peak discharge and amplitude of varia-
tions. On average the lowest mean juF2uQj are those of Si, Na1, K1, and Cl-, whereas the F peaks of Mg21,
Ca21, and SO224 fluxes have longer time lags with Q peaks. This pattern is consistent with the fact that Si
Table 4
‘‘b-exponent’’ Power Law Best Fits of C-Q Relationships for Each Solute and Station
Si SO224 Cl
- Na1 K1 Mg21 Ca21
Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d.
FOR 20.04 0.03 20.19 0.04 20.32 0.06 20.22 0.04 20.07 0.03 20.21 0.03 20.11 0.04
BOR 20.02 0.05 20.03 0.05 20.46 0.06 20.27 0.06 0.11 0.04 20.03 0.06 0.04 0.05
ATA 20.04 0.02 20.35 0.03 20.59 0.04 20.33 0.05 0.01 0.02 20.27 0.02 20.12 0.02
RUR 20.02 0.01 20.37 0.04 20.66 0.03 20.29 0.02 0.06 0.03 20.20 0.02 20.15 0.02
LAB 0.00 0.03 20.14 0.04 20.19 0.05 20.29 0.03 20.10 0.02 20.31 0.02 20.37 0.02
SER 20.17 0.03 20.48 0.19 0.22 0.08 0.02 0.14 20.03 0.11 0.10 0.06 0.05 0.05
CAI 20.11 0.01 20.23 0.08 20.13 0.02 20.15 0.02 20.07 0.03 20.01 0.02 0.01 0.02
ITA 20.06 0.01 0.05 0.06 20.36 0.06 0.02 0.03 0.06 0.02 20.24 0.04 0.07 0.02
TAB 0.07 0.05 20.11 0.06 20.15 0.11 20.52 0.06 0.01 0.03 20.22 0.05 20.08 0.05
MAN 20.19 0.02 20.66 0.08 20.84 0.05 20.56 0.06 20.12 0.04 20.22 0.05 20.14 0.04
PVE 0.01 0.01 20.30 0.02 20.38 0.04 20.29 0.02 0.06 0.01 20.24 0.02 20.26 0.02
FAZ 0.00 0.01 20.22 0.02 20.30 0.04 20.30 0.03 20.03 0.02 20.22 0.02 20.26 0.02
OBI 20.33 0.04 20.85 0.07 21.06 0.04 20.82 0.03 20.34 0.04 20.49 0.04 20.50 0.05
Table 5
Phase Offset Between Q and F (as a Measure of Hysteresis in C-Q Relationships) for Each Solute and Station
Si SO224 Cl
- Na1 K1 Mg21 Ca21
Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d.
days days days days days days days
FOR 17 12 12 43 255 500 24 62 26 54 22 13 8 101
BOR 4 3 35 2 228 68 29 8 16 13 25 1 36 7
ATA 22 1 25 0 219 2 21 2 4 2 23 0 4 1
RUR 25 4 4 2 11 41 1 6 21 18 21 2 22 5
LAB 10 1 18 4 3 50 23 3 8 5 25 1 23 5
SER 2 2 281 172 11 34 29 8 9 13 21 11 14 28
CAI 22 3 11 67 14 33 22 8 10 15 21 12 0 23
ITA 26 0 24 30 23 11 214 4 24 2 212 2 216 3
TAB 223 0 26 0 225 1 33 0 22 1 15 0 20 1
MAN 1 0 78 0 3 1 24 0 25 1 29 0 34 0
PVE 24 0 18 0 211 4 23 1 2 1 2 0 9 1
FAZ 4 0 11 0 10 2 11 1 6 1 8 0 12 0
OBI 3 0 89 0 128 2 20 0 35 0 51 0 44 0
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8677
Figure 8. Phase offset between Q and F (as a measure of hysteresis in C-Q relationships) for the different domains of the Amazon Basin, as defined in Figure 1b
(see supporting information section S5 and supporting information Figures. S45 and S46). A value of 0 days indicates that variation in Q and F are in phase, and
values of 182 or2182 days (half a year) correspond to the case where Q and F are in phase opposition. Between these values, phase offset indicate the presence
of hysteresis in C-Q relationships. Data points are color-coded following the proportion of the corresponding drainage area that is located in the Andes. The data
used in this figure are reported in Table 5.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8678
and K1 display a relatively homogenous concentration across the Amazon Basin regardless of the domain
(Table 3). In such conditions of homogenous concentration (i.e., rc  1), our model predicts chemostatic C-Q
relationships in the mixed waters regardless of the phase offset between the tributaries input (supporting
information section S3), meaning in turn that F and Q vary in phase in these mixed waters.
To our knowledge this is the first time that a systematic trend in the magnitude of C-Q hysteresis loops ver-
sus catchment heterogeneity is reported. This trend is compatible with the predictions based on our tribu-
tary mixing model presented in section 4.
Figure 9. Power spectral density (P.S.D) of discharge (Q) and solute fluxes (F) for the station OBI (at the mouth of the Amazon) for the 7 major solutes considered
in this study (see supporting information section S1). The frequency of 1 y21 is indicated as a vertical dotted line. The other dotted line is the power law best fit
(linear in this log-log plot) of the power density spectrum for frequencies above 1 y21, and the scaling factor a (exponent of this power law best fit) is provided in
each panels.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8679
5.3. Discussion and Implications
To summarize, our analysis shows that the C, Q, and F time series of 13 gauging stations of the SNO-HYBAM
network on the Amazon are fully compatible with our model of tributary mixing. In addition, the shifts in
the trend and shape of C-Q relationships are accounted for by our mixing model, for all solutes considered.
Obviously, alternative models can reproduce the data, but in any case tributary mixing does occur in large
river systems and therefore has to be taken into account. We also acknowledge that other factors and pro-
cesses are at play when comparing different stations across the Amazon basin. Solutes can be reactive such
that they are not only added downstream but can be lost, for example, by adsorption or mineral precipita-
tion or vegetation uptake e.g., in floodplains or alluvial deposits (Bouchez et al., 2012) or at confluences
(Guinoiseau et al., 2016). Although the list of elements selected for the present study most likely fulfill this
condition of solute conservativity, additional complexities (e.g., water and solute routing through flood-
plains) might affect the exact figures presented in this study. However, despite these additional complexi-
ties that cannot be taken into account for now, both by lack of data and difficulty of building up a simple
tractable model, the first-order analysis performed here supports a control of large-scale C-Q relationships
by mixing.
Hydrochemical time series often display variations at all time scales. Spectral analysis is therefore a powerful
approach to examine the behavior of these different frequencies contributing to the signal, and to explore
how the signal is modified by a given operator (e.g., a catchment or a river reach) as a function of frequency
(Aubert et al., 2013; Duffy & Gelhar, 1985, 1986; Godsey et al., 2010; Kirchner & Neal, 2013; Kirchner et al.,
2000, 2001; Neal et al., 2012). We computed the Fourier transform of the Q and F time series (see supporting
information section S1). Hydrochemical time series of relatively small systems have been reported to univer-
sally follow a ‘‘1=f ’’ scaling, i.e., their power spectral density varies as a function of 1=f2a (with a > 0) for fre-
quency higher than some threshold. This approach was hitherto applied to Q and C time series, but here
we expand its use to F time series (see supporting information section S1). A large value of a means in par-
ticular that the contribution of high frequencies is much smaller than that of low frequencies. Therefore we
use a as a measure of the dominance of the 1 y21-frequency over higher frequencies in the Q and F time
series. The power spectral density of Q and F are shown in Figure 9 and supporting information Figures S1–
S13. All spectra show the highest power for the 1 y21-frequency, with a strong decrease of the power for
higher frequencies. The spectra can be reasonably fitted by a power law for frequencies higher than 1 y21,
yielding the a-value. The Q a -values (Table 6) vary from 1.25 (CAI) to 3.48 (OBI), and span a wider ranger
than that reported for smaller basins by Kirchner et al. (2000, 2001), Godsey et al. (2010), Neal et al. (2012),
Kirchner and Neal (2013), or Aubert et al. (2013) with most values in the range 1–2. Values of a for F are
commensurate to those of Q, consistent with the fact that the variations of C are relatively small and that at
first order, F scales with Q across the set of gauging stations and solutes considered. Fluxes a values range
from 0.01 (Si at FOR, a value equivalent to white noise, i.e., same power of all frequencies) to 4.28 (Cl- at
Table 6
‘‘1/f-Scaling Factor’’ a of Power Density Spectra (e.g., Figure 9) of Discharge (Q) and Solute Fluxes (F)
Discharge Si SO224 Cl
- Na1 K1 Mg21 Ca21
Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d. Mean 1 s.d.
FOR 0.7 0.3 0 0.3 0.6 0.3 1.2 0.3 0.5 0.3 0.5 0.3 0.1 0.2 0.5 0.3
BOR 1.4 0.4 1.5 0.3 1.5 0.3 1.1 0.3 0.9 0.3 1.8 0.4 1.8 0.3 1.9 0.3
ATA 1.5 0.3 2 0.3 2.1 0.3 2.3 0.3 2.4 0.3 1.9 0.3 2.1 0.4 2 0.3
RUR 1.4 0.3 1.7 0.3 1.6 0.3 1.5 0.3 1.8 0.3 1.8 0.3 1.5 0.3 1.4 0.3
LAB 2.1 0.3 1.3 0.2 2 0.4 2.9 0.4 2.8 0.3 2.4 0.3 2.5 0.4 2.5 0.4
SER 2.9 0.3 2.3 0.3 2.9 0.4 3.1 0.3 3.4 0.3 2.6 0.4 2.3 0.4 2.4 0.3
CAI 1.3 0.3 1.7 0.3 2.5 0.3 2.6 0.3 2.1 0.3 1.9 0.4 1.8 0.4 1.6 0.4
ITA 2.5 0.3 1.8 0.4 3.5 0.4 3 0.4 2.7 0.4 2.4 0.4 1.9 0.3 3.1 0.4
TAB 2.3 0.3 4 0.3 2.6 0.3 2.6 0.4 3 0.3 2.1 0.3 2.7 0.3 1.7 0.3
MAN 3.2 0.3 1.8 0.3 2.8 0.4 3.5 0.3 2.8 0.4 2.7 0.3 2 0.3 2 0.3
PVE 2 0.3 2.9 0.3 2.4 0.4 4.3 0.4 3 0.4 3 0.3 2.2 0.3 2.8 0.3
FAZ 2.1 0.3 2.7 0.3 3 0.3 2.9 0.3 3 0.3 3.4 0.3 3.1 0.4 3.5 0.4
OBI 3.5 0.3 3.4 0.3 2.2 0.3 2.8 0.4 3.1 0.3 3.2 0.3 2.5 0.3 2.3 0.3
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8680
Figure 10. ‘‘1/f-scaling factor’’ a of power spectral density (e.g., Figure 9) of discharge (small open circles) and solute fluxes (large colored circles) for the different
domains of the Amazon Basin, as defined in Figure 1b. A value of 0 corresponds to ‘‘white noise’’ (meaning that all frequencies contribute equally to the signal);
higher values indicate 1/f-scaling. Data points are color-coded following the proportion of the corresponding drainage area that is located in the Andes. The a-val-
ues are reported in Table 6.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8681
PVE), but no systematic difference is observed between two given elements. However, there is a clear differ-
ence in a values between the different river types: from 0.01 (Si at FOR) to 2.37 (Na1 at ATA) for Andean sta-
tions; from 1.29 (Si at LAB) to 3.54 (SO224 at ITA) for pure Lowland rivers; and from 1.80 (Si at MAN) to 4.28
(Cl- at PVE) for the most downstream stations (Figure 10). To our knowledge, this is the first time that such a
spectral analysis of solute river fluxes is conducted, especially at this large scale, and the first time that a
change in the scaling factor of power spectral spectra of hydrochemical fluxes, as a function of basin hetero-
geneity is reported. A preliminary analysis suggests that this change in the a-value of F and Q time series
across our set of gauging stations (Figure 10) is not due to transport of water and solutes alone (see sup-
porting information section S4). One alternative explanation would be that mixing leads to this continental-
scale dampening of high-frequency signals. This would be possible if the high-frequency components of
the various inputs were likely to cancel out (e.g., were in phase opposition), which is plausible but remains
to be tested with a more systematic assessment of the phase lags of Q and F between different regions.
The present case study focuses on the sole Amazon Basin, but we suggest that our findings should be appli-
cable to other large river basins. Indeed, the world’s largest river basins also drain areas with contrasted
characteristics in terms of landscape, lithology, climate, and ecosystems. However, we also emphasize that
for a particular system, other factors might have to be taken into account and the model might require
some adjustments. For example, the downstream evaporative loss that occurs in semiarid systems will exert
an additional influence on solute concentration and more generally C-Q relationships at the most down-
stream stations.
We show that mixing of water and solutes between compositionally different tributaries produces down-
stream shifts in C-Q relationships, both in terms of trends (b-exponent) and shape (hysteresis). This fact chal-
lenges the use of C-Q relationships to infer weathering properties in large catchments, where aggregation
of such heterogeneity is most likely. This whole view could be ‘‘downscaled’’ to a smaller, more homoge-
nous catchment. Indeed, even in small catchments, packets of water follow different flowpaths acquire dif-
ferent chemical composition and mix before or when entering the stream where the gauging station is
located. One prominent example of such flowpath distribution would be the separation between ‘‘surface
runoff’’ and deeper groundwater (e.g., Calmels et al., 2011). The relative contribution of these different flow-
paths will vary through time depending on hydroclimatic conditions, just like for tributaries in a large river
basin, as shown by hysteresis loops in C-Q relationships of small basins recorded during storm events.
Therefore, characteristics of C-Q relationships, even in small catchments, could also simply result from mix-
ing of different flowpaths. To circumvent such issues one possibility could be to select solutes such that
they reflect the input of only one of the water reservoirs.
6. Conclusion
In this paper we addressed the downstream evolution of concentration-discharge (C-Q) relationships in
large systems, using the Amazon basin in a case study. In such large systems, as water and solutes move
downstream they mix with water and solutes from compositionally different tributaries. Therefore, we
explored how mixing between these different types of water could affect C-Q relationships and potentially
hamper their use as messengers of the Critical Zone in terms of weathering properties. Based on discharge
and solute concentration at a 10 day frequency recorded across 13 gauging stations throughout the Ama-
zon basin, we show the following downstream evolution:
 Increase in the exponent (‘‘b’’) of power law best fit of C-Q relationships, from a quasi ‘‘chemostatic’’ (con-
stant C) behavior in homogenous basins (i.e., those draining only the lowlands, or only the Andes) to
more dilutional behavior in more heterogeneous basins.
 Increase in the magnitude of C-Q hysteresis loops, especially for the most downstream stations, due to an
increase in the phase offset between discharge and solute flux signals of the different tributaries.
We built up a simple model of tributary mixing, which predicts, given the differences in water and solutes
input signals between tributaries draining different geomorphic/hydroclimatic subregions in the Amazon,
the above patterns. These findings have broader implications for C-Q relationships in other large river sys-
tems, but also for smaller systems where mixing of different flowpaths could lead to the same effects on
the C-Q relationships.
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BOUCHEZ ET AL. C-Q RELATIONSHIPS IN THE AMAZON 8682
From the methodological perspective, we also showed that C-Q hysteresis is an overlooked feature of C-Q
relationships that can be highly informative to detect mixing or more generally to discriminate between dif-
ferent models aiming at reproducing C-Q trends. We suggest that the phase offset between Q and C (or flux
time series F) could be used as a simple proxy for quantification of C-Q hysteresis loops. Finally, we show
how solute fluxes F, as additive properties like Q but unlike C, can be used in tributary mixing models and in
spectral analysis to retrieve transport properties - but also potentially reactivity - of chemical elements in
the Critical Zone.
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Acknowledgments
Additional information is available in
the supporting information of this
manuscript. Original data can be found
at on the SNO-HYBAM website (http://
www.ore-hybam.org). We are grateful
to J. West, M. Torres, J. Baronas,
J. Gaillardet, P. Floury, and the whole
HYBAM team for fruitful scientific
discussions. M. Albenque is thanked
for checking the mathematical
derivations. Data used in this study can
be dowloaded from http://www.ore-
hybam.org/. This work was partially
funded by the Programme
‘‘Emergences’’ of the City of Paris
‘‘Chemical weathering of sediments in
large tropical floodplains’’ (agreement
205DDEEES165). This is IPGP
contribution 3876.
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