Concentration–discharge relationships reflect chemostatic...
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HYDROLOGICAL PROCESSES Hydrol. Process. 23, 1844–1864 (2009) Published online 7 May 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.7315 Concentration–discharge relationships reflect chemostatic characteristics of US catchments Sarah E. Godsey, 1 * James W. Kirchner, 1,2,3 and David W. Clow 4† 1 Department of Earth and Planetary Science, 307 McCone Hall, University of California, Berkeley, CA 94720-4767, USA, 510-643-8559 2 Swiss Federal Institute for Forest, Snow, and Landscape Research (WSL), Birmensdorf, Switzerland 3 Department of Environmental Sciences, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland 4 U.S. Geological Survey, Water Resources Division, Denver Federal Center, MS 415, Box 25046, Denver, CO 80225 USA, 303-236-4882 x294 Abstract: Concentration–discharge relationships have been widely used as clues to the hydrochemical processes that control runoff chemistry. Here we examine concentration–discharge relationships for solutes produced primarily by mineral weathering in 59 geochemically diverse US catchments. We show that these catchments exhibit nearly chemostatic behaviour; their stream concentrations of weathering products such as Ca, Mg, Na, and Si typically vary by factors of only 3 to 20 while discharge varies by several orders of magnitude. Similar patterns are observed at the inter-annual time scale. This behaviour implies that solute concentrations in stream water are not determined by simple dilution of a fixed solute flux by a variable flux of water, and that rates of solute production and/or mobilization must be nearly proportional to water fluxes, both on storm and inter-annual timescales. We compared these catchments’ concentration–discharge relationships to the predictions of several simple hydrological and geochemical models. Most of these models can be forced to approximately fit the observed concentration–discharge relationships, but often only by assuming unrealistic or internally inconsistent parameter values. We propose a new model that also fits the data and may be more robust. We suggest possible tests of the new model for future studies. The relative stability of concentration under widely varying discharge may help make aquatic environments habitable. It also implies that fluxes of weathering solutes in streams, and thus fluxes of alkalinity to the oceans, are determined primarily by water fluxes. Thus, hydrology may be a major driver of the ocean-alkalinity feedback regulating climate change. Copyright 2009 John Wiley & Sons, Ltd. KEY WORDS concentration–discharge; dissolved load; solute; chemical weathering; Hydrologic Benchmark Network; water quality; catchment; watershed Received 27 August 2008; Accepted 19 February 2009 INTRODUCTION Chemical weathering and solute transport are coupled with hydrology in catchments, and this coupling is reflected in the relationships between solute concentra- tions and stream discharge. Catchment hydrologists often study concentration–discharge relationships because the necessary data are frequently available and the initial analysis is straightforward. However, despite decades of work, there are still open questions about what concentration–discharge relationships can tell us about catchment behavior. Concentrations of the major base cations and silica have generally been observed to decrease with discharge (Hem, 1948, 1985; Johnson et al., 1969 and references therein; Waylen, 1979; Walling and Webb, 1986; Clow and Drever, 1996). Typical concentration–discharge anal- yses have centered on mixing models of different source waters (e.g. event and pre-event water; old and new water; or soil water, groundwater and precipitation) * Correspondence to: Sarah E. Godsey, Department of Earth and Plane- tary Science, 307 McCone Hall, University of California, Berkeley, CA 94720-4767, USA, 510-643-8559. E-mail: [email protected] † The contribution of David W. Clow to this article was prepared as part of his official duties as a United States Federal Government employee. inferred from the shape of the concentration–discharge relationship for different solutes (Johnson et al., 1969; Hall, 1970, 1971). In other studies, researchers have inferred the relative timing of mixing from hysteresis loops observed in concentration–discharge plots (Evans and Davies, 1998; Evans et al., 1999; House and War- wick, 1998; Hornberger et al., 2001; Chanat et al., 2002). Evans and Davies (1998) proposed that the form and direction of hysteresis loops could uniquely identify the rank order of end-member concentrations in a three end-member mixing scenario. However, Chanat et al. (2002) showed that even the small number of assumptions required for this identification are not always valid, and that when the assumptions are violated, hysteresis loops cannot definitively distinguish the relative concentrations of different end-members. Nonetheless, other character- istics of concentration–discharge relationships may pro- vide insight into the coupling of chemical weathering and hydrological processes in catchments. In this paper, we identify common features of concen- tration–discharge relationships across a range of hydro- chemically distinct catchments with minimal human impacts, and discuss the implications of those features. We then compare several simple models against the Copyright 2009 John Wiley & Sons, Ltd.
Transcript of Concentration–discharge relationships reflect chemostatic...
HYDROLOGICAL PROCESSESHydrol. Process. 23, 1844–1864 (2009)Published online 7 May 2009 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/hyp.7315
Concentration–discharge relationships reflect chemostaticcharacteristics of US catchments
Sarah E. Godsey,1* James W. Kirchner,1,2,3 and David W. Clow4†
1 Department of Earth and Planetary Science, 307 McCone Hall, University of California, Berkeley, CA 94720-4767, USA, 510-643-85592 Swiss Federal Institute for Forest, Snow, and Landscape Research (WSL), Birmensdorf, Switzerland
3 Department of Environmental Sciences, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland4 U.S. Geological Survey, Water Resources Division, Denver Federal Center, MS 415, Box 25046, Denver, CO 80225 USA, 303-236-4882 x294
Abstract:
Concentration–discharge relationships have been widely used as clues to the hydrochemical processes that control runoffchemistry. Here we examine concentration–discharge relationships for solutes produced primarily by mineral weatheringin 59 geochemically diverse US catchments. We show that these catchments exhibit nearly chemostatic behaviour; theirstream concentrations of weathering products such as Ca, Mg, Na, and Si typically vary by factors of only 3 to 20 whiledischarge varies by several orders of magnitude. Similar patterns are observed at the inter-annual time scale. This behaviourimplies that solute concentrations in stream water are not determined by simple dilution of a fixed solute flux by a variableflux of water, and that rates of solute production and/or mobilization must be nearly proportional to water fluxes, both onstorm and inter-annual timescales. We compared these catchments’ concentration–discharge relationships to the predictions ofseveral simple hydrological and geochemical models. Most of these models can be forced to approximately fit the observedconcentration–discharge relationships, but often only by assuming unrealistic or internally inconsistent parameter values. Wepropose a new model that also fits the data and may be more robust. We suggest possible tests of the new model for futurestudies. The relative stability of concentration under widely varying discharge may help make aquatic environments habitable.It also implies that fluxes of weathering solutes in streams, and thus fluxes of alkalinity to the oceans, are determined primarilyby water fluxes. Thus, hydrology may be a major driver of the ocean-alkalinity feedback regulating climate change. Copyright 2009 John Wiley & Sons, Ltd.
KEY WORDS concentration–discharge; dissolved load; solute; chemical weathering; Hydrologic Benchmark Network; waterquality; catchment; watershed
Received 27 August 2008; Accepted 19 February 2009
INTRODUCTION
Chemical weathering and solute transport are coupledwith hydrology in catchments, and this coupling isreflected in the relationships between solute concentra-tions and stream discharge. Catchment hydrologists oftenstudy concentration–discharge relationships because thenecessary data are frequently available and the initialanalysis is straightforward. However, despite decadesof work, there are still open questions about whatconcentration–discharge relationships can tell us aboutcatchment behavior.
Concentrations of the major base cations and silicahave generally been observed to decrease with discharge(Hem, 1948, 1985; Johnson et al., 1969 and referencestherein; Waylen, 1979; Walling and Webb, 1986; Clowand Drever, 1996). Typical concentration–discharge anal-yses have centered on mixing models of different sourcewaters (e.g. event and pre-event water; old and newwater; or soil water, groundwater and precipitation)
* Correspondence to: Sarah E. Godsey, Department of Earth and Plane-tary Science, 307 McCone Hall, University of California, Berkeley, CA94720-4767, USA, 510-643-8559. E-mail: [email protected]† The contribution of David W. Clow to this article was prepared as partof his official duties as a United States Federal Government employee.
inferred from the shape of the concentration–dischargerelationship for different solutes (Johnson et al., 1969;Hall, 1970, 1971). In other studies, researchers haveinferred the relative timing of mixing from hysteresisloops observed in concentration–discharge plots (Evansand Davies, 1998; Evans et al., 1999; House and War-wick, 1998; Hornberger et al., 2001; Chanat et al., 2002).Evans and Davies (1998) proposed that the form anddirection of hysteresis loops could uniquely identify therank order of end-member concentrations in a threeend-member mixing scenario. However, Chanat et al.(2002) showed that even the small number of assumptionsrequired for this identification are not always valid, andthat when the assumptions are violated, hysteresis loopscannot definitively distinguish the relative concentrationsof different end-members. Nonetheless, other character-istics of concentration–discharge relationships may pro-vide insight into the coupling of chemical weathering andhydrological processes in catchments.
In this paper, we identify common features of concen-tration–discharge relationships across a range of hydro-chemically distinct catchments with minimal humanimpacts, and discuss the implications of those features.We then compare several simple models against the
Copyright 2009 John Wiley & Sons, Ltd.
CONCENTRATION–DISCHARGE RELATIONSHIPS 1845
1
10
100
0.1 10
Si x2CaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Cache Creek near Jackson, Wyoming
0.1
1
10
100
0.1 10
SiCaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Simple dilution: C~1/Q
Andrews Creek near Mazama, Washington
1 1
0.1
1
10
0.1 10
SiCa /2Na
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Hayden Creek near Hayden Lake, Idaho
0.1
1
10
100
0.1 10 100
Si x2CaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Elder Creek near Branscomb, California
1 1
1
10
100
0.01 0.1 10
Si x4CaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Washington Creek, Isle Royale, Michigan
1
10
100
0.1 10
SiCaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Sagehen Creek near Truckee, California
0.1
1
10
100
0.01 0.1 10 100
SiCaNa
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
North Sylamore Creek near Fifty Six, Arkansas
0.1
1
10
100
0.01 0.1 10
Si x2CaNa/4
Con
cent
ratio
n (p
pm)
Discharge (mm/d)
Merced River near Yosemite, California
1 1
1 1
Figure 1. Concentration–discharge relationships for Si, Ca and Na at eight Hydrologic Benchmark Network streams. Each plot has consistent axes(same number of log units for both concentration and discharge), so concentrations determined by dilution of fixed weathering fluxes would havethe same slope as the grey diagonal lines shown (log–log slope of �1). Instead, concentration–discharge relationships conform more closely to
chemostatic behaviour (log–log slope near zero).
observed patterns in concentration–discharge relation-ships and evaluate whether their assumptions are gener-ally valid across the study sites. Finally, we show that theconcentration–discharge relationships observed in manycatchments imply that changes in stream solute fluxes asa result of climatic forcing are, to first order, dependentupon changes in hydrology.
OBSERVATIONS
We plotted concentrations of the major weathering-derived cations (Mg, Na and Ca) and dissolved silica(Si) against instantaneous discharge on logarithmic axesfor 59 sites in the United States Geological Survey’s(USGS) Hydrologic Benchmark Network (HBN) (e.g.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1846 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
Figure 1, discussed further below). The HBN was estab-lished in the mid-1960s to provide a long-term databaseto track changes in the flow and water quality of undis-turbed streams and rivers, and to serve as a reference, or‘benchmark’, for discerning natural from human-inducedchanges in river ecosystems (Leopold, 1962). Most HBNsites have more than 30 years of hydrochemical data, andwater samples were typically collected five to seven timesper year using standard USGS methods (Wilde et al.,1998). The HBN is the only nationwide network of envi-ronmental monitoring sites that tracks the health of riversdraining mid-sized, undisturbed basins in the UnitedStates. The sites are located throughout the country, usu-ally in National Parks, National Forests or reserves whereminimal human influence is expected. Site characteristicscompiled from USGS circulars (Mast and Turk, 1999a,b;Clark et al., 2000; Mast and Clow, 2000) are summarizedin Table I. Drainage areas range from 6Ð1 to 5196 km2
(median D 146 km2) and average annual runoff variesfrom 0Ð7 to 400 cm/year (median D 40 cm/year). Manyenvironments in the United States are represented, includ-ing tropical forests, tundra and eight sites with <10-cmaverage annual runoff, which include arid and semi-aridgrasslands, shrublands and semi-desert areas. Most catch-ments are forested, but several of the basins have substan-tial alpine and grassland areas. The 59 sites encompassa wide range of lithologic settings (Table I). Solute con-centrations in precipitation are not available at all sitesfor the entire period of record, so the reported concen-trations are not corrected by the precipitation chemistry.Stream concentrations of Ca, Na and Mg are at least 10 to100 times higher than available mean precipitation con-centrations. Si concentrations in rainfall are not typicallymeasured, but are normally orders of magnitude lowerthan Si concentrations in streamflow.
We plotted the concentrations of each of the majorweathering-derived solutes against instantaneous dis-charge. We then calculated simple linear regression statis-tics for each of the sites and solutes, and, using Student’st test, determined whether the best-fit slope was sig-nificantly different from reference slopes of zero and�1 (whose meaning is discussed in detail below). Weidentified sites with non-linear relationships between logconcentration and log discharge by examining the resid-uals of the linear fit. Most of the catchments in the HBNexhibit much less variability in concentrations than indischarge (Figure 1). Figure 1 shows concentration as afunction of discharge on logarithmic axes, with the samenumber of log units shown on each axis to facilitate avisual comparison of the relative variability of concentra-tion and discharge. For example, between high flow andlow flow, concentrations at Elder Creek increase by fac-tors of 3 (Ca and Mg), 2Ð6 (Na) and 1Ð5 (Si), as dischargechanges by a factor of ¾6000. Although not plotted inFigure 1, similar relationships also hold for Mg acrossthe HBN sites.
As seen in Figure 1, concentration–discharge plots areoften linear on logarithmic axes, indicating that there isa power–law relationship between concentration, C, and
discharge, Q (i.e. C D aQb, where a and b are constants).The exponent in this power–law relationship (or equiva-lently, the slope of the concentration–discharge relation-ship on logarithmic axes) has a physical interpretation. Aslope of zero would indicate that the catchment behaveschemostatically, that is, the system keeps concentrationsconstant as discharge varies. A slope of �1, on the otherhand, would indicate that concentrations vary inverselywith discharge, as might be expected if dilution were thedominant process controlling concentrations, such thatapproximately constant fluxes of solutes were diluted byvariable fluxes of water.
Power-law concentration–discharge relationships likethose shown in Figure 1 can be usefully summarized bytheir log–log slopes. The best-fit log�C�- log�Q� slopesfor each solute can then be compared across all 59of the HBN sites (Figure 2) and can be compared toreference slopes of zero (chemostatic behaviour) and �1(dilution). Uncertainty in the best-fit slope of š1 standarderror (SE) is indicated by error bars, which are shownif they are larger than the plotting symbols. Across allsites, slopes are generally slightly less than zero. Themeans of the best-fit log–log concentration–dischargeslopes vary between approximately �0Ð05 and �0Ð15(SE of the slope ¾0Ð01–0Ð02) for each of the solutes.No slope is within two SEs of �1 and only four (Na),six (Mg), eight (Ca) and 14 (Si) of the 59 sites haveslopes within two SEs of zero. Slopes across all sitesand solutes are strikingly similar. In general, Si slopesare closer to zero than the slopes of the other solutes.With few exceptions (discussed below), nearly all ofthe concentration–discharge relationships in this studycan be described by power–law relationships with smallnegative exponents. The data in Figure 2 show that mostcatchments behave almost chemostatically for chemicalweathering products such as Ca, Mg, Na and Si.
Although this near-chemostatic behaviour is common,examination of residual plots for all sites and solutesreveals that the best fits for up to 10% of the sites maynot be power-law. Such sites are marked by open circlesin Figure 2 and the plotted slope of the linear fit onlyprovides a general sense of the concentration–dischargerelationship for these sites. We also calculated the ratioof the standard deviation of log discharge to log concen-tration for all sites and solutes (not shown) to quantifythe relative variability of discharge and concentrationwithout making any assumptions about the form of theconcentration–discharge relationship. Only Upper ThreeRuns, SC had Ca and Mg concentrations that were morevariable than its discharge, and only Castle Creek, SDand Dismal River, NE had more variable Na concen-trations. No sites had Si concentrations that were morevariable than discharge. The ratio of the standard devia-tions of log discharge to log concentration ranged fromjust below 1 to 21Ð5, with the median ratio equal to 4Ð1(Ca), 4Ð0 (Mg and Na), and 5Ð3 (Si). These observationsreinforce the inference drawn from the shallow power-law slopes, namely that these catchments behave almost
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1847
Tabl
eI.
Site
char
acte
rist
ics
for
the
59H
ydro
logi
cB
ench
mar
kN
etw
ork
site
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scus
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SC
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lars
1173
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indi
cate
dby
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iled
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ebo
ttom
ofth
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ble
Sta
tion
nam
eSt
atio
nnu
mbe
rD
rain
age
area
[km
2]
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rage
annu
alru
noff
[cm
]L
and
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r/us
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ogy
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nm
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lyte
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re[°
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tem
pera
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[°C
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ean
annu
alpr
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itat
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[cm
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And
rew
sC
reek
near
Maz
ama,
WA
1244
739
057
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balp
ine
fore
stQ
uart
zm
onzo
nite
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dior
ite
410
89a
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332
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2Ð320
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300
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0731
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327
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127
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son,
WY
1301
830
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rest
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ston
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1614
5
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sato
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near
Van
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340
300
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men
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mal
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0677
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23Ð7
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CA
1147
556
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631
215
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1848 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
Tabl
eI.
(Con
tinu
ed)
Sta
tion
nam
eSt
atio
nnu
mbe
rD
rain
age
area
[km
2]
Ave
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annu
alru
noff
[cm
]L
and
cove
r/us
eL
ithol
ogy
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mea
nm
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lyte
mpe
ratu
re[°
C]
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ean
mon
thly
tem
pera
ture
[°C
]M
ean
annu
alpr
ecip
itat
ion
[cm
]
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Cre
ekne
arD
ecat
urC
ity,
IA06
897
950
136
23Pr
airi
e,de
cidu
ous
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s,ag
ricu
ltur
eL
imes
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ale,
allu
vium
,lo
ess
�6Ð6
24Ð6
92
Enc
ampm
ent
Riv
erab
ove
Hog
Park
Cre
ekne
arE
ncam
pmen
t,W
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0662
380
018
850
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tane
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nefo
rest
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assl
and
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nite
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artz
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te,
gran
odio
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lsic
gnei
ss,
calc
areo
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ltst
one,
shal
e,m
udst
one,
till
�5Ð8
17Ð2
76
Eso
pus
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atSh
anda
ken,
NY
0136
219
816
574
Mix
edfo
rest
,re
side
ntia
lSh
ale,
sand
ston
e,co
nglo
mer
ate
�4Ð2
21Ð5
157
Fall
ing
Cre
ekne
arJu
liet
te,
GA
0221
260
018
730
Mix
edfo
rest
Gne
iss,
gabb
ro7Ð3
26Ð6
122
Hal
fmoo
nC
reek
near
Mal
ta,
CO
0708
300
061
40C
onif
erou
sfo
rest
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ndra
Bio
tite
gnei
ss,
schi
st,
mig
mat
ite
�8Ð9
12Ð2
53a
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den
Cre
ekbe
low
NFo
rkne
arH
ayde
nL
ake,
ID
1241
600
057
44C
onif
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sfo
rest
Qua
rtzi
te,
argi
llit
e,si
ltst
one,
dolo
mit
e,li
mes
tone
�221
89
Hol
iday
Cre
ekne
arA
nder
sonv
ille
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A02
038
850
2235
Mix
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rest
Phyl
lite
,sc
hist
,gn
eiss
325
105
Hon
olii
Stre
amne
arPa
paik
ou,
HI
1671
700
031
377
Lav
a,tr
opic
alve
geta
tion
Bas
alt
17Ð6c
28Ð8c
328c,
k
Kah
akul
oaSt
ream
near
Hon
okoh
au,
Mau
i,H
I16
618
000
819
7D
ense
trop
ical
vege
tatio
nB
asal
t17
Ð4d31
Ð1dN
/A
Kaw
ishi
wi
Riv
erne
arE
ly,
MN
0512
448
065
528
Mix
edfo
rest
Ano
rtho
site
,ig
neou
sin
trus
ives
,ti
ll�1
4Ð419
Ð270
Kia
mic
hiR
iver
near
Big
Ced
ar,
OK
0733
570
010
474
Mix
edfo
rest
Shal
e,si
ltst
one,
sand
ston
e4Ð9
27Ð2
130
Kin
gsC
reek
near
Man
hatta
n,K
S06
879
650
10Ð6
20Ta
llgr
ass
prai
rie
Shal
e,li
mes
tone
�1Ð8
26Ð6
84
Lim
pia
Cre
ekab
ove
Ft.
Dav
is,
TX
0843
170
013
51Ð6
Dec
iduo
usfo
rest
,de
sert
and
ripa
rian
plan
tsA
ndes
ite,
basa
lt7Ð1
c25
c45
c
Lit
tle
Riv
erab
ove
Tow
nsen
d,T
N03
497
300
275
93B
road
leaf
fore
stPh
ylli
te,
sand
ston
e,sl
ate,
lim
esto
ne4
2315
2i,j
McD
onal
dsB
ranc
hin
Leb
anon
Stat
eFo
rest
,N
J
0146
650
06Ð1
32B
road
leaf
fore
st/s
wam
pU
ncon
soli
date
dse
dim
ents
12Ð8
25f
116
Mer
ced
Riv
erat
Hap
pyIs
l.B
ridg
ene
arY
osem
ite,
CA
1126
450
046
966
Ð5M
ixed
fore
st,
mea
dow
Gra
nite
,gr
anod
iori
te3
2294
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1849
Tabl
eI.
(Con
tinu
ed)
Sta
tion
nam
eSt
atio
nnu
mbe
rD
rain
age
area
[km
2]
Ave
rage
annu
alru
noff
[cm
]L
and
cove
r/us
eL
ithol
ogy
Low
mea
nm
onth
lyte
mpe
ratu
re[°
C]
Hig
hm
ean
mon
thly
tem
pera
ture
[°C
]M
ean
annu
alpr
ecip
itat
ion
[cm
]
Min
amR
iver
atM
inam
,O
R13
331
500
622
65C
onif
erou
sfo
rest
,gr
assl
ands
,m
eado
wG
rani
te,
intr
uded
byap
lite
,pe
gmat
ite,
basa
lt�7
Ð214
Ð610
0h,i
Mog
ollo
nC
reek
near
Cli
ff,
NM
0943
060
017
917
Mix
edfo
rest
,m
eado
wR
hyol
ite,
quar
tzla
tite
,po
rphy
riti
cflo
ws,
plag
iocl
ase,
biot
ite,
sani
dine
3Ð624
Ð437
Nor
thFo
rkQ
uina
ult
Rne
arA
man
daPa
rk,W
A12
039
300
192
402
Con
ifer
s,al
pine
vege
tatio
n,ba
rero
ckB
asal
t,se
mi-
cons
olid
ated
sedi
men
ts,
slat
e1Ð4
c25
Ð4c32
8c,l
Nor
thFo
rkW
hite
wat
erR
iver
near
Elb
a,M
N05
376
000
262
17A
gric
ultu
re,
broa
dlea
ffo
rest
Lim
esto
ne,
dolo
mite
,sh
ale,
sand
ston
e,ti
ll�1
0Ð722
Ð784
Nor
thSy
lam
ore
Cre
ekne
arFi
fty
Six,
AR
0706
071
015
028
Har
dwoo
d/pi
nefo
rest
Sand
ston
e,do
lom
ite,
shal
e,li
mes
tone
1Ð926
Ð211
2
Popp
leR
iver
near
Fenc
e,W
I04
063
700
360
29M
ixed
fore
stG
rani
tean
dgr
anit
oid
gnei
ss�1
7Ð120
Ð978
Red
But
teC
kat
Ft.
Dou
glas
near
Salt
Lak
eC
ity,
UT
1017
220
018
Ð820
Mix
edfo
rest
,gr
assl
and
Qua
rtzi
te,
sand
ston
e,li
mes
tone
�0Ð7
25Ð4
74a
Rio
Mor
ane
arTe
rrer
o,N
M08
377
900
138
20C
onif
erou
sfo
rest
,m
eado
wG
rani
te,
amph
ibol
ite,
tona
lite
,qu
artz
dior
ite,
quar
tzit
e,sc
hist
,li
mes
tone
�2Ð2
15Ð8
45
Roc
kC
kbe
low
Hor
seC
kne
arIn
tlB
ound
ary,
MT
0616
950
085
01Ð5
Prai
rie,
agri
cult
ure,
graz
ing
Shal
e,un
cons
olid
ated
sedi
men
ts�1
3Ð518
Ð829
Sage
hen
Cre
ekne
arT
ruck
ee,
CA
1034
350
027
40C
onif
erou
sfo
rest
,m
eado
wG
rani
te,
ande
site
,br
ecci
a,ho
rnfe
ls,
till
,al
luvi
um�3
1794
Scap
eO
reSw
amp
near
Bis
hopv
ille
,SC
0213
530
024
938
Mix
edfo
rest
,ag
ricu
ltur
eA
rkos
ican
dar
gill
aceo
ussa
nds
5Ð525
Ð511
5
Sips
eyFo
rkne
arG
rays
on,
AL
0245
025
023
962
Mix
edfo
rest
Lim
esto
ne,
sand
ston
e,sh
ale,
mud
ston
e,cl
ayst
one,
nonp
ersi
sten
tco
albe
ds
525
Ð514
5
Sopc
hopp
yR
iver
near
Sopc
hopp
y,FL
0232
710
026
567
Mix
edfo
rest
Lim
esto
ne,
unco
nsol
idat
edm
arin
ese
dim
ent
12Ð2
27Ð4
145
Sout
hFo
rkR
ocky
Cre
ekne
arB
rigg
s,T
X08
103
900
8611
Ari
dgr
assl
and
Lim
esto
ne,
mar
l,do
lom
ite
8Ð428
Ð275
Sout
hH
ogan
Cre
ekne
arD
ills
boro
,IN
0327
670
099
39B
road
leaf
fore
st,
agri
cult
ure,
graz
ing
Lim
esto
ne,
shal
e,gl
acia
lti
ll,
loes
s�1
Ð922
Ð910
0
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1850 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
Tabl
eI.
(Con
tinu
ed)
Sta
tion
nam
eSt
atio
nnu
mbe
rD
rain
age
area
[km
2]
Ave
rage
annu
alru
noff
[cm
]L
and
cove
r/us
eL
ithol
ogy
Low
mea
nm
onth
lyte
mpe
ratu
re[°
C]
Hig
hm
ean
mon
thly
tem
pera
ture
[°C
]M
ean
annu
alpr
ecip
itat
ion
[cm
]
Sout
hTw
inR
iver
near
Rou
ndM
ount
ain,
NV
1024
930
052
11Ð4
Fore
st,
sage
brus
h,se
mi-
dese
rtFe
lsit
e,tu
ff,
quar
tzit
e,lim
esto
ne,
dolo
ston
e,qu
artz
�1Ð4
22Ð2
20h,
i
Step
toe
Cre
ekne
arE
ly,
NV
1024
495
029
22Fo
rest
,sc
rub,
sem
i-de
sert
Lim
esto
ne,
shal
e�4
Ð319
Ð624
h,i
Talk
eetn
aR
iver
near
Talk
eetn
a,A
K15
292
700
5196
70M
ixed
fore
st,
mix
edsh
rubs
,tu
ndra
Gla
cial
drif
t,co
lluv
ium
,ig
neou
sbe
droc
k�1
1Ð7g
14Ð9g
72g
Tall
ulah
Riv
erne
arC
layt
on,
GA
0217
840
014
611
6B
road
leaf
fore
stG
neis
s,sc
hist
,am
phib
olit
e3Ð3
21Ð6
182
Upp
erT
hree
Run
sne
arN
ewE
llent
on,
SC02
197
300
255
36M
ixed
fore
stA
rkos
icsa
nds
7Ð527
Ð112
0
Upp
erTw
inC
reek
atM
cGaw
,O
H03
237
280
31Ð6
38B
road
leaf
fore
stSa
ndst
one,
shal
e0
24Ð4
105
Val
leci
toC
reek
near
Bay
field
,C
O09
352
900
187
70C
onif
erou
sfo
rest
,al
pine
tund
raSl
ate,
schi
st,
quar
tzite
,br
ecci
a,li
mes
tone
,co
nglo
mer
ate,
quar
tzm
onzo
nite
,gr
anit
e
�5Ð9
17Ð6
117a
Was
hing
ton
Cre
ekat
Win
digo
Isle
Roy
ale,
MI
0400
100
034
42M
ixed
fore
stB
asal
t,an
desi
te,
sand
ston
e,gl
acia
ldr
ift
3Ð4e
N/A
77
Wet
Bot
tom
Cre
ekne
arC
hild
s,A
Z09
508
300
9415
Ð5Se
mid
eser
t,gr
assl
and,
woo
dlan
dsG
rani
te,
basa
lt,
ande
site
,si
ltst
one,
dolo
mit
e,li
mes
tone
7Ð729
Ð463
Ð5
Wil
dR
iver
atG
ilea
d,M
E01
054
200
180
89M
ixed
fore
stSa
ndst
one,
shal
e,si
ltst
one,
gnei
ss,
mic
asc
hist
s,qu
artz
ites
�919
119
You
ngW
oman
sC
reek
near
Ren
ovo,
PA01
545
600
120
56M
ixed
fore
stSa
ndst
one,
silt
ston
e,sh
ale,
cong
lom
erat
e,di
scon
tinuo
usco
albe
ds
�3Ð3
22Ð5
105
aH
ydro
clim
atic
Dat
aN
etw
ork
(HC
DN
).b
Mur
doch
and
Stod
dard
(199
2).
cD
eser
tR
esea
rch
Inst
itute
(DR
I).
dN
atio
nal
Wea
ther
Serv
ice
ofth
eN
atio
nal
Oce
anic
and
Atm
osph
eric
Adm
inis
trat
ion
(NW
S/N
OA
A).
eM
ean
annu
alte
mpe
ratu
re(n
otlo
wor
high
)an
dth
eref
ore
not
used
inem
piri
cal
mod
el.
fD
ata
from
Sila
sL
ittle
Exp
erim
enta
lFo
rest
.g
Nat
iona
lC
limat
icD
ata
Cen
tre
(NC
DC
/NO
AA
).h
Low
elev
atio
nes
timat
e.iV
arie
sst
rong
lyw
ithel
evat
ion.
jN
orth
Car
olin
aSt
ate
Uni
vers
ity(N
CSU
/NO
AA
).k
Mea
nan
nual
prec
ipita
tion
inH
ilo,
HI.
lM
ean
annu
alpr
ecip
itatio
nin
Am
anda
Park
,W
A.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1851
-1
0
And
rew
s C
reek
, WA
Bea
r D
en C
reek
, ID
Bea
uvai
s C
reek
, MT
Bea
ver
Cre
ek, N
DB
ig C
reek
, LA
Big
Jac
ks C
reek
, ID
Bis
cuit
Bro
ok, N
YB
lack
wat
er R
iver
, AL
Blu
e B
eave
r C
reek
, OK
Buf
falo
Riv
er, T
NC
ache
Cre
ek, W
YC
astle
Cre
ek, S
DC
atal
ooch
ee C
reek
, NC
Cos
sato
t Riv
er, A
RC
ypre
ss C
reek
, MS
Dis
mal
Riv
er, N
EE
lder
Cre
ek, C
AE
lk C
reek
, IA
Enc
ampm
ent R
iver
, WY
Eso
pus
Cre
ek, N
YF
allin
g C
reek
, GA
Hal
fmoo
n C
reek
, CO
Hay
den
Cre
ek, I
DH
olid
ay C
reek
, VA
Hon
olii
Str
eam
, HI
Kah
akul
oa S
trea
m, H
IK
awis
hiw
i Riv
er, M
NK
iam
ichi
Riv
er, O
KK
ings
Cre
ek, K
SLi
mpi
a C
reek
, TX
Littl
e R
iver
, TN
McD
onal
ds B
ranc
h, N
JM
erce
d R
iver
, CA
Min
am R
iver
, OR
Mog
ollo
n C
reek
, NM
N. F
ork
Qui
naul
tR.,
WA
N. F
ork
Whi
tew
ater
R.,
MN
Nor
th S
ylam
ore
Cre
ek, A
RP
oppl
e R
iver
, WI
Red
But
te C
reek
, UT
Rio
Mor
a, N
MR
ock
Cre
ek, M
TS
ageh
en C
reek
, CA
Sca
pe O
re S
wam
p, S
CS
ipse
y F
ork,
AL
Sop
chop
py R
iver
, FL
S. F
ork
Roc
ky C
reek
, TX
Sou
th H
ogan
Cre
ek, I
NS
outh
Tw
in R
iver
, NV
Ste
ptoe
Cre
ek, N
VT
alke
etna
Riv
er, A
KT
allu
lah
Riv
er, G
AU
pper
Thr
ee R
uns,
SC
Upp
er T
win
Cre
ek, O
HV
alle
cito
Cre
ek, C
OW
ashi
ngto
n C
reek
, MI
Wet
Bot
tom
Cre
ek, A
ZW
ild R
iver
, ME
You
ng W
oman
's C
r., P
A
Log-
log
slop
e of
[Ca]
-Q r
elat
ions
hip
Simple dilution: C~1/Q
Ideal chemostatic behavior: C~constant
Calcium(a)
-1
0
Log-
log
slop
e of
[Si]-
Q r
elat
ions
hip
Silica(d)
-1
0
Log-
log
slop
e of
[Na]
-Q r
elat
ions
hip
Sodium(c)
-1
0
Log-
log
slop
e of
[Mg]
-Q r
elat
ions
hip
Magnesium(b)
Figure 2. Log–log slopes of concentration–discharge relationships for 59 Hydrologic Benchmark Network streams arranged alphabetically by sitename. Grey lines indicate slope values expected for ideal chemostatic behaviour (concentration held constant; log – log slope D 0) and for simpledilution of a constant solute flux (concentration inversely proportional to discharge; log – log slope D �1). With few exceptions, these catchmentsexhibit nearly ideal chemostatic behaviour for Ca, Mg, Na and Si. Error bars indicate š1 standard error, and are shown where they are larger than
the plotting symbols. Open symbols indicate that the observed relationship appears to be non-linear in log–log space.
chemostatically. We also verified that the samples col-lected are representative of the range of flows at each site.At all sites, samples are collected across a range of flowsfrom the 5th to 97th flow percentiles (as calculated fromthe complete flow record reported in the USGS NationalWater Information System database, excluding dates onwhich flow is zero). Most sites include samples from the1st to 99th percentiles of flow, and median HBN samplescorresponded approximately to the 52nd flow percentile,suggesting that base flows are not oversampled.
A handful of sites have positive slopes for one ormore solutes, which indicate that concentration increaseswith increasing flow (e.g. Upper Three Runs, SC for Ca
and Mg and Steptoe Creek, NV for Na). In past studies,increased concentrations of more biologically active ionssuch as KC and NO3
� with increased flow have beenattributed to leaching from organic soil horizons duringhigher flows (Walling and Webb, 1986). At the HBNsites, the observed increase in concentration of the majorweathering products with discharge may be due to aweak correlation between discharge and concentration,or due to limited variability in the sampled discharge(e.g. discharge varies by only a factor of ¾4 at UpperThree Runs). At Upper Three Runs, it also may be aresult of analytical error or instrumentation changes invery dilute stream waters (Mast and Turk, 1999a, pp.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1852 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
2
3
456789
10
100 1000
Ca /6SiNa
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Cache Creek near Jackson, Wyoming
1
10
100 1000
SiCaNa
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Simpledilution:C~1/Q
Andrews Creek near Mazama, Washington
1
10
100 1000
SiCaNa
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Elder Creek near Branscomb, California
1
10
100 1000
SiCa /3Na
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Washington Creek, Isle Royale, Michigan
1
10
100 1000
SiCaNa
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Sagehen Creek near Truckee, California
1
10
100 1000
SiCa /10Na
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
North Sylamore Creek near Fifty Six, Arkansas
1
10
100 1000
SiCa x2NaA
nnua
l flo
w-w
eigh
ted
mea
nco
ncen
trat
ion
(ppm
)
Annual water yield (mm/yr)
Merced River near Yosemite, California
1
10
100 1000
SiCaNa x2
Ann
ual f
low
-wei
ghte
d m
ean
conc
entr
atio
n (p
pm)
Annual water yield (mm/yr)
Hayden Creek near Hayden Lake, Idaho
Figure 3. Annually averaged concentration–discharge relationships for Si, Ca and Na at the same eight Hydrologic Benchmark Network streamsshown in Figure 1. Each plot has consistent axes (same number of log units for both concentration and discharge), so concentrations determined bydilution of fixed weathering fluxes would have the same slope as the grey diagonal lines shown (log–log slope of �1). Even on inter-annual timescales, concentration–discharge relationships do not generally follow the predictions of a simple dilution model, instead conforming more closely to
chemostatic behaviour (log–log slope near zero).
108–110). However, most sites show little variability inconcentration with discharge and we focus on these sitesfor the rest of this paper.
One might postulate that concentrations are rela-tively constant across wide ranges of discharge simplybecause the volume of water stored in a catchment ismuch larger than the amount discharged during an indi-vidual storm event. Therefore we also tested whethercatchments behave chemostatically on inter-annual time
scales. Annualized concentration–discharge relations foreach of the HBN sites were plotted as mean annualflow-weighted concentrations against annual water yield.Water yield was calculated by summing all daily flowsin the water year, as available from the daily USGSstreamflow record, and dividing by catchment area.Mean flow-weighted concentrations were calculated as�QiCi�/Qi, where the subscript i indicates each sam-ple during the water year. We excluded years with fewer
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1853
-1
0
Log-
log
slop
e of
[Mg]
-Q r
elat
ions
hip
Magnesium (mean annual concentrations)(b)
-1
0
Log-
log
slop
e of
[Na]
-Q r
elat
ions
hip
Sodium (mean annual concentrations)(c)
-1
0
Log-
log
slop
e of
[Si]-
Q r
elat
ions
hip
Silica (mean annual concentrations)(d)
-1
0
1
And
rew
s C
reek
, WA
Bea
r D
en C
reek
, ID
Bea
uvai
s C
reek
, MT
Bea
ver
Cre
ek, N
DB
ig C
reek
, LA
Big
Jac
ks C
reek
, ID
Bis
cuit
Bro
ok, N
YB
lack
wat
er R
iver
, AL
Blu
e B
eave
r C
reek
, OK
Buf
falo
Riv
er, T
NC
ache
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ek, W
YC
astle
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ek, S
DC
atal
ooch
ee C
reek
, NC
Cos
sato
t Riv
er, A
RC
ypre
ss C
reek
, MS
Dis
mal
Riv
er, N
EE
lder
Cre
ek, C
AE
lk C
reek
, IA
Enc
ampm
ent R
iver
, WY
Eso
pus
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ek, N
YF
allin
g C
reek
, GA
Hal
fmoo
n C
reek
, CO
Hay
den
Cre
ek, I
DH
olid
ay C
reek
, VA
Hon
olii
Str
eam
, HI
Kah
akul
oa S
trea
m, H
IK
awis
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er, M
NK
iam
ichi
Riv
er, O
KK
ings
Cre
ek, K
SLi
mpi
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reek
, TX
Littl
e R
iver
, TN
McD
onal
ds B
ranc
h, N
JM
erce
d R
iver
, CA
Min
am R
iver
, OR
Mog
ollo
n C
reek
, NM
N. F
ork
Qui
naul
t R.,
WA
N. F
ork
Whi
te w
ater
R.,
MN
Nor
th S
ylam
ore
Cre
ek, A
RP
oppl
e R
iver
, WI
Red
But
te C
reek
, UT
Rio
Mor
a, N
MR
ock
Cre
ek, M
TS
ageh
en C
reek
, CA
Sca
pe O
reS
wam
p, S
CS
ipse
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ork,
AL
Sop
chop
py R
iver
, FL
S. F
ork
Roc
ky C
reek
, TX
Sou
th H
ogan
Cre
ek, I
NS
outh
Tw
in R
iver
, NV
Ste
ptoe
Cre
ek, N
VT
alke
etna
Riv
er, A
KT
allu
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AU
pper
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ee R
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Upp
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win
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ashi
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reek
, MI
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Bot
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ek, A
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ild R
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, ME
You
ng W
oman
's C
r., P
A
Log-
log
slop
e of
[Ca]
-Q r
elat
ions
hip
Simple dilution: C~1/Q
Ideal chemostatic behavior: C~constant
Calcium (mean annual concentrations)(a)
Figure 4. Log–log slopes of relationships between mean annual flow-weighted concentrations and mean annual discharge for 57 HydrologicBenchmark Network streams (slopes are not shown for Limpia Creek, TX or Tallulah River, GA because too few years of data are available).Grey lines indicate slope values expected for ideal chemostatic behaviour (concentration held constant; log – log slope D 0) and for simple dilutionof a constant solute flux (concentration inversely proportional to discharge; log – log slope D �1). Even on inter-annual time scales, almost all ofthese catchments exhibit nearly ideal chemostatic behaviour for Ca, Mg, Na and Si. Error bars indicate š1 standard error calculated from a weightedleast squares fit to the log annual flow-weighted concentration for each water year versus log annual water yield. The weighting function is equalto the inverse unbiased weighted variance of the log concentration. By unbiased, we mean that we account for a potential loss in the degrees of
freedom introduced by variable flow by calculating the effective number of solute measurements per water year.
than four available concentration measurements from theinter-annual concentration-discharge analysis.
The plots of mean annual flow-weighted concentra-tions versus annual water yield exhibit similar patterns tothose observed on an event basis (Figures 3 and 4). Meanannual concentrations vary much less than water yielddoes from year to year. The concentration–dischargeplots for inter-annual time scales (Figure 3) are similarto those for individual samples (Figure 1); both exhibitpower-law relationships between solute concentrationsand water yields, with small negative log–log slopes.
Across the HBN sites, these log–log slopes are gen-erally close to zero on both inter-annual time scales(Figure 4) and event time scales (Figure 2), although theerror bars in Figure 4 are larger because the annualizedconcentration–discharge plots (e.g. Figure 3) have fewerpoints and a smaller range of discharge. The near-zeroslopes imply that catchments exhibit near-chemostaticbehaviour over both event and inter-annual timescalesand across a wide range of hydrologically and geochem-ically diverse sites. Thus, an interesting first-order ques-tion is not why concentrations of weathering products
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1854 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
vary, but why they vary so little as discharge changes somuch.
ASSESSMENT OF ALTERNATIVE MODELS
Consistent near-zero log�C�- log�Q� slopes imply thatrates of solute production or mobilization must be nearlyproportional to water fluxes on intra- and inter-annualtimescales. Here we explore several simple quantitativeexplanations for the observed concentration–dischargepatterns to verify if they hold across all the HBN sites,and to compare different possible models at each site. Weevaluate these explanations on the basis of several keycriteria. They should generate power–law relationshipsbetween concentration and discharge with small negativeslopes and little hysteresis. Furthermore, they should havereasonable physical and chemical parameters and shouldmake plausible assumptions about catchment behaviour.They should be as simple as possible and explainobservations in a wide range of catchments. We examineseveral empirical, mixing and chemical models to gaininsight into the linkages between chemical weatheringand hydrologic processes that produce the observedconcentration–discharge relationships. We quantitativelyassess the models’ performance at each site using theAkaike and Bayesian information criteria (AIC and BIC)for each set of comparable models.
Empirical models
We first evaluated whether the slope of the concen-tration–discharge relationship can be predicted as a sim-ple empirical function of each catchment’s character-istics, or as a multivariate model of several charac-teristics. For each site, we plotted average low andhigh temperature, annual average runoff, area, and MeanAnnual Precipitation (MAP) against the event-basedconcentration–discharge slope for each solute, and cal-culated the correlation between each of these site charac-teristics and each solute’s concentration–discharge slope(using the non-parametric Spearman correlation coeffi-cient because the distributions of site characteristics arenot normal). Based on the site descriptions from theUSGS circulars (Mast and Turk, 1999a,b; Clark et al.,2000; Mast and Clow, 2000), we also coded each siteaccording to the presence or absence of carbonates andvolcanics in the underlying bedrock. We then used the F-test (Zar, 1984) to determine whether the presence ofeither of these broad rock types significantly affectedthe concentration–discharge slope for any solute. Forthe multivariate models, we systematically added andremoved characteristics from a series of multiple linearregression models and evaluated their performance.
The site characteristics that we tested generally donot satisfactorily explain the observed slopes of theconcentration–discharge relationships (Table II). Steeperconcentration–discharge slopes are associated with high-er average annual runoff (although, surprisingly, not
always with higher MAP); these correlations are sta-tistically significant for some solutes but not others.Sites with volcanic bedrock have significantly (p < 0Ð05)steeper concentration–discharge slopes for Ca, Mg andNa compared to sites where volcanics are absent. Sim-ilarly, log–log slopes for Na are significantly (p <0Ð02) shallower at sites with carbonate bedrock, com-pared to sites without carbonates. Although some sitecharacteristics show statistically significant effects onconcentration–discharge slopes, their predictive poweris weak because they explain only a small fraction ofthe variance; typical r2 values are 0Ð1 or less. Multi-variate models with combinations of these site charac-teristics could only explain <20% of the variability inlog�C�- log�Q� slopes. We conclude that the observedvariation in concentration–discharge slopes cannot bestraightforwardly predicted by any of the site charac-teristics tested here. Other site characteristics, such asbasin slope, soil permeability, or amount and type of soiland vegetation, might be useful explanatory variables,but they have not been quantified for most of the HBNbasins. Basin slope and soil permeability, for example,might be important because they influence hydrologicalflow paths and the residence time of water in the basins.
Mixing models
We turn from empirical models based on site charac-teristics to simple models based on the mixing of waterswith different compositions. Isotopic studies and otherhydrometric and hydrochemical evidence have shownthat in many catchments, typical residence times aremuch longer than the duration of individual storm events
Table II. Statistical tests of association between site character-istics and concentration–discharge slopes across the 59 HBN
streams
F ratio Ca Mg Na Si
Volcanics (present vs. absent) 5Ð32a 6Ð51a 4Ð59a 3Ð77Carbonates (present vs. absent) 0Ð07 0Ð02 12Ð47b 2Ð54
Spearman correlationcoefficient, rs
Ca Mg Na Si
Average annual runoff (cm) �0Ð27a �0Ð19 �0Ð21 �0Ð28a
Average low temperature(deg C)
�0Ð02 0Ð10 0Ð06 �0Ð30a
Average high temperature(deg C)
�0Ð06 0Ð06 0Ð25 �0Ð16
MAP, Mean annualprecipitation (cm)
0Ð03 0Ð05 �0Ð08 �0Ð28a
Area (km2) �0Ð11 �0Ð08 0Ð00 0Ð04
Superscripts a and b indicate statistical significance at the p < 0Ð05and p < 0Ð02 levels, respectively. Several of the tested relationships arestatistically significant but their explanatory power is low. Sites withvolcanic bedrock have steeper concentration–discharge slopes for Ca,Mg and Na than sites without volcanics, and slopes for Na are shallowerat sites with carbonate bedrock compared to sites without carbonates.Nonparametric (Wilcoxon) tests agree with the parametric (F-test) results.Average annual runoff is significantly correlated with Ca and Si slopes,and average low temperature and MAP with Si slopes, according to therobust Spearman rank correlation coefficient.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1855
(e.g. Buttle, 1994). One can imagine that residencetimes may be long enough for weathering reactions toapproach equilibrium, and that this explains the chemo-static behaviour observed across the HBN sites. Sufficientstorage capacity for the ‘old’ water must exist for thisoption to be physically plausible, and we explore stor-age requirements for different models below. In general,we find that mixing models that assume a constant rate ofsolute supply generally cannot reproduce the observationswell.
First, we present a simple ‘bucket’ mixing model thattreats the catchment as a single well-mixed reservoirwhose volume (V, [m3]) remains constant as discharge(Qw, [m3/s]) varies. The model likewise assumes thatthe solute flux (Qs, [ppm–m3/s]) produced by mineralweathering is constant through time. It further assumesthat concentrations in precipitation are very dilute com-pared to streamflow (as is typically the case for themajor weathering products), so that precipitation solutefluxes can be ignored in the mass balance. Under theseconditions, the solute concentration (C, [ppm]) in thewell-mixed reservoir, and thus in its outflow, will evolveaccording to the familiar mass-balance equation:
dC
dtD �Qs � QwC�
V�1�
where Qw is the average daily water flux through thereservoir, for which we use the USGS historical dailyflow record. We fix the solute flux (Qs) equal to theproduct of the average water flux and the flow-weightedaverage concentration, so that the average modeled soluteflux will equal the observed long-term average. Thereservoir volume (V) is the only free parameter, which isadjusted to reproduce the observed range of variation inoutflow concentrations.
Constraining the concentrations modeled using Equa-tion (1) within the bounds of observed variability inthe HBN data set requires a storage volume that iswell-mixed to depths of several meters throughout the
Table III. Subsurface storage depths required for a simple‘bucket’ model assuming a constant 5%, 10% and 30% porosity
for the model described in Equation (1) of the text
Site Storage Depth (m)
5% porosity 10% porosity 30% porosity
Andrews 10Ð8 5Ð4 1Ð8Cache 10Ð1 5Ð0 1Ð7Elder 83Ð7 41Ð9 14Ð0Hayden 6Ð0 3Ð0 1Ð0Merced 9Ð8 4Ð9 1Ð6N Sylamore 6Ð5 3Ð3 1Ð1Sagehen 21Ð7 10Ð9 3Ð6Washington 9Ð2 4Ð6 1Ð5
The porosity is a single average value for all soil and underlying bedrockto the specified depth. The specified storage depth indicates the volumerequired to maintain the observed variability in concentrations at a givensite (reported value is the median storage for all four solutes based onthe observed ratio of maximum to minimum concentrations at each site,the median of which is ¾4 across all sites and solutes).
catchment (Table III) for a porosity of 10% (averagedacross both soil and underlying rock). Higher porosi-ties require a smaller storage volume whereas lowerporosities require a larger storage volume. Even withsufficiently large storage volumes, however, modeledconcentration–discharge relationships do not match theobservations well; the model results exhibit large hys-teresis loops that are not present in the observations(Figure 5a). These loops result from the assumption thatthe catchment behaves as a simple well-mixed bucket: theintegro-differential relationship in Equation (1) implies a90-degree phase lag between changes in discharge andchanges in concentration, rather than the simultaneousvariations in concentration and discharge that are usu-ally observed. Hysteresis loops are frequently observed inconcentration–discharge relationships, but the extent oflooping modeled in Figure 5a is much greater than is typ-ically observed (e.g. Evans and Davies, 1998). To matchthe observed concentration–discharge relationships, thismodel would have to be modified to allow solutes tobe: (1) produced at a variable rate or (2) mobilized fromdifferent sources at variable rates.
Another well-known model that we consider is theHubbard Brook ‘working model’ (Johnson et al., 1969)which assumes that discharge is proportional to stor-age volume (V, [m3]), and that solute concentrations(C, [ppm]) associated with each storage volume arefixed. Storage volume is defined as the subsurface porespace available above an ‘impermeable’ layer. C is theninversely related to Q according to:
C D [d
�1 C bQ�] C a �2�
where a D concentration of solute in the low-concentra-tion end-member [ppm], b D mean residence time/VQD0
(flows at the catchment outlet are assumed to ceaseat some non-zero storage volume, VQD0) (s/m3), andd D concentration difference between high and low con-centration waters [ppm]. Any non-zero flow is directlyproportional to water in storage above the minimum vol-ume. One can immediately see that this model does notrepresent a power-law relationship between C and Q,but instead that the relationship is a hyperbolic function.However, because it has an additional free parameter,Equation (2) can also fit the data well (Figure 5b). Anon-linear fitting algorithm which minimizes the sum ofsquared error between the model results and observeddata is used to select best-fit values of a, b, and d. Theparameter b depends only on the mean residence time ofwater in the catchment and the minimum volume requiredfor flow in the stream. Importantly, b does not depend onthe solute of interest; therefore, it should have the samevalue for all the solutes at an individual site. This con-sistency in b can be ensured by simultaneously fittingall solutes in a given catchment. Johnson et al. (1969)suggest that a should be thought of as the ‘rainwater’ con-centration of each solute. However, the best-fit a valuesexceed measured volume-weighted precipitation concen-trations for Ca, Mg and Na [as reported in Mast and Turk
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1856 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
10
1simple bucket modelobservations
Con
cent
ratio
n, C
(pp
m)
10
1Con
cent
ratio
n, C
(pp
m)
0.01
Discharge, Q (m3/s)
0.1 1 10 100
10
1Con
cent
ratio
n, C
(pp
m)
10
1Con
cent
ratio
n, C
(pp
m)
0.01
Discharge, Q (m3/s)
0.1 1 10 100
0.01
Discharge, Q (m3/s)
0.1 1 10 100
0.01
Discharge, Q (m3/s)
0.1 1 10 100
(a) (b)
(c) (d)
Hubbard Brook ''working model''observations
Langbein-Dawdy chemical modelobservations
permeability-porosity-aperturemodelobservations
Figure 5. Concentration (C)–discharge (Q) relationships for several models plotted with the observed values for Ca at Andrews Creek in Mazama,WA. (a) A well-mixed reservoir model (Equation (1)) with a very large storage volume can reproduce the observed range in concentration variability,but only with excessively large hysteresis loops that are not observed in the real data. (b) Inverse relationship between C and Q as specified by theHubbard Brook ‘working model’ (Johnson et al., 1969) generally matches the form for each solute, but best-fit a parameters (Eq. 2) exceed observedrainfall concentrations (Table IV). (c) Langbein and Dawdy’s (1964) chemical mixing model can fit the data well but allows no storage of water ormixing of waters of different ages in the catchment. (d) The permeability-porosity-aperture model (see text) can also fit the data reasonably well,although its assumptions still need to be tested in the field. The slight overestimate of concentration at high discharges is due to the simultaneousfitting to all four solutes with one hydrologic parameter, b0, which is the best-fit slope to all four solutes’ concentration–discharge data. See text for
more detailed discussion.
(1999a,b); Clark et al. (2000); Mast and Clow (2000)] byup to two orders of magnitude (Table IV). Concentrationsof Si in precipitation are not generally reported, but thebest-fit a for Si is in the range of ¾10 ppm. Instead of
representing ‘rainfall’ concentrations, the high best-fit avalues could perhaps be thought of as ‘soil water’ con-centrations. In any case, the fact that the best-fit valuesof a are much higher than rainfall concentrations implies
Table IV. Best-fit parameter a in the Hubbard Brook ‘working model’ and observed precipitation concentrations for the major basecations at eight representative sites
All (ppm) Best-fit aCa Observed Ca inprecipitation
Best-fit aMg Observed Mg inprecipitation
Best-fit aNa Observed Na inprecipitation
Andrews 2Ð04 0Ð03 0Ð31 0Ð02 0Ð97 0Ð07Cache 44Ð1 0Ð19 12Ð3 0Ð02 2Ð20 0Ð04Elder 7Ð21 0Ð04 2Ð34 0Ð04 4Ð32 0Ð18Hayden 4Ð23 0Ð11 1Ð31 0Ð04 1Ð13 0Ð01Merced 0Ð72 0Ð05 0Ð10 0Ð01 0Ð56 0Ð04N Sylamore 28Ð6 0Ð14 1Ð89 0Ð02 0Ð82 0Ð05Sagehen 3Ð56 0Ð05 1Ð36 0Ð01 1Ð75 0Ð04Washington 7Ð99 0Ð16 2Ð44 0Ð03 1Ð14 0Ð02
Observed Si concentrations in precipitation were not available. The best-fit concentrations, a, are observed to be one to two orders of magnitudehigher than the observed concentrations. This suggests that the model is unable to represent some mixing or reaction processes occurring in thecatchment. In all cases, all three parameters in Equation (2) were fit simultaneously for all four solutes; a and d were constrained to be ½0, and b,which depends only on hydrology and is independent of the solute, was forced to be a single value for all four solutes at each site.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1857
that important solute sources within the catchment arenot captured by the model.
Chemical models
On the other end of the modeling spectrum, one couldassume that waters react so rapidly with soil and rockthat one can ignore mixing of waters of different agesaltogether. If changes in chemical reactions were solelyresponsible for the concentration–discharge patterns wesee, what would this imply? The simplest way to maintainchemical concentrations as discharge increases is througha combination of increasing reactive surface area andincreasing reaction rates.
A simple approach was proposed by Langbein andDawdy (1964) who assumed that dissolution rate variesbetween zero and the maximum dissolution rate, D,depending upon distance from equilibrium. They alsoassumed that the dissolved load is removed as quickly asit is formed, which is equivalent to assuming that mixingof water of different ages is negligible. To generalizetheir model from 1st order to nth-order reactions, one canlet the forward reaction be equal to DA [mol/s] where A[m2] is the reactive surface area, and let the back reactionbe equal to DA �C/Cs�n where n [dimensionless] is theorder of the reaction, C [mol/m3] is the concentrationand Cs [mol/m3] is the concentration at saturation. Theload, L [mol/s], would then be equal to the balance ofthe forward and backward reactions:
L D DA[1 � �C/Cs�n] �3�
Assuming that the dissolved load is removed as quicklyas it is formed, one can also write:
L D Q�C � Co� �4�
where Co [mol/m3] is the initial (i.e. rainfall) concentra-tion, C [mol/m3] is the final concentration and Q [m3/s]is the discharge. For first- and second-order reactions(n D 1 or 2, respectively), Equations (3) and (4) can besolved for C as a function of Q:
�for n D 1� C D DA C QCo
Q C DA/
Cs
�5a�
or
�for n D 2� C DQ �
√Q2 C 4DA
S2 �DA C QCo�
�2DA/
Cs2
�5b�where the negative root for n D 2 gives the concentra-tion that has physical meaning. Using these equationsand the observed concentration–discharge relationships,one can determine the values of DA, Cs and Co thatare required to fit the data. With three adjustable param-eters, the model approximately reproduces the observedconcentration–discharge relationship with either n D 1 orn D 2, although the log–log relationship is curved rather
than linear (Figure 5c). The order of the reaction doesnot strongly affect the overall quality of fit, but the dis-solution rate times area (DA) parameter for the best-fit2nd order scenario differs from the 1st order case. Onecannot distinguish among models of different reactionorder given only concentration–discharge data; additionala priori information about the order of the reaction or thedissolution rate and reaction area is needed to distinguishamong the models. The best-fit saturated concentration islower than the observed maximum concentration and thebest-fit initial concentration is higher than the observedminimum concentration; the best-fit concentrations varylittle as reaction order changes. In either case, the best-fit initial concentration is usually much higher than isrealistic for rainfall. Rather than representing rainfall,the best-fit Co probably reflects solutes acquired dur-ing infiltration through the soil column before reachingthe groundwater system. Although multiple versions ofthe Langbein and Dawdy model fit the data reasonablywell, the key assumption that there is no storage of waterbetween storms and therefore negligible mixing of waterof different ages is generally not valid (e.g. Buttle, 1994).
Permeability-porosity-aperture model
Here we suggest another model that is the only generalmodel that we know of that produces a power–law rela-tionship between concentration and discharge, assumesa variable solute flux that is proportional to the reactivesurface area, and allows for mixing of waters of differentages. This final point distinguishes this model from theLangbein-Dawdy model.
The full derivation is shown in the appendix. Thekey assumptions are that permeability, porosity andaverage pore aperture or width, all decrease exponentiallywith depth; that Darcy’s Law describes flow throughthe catchment; that the effective precipitation rate isapproximately spatially uniform across a hillslope; thatflows originating near the divide and stream are minimal;and that solute flux is proportional to reactive surface areasuch that secondary and back-reactions do not controlsolute fluxes.
One can show that under these assumptions, thevolume-weighted mean concentration (C, [mol/m3])draining a one-dimensional hillslope will be a powerfunction of the water flux (Qw),
C D aoQb0w , b0 D
(�k
/�� � �k
/�p � 1
)�6�
where the constant ao is a function of solute and catch-ment characteristics, and the power-law exponent b0
(dimensionless) depends on the rate at which perme-ability, porosity and pore aperture decrease exponentiallywith depth in the subsurface (represented by the e-foldinglength scales �k , �� and �p, all in [m]). Because �k ,�� and �p are geometric properties of the subsurfaceand are not specific to individual solutes, equation (6)implies that bo should be the same for all weathering-derived solutes at an individual site (see, for one example,
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1858 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
Figure 5d showing Ca at Andrews Creek). We fitted a sin-gle value of bo for each site, and a value of ao for eachsite and solute, using a non-linear fitting algorithm thatminimizes the error between the modeled and observedconcentrations for all solutes simultaneously.
Because we fit some models to all the solutes simul-taneously whereas we fit other models to each soluteindependently, comparing the models is not a straight-forward exercise. Using the AIC or BIC, or by look-ing at the example in Figure 5, it is clear that theLangbein-Dawdy model (5b) outperforms the simplemixing model (5a). The permeability-porosity-aperturemodel (5d) outperforms the Hubbard Brook Experimen-tal Forest ‘working model’ (5c) according to the AICand BIC at each of the eight featured sites. Comparisonsbetween the Langbein-Dawdy and permeability-porosity-aperture model cannot be made using the AIC or BICbecause the models are based on different sets of data(fitting each solute independently vs. all solutes simul-taneously). Regardless of model performance, a key testof each of these models is whether the best-fit parame-ters make physical and chemical sense in the real-worldcatchments in which they are applied.
Contrary to the prediction of the permeability-porosity-aperture model, that the log-log slope b0 of the concen-tration–discharge relationship should be the same fordifferent weathering products at an individual catchment,the log�C�- log�Q� slopes for different solutes at the samesite often differ by more than their uncertainties. Thisresult could be explained by different depth profiles in theabundances of different minerals (and thus their reactivesurface areas per unit pore surface). Mathematicallythis could be represented as �p taking on differentvalues for different solutes, although this would implythat p would reflect not only pore aperture but alsothe relative abundances of different minerals. However,because b0 is usually close to zero for different solutes, itis possible that the variation in �p may be small. Anotherparameter, b1, has been shown to relate storage anddischarge in catchments, and is expressed as a functionof the parameters �k and �� (Kirchner, 2009; and seeAppendix, Equation A12 and A13). The two log-logslopes b0 and b1 are not sufficient to uniquely constrainthe three parameters �k , �� and �p, so the individuale-folding depths (as well as values of the reactivityparameter kR for each solute, see Appendix) could onlybe determined by direct measurement. Any such attemptat direct measurement, however, would be complicatedby the spatial heterogeneity in subsurface properties, aswell as the large differences between field and laboratoryweathering rates (Schnoor, 1990; Brantley, 1992; Whiteet al., 1996).
The mechanism proposed here is in some ways anal-ogous to that proposed by Clow and Drever (1996) toexplain the relatively constant concentrations of weather-ing products in runoff from an alpine soil under widelyvarying rainfall. Clow and Drever (1996) argued that Siconcentrations in their study catchment may be controlledby flushing and diffusion from micropores and seasonal
precipitation/dissolution of metastable amorphous alumi-nosilicates. They argued that because both flushing andreaction rates increase with increasing discharge, thiscombination of mechanisms would allow concentrationsto remain relatively constant with fluctuating discharge.But whereas Clow and Drever (1996) argued that therate of Si dissolution from the mineral phase shouldbe controlled by Si concentrations in solution (and thusby the fluid flushing rate), here we assume that silicateweathering reactions are always far from equilibrium andthus are unaffected by changes in solute concentrations.Instead, in the permeability-porosity-aperture model out-lined in Equation 6 and in the appendix, reaction ratesincrease at higher discharges because the wetted min-eral surface area increases. Others have also argued thatbecause Si is retained by secondary minerals to vary-ing degrees, its concentration may be controlled in largemeasure by equilibration with respect to the secondaryminerals that form (Drever and Clow, 1995; Godderiset al. 2006). Secondary mineral formation should pref-erentially affect Si and may explain why the power-lawconcentration–discharge slopes for Si are generally shal-lower than for the other major weathering products. Theseexamples illustrate that different mechanisms can poten-tially generate similar observed concentration–dischargerelationships, so it is important to verify the site-specificplausibility of any proposed mechanism.
FURTHER POSSIBILITIES
Many catchment-based chemical weathering models,such as Birkenes, ETD (Enhanced Trickle-Down),ILWAS (Integrated Lake Watershed Acidification Study),PROFILE/SAFE (Soil Acidification in Forested Ecosys-tems), MAGIC (Model of Acidification of Groundwaterin Catchments), or WITCH [see review in Nordstrom(2004), pp. 60–62; Godderis et al., (2006)], and othermineral weathering models such as PHREEQC (Parkhurstand Appelo, 1999), allow the simulation of weatheringprocesses. We do not use these models in this studybecause most of our sites have insufficient informationto apply them. Instead we explore in this paper whethera simple general modeling approach is possible basedprimarily on the observed concentration–discharge rela-tionship. We show that several simple models can fitthe observations well, but often they require unrealisticor internally inconsistent parameter values. Because datalimitations have precluded us from testing more complexmodels, we cannot say whether their added complex-ity, and additional data requirements, would be helpfulin understanding catchments’ concentration–dischargebehaviour. Future studies could also examine the geomor-phic features, soil or regolith depth, presence of organicmatter, or perennial/ephemeral status of the catchments,which may control weathering rates or fluxes in somecatchments (e.g. Drever, 1994; Oliva et al., 1999; John-son et al., 2001). These characteristics have not beenconsidered in this study.
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1859
The relative constancy of concentrations across wideranges of discharge requires solute production or solutemobilization at rates nearly proportional to the water flux.Depending on the relationship between reaction times andwater transit times, one can infer the relative importanceof production and mobilization. Based on field experi-ments, reaction rates have sometimes been inferred to befast enough that the time to equilibrium is much shorterthan average water transit times, such that waters areeffectively always near equilibrium (e.g. Anderson andDietrich, 2001). Buttle (1994) and many others have doc-umented that most water reaching a stream during a stormevent is so-called ‘old’ (i.e. pre-storm) water. Mean tran-sit times in many catchments are on the order of ¾1 year(McGuire and McDonnell, 2006, Table I), so this ‘old’water may be near equilibrium with respect to certainminerals, with the consequence that solute fluxes mobi-lized by this ‘old’ water must be nearly proportional towater flux. However, studies of silicate weathering havetypically found that reaction rates are slow (even relativeto mean transit times of a year) and have assumed thatcatchments are kinetically-limited systems (see discus-sion in Brantley, 2004). Because the chemical weatheringkinetics of silicates and carbonates differ (e.g. Brant-ley, 2004), one might expect different log�C�- log�Q�behaviour or explanatory models for catchments domi-nated by the distinct lithologies. However, for most of thesolutes considered, the concentration–discharge slopesare not significantly different between the different litho-logic settings. This suggests that differences in lithologyand in reaction kinetics between carbonates and silicatesdo not significantly alter the relationship between con-centration and discharge across the study sites. Time toequilibrium in the laboratory for mineral-water equilib-rium reactions has been reported as a week to a year orlonger (e.g. Bricker, 1968; Langmuir, 1997, Table 2.1)and field rates are even slower (Schnoor, 1990; Brant-ley, 1992; White et al., 1996). This suggests that mineralweathering reactions would be unlikely to stay close toequilibrium, particularly during high flows. Determiningthe average time to equilibrium in the field is challenging:flow paths are heterogeneous and reactions and reactivesurface areas are difficult to define. Nonetheless, as thepreceding discussion illustrates, there would be much tobe gained from determining how close weathering reac-tions are to equilibrium in the field.
IMPLICATIONS
Because concentrations are relatively constant with dis-charge across these diverse study catchments (Figures 1and 2), solute fluxes (defined as concentration times dis-charge) from these catchments change nearly proportion-ally to discharge, both on an event basis and on aninter-annual basis (e.g. Figure 6). Although our obser-vations are drawn from US catchments, these results arelikely to be broadly generalizable because the HBN catch-ments include a broad range of climatic and lithologic
environments (Table I). A similar relationship betweenconcentration and discharge is also seen in granitic borealcatchments in permafrost regions of Russia (Zakharovaet al., 2005), suggesting these observations and impli-cations may be globally applicable. Climate change maysubstantially alter stream flows (especially if precipitationand evapotranspiration change in opposite directions).Such hydrologic changes will have little effect on con-centrations but will alter fluxes almost proportionally.
Alkalinity flux to the ocean can be estimated as the sumof the Ca and Mg concentrations (McSween et al., 2003,p. 147), or the hardness of the water. Raymond and Cole(2003) used this approach to estimate alkalinity for theMississippi River for years before 1973 using hardness.They showed that overlapping alkalinity and carbonatehardness (Dtotal hardness � noncarbonate hardness)measurements are accurate to 99Ð8 š 0Ð8%. Thus, wecan estimate alkalinity fluxes based on the relationshipsbetween Ca, Mg and discharge presented in Figures 2and 4. Raymond and Cole (2003) found that dischargeand alkalinity flux in the Mississippi River increased byapproximately 44% and 70%, respectively, over 48 years(1953–2001), attributed partially to changes in climate aswell as changes in cropping patterns and additions fromgroundwater pumping. Based on the patterns in Figure 6(and similar results for other solutes), one would expectsolute flux from most watersheds across North Americato vary nearly proportionally to water fluxes. Unlikethe Mississippi River, most of the streams and riversin the HBN are minimally affected by land use changeor large amounts of groundwater pumping. Nonetheless,the nearly chemostatic behaviour demonstrated acrossso many diverse sites implies that alkalinity flux islargely determined by stream discharge. Streamflows athigh latitudes are expected to increase by 10 to 40%over the next century, according to the latest report ofthe Intergovernmental Panel on Climate Change (IPCC,Core Writing Team, 2007). For a 10% increase inaverage annual flows, we would expect an approximatelyproportional 10% increase in alkalinity fluxes to theoceans. Carbonic acid weathering is one of the mainmechanisms generating alkalinity; thus, an increase inalkalinity fluxes implies a concomitant increase in CO2
consumption.On the other hand, if discharges were to drop sharply,
solute fluxes would also be expected to decrease. As anexample, the seaward total dissolved solids flux from theHuanghe (Yellow) River, China has decreased by morethan half over the last approximately 40 years because ofa sharp decrease in water discharge in the lower reachesof the river (Chen et al., 2005). Diversions, irrigation andreservoir use are primarily responsible for the decreasein water discharge, and despite an increase of ¾5 to10 mg/l/year in total dissolved solids in the middle andlower reaches of the river, there has been an overalldecrease in solute flux (Chen et al., 2005). Climatechange models predict that flows in the southwesternUnited States and other dry areas are likely to decreaseby 10 to 40% by 2090–2099 relative to 1980–1999 in a
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1860 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
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ual S
i flu
x (k
g S
i/ha-
yr)
Annual water yield (mm/yr)
Chemostatic behavior(C~constant; flux~discharge)
Andrews Creek near Mazama, Washington
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Hayden Creek near Hayden Lake, Idaho
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Elder Creek near Branscomb, California
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yr)
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Merced River near Yosemite, California
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yr)
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North Sylamore Creek near Fifty Six, Arkansas
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Sagehen Creek near Truckee, California
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Washington Creek, Isle Royale, Michigan
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ual S
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yr)
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Cache Creek near Jackson, Wyoming
Figure 6. Annual silica fluxes in streamflow as a function of annual water yield for the eight Hydrological Benchmark Network streams shown inFigures 1 and 3. The diagonal grey line indicates solute fluxes proportional to discharge (i.e. constant concentration).
scenario in which global average temperatures increaseby 2Ð8 °C over the same period (IPCC, Core WritingTeam, 2007). One would expect solute flux of the majorbase cations and silica from streams and rivers in theseregions to decrease by an amount that is approximatelyproportional to the decrease in flows.
The stability of concentrations of weathering productsacross a wide range of flow regimes may be important
both for the health of individual organisms and for thediversity of aquatic ecosystems. At an organismal level,mortality from acidification-induced aluminum toxicity isclosely linked to decreases in alkalinity and base cationconcentrations (Jeffries et al., 1992; Thornton and Dise,1998). Metals and other contaminants are more toxic tojuvenile fish in waters with lower hardness, which asnoted above, is usually equal to sum of the concentration
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1861
of Ca and Mg (e.g. Hall, 1991). The greater the stabilityof base cation concentrations across a wide range offlows, the smaller the likelihood that toxicity thresholdswill be crossed during hydrological extremes becauseconcentrations will not drop precipitously when flowsincrease.
At a species to community level, site-to-site compar-isons reveal that benthic macroinvertebrate and diatomspecies diversity is a function of total dissolved solidconcentration, conductivity or salinity in some streams,lakes and fjords (Metzeling, 1993, Ryves et al., 2004).Solute fluxes, especially the amount of Si relative to otherpotentially limiting nutrients, can be a predictor of phyto-plankton blooms and diatom growth (LePape et al., 1996;Grenz et al., 2000). Si concentrations relative to concen-trations of Fe, N and P strongly control diatom nutrientuptake and growth rates. Increased N and P concentra-tions relative to Si concentrations and decreased Si fluxdue to river regulation have shifted ecosystems dominatedby siliceous phytoplankton or diatoms to those dominatedby non-siliceous algae or flagellate communities (e.g.Officer and Ryther, 1980; Egge and Aksnes, 1992; Turneret al., 1998; Billen and Garnier, 2007). Oceanic Si limi-tations have been seen in the eastern equatorial Pacificand North Atlantic as well as in coastal communitiesthat have experienced eutrophication problems due toincreased N and/or P loads relative to Si loads (Rague-neau et al., 2000). Furthermore, rare taxa are more sen-sitive to changes in salinity than common taxa in site-to-site comparisons (Metzeling, 1993). Species that aremore tolerant to salinity variations may be able to dis-perse widely into different rivers feeding an estuary orcoastal region and negatively affect native populations,which may be adapted to the narrow range of concen-trations seen in their particular chemostatic catchment(Bringolf et al., 2005). Therefore, the chemostatic behav-ior of catchments may enhance hydrochemical stabilityacross a wide range of flows and thus promote biodi-versity, but also suggests that streams subject to anthro-pogenic salinity variations or estuarine influences aremore vulnerable to invasive species.
CONCLUSIONS
Concentrations of weathering-derived solutes vary lit-tle with discharge for 59 USGS HBN streams, indi-cating that these catchments exhibit nearly chemostaticbehaviour. Concentrations vary by only a factor of 3 to20 as discharge varies by several orders of magnitude,and annual mean concentrations are similarly insensi-tive to changes in annual water yield. Concentrationsof the major base cations and silica typically exhibitpower-law relationships with discharge, with small nega-tive exponents. Broad qualitative lithologic differences,such as the presence or absence of volcanics or car-bonates, are associated with statistically significant dif-ferences in the power-law concentration–discharge slopeamong different sites. Other site characteristics such as
area, low temperature and average annual runoff aresignificantly rank-correlated with site-to-site variationsin concentration–discharge slopes for some solutes, butthe predictive power of site characteristics to quantifyconcentration–discharge slopes is limited. The narrowrange of concentration variability with discharge impliesthat rates of solute production and mobilization must benearly proportional to water fluxes. Because concentra-tions remain nearly constant across wide ranges in dis-charge, solute yield from catchments (and, at continentalscale, alkalinity fluxes to the oceans) are predominantlydetermined by water yield.
It is difficult to find simple generalizable mod-els that accurately represent the typical form of theconcentration–discharge relationship, that are internallyconsistent, and that make plausible assumptions aboutcatchment behaviour. A simple ‘bucket’ model can-not reproduce the observed concentration–dischargebehaviour. The Hubbard Brook ‘working model’ canfit the observations, but the required best-fit ‘rainfall’concentrations are much higher than observed concentra-tions, suggesting that key physical or chemical processesare not captured by the model. A Langbein-Dawdy chem-ical reaction model that assumes that reaction rates varywith distance from equilibrium can also fit the observedconcentration–discharge relationships, but requires rel-atively high input concentrations (and therefore likelyrequires an important role for soil water) and assumesthat no mixing of waters of different ages can occur.Finally, a new chemical and mixing model can explainthe observed power-law relationships between concen-tration and discharge in terms of depth profiles of poros-ity, characteristic pore size and hydraulic conductivityin the catchment. In this model, changes in dischargecorrespond to changes in the depth of saturation in thesubsurface. These in turn alter solute fluxes by alter-ing the reactive wetted surface area. The assumptionsof this new model should be tested at a subset of theHBN sites. An internally consistent model of hydrology,chemical weathering and transport is needed to explainthe power-law concentration–discharge relationships thatare observed across hydrochemically diverse catchments.
ACKNOWLEDGEMENTS
We thank Tom Meixner, Madeline Solomon, SteveSebestyen, Jake Peters and Kathleen Lohse for their help-ful discussions, and thank the 2003 Sagehen WatershedSymposium for stimulating the collaboration that led tothis work. This work was supported by National ScienceFoundation (NSF) grant EAR-0125550, the Miller Insti-tute for Basic Research in Science and an NSF GraduateResearch Fellowship.
APPENDIX
Here we develop the new permeability-porosity-aperturemodel, briefly discussed in the main text, in which
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
1862 S. E. GODSEY, J. W. KIRCHNER, D. W. CLOW
chemical weathering and hydrological mixing jointlycontrol the relationship between solute concentrations andwater fluxes.
First, we assume that permeability (k, [m/s]), porosity(�, [dimensionless]) and average pore aperture or width(p, [m]) all decrease exponentially with depth (z, [m])from their values at the soil surface (ko, �o and po,respectively):
k D koe�z
/�k �A1�
� D �oe�z
/�� �A2�
p D poe�z
/�p �A3�
The rates at which these variables decrease with depth,as expressed by the e-folding distances (�k , ��, �p all[m]), do not need to be equal to one another. Permeabilityand porosity are known to decrease roughly exponentiallywith depth at many sites. An exponential decrease inaperture with depth is less well characterized, but thisassumption is plausible and testable. Invoking Darcy’slaw with an approximately constant head gradient alongthe hillslope, and using Equation A1, one can expressspecific discharge (q, [m/s]) in a similar manner as anexponential function that varies as permeability decreaseswith depth:
q D qoe�z
/�k �A4�
where qo is a reference discharge that equals ko times thehillslope gradient. Thus water discharge (Qw, [m2/s]) perunit hillslope width can be calculated as:
Qw D∫ 1
zqdz D �kqoe
�z/
�k �A5a�
One can also assume that water discharge is propor-tional to effective precipitation inputs:
Qw D Rx �A5b�
where R [m/s] is the effective precipitation rate and x[m] is distance from the hill crest to any point along thehillslope. By setting Equations (A5a) and (A5b) equal toone another, one can solve for depth of the hydrologicallyactive region (i.e. soil and bedrock) as a function ofdistance along the hillslope:
z�x� D ��k ln(
Rx
�kqo
)�A6�
Equation A6 assumes that effective precipitation isspatially uniform along the hillslope and that flow orig-inating near the divide and near the stream (i.e. in thenon-linear portions of the hillslope) is negligible, and thusthat the approximation of a constant hillslope gradient inEquation A4 is valid.
We can also define the solute flux per unit hillslopewidth (Qs, [mol m�1 s�1]) so that it is proportionalto the reactive surface area per unit land area, AŁ[dimensionless]. Pore volume is the product of porosity
and total volume (V, [m3]), and reactive surface area perunit volume of pores and medium can be defined as
surf ace area/�volume of pores C medium�
D V��
pVD ��
p�A7�
where � [dimensionless] is a shape factor that is assumedto be constant with depth. We then integrate to getreactive surface area per unit land area, AŁ:
AŁ D∫ 1
z�
�
pD
∫ 1
z�
�o
poe
�z
(1/
��� 1/
�p
)
D ��o
po(
1/
�� � 1/
�p
)e�z
(1/
��� 1/
�p
)�A8�
Note that there is a larger wetted reactive surfacearea per unit land area, AŁ, at higher flow rates in thismodel as a result of the relationship between z andR (Equation (A6)). For a surface dissolution reactionin which secondary and back-reactions can be ignored,solute flux per unit hillslope width (Qs) is the product ofthe reaction constant, kR [mol m�2 s�1], and the reactivesurface area per unit land area, AŁ, integrated along thelinear segment of the hillslope from the base (x D 0) tonear the ridge (x D L) using the relationships derived inEquations (A6) and (A8):
Qs D∫ L
0kRAŁdx D
∫ L
0
kR��o
po(
1/
�� � 1/
�p
)e
(�k/
��� �k/
�p
)ln
(Rx
�kqo
)dx
D∫ L
0
kR��o
po(
1/
�� � 1/
�p
)(
Rx
�kqo
)(�k/
��� �k/
�p
)dx
D kR��o
po(
1/
�� � 1/
�p
)(
1
1 C �k/
�� � �k/
�p
)
(�kqo
R
) (Rx
�kqo
)(�k/
��� �k/
�pC1
)∣∣∣∣∣∣∣L
0
D LkR��o
po(
1/
�� � 1/
�p
) (1 C �k
/�� � �k
/�p
)(
RL
�kqo
)(�k/
��� �k/
�p
)�A9�
The volume-weighted mean concentration in the waterflux (C, [mol/m3]) is equal to the ratio of the solute flux(Equation A9) to the water flux (D RL), both expressedper unit hillslope width:
C D Qs
QwD
∫ L
0kRAŁdx
Qw
Copyright 2009 John Wiley & Sons, Ltd. Hydrol. Process. 23, 1844–1864 (2009)DOI: 10.1002/hyp
CONCENTRATION–DISCHARGE RELATIONSHIPS 1863
D LkR��o
po(
1/
�� � 1/
�p
) (1 C �k
/�� � �k
/�p
)(
1
�kqo
)(�k/
��� �k/
�p
)�RL�
(�k/
��� �k/
�p�1
)
D aoQb0w �A10�
A power-law concentration–discharge relationshipcan thus be expressed for a one-dimensional hill-
slope, with the constant ao
mol Ð s
(�k��
� �k�p
�1)
/
m
(2�k��
� 2�k�p
C1)
equal to all of the other constants
before the RL term, and the power-law exponent b0
[dimensionless] equal to:
b0 D (�k/
�� � �k/
�p � 1)
�A11�
Furthermore, storage (S, [m]) can be defined as theintegral of porosity over total depth:
S D∫ 1
z� D ���oe
�z/
�� �A12�
By combining Equations A5a and A12, one can seethat discharge (Qw) varies as a power function of storage:
Qw/Qo D �S/So�b1 �A13a�
where Qo and So are reference values of discharge andstorage, such that Qw D Qo at S D So, and the power-lawexponent, b1 [dimensionless] is:
b1 D ��
�k. �A13b�
The value of b1 can be determined from recessionanalysis, because Equation A13a implies that a plot of�dQw/dt as a function of Qw should have a log–logslope of 2 � �1/b1�, derived during periods when bothevapotranspiration and precipitation are small relative todischarge (e.g. during rain-free nights) (Kirchner, 2009).
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