Scaling and predicting solute transport processes in streams

Authors

  • Ricardo González-Pinzón,

    Corresponding author
    1. College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, USA
    • Corresponding author information: R. González-Pinzón, College of Earth, Ocean, and Atmospheric Sciences, 104 Wilkinson Hall, Oregon State University, Corvallis (OR), 97331-5506, USA. (gonzaric@geo.oregonstate.edu)

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  • Roy Haggerty,

    1. College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, USA
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  • Marco Dentz

    1. Department of Geosciences, Institute of Environmental Assessment and Water Research (IDAEA-CSIC), Barcelona, Spain
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Abstract

[1] We investigated scaling of conservative solute transport using temporal moment analysis of 98 tracer experiments (384 breakthrough curves) conducted in 44 streams located on five continents. The experiments span 7 orders of magnitude in discharge (10−3 to 103 m3/s), span 5 orders of magnitude in longitudinal scale (101 to 105 m), and sample different lotic environments—forested headwater streams, hyporheic zones, desert streams, major rivers, and an urban manmade channel. Our meta-analysis of these data reveals that the coefficient of skewness is constant over time ( math formula, math formula). In contrast, the math formula of all commonly used solute transport models decreases over time. This shows that current theory is inconsistent with experimental data and suggests that a revised theory of solute transport is needed. Our meta-analysis also shows that the variance (second normalized central moment) is correlated with the mean travel time ( math formula), and the third normalized central moment and the product of the first two are very strongly correlated ( math formula). These correlations were applied in four different streams to predict transport based on the transient storage and the aggregated dead zone models, and two probability distributions (Gumbel and log normal).

1. Introduction

[2] Two of the most challenging problems in surface hydrology are scaling and predicting solute transport in streams [Young and Wallis, 1993; Jobson, 1997; Wörman, 2000; O'Connor et al., 2010]. We must resolve these challenges to wisely manage water resources because there is a need to understand controls on stream ecosystems at local, regional, and continental scales, and because we need to predict transport in environments and conditions that do not have supporting tracer test data.

[3] Quantitative representations of hydrobiogeochemical processes are based on mathematical and numerical simplifications. Each simplification, the need to parameterize and integrate spatial and temporal processes, and the limitation of available observations to constrain models introduce structural errors and uncertainty in the predictions derived from such models [Beven, 1993; Wagener et al., 2004]. On the other hand, the transferability of empirical relationships from intensely instrumented catchments (mainly located in developed countries) to ungauged catchments relies on the similarity of hydrobiogeochemical characteristics [Sivapalan, 2003], thus limiting their practical application in regions where they are more needed.

[4] Solute transport and nutrient processing have been analyzed from different modeling perspectives, i.e., physically based, stochastic [Botter et al., 2010; Cvetkovic et al., 2012] and data-based mechanistic approaches [Young and Wallis, 1993; Young 1998; Ratto et al., 2007]. Although these approaches have increased our awareness about key compartments and hydrologic conditions that exert important influence on biogeochemical processes, i.e., identification of hot spots and hot moments [McClain et al., 2003], there is not yet a unified approach that has proven successful to scale and predict solute transport and nutrient processing.

[5] In the last three decades, research on solute transport and nutrient processing has revealed complex interactions between landscape and stream ecosystems, and attempts to scale and predict these processes have been limited by the difficulty of measuring and extrapolating hydrodynamic and geomorphic characteristics [Scordo and Moore, 2009; O'Connor and Harvey, 2008; O'Connor et al., 2010], and by the qualitatively confusing analyses derived from poorly constrained parametric interpretations of model-based approaches. A literature review presented hereafter (chronologically organized) shows contradictory evidence about the relationship between transient storage (TS) [Bencala and Walters, 1983; Beer and Young, 1983], the theory most frequently used to explain solute transport and in-stream processing. Valett et al. [1996] found a strong correlation ( math formula) between TS and NO3 retention in three first-order streams in New Mexico. Mulholland et al. [1997] found larger PO4 uptake rates in a stream with higher TS, when they compared two forested streams. Martí et al. [1997] found no correlation between NH3 uptake length and math formula (TS to main channel sizing ratio) in a desert stream. Hall et al. [2002] found a very weak correlation ( math formula) between TS parameters and NH4 demand in Hubbard Brook streams. In the 11 stream LINX-I data set, Webster et al. [2003] found no statistically significant relationship between NH4 uptake and TS. Thomas et al. [2003] showed that TS accounted for 44%–49% of NO3 retention measured by 15N in a small headwater stream in North Carolina. Niyogi et al. [2004] did not find significant correlations among soluble reactive phosphorous (P-SRP) and NO3 uptake velocities, and TS parameters. Ensign and Doyle [2005] found an increase of math formula and uptake velocities for NH4 and PO4, after the addition of flow baffles to the streams studied. Ryan et al. [2007] found strong relationships in two urban streams between P-SRP retention and TS when the variables were measured at different regimes in the same stream. Lautz and Siegel [2007] found a modest correlation ( math formula) between NO3 retention efficiency and TS in the Red Canyon Creek watershed (WY). Bukaveckas [2007] reported an indefinite relationship between TS and NO3 and P-SRP retention efficiencies. Lastly, the LINX-II data set from 15N-NO3 injections in 72 streams showed no relationship between NO3 uptake and TS [Hall et al., 2009].

[6] One factor that might contribute to the absence of strong relationships between TS and nutrient processing is the use of metrics that obscure the importance of TS across study sites (see discussions by Runkel [2002, 2007]). Also, it has become apparent that there are important limitations to identifying TS parameters with current techniques [Wagener et al., 2002; Schmid, 2003; Camacho and González-Pinzón, 2008], i.e., multiple sets of parameters might represent field observations “equally well” [Beven and Binley, 1992], and choosing a unique set of parameters to describe the behavior of a system might lead to misinterpretations of their physical meaning (if any), especially when those parameter sets are used to compare streams from different ecosystems and/or hydrologic conditions.

[7] In spite of the observed complexity of solute transport processes in streams, it is surprising that systems governed by physical processes that are considered “well understood” and by reasonably predictable biochemical interactions behave so unpredictably when combined. More robust methods are required to deconvolve signal imprints of solute transport and nutrient processing, thus allowing the development and implementation of improved decision-making approaches for stream management.

[8] In this paper, we investigated the existence of temporal patterns that can be used to scale and predict solute transport processes using an extensive database of tracer experiments that span 7 orders of magnitude in discharge, 5 orders of magnitude in longitudinal scale, and sample different lotic environments on five continents–forested headwater streams, hyporheic zones, desert streams, major rivers, and an urban manmade channel. From this meta-analysis, which is only implicitly dependent on hydrogeomorphic characteristics, we have proposed an approach to perform uncertainty analysis on solute transport processes and discussed some inconsistencies of the classic solute transport theory.

2. Methodology

2.1. Temporal Moments From Time Series

[9] We investigated conservative solute transport using temporal moments of the histories of multiple conservative tracer tests. Our analysis is based on an Eulerian approach, where the time series have been collected at different fixed spatial locations in each stream. Temporal moments have been widely used in the study of solute transport and biochemical transformations. Das et al. [2002] and Govindaraju and Das [2007] presented an extensive review of the theory and applications of temporal moment analysis to study the fate of conservative and reactive solutes. Recently, Leube et al. [2012] discussed the efficiency and accuracy of using temporal moments for the physically based model reduction of hydrogeological problems.

[10] Moments of distributions are commonly expressed as measures of central tendency. The nth absolute moment (also referred to as the nth raw moment or nth moment about 0), math formula, of a concentration time series, math formula, is defined as

display math(1)

[11] The nth normalized absolute moment (also referred to as the nth normalized raw moment or nth normalized moment about 0), math formula, is defined as

display math(2)

and the nth normalized central moment (also referred to as the nth normalized moment about the mean), math formula, is defined as

display math(3)
display math(4)

where math formula is an index. Note that (4) is an inverse binomial transform that can be easily used to calculate the normalized central moments of order 1 (mean travel time), 2 (variance), and 3 (skewness):

display math(5)

[12] Temporal moments are also related to residence time distributions and transfer functions of linear dynamic systems [Jury and Roth, 1990; Sardin et al., 1991]. Aris [1958] developed a method to compute the theoretical temporal moments of linear functions, thus allowing the use of experimental temporal moments (i.e., those estimated from observed time series) to estimate the parameters of linear dynamic models, i.e.,

display math(6)

where math formula is the Laplace transform of math formula and math formula is the longitudinal distance in one-dimensional approximations.

[13] Theoretical temporal moments for most solute transport models have been estimated for different types of boundary conditions. A few examples of the progress on this topic are the development of temporal moment-generating equations to model transport and mass transfer [Harvey and Gorelick, 1995; Luo et al., 2008], and the calculation of temporal moments for the TS model [Czernuszenko and Rowinski, 1997; Schmid, 2002], equilibrium and nonequilibrium sorption models [Goltz and Roberts, 1987; Cunningham and Roberts, 1998], the aggregated dead zone model [Lees et al., 2000], and the metabolically active TS model [Argerich et al., 2011].

[14] Matching (or equating) experimental and theoretical temporal moments is a useful technique to parameterize linear models [Nash, 1959]. The advantages of using experimental moments to match theoretical moments come with the challenge to completely recover the tracer experiment signals, as it has been shown that truncation errors affect the estimation of higher-order temporal moments. Using experimental data, Das et al. [2002] and Govindaraju and Das [2007] showed that when the error in mass recovery is 16%, the errors in absolute nth moments can be as high as approximately math formula for math formula through math formula. This problem is related to the early cutoff of data measurement or the lack of instrumental resolution to detect low concentrations of tracers, and is not related to the apparent incomplete mass recovery due to dilution effects (e.g., groundwater contributions). Note that correcting the observed breakthrough curves (BTCs) uniformly (with a steady-state gain factor) for dilution only affects the magnitude of the absolute moments but does not modify the magnitude of the normalized absolute moments or that of the normalized central moments.

2.2. Experimental Database

[15] We created a database that includes 384 concentration time series, or BTCs, from 98 conservative tracer experiments conducted in 44 streams under different quasi-steady hydrologic conditions (10−3 to 103 m3/s), different experimental conditions (BTCs observed from 101 to 105 m downstream the injection point), and different types of lotic environments (Table 1). We grouped the database by the orders of magnitude of discharge (Table 2) to facilitate the analysis and presentation of the statistical regressions in Figures 1 and 2. All BTCs were zeroed to background concentrations and corrected by discharge changes during the experiments as specified in the references or recorded in experimental notes.

Table 1. Conservative Solute Transport Databasea
StreamReach Length (km)Discharge (m3/s)State, Country, (Continentb)References
  1. a

    A total of 98 tracer experiments with 384 BTCs were used in this meta-analysis.

  2. b

    SA: South America; NA: North America; EU: Europe; AUS: Australia; AN: Antarctica.

Canal Molinos0.20.2–0.4Colombia (SA)As referenced by González-Pinzón [2008]
Quebrada Lejía0.30.1–0.5Colombia (SA)
Subachoque 10.3–0.40.2–1.3Colombia (SA)González-Pinzón [2008] and Camacho and González-Pinzón [2008]
Subachoque 20.1–0.20.3–1.9Colombia (SA)
Teusacá 10.1–0.20.3–0.4Colombia (SA)
Teusacá 20.3–0.40.2–1.4Colombia (SA) 
Rio Magdalena36–2071200–1390Colombia (SA)Torres-Quintero et al. [2006]
Shaver's Cr.0.1–0.40.2PA, USA (NA)Unpublished data
Cherry Cr.0.7–1.30.2WY, USA (NA)Briggs et al. [2013]
Oak Cr.0.04–0.30.02OR, USA (NA)Experiments conducted during the Ph.D. dissertation of the first author.
Fuirosos 10.2–0.30.01Spain (EU)
Fuirosos 20.2–0.30.01Spain (EU) 
Antietam Cr.2.6–671.2–12.7MD, USA (NA)As referenced by Nordin and Sabol [1974, Appendix A].
Monocacy River7.5–3412.7–22.1MD, USA (NA)
Conococheague Cr.4.4–342.6–30.6MD, USA (NA)
Chattahoochee River10.5–104108–180GA, USA (NA)
Salt Cr.9.3–522.5–4.1NE, USA (NA)
Difficult Run0.6–20.9–1.1VA, USA (NA)
Bear Cr.1.1–10.910.2–10.5CO, USA (NA)
Little Piney Cr.0.6–7.31.4–1.6MO, USA (NA)
Bayou Anacoco11–382.0–2.7LA, USA (NA)
Comite River6.8–790.8–1.0LA, USA (NA)
Bayou Bartholomew3.2–1174.1–8.1LA, USA (NA)
Amite River10–1485.7–8.9LA, USA (NA)
Tickfau River6.4–502.0–2.9LA, USA (NA)
Tangipahoa River8.2–943.5–18.7LA, USA (NA)
Red River5.7–199108–249LA, USA (NA)
Sabine River7.9–209127–433LA, USA (NA)
Sabine River17–1210.7–9.5TX, USA (NA)
Mississippi River35–2941495–6824LA, USA (NA)
Wind/Bighorn River9.1–18155–255WY, USA (NA)
Copper Cr.0.2–8.41.0–8.7VA, USA (NA)
Clinch River0.7–6.65.7–110VA, USA (NA)
Powell River1.0–7.13.9–4.1TN, USA (NA)
Coachella Canal0.3–5.525.4–26.9CA, USA (NA)
Missouri River66–227883–977IA, USA (NA) 
WS10.02–0.31 l/s–0.06OR, USA (NA)Gooseff et al. [2003, 2005];Haggerty et al. [2002], unpublished
WS30.04–0.71 l/s–0.03OR, USA (NA)
Lookout Cr.0.2–0.40.3OR, USA (NA)Gooseff et al. [2003]
Huey Cr.0.5–1.00.1ANRunkel et al. [1998]
Swamp Oak Cr.0.1–0.30.1AUSLamontagne and Cook [2007]
Clackamas River9.336.8OR, USA (NA)Lee [1995]
Uvas Cr.0.04–0.40.01CA, USA (NA)Bencala and Walters [1983]
River Mimram0.1–0.20.3UK (EU)Lees et al. [2000]
Table 2. Conservative Solute Transport Database Grouped by the Orders of Magnitude of Dischargea
Discharge Group Q Gr.Discharge Order of Magnitude (m3/s)Number of Experiments
  1. a

    The regressions presented in Figures 1 and 2 were labeled as described hereafter.

110−319
210−237
310−168
4100131
510159
610253
710317
Figure 1.

Meta-analysis (n = 384 BTCs) of conservative solute transport experiments in streams demonstrates the general occurrence of non-Fickian dispersion processes. (a) The growth rate of the variance is nonlinear (therefore non-Fickian) with respect to the mean travel time; the thick dashed line represents the slope pattern of Fickian dispersion. (b) Skewness as a function of the mean travel time. Coefficients were fitted with 95% confidence bounds. Thin dashed lines represent 95% prediction bounds.

Figure 2.

(a) Meta-analysis (n = 384 BTCs) of conservative solute transport experiments from contrasting stream ecosystems suggests that the coefficient of skewness holds statistically constant. Fitted coefficients defined math formula. (b) The factor [ math formula] is a quasi-linear estimator of math formula. However, using math formula to define the ratio [ math formula] yields an math formula, showing that a satisfactory bottom-up estimation of normalized central moments is restricted to one level, at most. Coefficients were fitted with 95% confidence bounds. Thin dashed lines represent 95% prediction bounds.

3. Results and Discussion

3.1. Statistical Relationships Derived From Temporal Moment Analysis

[16] Information regarding longitudinal mixing and exchange processes can be found in the normalized central moments (moments about the mean). Figure 1a shows that the variance scales in a nonlinear (non-Fickian) form with the mean travel time. If dispersion processes in streams were Fickian, the regression presented in Figure 1a would have a slope of ∼1.0, still preserving a scatter pattern that would be associated with the magnitudes of the dispersion coefficient for each experiment (i.e., different intercepts). Non-Fickian dispersion processes have been widely observed in stream ecosystems [e.g., Fischer, 1967; Nordin and Sabol, 1974; Nordin and Troutman, 1980; Bencala and Walters, 1983, and references therein], and in heterogeneous porous media [e.g., Rao et al., 1980; Haggerty and Gorelick, 1995; Dentz and Tartakovsky, 2006]. A non-Fickian behavior is, broadly defined, the result of the presence of multiscale heterogeneities that cannot be integrated into a singular dispersion coefficient [Neuman and Tartakovsky, 2009]. To date, several approaches have been proposed to better represent non-Fickian transport, which are largely based on the conceptualization of TS processes and/or the definition of smaller representative elementary volumes, where local homogeneities can be integrated in space and time.

[17] We also correlated math formula versus math formula and math formula versus math formula. Figure 2a suggests that solute transport data have a small range in their coefficient of skewness ( math formula, equation (7)). The coefficient of skewness is a measure of asymmetry, i.e., when math formula the data is perfectly symmetrical (no tailing), but it is known that solute transport experiences tailing effects due to surface and hyporheic TS, regardless of the type of stream ecosystem. For the 98 tracer tests (384 BTCs), math formula (95% confidence bounds). In Figure 2b, we show that the product math formula is a quasi-linear estimator of math formula ( math formula). This result, although not representing a predefined statistical descriptor on its own, will be later used to define objective functions for predictive solute transport models (see section 3.3.). Not unexpectedly, based on the results from Figure 1, math formula is a much weaker predictor of the ratio math formula ( math formula, results not shown), suggesting that a satisfactory bottom-up estimation of normalized central moments is restricted to one level at most.

display math(7)

3.2. Observed Scale Invariance in Streams and Solute Transport Models

[18] Nordin and Sabol [1974] first reported observations revealing persistent skewness (longitudinally) from Eulerian observations of solute transport time distributions. Nordin and Troutman [1980] investigated the performance of the Fickian-type diffusion equation (advection dispersion equation (ADE)), and the inclusion of dead zone processes (i.e., TS model (TSM)) to account for the persistence of skewness, concluding that “…the observed data deviate consistently from the theory in that the skewness of the observed concentration distributions decreases much more slowly than the Fickian theory predicts,” and that although the inclusion of dead zones “…yields a theoretical skewness coefficient [ math formula] considerably larger than that given by the ordinary Fickian diffusion equation,” “…the skewness of the observed concentrations does not appear to be decreasing as rapidly as the theory predicts.” The skewness of BTCs also do not begin with values as high as those predicted by the TSM (cf. Nordin and Troutman, 1980, Figure 3).

Figure 3.

Predicted results using empirical relationships derived from normalized central moment meta-analysis (n = 384 BTCs) and the moment-matching technique for the TSM. The known variables were math formula, math formula and math formula, and all others were predicted from 1000 Monte Carlo simulations. The effects of uncertainty in estimating math formula (i.e., math formula, with math formula), the parameters of the TSM and the fitting coefficients from our meta-analysis are shown as uncertainty bounds. (a) River Brock, (b) River Conder, (c) River Dunsop, and (d) River Ou Beck. Experimental observations from Young and Wallis [1993]. The best parameter sets from the simulations are presented in Table 3. Goodness of fit was estimated with the Nash–Sutcliffe model efficiency coefficient ( math formula).

[19] The work by van Mazijk [2002] reported that tracer experiments conducted to develop the River Rhine alarm model also showed time distributions with persistent math formula along the extensive reach studied (100 km < L   < 1000 km; math formula; cf. van Mazijk, 2002, Figure 6), i.e., math formula. These observations justified the use of the Chatwin-approximation (Edgeworth series) [Chatwin, 1980] to predict solute concentrations in space and time, by fixing math formula for the whole river. Further tracer experiments in the River Rhine ( math formula) supported the existence of a persistent math formula [van Mazijk and Veling, 2005].

[20] Schmid [2002] investigated the conditions under which the TSM could represent the persistence of skewness in solute transport processes. Schmid [2002] examined the case of a slug injection into a uniform channel and concluded that a small parametric region (a loop right bounded by math formula; cf. Schmid [2002, Figure 1]) could generate a nondecreasing math formula. However, this condition was hypothetical and does not play a major role in practice. Such conditions, if they exist, would be logically inconsistent because tailing effects would be inversely proportional to TS. Schmid [2002] also examined a more general scenario with a time-varying concentration distribution as an upstream boundary condition, the division of long reaches into hydraulically uniform subreaches and a routing procedure to link temporal moments at both ends of the subreaches. This analysis suggested that “…the TS model has the potential to explain persistent or growing temporal skewness coefficients, if applied to a sequence of subreaches with respective parameter sets different from each other.” However, predicting solute transport meeting these conditions is rather impractical.

[21] If a transport theory is to be capable of scaling and predicting solute transport processes, it will have a persistent and statistically constant math formula. Our observations of math formula being statistically constant for widely different hydrodynamic conditions suggest that math formula is not only persistent for a given stream (with distance traveled downstream), but can also be used to scale and predict solute transport processes across ecosystems. At a minimum, a persistent value of math formula is a test that a theory of solute transport must pass.

[22] We used the theoretical temporal moments of three models commonly used for the analysis of in-stream solute transport (ADE, TSM, and the aggregated dead zone model (ADZM)) to calculate their theoretical math formula. If these models were systematically capable of representing the scale-invariant patterns observed in our meta-analysis, the parameters would be self-consistent when describing math formula. The model equations and the theoretical temporal moments and math formula (calculated for an impulse-type boundary condition, e.g., Cunningham and Roberts [1998]) are shown below, along with the consequences of the invariance of math formula on the model parameters. We also included in our analysis (see section 3.2.4) three additional transport models less commonly used to describe solute transport in streams, but that have been used in groundwater systems.

3.2.1. Advection Dispersion Equation

[23] 

display math(8)
display math(9)

where math formula [ML−3] is the concentration of the solute in the main channel; math formula [L3T−1] the discharge; math formula [L2] the cross-sectional area of the main channel; math formula [LT−2] the dispersion coefficient; math formula [L] the reach length; math formula [T] time; math formula [T] is the conservative mean travel time; math formula the Peclet number; and math formula the mean velocity in the main channel [LT−1].

[24] Equation (9) suggests that if math formula is constant, the Peclet number should also be constant. This implies that, under steady-state flow conditions, the dispersion coefficient must scale linearly with the distance traveled. This violates the assumption of spatially uniform coefficients. Therefore, the ADE with spatially uniform coefficients is incapable of representing the experimental observations. Dispersion coefficients scaling with distance have been widely observed in porous media [e.g., Pickens and Grisak, 1981; Silliman and Simpson, 1987, Pachepsky et al., 2000, and references therein]. Note that the ADE with constant coefficients predicts BTCs with longitudinally decreasing skewness ( math formula), becoming asymptotically Gaussian (i.e., math formula).

3.2.2. Transient Storage Model

[25] 

display math(10a)
display math(10b)
display math(11)

where math formula [ML−3] is the concentration of the solute in the storage zone; math formula [L2] is the cross-sectional area of the storage zone; math formula [T−1] is the mass-exchange rate coefficient between the main channel and the storage zone; and math formula. Other variables are as defined for the ADE. The TSM in equation (10a) is the same presented by Bencala and Walters [1983] and Runkel [1998] for a reach without lateral inputs, with a slightly different definition of math formula. Note that math formula when math formula.

[26] If dispersion effects were assumed negligible [e.g., Wörman, 2000; Schmid, 2002], math formula in equation (11) would simplify to

display math(12)

[27] Using the math formula value found in our meta-analysis, the mean residence time in the storage zones ( math formula) normalized by math formula scale linearly with travel time ( math formula), i.e.,

display math(13)

[28] Equations (11) and (12) suggest that the standard TSM generates BTCs with longitudinally decreasing skewness ( math formula), becoming asymptotically Gaussian (i.e., math formula). The physical meaning of the parameters describing math formula is unclear unless dispersion is assumed negligible ( math formula). In this case, equation (13) suggests that the TSM model parameters are not independent and that their ratio grows with distance traveled. This analysis supports the results of other studies showing problems of equifinality for the TSM [e.g., Wagner and Harvey, 1997; Wagener et al., 2002; Camacho and González-Pinzón, 2008; C. Kelleher et al., Stream characteristics govern the importance of transient storage processes, submitted to Water Resources Research, 2012]. Equations (11) and (13) suggest that the physical meaning of the TSM parameters is limited, and that relationships between TSM parameters and biogeochemical processing may be site dependent (as was discussed in section 1) or even experiment dependent.

3.2.3. Aggregated Dead Zone Model

[29] 

display math(14)
display math(15)

where math formula [T] is the lumped ADZ residence time parameter representing the component of the overall reach travel time associated with dispersion; math formula [ML−3] is the known concentration at the input or upstream location; and math formula [T] is the time delay describing solute advection due to bulk flow movement.

[30] Equation (14) describes the mass balance of an imperfectly mixed system (ADZ representative volume), where a solute undergoes pure advection, followed by dispersion in a lumped active mixing volume [Lees et al., 2000]. In the ADZM, the distance math formula implicitly appears in the model description through the time parameters. Note that when math formula, the mean travel time ( math formula) could be written as math formula. In equation (15), the parameter math formula represents the number of identical ADZ elements serially connected ( math formula for a single ADZ representative volume) to route the upstream boundary condition. The serial ADZM, although capable of representing a persistent math formula, would require the specification of the nonphysical parameter math formula. More complex ADZM structures can be defined under the database mechanistic approach [e.g., Young, 1998], but we restricted our discussion to those that have been more commonly used in stream solute transport modeling [Young and Wallis, 1993; Lees et al., 2000; Camacho and González-Pinzón, 2008; Romanowicz et al., 2013].

3.2.4. Alternative Solute Transport Models

[31] Similar sets of calculations also show that the multirate mass transfer (MRMT) model [Haggerty and Gorelick, 1995; Haggerty et al., 2002] (Appendix A) and a decoupled continuous time random walk (dCTRW) model [e.g., Dentz and Berkowitz, 2003; Dentz et al., 2004; Boano et al., 2007] (Appendix B) are equally incompatible with observations of persistent skewness. The math formula in both of these models also scales as math formula.

[32] We also explored a Lévy-flight dynamics model (LFDM) (Appendix C) [e.g., Shlesingerm et al., 1982; Pachepsky et al., 1997, 2000; Sokolov, 2000], which describes the motion of particles behaving similarly to Brownian motion, but allowing occasional clusters of large jumps (significant deviations from the mean). Lévy-flight models have constant transition times, combined with transition length distributions that are characterized by power-law behaviors for large distances. Therefore, such models represent processes characterized by large velocities for long transitions and low velocities for short transitions, and would account for transport in the continuum of river and storage, with the high velocities present in the stream. We were able to generate an LDFM with persistent math formula for a Lévy distribution parameter math formula (this math formula is different from the mass-exchange rate coefficient used in the TSM and MRMT model, (cf. (C2) and (C31)). However, math formula gives an inconsistent scaling of the variance with distance, i.e., math formula (cf. (C25)). Furthermore, this distribution parameter would imply a velocity distribution in the stream that scales as math formula at large velocities, which does not appear realistic.

3.2.5. Remarks on Existent Solute Transport Models

[33] To preserve math formula, the parameters in the solute transport models, including common versions of the CTRW and MRMT, must change with travel distance. Solute transport parameters therefore have some degree of scale dependence (and arbitrariness) imposed by the constant math formula. Furthermore, these parameters have scaling patterns that are unrelated to anything that can currently be measured in the field. These inconsistencies might be because (1) the common solute transport models and assumptions are partly incorrect or (2) we (the stream research community) have collected erroneous observations for decades. The latter condition is possible, but is not likely the explanation for a problem that has been observed across so many data sets. The worst-case scenario in our meta-analysis is that all BTCs were truncated prematurely, due to lack of instrument sensitivity or other reasons. However, this would generate BTCs with larger math formula and would contradict the asymptotic behavior shown for math formula in the transport models discussed above. Consequently, we suspect that our models do not correctly represent one or more aspects of solute transport processes from the field.

3.3. Use of Moments Scaling Properties to Predict Solute Transport

[34] While the models contain an error that needs correction, it may be possible (in the meantime) to adjust the parameters in a way that is predictive of field behavior. In this section, we use the regressions from the temporal moment analysis (section 3.1.) to predict solute transport. We provide the parameterization of the TSM, ADZM, and two probability distributions. We then provide an example using data from tracer experiments that were conducted in the River Brock, River Conder, River Dunsop, and River Ou Beck in the United Kingdom [Young and Wallis, 1993, pp. 160–165]. The first three rivers are natural, and River Ou Beck is a concrete urban channel.

[35] The methodology requires an independent estimation of the mean travel time ( math formula). One way to do this is to regress math formula against discharge ( math formula) using a power law or an inverse relationship in math formula [Young and Wallis, 1993; Wallis et al., 1989; Pilgrim, 1977; Calkins and Dunne, 1970]. Once math formula is estimated, the results from our temporal moment analysis can be used to constrain predictive (forward) simulations of solute transport models. We exemplify this methodology using the experiments by Young and Wallis [1993], which were not used in the previous moment analysis, because they show the technique to estimate mean travel times from discharge.

3.3.1. Predicted Solute Transport With Classic Solute Transport Models

[36] The parameters of solute transport models can be determined by matching theoretical and experimental moments. Here, we show how the empirical scaling relationships described in section 3.1 can be used to direct the search of the parameters of the TSM and the ADZM in predictive simulations.

3.3.1.1. Predicted Solute Transport With TSM

[37] We used the empirical relationships derived for math formula versus math formula and math formula versus math formula (Figure 2) to match the theoretical moment equations presented by Czernuszenko and Rowinski [1997]. These theoretical equations have been developed for a general upstream boundary condition with tracer distribution math formula. The parameters for the TSM are those defined by Bencala and Walters [1983] and Runkel [1998].

display math(16)
display math(17)
display math(18)

[38] We have eight variables, i.e., the dispersion coefficient math formula, math formula ( math formula), the mass-transfer rate math formula, the length of the reach math formula, the discharge math formula ( math formula), and the normalized central moments math formula, math formula, math formula. We have five equations: three for the theoretical moments (equations (16)-(18)) and two empirical relationships (derived from Figure 2). To balance the degrees of freedom ( math formula), we therefore need to specify three ( math formula) variables, namely math formula, math formula, and math formula. We used a Newton-Raphson algorithm to solve for the five unknowns by minimizing the objective function ( math formula) shown in equation (19). We estimated the mean travel time as: math formula, with math formula, and randomly varied the regression coefficients of our meta-analysis within the 95% confidence bounds.

display math(19)

[39] In the optimization routine, we allowed the TSM parameters to vary within ranges typically found in similar streams, i.e., math formula (m2/s), math formula (m2), math formula (m2), math formula (s−1). Once the system of equations was optimized for each random set of estimated mean travel time and fitting coefficients (n = 1000), we ran a forward simulation using the optimum parameters. Results from the Monte Carlo simulations are presented in Figure 3 and Tables 3 and 4. We used the Nash–Sutcliffe model efficiency coefficient ( math formula) [Nash and Sutcliffle, 1970] to estimate the goodness of fit of the predictions, i.e., how well the plot of observed versus simulated data fits a 1:1 line.

Table 3. Best Parameter Sets From 1000 Monte Carlo Simulations Using Empirical Relationships Derived From Normalized Central Moment Meta-Analysis (n = 384 BTCs) and the Moment-Matching Techniquea
River  TSMADZM
Q (m3/s)L (m) math formula (m2/s) math formula math formula × 105 (s−1) math formula math formula(s) math formula
  1. a

    Study case of four rivers located in the United Kingdom [Young and Wallis, 1993; pp. 160–165]. Goodness of fit was estimated with the Nash–Sutcliffe model efficiency coefficient ( math formula).

Brock4.5 × 10−11282.331.31 × 10−29.770.96218.010.98
Conder1.01162.208.12 × 10−38.080.99151.950.97
Dunsop5.4 × 10−11301.331.45 × 10−27.890.98332.551.00
Ou Beck3.5 × 10−21270.674.40 × 10−38.921.00135.950.76
Table 4. List of Estimated Parameters and Prediction Efficiencies for Each Predictive Model Exploreda
Predictive ModelEstimated Parameters Besides math formulaPrediction Efficiency ( math formula)
River BrockRiver ConderRiver DunsopRiver Ou Beck
  1. a

    The 1000 Monte Carlo simulations were run per model using empirical relationships derived from normalized central moment meta-analysis (n = 384 BTCs). Study case of four rivers located in the United Kingdom [Young and Wallis, 1993, pp. 160–165]. math formula, with math formula

  2. b

    In the predictive TSM simulations, we entered the actual discharge math formula and reach length math formula.

TSM math formula, math formula, math formula, math formulab, math formulab0.74–0.960.71–0.990.39–0.990.26–1.00
ADZM math formula 0.50–0.980.21–0.970.48–1.00−0.26–0.76
Gumbel dist. math formula 0.39–0.960.45–0.950.38–0.990.18–0.77
Lognormal dist. math formula 0.42–0.940.47–0.920.45–0.970.18–0.74
3.3.1.2. Predicted Solute Transport With ADZM

[40] The two parameters of this model are the advection time delay, math formula, and the residence time, math formula, where math formula is the mean travel time ( math formula). The theoretical moments of the ADZM for one first-order ADZ element ( math formula) were presented in equation (15). Since the mean travel time is a measured or estimated quantity, we only need to solve for the advection time delay, math formula. We applied the same optimization routine described for the TSM, and the results obtained are presented in Figure 4 and Tables 3 and 4.

Figure 4.

Predicted results using empirical relationships derived from normalized central moment meta-analysis (n = 384 BTCs) and the moment-matching technique for the ADZM. The known variable was math formula (or math formula), and math formula was predicted from 1000 Monte Carlo simulations. The effects of uncertainty in math formula (i.e., math formula, with math formula) and the fitting coefficients from our meta-analysis are shown as uncertainty bounds. (a) River Brock, (b) River Conder, (c) River Dunsop, and (d) River Ou Beck. Experimental observations from Young and Wallis [1993]. The best parameter sets from the simulations are presented in Table 3. Goodness of fit was estimated with the Nash–Sutcliffe model efficiency coefficient ( math formula).

3.3.2. Predicted Solute Transport With Probability Distributions

[41] Time series described by probability distributions can be used to predict solute transport processes. Here, we show how the empirical scaling relationships described in section 3.1 can be used to estimate the temporal moments of two probability distributions and then to perform predictive simulations.

3.3.2.1. Predicted Solute Transport With Gumbel Distribution

[42] We chose the Gumbel (Extreme Value I) probability distribution because of its constant math formula, which closely agrees with the empirical relationships derived from our meta-analysis ( math formula). This distribution is typically used to describe hydrologic events pertaining to extremes [Brutsaert, 2005]. The concentration distribution of a solute BTC using this distribution takes the form:

display math(20)

where math formula and math formula are the location (mode) and scale parameters, respectively. Note that these parameters, and those of any other probability distribution, have no direct physical interpretation.

[43] The use of probability distributions requires the explicit definition of moments beyond the mean travel time, i.e., variance and in some cases the skewness. Therefore, we would need to use empirical relationships such as those derived in Figure 1, even though math formula. In our predictive analysis, we used math formula, with math formula to estimate the uncertainty of math formula, and math formula, with math formula, as it was suggested by our meta-analysis (i.e., math formula, math formula, regression not shown in Figure 1). The results obtained are presented in Figure 5 and Table 4.

Figure 5.

Predicted results using empirical relationships derived from normalized central moment meta-analysis (n = 384 BTCs) and the Gumbel distribution, which has a constant math formula. Uncertainty bounds represent 1000 Monte Carlo simulations where math formula, with math formula, and math formula, with math formula. The “Gumbel=f(Obs.)” simulation uses the actual math formula and math formula moments derived from the observed data. (a) River Brock, (b) River Conder, (c) River Dunsop, and (d) River Ou Beck. Experimental observations from Young and Wallis [1993]. Goodness of fit was estimated with the Nash–Sutcliffe model efficiency coefficient ( math formula).

Figure 6.

Predicted results using empirical relationships derived from normalized central moment meta-analysis (n = 384 BTCs) and the lognormal distribution. Uncertainty bounds represent 1000 Monte Carlo simulations where math formula, with math formula, and math formula, with math formula. The “L-N=f(Obs.)” simulation uses the actual math formula and math formula moments derived from the observed data. (a) River Brock, (b) River Conder, (c) River Dunsop, and (d) River Ou Beck. Experimental observations from Young and Wallis [1993]. Goodness of fit was estimated with the Nash–Sutcliffe model efficiency coefficient ( math formula).

3.3.2.2. Predicted Solute Transport With Lognormal Distribution

[44] A random variable described by a lognormal distribution comes from the product of n variables, each with its own arbitrary density function with finite mean and variance. This distribution has been widely used in hydrologic modeling of flood volumes and peak discharges, duration curves for daily streamflow, and rainfall intensity-duration data [Chow, 1954; Stendinger, 1980]. Applications in solute transport suggested that the solute velocity, saturated hydraulic conductivity, and dispersion coefficient are lognormally distributed [Rogowski, 1972; Van De Pol et al., 1977; Russo and Bresler, 1981]. The concentration distribution of a solute BTC with this distribution takes the form:

display math(21)

where math formula and math formula are the mean and the standard deviation of math formula. In our predictive analysis, we followed the same procedure described for the Gumbel distribution. The results obtained are presented in Figure 6 and Table 4.

3.3.3. Analysis of Predictive Solute Transport Modeling

[45] In our predictive analyses, we used two classic models (TSM and ADZM) and hypothesized that these models could adequately predict solute transport if the results of our meta-analysis were defined as objective functions to minimize the differences between the theoretical and empirical temporal moments. Our main goal therefore was to fix a constant math formula regardless of the longitudinal positioning. The predictive results presented in Figures 3 and 4 and Tables 3 and 4 show that this approach required only basic information (i.e., math formula, math formula, and an estimation of the mean travel time) to adequately predict the behavior of the solute plumes traveling downstream. For the TSM (four parameters), the best predictions in the uncertainty analysis had math formula for the four rivers. For the ADZM (two parameters), the best predictions had math formula for all natural rivers, and math formula for the concrete channel. Although satisfactory results can be achieved with this predictive methodology, it is important to bear in mind that good fittings do not necessarily come from adequate interpretations of mechanistic processes and, therefore, the physical meaning of the parameters should not be taken literally in both inverse (used for calibration) and forward (predictive) simulations.

[46] Besides from predicting solute transport with classic models, we explored the use of probability distributions. We developed predictive models through the parameterization of the Gumbel and lognormal probability distributions, using the results from our meta-analyses and performing uncertainty estimations. The results of our predictive simulations can be summarized as (Table 4): (1) the Gumbel distribution ( math formula) yielded better predictions when the distributions were parameterized with the observed math formula and math formula, suggesting that math formula is a consistent pattern derived from our meta-analysis and (2) estimating the variance ( math formula) of the distributions from the mean travel time ( math formula) can be highly uncertain, and it is explicitly required for using probability distributions in predictive mode; therefore, uncertainty analysis must be always included. Importantly, the parameters of these distributions do not have direct physical meaning, and this has two main consequences: (1) solute transport understanding cannot be mechanistically advanced and (2) erroneous parametric interpretations from physically based, but poorly constrained models are explicitly avoided.

[47] In summary, we found that the regressions from our meta-analysis can be used to adequately predict solute transport processes using either transport models (fixing math formula) or probability distributions. We consider this a transitional methodology (“a patch solution”) between our current understanding and an improved transport theory that better represents the experimental results.

3.4. Implications for Scale-Invariant Patterns

[48] Other experimental findings reveal intriguing similarities to the scale-invariant patterns that we have highlighted here. These include the linear relationship between cross-sectional maximum and mean velocities [Chiu and Said, 1995; Xia, 1997; Chiu and Tung, 2002], and the relatively constant behavior of the dispersive fraction (a parameter derived from the ADZM) in alluvial and headwater streams [Young and Wallis, 1993; González-Pinzón, 2008]. These observations suggest that stream cross sections establish and tend to maintain a quasi-equilibrium entropic state by adjusting the channel characteristics, i.e., erodible channels adjust their geomorphic characteristics with discharge (bedform and type of sediment transported, slope, alignment, etc.) and nonerodible channels adjust their velocity distributions by changing the maximum velocity and flow depths [Chiu and Said, 1995; Chiu and Tung, 2002]. An improved solute transport theory should address these observed scale-invariant hydrodynamic patterns and explore the physical meaning of the persistence of skewness, which perhaps could be based on principles of thermodynamics and fluid dynamics.

[49] The coefficient of skewness of the classic solute transport models discussed in section 3.2 shows that Fickian dispersion is inconsistent with the experimental results. The inclusion of macroscopic Fickian dispersion generates a system where the variance of a dispersing solute grows linearly with the distance traveled, generating skewed distributions that later become asymptotically Gaussian [Fisher et al., 1979; Nordin and Troutman, 1980]. This behavior is independent of the assumption of hydraulically uniform stream reaches, suggesting that a revised dispersion approach would be needed unless other mechanisms included in the transport theory (e.g., TS) were capable of counteracting the ever decreasing skewness represented by Fickian dispersion.

[50] Although we have not yet investigated scale-invariant behaviors of temporal distributions in processes other than solute transport, we predict that similar patterns can be derived from meta-analysis of flow routing BTCs. We ground this prediction in the fact that the conservative tracers used in our analyses have marked up how water flowed through the different stream ecosystems considered, experiencing similar physical characteristics and processes involved in flow routing (i.e., shear effects, heterogeneity and anisotropy, and dual-domain mass transfer). Regardless of the adequacy of current transport and flow routing modeling approaches, clear similarities appear when comparing the BTCs of these hydrologic processes, and the temporal moments of (for example) the ADZM and those of the Nash cascade [Nash, 1960] and the Linear (and Multilinear) Discrete (Lag) Cascade channel routing models [O'Connor, 1976; Perumal, 1994; Camacho and Lees, 1999]. If similar patterns were found with respect to the persistence of skewness in solute transport and flow routing, this could be advantageously used to better understand, scale, and predict solute transport processes under flow dynamic conditions, which is a problem that still remains largely unresolved [Runkel and Restrepo, 1993; Graf, 1995; Zhang and Aral, 2004].

4. Conclusions

[51] Despite numerous detailed studies of in-stream transport processes [e.g., Bencala and Walters, 1983; Harvey and Bencala, 1993; Elliott and Brooks, 1997a, 1997b; Gooseff et al., 2005; Wondzell, 2006; Cardenas et al., 2008], scaling and predicting solute transport can be highly uncertain. This is primarily due to the difficulties of measuring and incorporating stream hydrodynamic and geomorphic characteristics into models. A consequence of these simplifications is that parameters cannot be obtained uniquely from physical attributes. The parameters are functions of a combination of several processes and physical attributes. Therefore, model parameters interact with each other, and the overall model response to different parameter sets might be numerically “equal” and mechanistically misleading.

[52] Our (model-free) meta-analysis of the BTCs from conservative tracer experiments conducted in a wide range of locations and hydrodynamic conditions suggests that the coefficient of skewness ( math formula) is scale invariant and equal to approximately 1.18. Considering the limited information that is currently available on solute transport processes in different catchments around the world, this methodology is perhaps the least biased (different personnel and instrumentation were used to collect the data) and most informative (BTCs sampled a wide range of multiscale heterogeneities) to investigate scaling patterns in stream ecosystems. The self-consistent relationships derived from our extensive database for normalized central temporal moments can be used to adequately predict solute transport. Such relationships also revealed systematic limitations of the solute transport models currently used in hydrology and suggest that we need a revised solute transport theory that is capable of representing the observed scaling patterns.

[53] Because solute transport is the foundation of biogeochemical models, if transport models with unidentifiable parameters are used to investigate the coupling between TS and biochemical reactions across ecosystems, it is not unexpected that the relationships derived are inconclusive, as it has been extensively shown to date. Ultimately, model structural errors generate equifinal systems that can lead to biased conclusions with respect to the nature of mechanistic relationships.

Appendix A: MRMT Model

display math(A1)
display math(A2)

[54] The theoretical temporal moments were computed in a manner similar to Cunningham and Roberts [1998]:

display math(A3)

where math formula [ML−3] is the concentration of the solute in the storage zone; math formula is the probability density function of mass transfer exchange rates; and math formula and math formula are the mean and variance of the distribution of TS residence times [cf., Haggerty and Gorelick, 1995; Cunningham and Roberts, 1998]. Other variables are as defined for the TSM. When math formula, math formula. If dispersion is negligible ( math formula):

display math(A4)

[55] If math formula is not fixed, the MRMT model will represent BTCs with longitudinally decreasing skewness ( math formula), becoming asymptotically Gaussian (i.e., math formula).

Appendix B: The dCTRW Model

[56] The Laplace transform (LT) of math formula for a dCTRW model is given by Dentz et al. [2004]:

display math(B1)

where math formula is the LT variable. Other variables have been defined previously in the ADE. The memory function math formula is defined by

display math(B2)

where math formula is the LT of the time transition probability density function; math formula is the LT of a joint space ( math formula) and time transition probability density function; and math formula is a median transition time. We estimated the temporal moments using the method by Aris [1958].

display math(B3)

[57] The solution for the Fickian case is found when math formula, which yields math formula, as it was shown for the ADE (section 3.2.1). A general pattern for the math formula can be inferred from this particular condition, and the specifics will depend on the memory function defined for the model. In summary, if math formula is not fixed, a dCTRW model will represent BTCs with longitudinally decreasing skewness ( math formula), becoming asymptotically Gaussian (i.e., math formula).

Appendix C: Lévy-Flight Dynamics Model

[58] We consider here a Lévy-flight type dynamics model, which has a fractal dependence on the sampling position and takes the form:

display math(C1)

where math formula is a constant time increment, and math formula are independent identically power law distributed random variables such that:

display math(C2)

[59] For large math formula (Lévy-flight variable), math formula could be a Pareto distribution, for example. The spatial Laplace transform of math formula for math formula then would be

display math(C3)

[60] We are interested in the distribution of arrival times math formula at a position math formula, which is given by

display math(C4)

where math formula is the number of steps needed to arrive at position math formula by the Lévy process shown in equation (C1). It is equivalent to math formula. Thus, we obtain for the arrival time density:

display math(C5)

where math formula denotes the Dirac delta distribution and the angular brackets denote the noise average over math formula. Expression (C5) can be written as

display math(C6)

where math formula is an indicator function that is 1 if the condition in its argument is true and 0 otherwise. The latter equation can be further developed as

display math(C7)

[61] Computing the second average we get:

display math(C8)
display math(C9)

[62] The latter satisfies the Kolmogorov type equation:

display math(C10)

[63] Combining equations (C8) and (C10) in Laplace space, we get

display math(C11)
display math(C12)

[64] The time increment math formula is supposed to be small compared to the observation time, so that we can write (C11) as

display math(C13)

[65] In real space, it reads as

display math(C14)

[66] Defining the moments of math formula by

display math(C15)

[67] We obtain from equation (C14) the moment equations

display math(C16)

where math formula for math formula. This equation can, again, be solved in Laplace space:

display math(C17)

[68] For math formula we obtain:

display math(C18)

because math formula. We are interested in the behavior at large distances, which means at small math formula. Inserting equation (C12) above gives

display math(C19)

[69] Inserting now equation (C3) and expanding up to leading order gives

display math(C20)

[70] Thus, the first moment is given by

display math(C21)

[71] For the second moment, we have

display math(C22)

[72] Inserting equation (C3) and expanding up to leading orders we have

display math(C23)

[73] Inversion of this expression gives

display math(C24)

[74] The second normalized central moment is

display math(C25)

[75] For the third moment, we have

display math(C26)

[76] Inserting equation (C3) and expanding up to leading orders, we have

display math(C27)

[77] Inversion of this expression gives:

display math(C28)

[78] The third normalized central moment is

display math(C29)

[79] We can now estimate the scaling of math formula as

display math(C30)

[80] For math formula to be independent of math formula (or persistent) we need:

display math(C31)

Acknowledgments

[81] This work was funded by NSF grant EAR 08–38338. Funding was also available from the HJ Andrews Experimental Forest research program, funded by the National Science Foundation's Long-Term Ecological Research Program (DEB 08–23380), U.S. Forest Service Pacific Northwest Research Station, and Oregon State University. We thank the Associate Editor, Olaf Cirpka, Adam Wlostowski, and an anonymous reviewer for providing insightful comments that helped to improve this manuscript.

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