Estimating uptake lengths: transport model used by Hall et al. (2013)
Hall et al. (2013) analyzed a data set of in-stream nutrient uptake
experiments performed using plateau tracer injections. The basis of these
experiments and estimation of nutrient uptake metrics come from the
advection–dispersion equation (Eq. 1), with the addition of a
first-order uptake rate coefficient (Stream Solute Workshop, 1990; Runkel,
2007):
dcdt=-udCdx+Dd2Cdx2-Kcc,
where c (M L-3) is the concentration of the reactive solute at a
cross section located downstream of the solute injection site; u
(LT-1) is the mean flow velocity; D (LT-2) is the dispersion
coefficient; Kc (T-1) is the first-order rate coefficient
representing nutrient uptake; x (L) is longitudinal distance; and t (T)
is time. Assuming that dispersion is negligible at plateau concentrations (i.e.,
when dc/dt=0), Eq. (1) can be solved for
downstream solute concentration:
c=coexp(-Kc/ux),
where co (M L-3) represents the initial (or upstream)
concentration. The form of this solution motivated the introduction of the
uptake length metric, Sw=u/Kc, which is a representation of
the average distance traveled by a nutrient molecule in inorganic phase prior
to uptake (Ensign and Doyle, 2006). Due to experimental simplicity, Eq. (2)
has guided data collection efforts on nutrient cycling where an
experimentalist estimates Sw by measuring the plateau concentrations
upstream (cup) and downstream (cdn) of a study reach of
length L:
Sw=u/Kc=L/lnCup/Cdn.
Note that Eqs. (1–3) support estimates of Sw, given that stream conditions
satisfy model assumptions, i.e., that stream reaches have constant discharge and
that dispersion and transient storage do not play important roles (Runkel,
2007). The uptake length derived from Eqs. (1) to (3) is equivalent to
SwI in Runkel (2007), who derived four different uptake lengths
(SwI, SwII, SwIII, SwIV)
from solute transport models with increased complexity (i.e., with added
transient storage, lateral inflows and dispersion). Following Runkel (2007),
uptake lengths can be generally represented by a velocity term and an uptake
term.
It is important to keep in mind that Sw is an abstract variable
represented by model parameters that cannot be simultaneously measured. Since
u and Kc are likely to be highly variable along a stream reach,
measurements of longitudinal decline in tracer concentrations (cup,
cdn) and stream length (L) offer a more tractable approach to
estimating Sw through the use of Eq. (3). While the use of Eq. (3)
circumvents errors associated with estimating u and Kc on the
reach scale, the estimation of Sw using cup, cdn,
and L must be numerically equivalent to u/Kc on the reach
scale. As is the case with any abstract variable derived from a mathematical
model, using Sw to infer stream processes entails acknowledging the
quantitative role of the model parameters u and Kc from where
Sw was derived.
Critique of the scaling approach used by Hall et al. (2013)
The analysis presented by Hall et al. (2013) was based on plateau experiments
conducted in multiple stream ecosystems, where Sw was estimated for each
experiment using Eq. (3). Hall et al. correlated nutrient uptake length,
Sw (L), with specific discharge, Q/w (L2 T-1), to test the
hypothesis that nutrient uptake demand is constant across stream orders.
Sw∝Q/wa,vf=Q/wSw,
where Q (L3 T-1) is stream discharge, w (L) is stream width,
a is a scaling exponent and vf (L T-1) is the nutrient uptake
demand (or nutrient uptake velocity, as it has been traditionally called).
In their hypothesis testing, the existence of a constant nutrient uptake
demand (constant vf) was implied by a scaling exponent a=1 (isometric
scaling), whereas a scaling exponent a≠1 (allometric scaling) would
imply the reverse. Note that in this context, the existence of a constant
nutrient uptake demand would be useful to scale and predict nutrient uptake
in stream ecosystems.
In Table 1 we present the different forms that Sw vs. Q/w from Hall et
al. (2013) would take if such a relationship was estimated for two general
types of natural-channel geometries.
The relationship Sw vs. Q/w for natural-channel
geometries.
Quantity or
Rectangular
Non-rectangular
relationship
channel
channel
A
w×h
f(w,h)
Q
u×A
u×A
Sw
u/Kc
u/Kc
Q/w
u×h
u×(fw,h/w)
Sw vs. Q/w
u/Kc vs. u×h
u/Kc vs. u×(fw,h/w)
Note that each side of Sw vs. Q/w shares the common (hidden) variable
u. Therefore, an increase in u (e.g., with stream order or increasing
discharge) would increase both sides of the proportion, likely forcing a
strong correlation between the variables. This would happen regardless of
whether u is measured in the field or not because Sw is an abstract
quantity derived from u and Kc (cf. Eqs. 2–3), and, by
definition, Q=u×A. The fact that Hall et al. (2013) used only
estimates of Sw, and field measurements of Q and w to seek a
mechanistic relationship from Sw vs. Q/w (cf. Eq. 4) does not change
the induced correlation created by having the factor u playing a key
quantitative role on both sides of the relationship. Since the form of
Sw is dependent on the transport model presented in Eqs. (1–2), the
only way to negate the role of u in Sw (note that it cannot be negated
in Q/w) is to select a completely different transport model and perform a
completely different set of field experiments. Also, under the ideal scenario
in which we could actually measure Sw in streams (i.e., if Sw was
not an abstract variable), the regression Sw vs. Q/w would mainly
support the development of conceptual models for Sw, which already
exist.
We propose that if a meaningful, significant correlation exists between
Sw and Q/w, there should be a significant correlation between the
underlying parameters (i.e., 1/Kc vs. h in rectangular channels
or 1/Kc vs. fw,h/w in other types of natural
channels). However, if there is not a corresponding correlation in both of
these cases, then the correlation between Sw and Q/w would be falsely
influenced by the presence of u in both products. Benson (1965) and
Kenney (1982) demonstrated that spurious correlations can result from the use
of ratios or products that share a common factor and are more likely when
working with complex variables and dimensional analysis. The relationship
from Hall et al. (2013) that we deem spurious is analogous to that of Model
II presented by Benson (1965) for the spurious correlation of products
sharing a common factor (i.e., X1×X2 vs. X3×X2;
where X1=1/Kc, X2=u, X3=h or X3=fw,h/w; cf. Table 2 in Benson, 1965). As shown by Benson (1965), the
correlation of complex variables (i.e., Sw and Q/w) is dependent on
the coefficients of correlation and variation of the three original component
variables. Due to the presence of a common factor in the scaling relationship
proposed by Hall et al. (2013), we hypothesize that it is a spurious
correlation (u influences both Sw and Q/w) that may be
mechanistically irrelevant for scaling in-stream nutrient uptake.
We tested our hypothesis using the data set published by Tank et al. (2008),
another meta-analysis of nutrient addition experiments, which was included in
the Hall et al. (2013) meta-analysis. This data set was chosen because it
reports values for Sw, Q, w, and h for nutrient experiments with
NH4 and NO3 (SRP – soluble reactive phosphorus – not included),
even though these values were not reported for all the studies (n=143 for
NH4; n=210 for NO3). Note that since we do not know the
particular geometry for each channel where the tracer experiments were
conducted (i.e., we do not know f(w,h)), we assumed a rectangular channel
geometry (i.e., A=fw,h=wh), which is the same assumption
as that made by Hall et al. (2013), while defining their equations for uptake
length and uptake velocity (cf. Eqs. 1–2 in Hall et al., 2013). The data set
published by Hall et al. (2013) does not include values of h; hence, we
were not able to use it for our analysis. While the assumption of having
rectangular channels might be seen as an overgeneralization, it is the only
one that allows us to see trends given the scarce information available on
the channel geometries of the headwater streams where the experiments were
conducted. Furthermore, the transport model implicitly used by Hall et
al. (2013) assumes uniform flow (i.e., dh/dx=0;
dw/dx=0), which supports our assumption of a
prismatic channel for testing our spurious-correlation hypothesis.
NH4 scaling relationship with and without shared velocity term.
The original relationship is represented by Sw vs. Q/w and the
null condition by 1/Kc vs. h.
We proposed a null condition in which we removed the common variable u from
the scaling relationship and compared the correlation with that of the
original scaling relationship (i.e., we compared 1/Kc vs. h
and Sw vs. Q/w). We calculated mean stream velocity as u=Q/(w×h). This allowed us to produce values for the relationship 1/Kc
vs. h, by dividing Sw and Q/w by u (cf. Table 1). By doing so, we
were able to evaluate the scaling relationship with and without the common
term u to compare the coefficient of determination, r2, for both
relationships. Results of this analysis are shown for NH4 and NO3
in Figs. 1 and 2.
Our results show that 1/Kc vs. h are weakly correlated
(r(NH4)2=0.029, p(NH4)=0.042;
r(NO3)2=0.036, p(NO3)=0.0057). However, the
correlation Sw vs. Q/w is higher (r(NH4)2=0.161,
p(NH4) < 0.00001; r(NO3)2=0.151,
p(NO3) < 0.00001), i.e., r2 is improved by 452 and
317 % for NH4 and NO3, respectively. These findings suggest
that the correlation Sw vs. Q/w is spurious because it is driven by
the shared velocity (u) term rather than by an inherent correlation
between the inverse of the nutrient uptake rate constant (1/Kc)
and stream depth (h). The correlations shown in Figs. 1 and 2 are
comparable to those reported by Hall et al. (2013). However, we note that the
r2 values do not match because of different data sets (we were limited by
the number of studies reporting all parameters Sw,Q,w,h) and our
aggregation of reference and altered streams. Regardless, our analysis
suggests that the inclusion of the parameter u falsely improves the
correlation of the investigated relationships.
The mechanism producing spurious correlation in the data set by Hall et
al. (2013) can be viewed more clearly using three arbitrary and uncorrelated
variables to represent the relationship between X1×X2 and
X3×X2. We gathered mean daily values for specific conductance
(X1, µS cm-1) in the Potomac River (DC) (USGS, 2008a),
turbidity (X2, FNU) in the Little Arkansas River (KS) (USGS, 2008b), and
temperature (X3, ∘C) in the Rio Grande (NM) (USGS, 2008c) for
the year 2008. First, we isolated the common factor X2 and plotted
X1versus X3, as shown in Fig. 3 (r2=0.020, p=0.012). As
expected, there was no statistically significant correlation between these
water quality parameters. However, when we incorporated the turbidity
(X2) from a remote location by plotting X1×X2 vs.
X3×X2 (n=313), we found a positive correlation (Fig. 4) with
a drastic improvement in r2 (r2=0.846, p < 0.00001).
Despite the evident correlation in this relationship, the result is
mechanistically irrelevant. Analogous to this case example where the
correlation is driven by X2 (turbidity), the correlation Sw vs.
Q/w seems to be driven by u (recall Sw=u Kc and
Q/w=u×h or Q/w=u×(fw,h/w)). Thus, our
findings suggest that the results produced by Hall et al. (2013) regarding
the isometric scaling (a=1) of NH4, and allometric scaling
(a > 1) of NO3 and SRP, resulted from an unintentional spurious
correlation of Sw vs. Q/w.
NO3 scaling relationship with and without shared velocity term.
The original relationship is represented by Sw vs. Q/w and the
null condition by 1/Kc vs. h.
In addition to scaling nutrient uptake length with specific discharge, Hall
et al. (2013) also provide a method for scaling nutrient uptake with stream
length using several parameters, including the scaling exponent a obtained
from the analysis of the scaling relationship shown in Eq. (2). Our findings
have implications for these results as well. While Hall et al. (2013)
commented that their results for scaling uptake with stream length were most
influenced by b (hydraulic geometry exponent), their analysis still relies
on the spurious correlation Sw vs. Q/w not only for parameter a but
also for the subsequent derivations (cf.
Eqs. 3–10 in Hall et al., 2013). Therefore, we also find those results
debatable.
Synthetic data correlation, type X1 vs. X3,
without common parameter, X2. There is a weak correlation between
these water quality parameters.
Concluding remarks
The majority of nutrient addition experiments have been performed in
headwater streams because they are more experimentally tractable (Tank et
al., 2008). Consequently, the dearth of empirical evidence of nutrient
processing in large rivers limits our understanding of the role of these
rivers in nutrient processing on the catchment scale. While empirical and
theoretical advances are being made toward performing nutrient addition
experiments in large rivers (Tank et al., 2008; Covino et al., 2010), the
need to understand and quantify nutrient export from these systems has driven
the development and use of scaling relationships. This motivated the work by
Hall et al. (2013), and their results after correlating Sw vs. Q/w for
a large data set of field nutrient experiments suggest that uptake demand
(vf) for NH4 is relatively constant across stream orders, whereas
that for soluble reactive phosphorous (SRP) and NO3 declines with
increasing specific discharge. Here, we demonstrated that these conclusions
are subject to debate due to unintentional spurious correlations present in
their scaling relationships.
Synthetic data correlation, type X1×X2 vs.
X3×X2, with common parameter X2. This
spurious correlation results simply because X2 is common to both
quantities.
We also suggest that Sw should be used with extreme caution to scale
nutrient uptake because, even though its magnitude can be directly estimated
from relatively simple field measurements, its mechanistic interpretation
strongly depends on the type of model assumed to describe the real-world
system (cf. Table 1 in Runkel, 2007). This is because the same estimate of
the magnitude of Sw may be arbitrarily used to co-estimate or constrain
the magnitude of parameters describing different (arbitrary) sets of
processes (see Cases I–IV in Runkel, 2007). Finally, when a model describing
a given set of processes is chosen to interpret how nutrient uptake scales
along a river continuum, the main assumption is that such processes operate
analogously along the continuum. For example, if the model of
advection–decay chosen by Hall et al. (2013) to interpret Sw across
stream orders were correct, our analysis presented in Figs. 1 and 2 would
suggest that headwater streams tend to have higher nutrient uptake rate
coefficients, which might be mechanistically supported by their higher ratio
of benthic area to cross-sectional area. However, this (biased) analysis
would not provide insight into how mass-transfer processes between the
main-channel and transient storage zones may control nutrient uptake and
retention along the river continuum. Paradoxically, increasing the complexity
of the transport models used to derive Sw (e.g., Cases II–IV in Runkel,
2007) does not necessarily improve the mechanistic understanding gained on
how nutrient uptake scales along the river continuum because such models are
poorly constrained, i.e., the number of parameters introduces more degrees of
freedom than the data collected (from field and remote measurements) can
constrain.