Understanding constraints on the fate and transport of nutrients through environmental systems is necessary for the preservation, remediation, and restoration of water quality [1]. Nutrients are components of global biogeochemical cycles and many have become pollutants as a consequence of widespread anthropogenic application. Terrestrial nutrient application has profoundly influenced ground and surface water quality, having near-term and legacy effects [2] as nutrients have been transported into groundwater, streams, and coastal zones. Consequences of this loading have become evident at varying scales and magnitudes, from localized, harmful algal blooms to continental-scale coastal dead zones [3], [4]. Thus, estimating how nutrient loads propagate through Earth systems requires the ability to predict how nutrients will be processed as they are transported along hydrologic gradients.
Earth systems have been discretized into geomorphic process domains, defined as "predictable areas of a landscape within which distinct suites of geomorphic processes govern physical habitat type, structure, and dynamics" [5]. More broadly, soils, aquifers, and stream corridors are examples of biogeochemical process domains, each with distinct natural boundaries and suites of physical attributes defining their capacity to attenuate or exacerbate nutrient pollution. Although the specific physical characteristics of soils, aquifers, and stream corridors differ, reactive transport within each of these systems shares common elements with the function of chemical reactors. Thus, the reactive transport of nutrients within and among process domains can be abstracted as a collection of linked biogeochemical reactors distributed across the landscape.
Nutrients transported into or produced within a process domain are often stored for a period of time but ultimately experience one of two potential fates: export or reaction. For nutrients that are ultimately exported, transport dynamics dictate the amount of time nutrients spend in a process domain. Reaction kinetics and thermodynamic constraints (i.e., energetic favorability) establish the probability of transformation occurring within a given amount of time. Thus, like a chemical reactor, the probability that a nutrient will be transformed prior to export from the system determines the net behavior of that nutrient within a process domain. Because process domains behave like environmental reactors, researchers have characterized the tendency for process domains to transform and export nutrients using Damköhler indices [6]- [12]. While two distinct Damköhler indices have been defined to describe advection-and diffusion-dominated systems [8], we focus on applications for systems with advection-dominated transport, as introduced in [13] and translated by [14]
TABLE I VARIABLES USED IN REACTOR REFERENCE FRAME MATHEMATICS (ALPHABETICAL ORDER ENGLISH TO GREEK) Variable Dimensions Description Ar [L 2 ] Area of an active surface within a reactor Fr [L 3 L -2 T -1 ] Equivalent water flux across the active surface required to remove solutes from the reactor at a given rate F Q [L 3 L -2 T -1 ] Influence of hydraulic export of solute mass expressed as an effective volumetric flux of water removing solute across an active-surface area I Da [1] Damköhler index calculated as τ t τr kr [T -1 ] Effective first-order rate constant for a rate-limited chemical transformation within a reactor k r,S [T -1 ] Effective first-order rate constant for a rate-limited chemical transformation within the reactive storage zone of a storage-exchange reactor Q [L 3 T -1 ] Steady-state flow of water through a reactor (equal to flow at input and output) τt [T]
Average travel time from input to output along a reactor a All definitions assume constant first-order reaction-rate kinetics.
where τ t is the characteristic advective transport time between the inlet and outlet of the reactor, and τ r is the characteristic reaction time between the input of a reactive solute to the reactor and the time of its transformation. This Damköhler index is thus a metric of the probability that reactive solute in the reactor will be transformed before it reaches the outlet.
Here, we apply a Damköhler perspective to the reactive transport behavior of a process domain, using a reach of stream corridor as an exemplar. We abstract a stream reach as a "storage-exchange reactor", which consists of a non-reactive preferential flow compartment that exchanges water and solutes across a permeable surface area into a biogeochemically reactive storage zone. The reactive storage zone consists of a collection of flow paths through porous substrate in which water moves and ultimately returns back to the non-reactive preferential flow compartment; solute transported into reactive storage is either transformed therein or transported back to the transport compartment.
We investigate how transport and reaction constraints interact to govern nutrient transformation and transport in a simulated agricultural stream, using nitrate as an example nutrient. In the simulated stream reach, the channel is conceptualized as a transport compartment and the hyporheic zone as a reactive storage zone. While studies exist that investigate similar concepts and also characterize their results with a Damköhler index, e.g., [15], [16], our study differs in that we are exploring an approach that is both more parsimonious than existing models and can be extended to other process domains, such as soils and groundwater aquifers. We prioritize parsimony and extensibility as each benefits the sustainability and broader usability of our simulation model.
Reference frames are abstract conceptualizations of process domain behavior that can be codified based on common suites of numerical simplifications. Here, we explore a storageexchange reference frame, which relates closely to common dual-or multi-compartment constructs for process domains such as aquifers and streams. In aquifers and heterogeneous porous media, these constructs are governed by differences in the hydraulic properties and reactivity of mobile and immobile compartments [17], [18]. Likewise, in streams, the channel is represented as the nonreactive transport compartment that exchanges water and solutes with one or more reactive storage compartments, such as a hyporheic zone [19]- [22].
A formulation of the Damköhler index for this reference frame can be presented as the ratio of the effective reaction flux to the effective export flux.
For stream process domains [23] and treatment wetlands [6], a conceptually identical formulation for this Damköhler index is the ratio of the vertical mass transfer coefficient to the hydraulic load.
The effective reaction flux F r in the storage-exchange reference frame is calculated as
where P R,S is the dimensionless fraction of solute removed from water passing through reactive storage, and
) of water exchange between the transport compartment and reactive storage compartments. Q in is calculated as the water volume in the storage compartment divided by the mean transit time through the storage compartment:
In this case, we calculate the the mean transit time as the integral of the "washout function," W (τ ) the complementary CDF of the exit age distribution [24]. Assuming first-order reaction-rate kinetics and advective solute transport into the reactive storage, the effective firstorder reaction rate constant for the reactor as a whole is calculated as follows.
Thus, P R,S is a critical control on the effective first-order reaction rate constant, the effective reaction flux, and the associated Damköhler index for an environmental reactor.
In a storage-exchange reactor, P R,S for the reactive storage zone can be simulated using a defined distribution of hydraulic transit times through the reactive storage compartment. Variable flowpath lengths can be approximated from transit times with the assumption that longer transport times correspond to longer flowpath lengths. With this construct, the biogeochemical behavior along the first part of a "long" flowpath is identical to the behavior of a "short" flowpath. All water entering reative reactive storage (e.g., hyporheic zone) at time t will have a solute concentration equal to that of the transport compartment at time t. Using a rate constant for each solute, solute concentration will increase or decrease by a fraction calculated from transit time.
The shape of the transit time distribution through the reactive storage compartment can vary, but is commonly represented with a power law [24]- [27]. This representation is well supported when the reactive storage represents a hyporheic zone that exchanges water and solutes with a channel (transport compartment) across the stream bed (active surface).
Following [24], we assume the function f (τ S ) defines the shape of the transit time distribution of flow paths in reactive storage and define a probability density function, E(τ S ), representing the transit time distribution for water exiting reactive storage (the "exit age" distribution)
τ is a dummy variable representing transit times from τ S,0 to τ S,n . The shortest transit time for water considered to have entered reactive storage is τ S,0 . Flow paths through reactive storage with transit times shorter than τ S,0 are determined to be too brief to be considered "stored". The longest transit time considered in reactive storage is τ S,n . Longer flow paths are considered to be lost from the system. Because E(τ S ) is a probability density function accounting for all the water passing through storage and returning to the transport compartment, the area under the entire distribution equals 1.
The washout function, W (τ ), is the complementary cumulative distribution of E(τ ):
where x is a dummy variable. When considering a parcel of water that enters reactive storage at time t, W (τ ) calculates the fraction of the parcel that remains in storage at time t + τ . W (τ ) can be used to determine the total flux of water across the active-surface area and through storage. For the example of a stream channel in exchange with hyporheic storage, q in can be calculated as the saturated thickness of water in reactive storage (z S ; Fig. 1) divided by the mean transit time through the storage zone (calculated as the integral of the washout function between the min and max flow-path transit times).
Importantly, the flux of water through storage (q in ) is not parameterized directly. Rather, this flux is an emergent property of the configuration of the shape of the transport time distribution and volume of water in the storage zone [24].
In order to calculate the proportion of solute removed (P R,S in equations 3 and 5), reaction kinetics are incorporated into the hydraulic model of the reactive storage zone as
where k r,S is the instantaneous first-order reaction rate constant governing solute transformation in storage. Note that k r,S is distinct from k r . In a storage-exchange reactor, k r incorporates the effect of exchange with the reactive storage zone on the apparent behavior of solute in the reactor (Equation 5). If the assumption is made that the reactive storage is operating under steady state, the effective Damköhler number for the reactive storage (I Da,S ) can be expressed as an exponential decay function.
Thus, an effective Damköhler number can be calculated for both reactive storage (I Da,S ) and the storage-exchange reactor as a whole (I Da ).
We explore the numerical behavior of a storage-exchange reactor by applying this reference frame to simulation models of a stream channel in exchange with reactive storage in the hyporheic zone using equations 2-11. Models were solved numerically, see Subsection III-A, and had the following assumptions.
• We consider the reactive solute as a metabolicallylimiting nutrient and simulate nutrient processing as a consequence of exchange across the stream bed with reactive storage (see nutrient spiraling and hyporheic flow theory [28]). • We treat both the channel and hyporheic zones as advection-dominated systems where transport. Thus, our conceptual model is relevant to systems with Peclet numbers > 1 sensu [8]. We do not account for diffusion or dispersion. • We assumte that rate limitation based on first-order kinetics sufficiently captures the biogeochemistry of nutrient transformation (e.g., denitrification, sulfate reduction).
This assumption dictates that a Damköhler index has an exponential relationship with the total fraction of nutrient processed by a reactor or its components. • We simulate steady-state hydrology at the scale of the whole reach and in the hyporheic zone. Flow (Q) into and out of each stream corridor reactor is equal and constant, as is the flow of water into and out of reactive storage (Q in = Q out ). • To simulate transport through the hyporheic zone, we employed a collection of flow paths with transit times determined by a power law distribution. This distribution is described by the function f
, where the value of α describes the shape of the transit time distribution (and thus degree of tailing) of water in hyporheic flowpaths [24]. A larger α indicates a shorter mean transit time (τ S ), less tailing, and higher hydraulic turnover through the storage zone than a smaller α. The value of τ S,n is the max transit time considered in the reactive storage. The max transit time is "capped" to prevent water in the model from being stored infinitely. Like the TSM, our model is a special case of a storageexchange reactor reference frame. In addition, our model
TABLE II MODEL CONSTANTS Parameter Units Value Surface water discharge m 3 s -1 0.069 Solute concentration of incoming water g NO 3 -N m -3 19.3 Length of channel (reach) m 250 Depth of channel m 0.3 Width of channel m 0.7 Stream bed area m 2 175 Channel volume m 3 52.5 (d) (c) shares notation with the TSM [29]. However, an important difference between our model and TSM is that, in our model, the exchange rate of water between the surface water and the reactive storage zone is not directly parameterized. Rather, this exchange rate is an emergent property, which is dependent on the transit time distribution of water in reactive storage and the volume of water the reactive storage zone.
We constructed a simulation model of a stream reach as a storage-exchange reactor with an abstract software framework. The framework was built using modular object-oriented design, and implemented as R6 classes [30] in R [31]. With this model, we conducted a series of numerical experiments on a simulated stream reach (Fig. 1). The full suite of numerical experiments can be run by following the instructions and executing the code found at https://reinholdlab.github.io/ systERS numExpt WRR/.
Physical characteristics of the simulated stream reach are consistent with an agricultural headwater stream in the Judith River watershed (Porter Creek in north central Montana, USA; Table II). Our model is agnostic to solute type, but parameterized for nitrate as nitrogen with an incoming surfacewater concentration of 19.3 mg NO 3 -N m -3 from [32]. These high concentrations reflect nitrate loading as a function of agricultural land use and hydrogeomorphic setting [32].
Numerical experiments explore the response of the stream reactor to variation in the shape of the hyporheic transit time distribution, the extent of the hyporheic volume, and the first-order reaction-rate constant governing the biogeochemical reactivity of the hyporheic zone. The three values for the α exponent of the power law describing the shape of
TABLE III PARAMETER VALUES USED IN NUMERICAL EXPERIMENTS Description Value(s) Power law shape parameter (α) describing the transit time distribution of water in reactive storage: Lowest 1.2 Moderate 1.4 Highest 1.6 Volume of water in reactive storage relative to volume of water in channel: 0.1x 5.25 m 3 0.2x 10.50 m 3 1x 52.50 m 3 5x 262.50 m 3 10x
525.00 m 3 First-order reaction-rate constant for solute transformation in the reactive storage (kr,S):
the residence distribution (Table III) were selected based on literature values compiled in Table 1 of [24]. These values were calculated from tracer releases for streams with Strahler orders generally ranging from 2 nd to 4 th , encompassing a range of geomorphological contexts and extents of agricultural influence. For all values of α, we set the shortest hyporheic transit time to be 10 seconds (τ S,0 ) and the longest to be 1 year (τ S,n ). Solute processing begins in the hyporheic zone at τ S,0 . Together, τ S,0 and τ S,n define the minimum and maximum transport times of reactive flow paths in storage, respectively. We used multiplication factors to describe scenarios with varying hyporheic volume and the reaction-rate constants (Table III). The factor for hyporheic volume is the ratio of the volume of water stored in the hyporheic zone relative to the volume of water in the channel in a simulated reach. A factor of 1 indicates that the volume of water in the hyporheic zone is equal to the volume of water in the channel, and a factor of 0.1 indicates that the volume of water in the hyporheic zone is one-tenth that in the channel. The range in first-order reaction-rate constants corresponds to published values for nitrate consumption in environmental settings [33], [34]. We conducted our numerical experiments with a fullfactorial design, including all possible combinations of the transit time distribution shape (n = 3), hyporheic volume (n = 5), and first-order reaction-rate constant (n = 5; Table III). All other parameters were held constant within and across simulations; these included discharge and solute load into the stream reach, stream channel geometry, the volume of surface water, and mean velocity of the surface water (Table II).
The Damköhler indices for the reactive storage zones varied as a function of the shape of the transit time distribution of the reactive storage zone. Reactive-storage Damköhler indices
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.00 0.05 Reactive storage Damkohler index 0.00 0.01 0.00 0.01 0.02 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Reactor Damkohler index 0 0.000 0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Fraction solute entered reactive storage 0e+00 1e-04 0e+00 1e-04 2e-04 0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 6e-04 7e-04 8e-04
Fraction solute processed in reactive storage 0.1x 0.2x 1x 5x 10x 0.1x 0.2x 1x 5x 10x Reaction-rate constant multiplier 0.1x 0.2x 1x 5x 10x 0.1x 0.2x 1x 5x 10x Reactive-storage size mul 1.2 1.4 1.6 1.2 1.4 1.6 Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x 1.2 1.4 1.6 (a) (b) (a) (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.00 0.05 0.10 0.00 0.05 Reactive storage Damkohler index 0.00 0.01 0.00 0.01 0.02 0.00 0.01 0.02 0.03 0.04 0.05 0.06 Reactor Damkohler index 0 0.000 0.002 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Fraction solute entered reactive storage 0e+00 1e-04 0e+00 1e-04 2e-04 0e+00 1e-04 2e-04 3e-04 4e-04 5e-04 6e-04 7e-04 8e-04 Fraction solute processed in reactive storage 0.1x 0.2x 1x 5x 10x 0.1x 0.2x 1x 5x 10x Reaction-rate constant multiplier 0.1x 0.2x 1x 5x 10x 0.1x 0.2x 1x 5x 10x
Reactive-storage size mul were greatest where α was smallest and mean transit times longest (Fig. 2a). The reaction-rate constant in storage (k r,S ) was important for determining the Damköhler indices for the reactive storage zone. After accounting for the shape of the transit time distribution, the reactive-storage Damköhler was greatest where reaction-rate constant for storage was greatest. Notably, the size of the reactive storage zone had no effect on the Damköhler indices for the reactive storage zones. In the simulated streams, reactive storage zone size was independent of α and corresponding exit age function; therefore, storage zone size was also independent of the probability of solute being processed in the reactive storage zone (Equation 10). Consequently, the percent of solute that entered the reactive storage zone which was processed therein was unchanged by varying the size of the reactive storage zone (Fig. 4a).
Damköhler indices for the whole reach reactors varied as a function of the shape of the transit time distribution of the reactive storage zone, the size of the reactive storage zone, and the reaction rate constant in storage (Fig. 2b). Similar to what was observed for reactive-storage Damköhler indices, higher k r,S values resulted in higher whole-reactor Damköhler indices. However, the shape of the transit time distribution and the size of the reactive storage zone had very different influences on the Damköhler indices for the whole reactors as compared with the Damköhler indices for the reactive storage zones (compare Fig. 2a to 2b and Fig. 4b to 4c). In contrast to the Damköhler indices for the reactive storage zones, the Damköhler indices for whole stream reactors were greatest where turnover in the reactive storage was highest (largest α) and the size of the reactive storage zone was largest. Reactor Damköhler indices scaled directly with the size of the reactive storage zones. All else being equal, a stream reactor having a storage zone with a size multiplier of 10x had a 10-fold greater whole-reactor Damköhler index than a stream with a 1x multiplier.
The fraction of solute advected from a transport compartment into a reactive storage zone was an important driver of the Damköhler index for the reactor, and was determined by the shape of the transit time distribution and size of the reactive storage zone (Fig. 3a). The fraction of solute entering the reactive storage zone scaled directly with the size of the reactive storage zone; all else being equal, a storage zone with a 10x multiplier received 10 times more solute than a storage zone with a 1x multiplier. Where α and reactive storage zone size were constant, the amount of solute entering the reactive storage zone was constant (Fig. 3a), and the reaction-rate constant for storage (k r,S ) determined the fraction of solute in the reactor that was processed by the reactive storage zone (Fig. 3b). Higher Damköhler indices for the whole stream reactors corresponded to lower concentrations in the outflow of the reach (Fig. 3c), reflecting a greater percentage of solute in the reach having been processed in the reactive storage zone (Fig. 4b). Reach reactors with low outflow concentrations and high whole-reactor Damköhler indices also had low reactivestorage Damköhler indices (Fig. 4b vs. c).
In exploring the behavior of a stream corridor represented as a storage-exchange reactor, our results make clear that whole system behavior is determined by a hierarchy of controls on the interactions among rates of storage, transformation, and transport. The influence of biogeochemical reactions occurring in storage on the whole reactor is constrained by the fluxes of water and solute between the transport compartment and the reactive storage. Because these fluxes are driven by the volume of reactive storage and the shape of the transport time distribution of flowpaths in it (Equation 9), solute concentrations in simulated stream channels correspond to the extent of hyporheic exchange and the probability of reaction while water and solute are "stored" in hyporheic flow paths.
Controls of hydraulic exchange on the potential influence of reactive storage are illustrated by the pattern that higher wholereactor Damköhler Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x
Reactive-storage size multiplier 0.1x
Reaction-rate constant multiplier 0.1x 0.1x 0.2x 1x 5x 10x 0.1x 0.2x 1x 5x 10x Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x 1.2 1.4 1.6 1.2 1.4 1.6 1.2 1.4 1.6 Reaction-rate constant multiplier 0.1x 0.2x 1x 5x 10x (a) (b) (c) equation 10; Fig. 4a). Although a Damköhler index is a metric of the efficiency with which a given reaction can change concentration, it contains no information about how much water and solute is being processed. A larger abundance of flow paths with shorter residence times (higher α) leads to lower Damköhler numbers in storage, but corresponds with a higher volumetric flow of water-and quantities of solutespassing through storage.
Because solute must be delivered to the reactive storage zone in order to be processed, the fraction of solute in a reactor that enters the reactive storage is a hierarchical control on solute processing. A higher α reduces the efficiency of the effect of the reactive storage, but exerts a disproportionately greater effect on transport-compartment water concentrations due to the higher exchange of water and solute from the transport compartment with reactive storage (Fig. 3a). Through substitution, we present the equation for the reach-scale Damköhler index to show how the turnover of channel water through the reactive storage controls the magnitude of the processing effect of the reactive storage.
Increasing α or the reactive storage zone size increases Qin Q , which increases the effect of P R,S on the I Da . The result is that the effect of varying the reaction-rate constant for storage (k r,S ) on whole reactor solute processing is greater for simulated reactors with larger and more transmissive (larger α) reactive storage zones (Fig. 4b). Thus, larger reactive storage zones have the potential to store more water and react more solute than smaller reactive storage zones (Fig. 4c).
Our characterization of stream corridors as storageexchange reactors with efficiencies that can be described with a net Damköhler index is a complement to established, contemporary approaches such as the calculation of a Reaction Significance Factor (RSF) [7], [35], which has been widely employed in recent literature, e.g., [36]- [38]. The Damköhler index for the reactive storage zone in the RSF is similar-in concept-to the Damköhler index that we calculate for reactive storage. However, we do not assume storage is a well-mixed reservoir and take a different approach to incorporating the effects of solute processing in the reactive storage zone on stream corridor function. Instead of calculating a reach scale RSF as our reach-scale comparative metric, we calculated a reach (reactor) scale Damköhler index (Equation 2). While complementary and similar to the RSF in the case of river corridors, we implemented our approach because it is extensible to process domains beyond rivers and streams and can scale spatially.
Our simulations are heuristic representations of stream systems wherein organic carbon (and nutrient) subsidies to the stream are not limiting and are likely to drive down dissolved oxygen concentrations rapidly. Simulated reaches with smaller I Da,S had larger reach scale I Da compared with reaches wherein the storage zones had larger I Da,S . The reasons for this are twofold. One, these reaches had reactive-storage zones with shorter average residence times, which resulted
0 10 20 30 Percent of solute entering reactive storage that is processed (%) 0.000 0.025 0.050 0.075 Percent of solute in reactor processed in reactive storage (%) 0.000 0.025 0.050 0.075
Percent of solute in reactor processed in reactive storage (%)
1.2 1.4 1.6 1.2 1.4 1.6 1.2 1.4 1.6 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 Reactive-storage Damkohler 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 Reactor Damkohler 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 Reactive-storage Damkohler Reaction-rate constant multiplier 0.1x 0.2x 1x 5x 10x Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x (a) (b) (c) 0 10 20 30 Percent of solute entering reactive storage that is processed (%) 0.000 0.025 0.050 0.075 Percent of solute in reactor processed in reactive storage (%) 0.000 0.025 0.050 0.075 Percent of solute in reactor processed in reactive storage (%) 1.2 1.4 1.6 1.2 1.4 1.6 1.2 1.4 1.6 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 Reactive-storage Damkohler 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 0.00 0.02 0.04 0.06 Reactor Damkohler 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 Reactive-storage Damkohler Reaction-rate constant multiplier 0.1x 0.2x 1x 5x 10x Reactive-storage size multiplier 0.1x 0.2x 1x 5x 10x in a larger amount of water and solute delivered to reactive storage. Two, in our simulations, solute processing begins the instant that solute enters reactive storage, τ S,0 . Parameterizing solute processing to begin as soon as solute enters reactive storage may be inconsistent with what would be expected for anaerobic processes like denitrification, e.g, [9], but may be reasonable in systems wherein Liebig's "Law" of the minimum driving threshold effects and "late start" patterns [39] are less of a concern.
In the Roter Main River, Germany, [40] found-similar to our simulation results-that Damköhler indices for nitrate processing were << 1 in the hyporheic zone, with smaller values of I Da corresponding to shorter transit times and greater hydraulic turnover. Yet, because so much water passes through the hyporheic zone in these shorter transit times, the net effect is that the hyporheic zone processes large quantities of nitrate; the authors estimate that the hyporheic zone processes 20% of the nitrate flux for the catchment over a 32-km reach. Thus, the Roter Main River appears to behave similarly to the streams with transmissive hyporheic zones simulated herein.
In environments where first-order reaction rate kinetics or "early start" patterns do not capture key system dynamics, reactor reference frames remain useful, but require additional considerations. For instance, coupling the effects of linked elemental cycles with the reactor reference frames will be particularly useful where multiple elemental cycles exert strong constraints on whole system biogeochemistry. Within reactive storage, additional constraints can be imposed by physical limitations of pore network structures [41]; by kinetic favorability or variable redox conditions [42]; or by thermodynamic limitations on solubility [11]. In our future research, we aim to approximate the effects of thermodynamics with additional kinetic constraints or to model coupled thermodynamic and stoichiometric constraints directly (e.g. GANGSTA [43], [44]).
We see promise in extending the storage-exchange reference frame from streams to other process domains, such as soils and groundwater aquifers. This frame offers a parsimonious approach to simulating a relatively non-reactive preferential flow in hydraulic connectivity with a distribution of flow paths with variable transport times.
Reactor reference frames link transport and reaction dynamics, and are founded on the idea that hydrophysical controls are hierarchical constraints on solute processing. In simulating stream corridors as storage-exchange reactors, we find that the physical template (reactive storage size) and solute delivery to the reactive storage zone (shape of the transit time distribution [α]) constrain the influence of reaction probabilities. As such, the pattern that simulated streams with low reactive-storage Damköhler indices had high whole-reactor Damköhler indices illustrates how the processing capacity of an "environmental reactor" is constrained by solute delivery to reactive storage and hydraulic connectivity of storage with the nonreactive transport compartment.