1. Introduction
Understanding the fate and transport of introduced pollutants and substances within river courses is relevant for public health, ecological diversity, and the administration of water resources [
1,
2,
3,
4]. Fundamental to this understanding is the accurate prediction of substance concentrations and their distribution, owing to mechanisms such as advection, molecular diffusion, and dispersion, where the former is the most dominant process in natural rivers [
5]. Dispersion in the longitudinal, lateral, and vertical directions accounts for the effects of spatial differences in velocities over the channel cross-section, and consequently its magnitude depends upon the scales of turbulent diffusion and mixing, owing to channel irregularities [
6,
7,
8,
9,
10]. Prediction of longitudinal mixing is complicated in natural rivers as the channel morphology increases in complexity (e.g., planform curvature, bed irregularity, variable roughness provided by macroforms, substrate, and vegetation) [
11,
12,
13,
14,
15,
16]. Under such circumstances the inertial terms in the hydrodynamic equation become increasingly important for mixing and pollutant transport. Therefore, the morphological features within a natural stream reach, such as contractions, expansions, and bed macroforms, must be accounted for in the fate and transport of dissolved and suspended constituents.
Different mathematical tools are currently used to characterize these highly complex three-dimensional (3D) mixing processes, all of which are based on the three-dimensional advection–dispersion–reaction equation (Equation (1)).
where
t represents time;
C is the concentration of the substance; and
u,
v, and
w correspond to the velocities in the directions
x,
y, and
z.
,
, and
are the dispersion coefficients in each associated direction. The term
R represents the sources and sinks that consume or contribute mass to the system (i.e., null value in the case of a conservative substance). The velocity field values (
u,
v,
w,
t) are obtained by solving hydrodynamic equations (i.e., Saint-Venant equations, or variant equations) and the dispersion coefficients (
,
,
,
t) characterize the directional mixing that are necessary to complete the mathematical scheme [
11,
17,
18,
19].
Therefore, to solve this system of equations, the necessary data requirements are high, given the complexity of the numerical solution when coupling both the hydrodynamic and transport processes. In practice, several simplifications can be made to reduce the complexity of Equation (1) [
6,
20,
21,
22,
23,
24,
25]. For example, it is frequently assumed that mixing in the vertical and transverse directions occurs instantaneously, which allows the estimation of the time–space variation of the concentration in the longitudinal direction, via a one-dimensional (1D) model [
26,
27]. Several empirical equations are available in the literature following this one-dimensional simplification and a sample of them is listed in
Table 1. None of the equations in
Table 1 explicitly account for channel irregularities or morphological complexity outside of general channel dimensions.
To obtain these estimates of reach-averaged dispersion coefficients, many authors have carried out tracer studies [
26,
33,
36,
37,
38,
39,
40,
41]. Many of these empirical models have been validated in the laboratory or simple stream reaches, whereas natural alluvial rivers present variations in planform and bed topography, which deviate from a plane configuration [
4] and rarely satisfy the implicit assumptions (
Table 1). Therefore, these empirical equations offer limited use because their application is only valid in scenarios that are similar to those used in their derivation. Predicted longitudinal dispersion for the same river reach might vary widely depending on the equation used [
26,
35,
42,
43]. This stems from various local morphological complexities, such as step-pools, sinuosity, and pool-riffle macrostructures being lumped into one value of reach-averaged longitudinal dispersion. When the distribution of specific morphological features like pool-riffle macrostructures (i.e., residual pool depth, riffle height, length of bars, widths, etc.) differ from one reach to another, the resultant dispersion coefficient might be less appropriate. Hence, there is still uncertainty regarding the best method for representing the complex three-dimensional mixing processes in a single, one-dimensional parameter, especially in non-uniform channels [
43].
This study accounts for morphological complexities arising directly from pool-riffle macrostructures within a river channel, which results in improved predictions of the longitudinal dispersion coefficient. A three-dimensional coupled hydrodynamic and transport model that includes turbulence fluctuations and variations in bathymetry was developed to provide output data necessary to characterize the complex velocity field through pool-riffle macrostructures. Ten different synthetically generated pool-riffle bathymetries provide a physical basis for the model runs. The numerical results were subject to a dimensional analysis and an empirical equation capable of predicting the longitudinal dispersion coefficient through pool-riffle structures was proposed for an improved one-dimensional simplification. Five field tracer studies were additionally used to validate the predictive equation.
3. Results
The simulated concentration time curves for a constituent passing through the pool-riffle macro-structure in scenario 1 are shown in
Figure 5. There was one curve per observation point illustrating the gradual decrease in peak concentration along the pool-riffle sequence. Integrating the concentration over time showed that the mass of each curve maintained a constant value of 15 g s m
−3, fulfilling conservation of mass within the model domain. Additionally, the shape of the curves indicated that the constituent was normally distributed, owing to its uniform distribution at the upstream boundary, with increasing variance, as it progressed downstream.
Longitudinal dispersion coefficients were calculated from the simulated concentration time curves for each one of the considered synthetic bathymetry scenarios, using the TM and MM methods. Both TM and MM methods estimated values of the dispersion coefficient of the same order of magnitude (
Figure 6), which were reflective of small streams and canals. It was also clear that the behavior of the curves was similar in shape, reaching a minimum around a residual pool depth of 0.4 m. The dispersion coefficient in both methods showed an inverse relation to the hydraulic radius through the residual pool depth.
To explore the relation between turbulent fluid motion through complex bathymetry and its impact on the dispersion coefficient,
Figure 7 shows the simulated range of variation in turbulent kinetic energy (TKE) for each one of the studied pool-riffle macrostructures. The magnitude of the TKE variation presented a minimum range between the residual pool depths of 0.14 and 0.20 m, with increasing variance in both directions. As the residual pool depth increased, an increased variance in TKE occurred, owing to the sudden fluid acceleration/deceleration, as the flow entered and exited the pool. The median values of TKE showed a decreasing trend with increasing residual pool depth, with a minimum value around 0.4 m. It could be noted that the trend in median TKE values presented the same tendency as the dispersion coefficient, reinforcing the direct relation between the dispersion coefficient and fluid turbulence. These findings support the appropriateness of the three-dimensional numerical modeling results.
Table 5 shows the 16 input sets used in the sensitivity analysis and applied to the 3D transport model, for Scenario 1. Results indicated that smaller initial dispersion values resulted in a greater difference between the initial and estimated dispersion coefficient. This difference decreased as initial coefficients increased, suggesting that the estimated dispersion coefficients were more sensitive to smaller initial values. Results were within an order of magnitude, which was an improvement over the predictions from existing empirical relations, when considering the bed macrostructures. Additionally, the null dispersion scenario resulted in a positive estimated longitudinal dispersion. This implied the presence of other dispersive processes that related to an unknown combination of the complex flow field within a pool-riffle macrostructure and numerical diffusion. Collectively, these processes resulted in dispersion on the order of 0.44 m
2/s, which was expected to be present throughout all simulations.
Results from the dimensional analysis indicated that
as function of
presented the highest correlation value
and was, therefore, proposed as the predictive equation. The proposed equation for the longitudinal dispersion coefficient through pool-riffle macrostructures is reported in Equation (8).
Mathematically, Equation (8) was only valid when
; however,
was physically quite unlikely since pool-riffle structures would have a non-zero water surface slope from the upstream riffle crest to the downstream pool. Additionally, pool-riffle sequences only form in relatively steep fluvial environments with 0.002 <
Sw < 0.04, where Equation (8) is applicable [
63,
64].
Equation (8) includes reach-averaged variables similar to those proposed by earlier empirical equations in the literature, but, directly considers the vertical and horizontal expansion–contraction geometry of pool-riffle macrostructures through the terms
SB and
SW. For example, in a case where there is no horizontal expansion, S
B is zero (i.e., there is no pool-riffle structure), Equation (8) reverts to the simplified version of the equation proposed by Elder [
28] for prismatic channels. It is interesting to note that subsequent research found Elder’s equation (
Table 1) to underestimate the value of the longitudinal dispersion coefficient [
26,
32,
34,
42]. When the proposed Equation (8) was simplified to the form of Elder’s equation (
SB = 0), the coefficient within the parentheses was less subject to underestimation as it predicted a 41% greater longitudinal dispersion coefficient.
The validation of Equation (8) was carried out using the remaining three numerical validation scenarios and the five field tracer experiments. Comparison between the predicted longitudinal dispersion coefficients and those in the validation data is shown in
Figure 8 The coefficient of determination for Equation (8) is
R2 = 0.64, providing an acceptable level of overall accuracy. This implied that by including pool-riffle expansion ratios in addition to the reach average values, 64% of the variability was explained by Equation (8). The resulting equation was an improvement from previous work that did not include bed complexity, which reported
R2 values of 0.55, 0.25, 0.5, and 0.55 [
14,
26,
29,
31]. In addition, 75% of the data were within the acceptable range defined by Seo and Cheong [
26], which was higher than the 34%, 47%, and 31% obtained by Antonopoulos et al. [
65].
The two points located farthest from the predictive line were the two most extreme ends of the morphological spectrum examined in the field tracer tests. These points represented field experiments 2 and 3, with the largest and smallest relative pool depths, in addition to the lowest ratio of Br/Bp from the field data. Therefore, these were the least well-developed pool-riffle macro-structures that were evaluated in the field sites. From the available data it is uncertain why this occurred, but both of these points fell within a predictive factor of 2. This indicated that further data near the extremes of the analyzed geometries needed to be generated to refine the coefficients in the proposed Equation (8).
4. Discussion
The predicted dispersion coefficients in this study ranged between 0.86 and 2.59 m
2/s (
Table 6), which were substantially less than those reported in natural rivers [
31,
35]. This was because the processes that were being captured in the 3D numerical model corresponded to a single bed macro-structure that did not include components like sinuosity, streambank dead zones, and vegetation that were usually lumped together in reach-averaged equations. Hence, these results represented the physically-based processes through an isolated pool-riffle sequence. This range of values indicated a minimum level of influence imposed by pool-riffle macroforms for longitudinal mixing processes. When examined against data from real rivers [
26,
33] that are known to have pool riffle structures, the influence of bed macrostructures calculated herein could constitute up to 100% of the dispersion values seen in small rivers and less than 2% in larger streams.
Specifically, it could be seen that the bed macro-structure imposed changes to the flow field as it progressed through the pool-riffle feature, with resultant impacts on the dispersion coefficient. Recent work suggests that the longitudinal dispersion coefficients should be directly related to the geometric properties of the river, specifically a direct relation to the hydraulic radius [
35]. However, this study identified an inverse relationship (
Figure 6), suggesting that the dispersion coefficient was minimal when the hydraulic radius was maximal. The numerical model results illustrated that this occurred because as the flow depth and hydraulic radius increased through the deepest part of the pool, there was an increase in flow divergence, decrease in velocity, and overall reduction in advective transport. The reduced advection was directly related to the dispersion coefficient and resulted in a decreased value. Additionally, the bed macroform affected the turbulent fluid exchange, owing to divergence through the pool and convergence over the riffle, which has been found to be related to the formation of unstable turbulent structures [
66]. As dispersion corresponds to the scattering of particles due to turbulent fluid movement, it was suggested that the variation of the dispersion coefficient must be linked to turbulence [
29]. Results showed that as flow divergence occurred in the pool and velocity decreased, the distribution of TKE responded with a proportional decrease (
Figure 7). This reduction in TKE and fluid exchange effectively reduced the dispersion coefficient supporting previous research in the literature [
6,
54,
56,
57,
58,
66].
Equation (8) attempts to characterize the response of the dispersion coefficient to the complex flow field through bed macro-structures seen in real rivers. For context, the equations presented in
Table 1 were applied to the numerical results and evident differences were apparent in the predicted dispersion coefficients (
Figure 9). The box plots showed the wide range of values in the predicted dispersion coefficients spanning two orders of magnitude, when applied to pool-riffle macrostructures. Theoretically, these variations could be attributed to their variable formulations (e.g., non-dimensional numbers like Fr, B/h, U/U*, etc.) and the morphology of the streams (or flumes) that they were developed for. For example, Zeng and Huai [
34] achieved good results for trapezoidal channels, yet the same formulation underestimated the dispersion coefficient values for rectangular channels. Such differences illustrated that channel morphology is a principle component of the variation that we see in the theoretical formulations and should be included more directly, in the estimates of the dispersion coefficient.
The proposed methodology to develop process-based mixing equations based on virtual morphological features provide an exciting new avenue of research. This methodology allowed for the simulation of specific morphological features and characterization of their individual impact on the hydrodynamic mixing processes. Future work using simulated river bathymetry and numerical modeling can identify the physically-based influence of other morphological structures like bars, width expansion/contraction, step-pools, sinuosity, etc., on river mixing. Ultimately, these individual dispersion relations would allow various combinations of features to be combined in reach-averaged values, via a partitioning approach seen in other branches in hydraulics, such as roughness and shear stress [
67,
68,
69,
70]. Current work is exploring the combinations of geomorphic features to guide restoration activities.
5. Conclusions
Reliable estimation of mixing in fluvial systems is critical for evaluating constituent and pollutant transport within rivers. Different equations applied to the same hydraulic conditions through a pool-riffle macroform can result in dispersions coefficients ranging in two orders of magnitude. The incorporation of local morphological variables and dimensional analysis for predicting longitudinal dispersion coefficients, like that proposed in this study, can reduce the variability in predicted results to within a factor of two. Therefore, an improved equation for calculating the longitudinal dispersions coefficient that accounts for flow expansion and the contraction seen in pool-riffle macrostructures is presented in Equation (8). This equation is proposed for application in one-dimensional water quality models for reaches of rivers with pool-riffle macrostructures. However, the use of the equation should be evaluated more extensively, in order to understand the valid ranges for its use.
More importantly, this study showed that employing a 3D hydrodynamic and transport model provided a process-based understanding of hydrological mixing through a single morphological macroform. In combination with synthetic bathymetry, these models would allow for direct evaluation of other channel features like alternating bars, varying sinuosity, and streambank vegetation and their respective significance in the mixing and transport phenomena. Such an approach could be further applied to engineering structures such as agricultural diversions, bank protection, and embankments, to understand the impact of river infrastructure. This approach is particularly powerful when combined with dimensional analysis for the identification of more robust process-based results.