# The Influence of Pool-Riffle Morphological Features on River Mixing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Syntethic Pool-Riffle Generation

#### 2.2. Numerical Modelling: Hydrodinamic and Transport Simulations

^{−1/3}s). A singular Manning’s roughness parameter was used to isolate the impacts of the pool-riffle bed macro-structure on the velocity field. Discharge increased gradually in the model in order to facilitate stability and convergence toward a stationary hydraulic stage.

_{x}= D

_{y}= D

_{z}) and the initial dispersion coefficients were defined as 1 m

^{2}/s. Solute transport in the WQ model used the same spatial discretization as the hydrodynamic model. The numerical transport solution was obtained by the minimal residual method [53], which corresponded to an unconditionally stable implicit method.

^{2}/s in increments of 0.1 m

^{2}/s. Additionally, a scenario with a null dispersion coefficient was considered, to test for numerical dispersion.

#### 2.3. Dimensional Analysis

#### 2.4. Validation: Numerical and Field Experiments

## 3. Results

^{−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.

^{2}/s, which was expected to be present throughout all simulations.

_{B}and S

_{W}. 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 (S

_{B}= 0), the coefficient within the parentheses was less subject to underestimation as it predicted a 41% greater longitudinal dispersion coefficient.

^{2}= 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 R

^{2}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].

_{r}/B

_{p}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

^{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.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Pool-riffle structure and the geometric variables considered in the analysis (modified from Caamaño et al. [47]).

**Figure 4.**(

**a**) Location of the Biobío region in Chile; (

**b**) the Bellavista watershed location within the Biobío region; and (

**c**) the tracer study location within the Bellavista watershed.

**Figure 5.**(

**a**) Plan view of the sampling points in the synthetic bathymetry domain for the time concentration curves. Pont spacing was 10 m; and (

**b**) time-concentration curves for bathymetric setting with ${\u2206}_{z}=0.494$.

**Figure 6.**Comparison of the method of Thomann and Mueller [54] and the method of the moments [55] for estimation of the longitudinal dispersion coefficient for the bathymetries described in Table 2. Variation of the dispersion coefficient (blue) and the hydraulic radius in the pool (orange) with respect to the residual pool depth for (

**a**) method of Thomann and Mueller (box); and (

**b**) method of the moments (circles).

**Figure 7.**Variation of turbulent kinetic energy for each scenario (expressed as the residual pool depth).

**Figure 8.**Comparison of values predicted by Equation (8) and those in the validation set. The orange points represent the field experiments and the purple x represent the numerical values.

**Figure 9.**Variation of dispersion coefficient values calculated in the numerical pool-riffle sequence through the empirical equations of Table 1, for each scenario.

Reference | Formula | Simplifications |
---|---|---|

Elder [28] | ${D}_{x}=5.93H{u}_{\ast}$ | Uniform flow in an infinitely wide channel. |

Fischer [29] | ${D}_{x}=0.011\left(\frac{{B}^{2}}{H}\right)\left(\frac{{U}^{2}}{{u}_{\ast}}\right)$ | Validated using measurements in straight prismatic channels of various regular cross-sectional shapes. |

Seo and Cheong [26] | ${D}_{x}=5.915{\left(\frac{B}{H}\right)}^{0.620}{\left(\frac{U}{{u}_{\ast}}\right)}^{1.428}H{u}_{\ast}$ | Developed using dimensional analysis and the one-step Huber method [30]. |

Kashefipour and Falconer [31] | ${D}_{x}=10.612\left(\frac{U}{{u}_{\ast}}\right);forB/H50$ $\begin{array}{c}{D}_{x}\hfill \\ =\lceil 7.428\hfill \\ +1.775{\left(\frac{B}{H}\right)}^{0.620}{\left(\frac{{u}_{\ast}}{U}\right)}^{0.572}\rceil HU\left(\frac{U}{{u}_{\ast}}\right);forB/H\hfill \\ 50\hfill \end{array}$ | Calibrated and validated using data from 30 streams in USA; previously used by Fischer [32], McQuivey and Keefer [33], and Seo and Cheong [26]. |

Zeng and Huai [34] | ${D}_{x}=5.4{\left(\frac{B}{H}\right)}^{0.7}{\left(\frac{U}{{u}_{\ast}}\right)}^{0.13}HU$ | Calibrated and validated using data from 50 rivers in the USA. |

Sahin [35] | ${D}_{x}=\mathsf{\beta}{R}_{h}U$ | Developed using dimensional analysis. This equation includes the hydraulic radius and the shape of the cross-section |

Bathymetry Scenario | Residual Pool Depth, m |
---|---|

1 | 0.494 |

2 | 0.444 |

3 | 0.394 |

4 | 0.344 |

5 | 0.294 |

6 | 0.244 |

7 | 0.194 |

8 | 0.154 |

9 | 0.094 |

10 | 0.044 |

Combination | Correlation |
---|---|

${\pi}_{1}-{\pi}_{2}$ | 0.8383 |

${\pi}_{1}-{\pi}_{3}$ | 0.8909 |

${\pi}_{1}-{\pi}_{4}$ | $1\times {10}^{-14}$ |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}$ | 0.9205 |

${\pi}_{1}-{\pi}_{2}{\pi}_{4}$ | 0.8304 |

${\pi}_{1}-{\pi}_{3}{\pi}_{4}$ | 0.8798 |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}^{-1}$ | 0.9495 |

${\pi}_{1}-{\pi}_{2}^{-1}{\pi}_{3}$ | 0.8556 |

${\pi}_{1}-{\pi}_{2}{\pi}_{4}^{-1}$ | 0.8304 |

${\pi}_{1}-{\pi}_{2}^{-1}{\pi}_{4}$ | 0.8096 |

${\pi}_{1}-{\pi}_{3}{\pi}_{4}^{-1}$ | 0.8798 |

${\pi}_{1}-{\pi}_{3}^{-1}{\pi}_{4}$ | 0.9532 |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}{\pi}_{4}$ | 0.9065 |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}{\pi}_{4}^{-1}$ | 0.9065 |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}^{-1}{\pi}_{4}$ | 0.9624 |

${\pi}_{1}-{\pi}_{2}^{-1}{\pi}_{3}{\pi}_{4}$ | 0.8556 |

${\pi}_{1}-{\pi}_{2}{\pi}_{3}^{-1}{\pi}_{4}^{-1}$ | 0.9499 |

${\pi}_{1}-{\pi}_{2}^{-1}{\pi}_{3}^{-1}{\pi}_{4}$ | 0.9590 |

${\pi}_{1}-{\pi}_{2}^{-1}{\pi}_{3}{\pi}_{4}^{-1}$ | 0.8556 |

Variable | Bathymetry Scenario | ||||
---|---|---|---|---|---|

Bellavista 1 | Bellavista 2 | Bellavista 3 | Bellavista 4 | Bellavista 5 | |

${h}_{pt}\left(\mathrm{m}\right)$ | 0.48 | 1.17 | 0.39 | 0.81 | 0.57 |

${h}_{rt}\left(\mathrm{m}\right)$ | 0.14 | 0.09 | 0.14 | 0.07 | 0.05 |

${\u2206}_{z}\left(\mathrm{m}\right)$ | 0.21 | 0.96 | 0.12 | 0.57 | 0.39 |

${L}_{PR}\left(\mathrm{m}\right)$ | 15.81 | 11.30 | 14.97 | 8.10 | 11.82 |

${B}_{r}\left(\mathrm{m}\right)$ | 8.61 | 6.46 | 16.40 | 5.20 | 12.00 |

${B}_{p}\left(\mathrm{m}\right)$ | 7.25 | 5.80 | 15.10 | 4.50 | 8.22 |

${u}^{\ast}\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ | 0.16 | 0.26 | 0.15 | 0.30 | 0.21 |

$U\left(\frac{\mathrm{m}}{\mathrm{s}}\right)$ | 0.38 | 0.51 | 0.42 | 0.26 | 0.30 |

${S}_{w}(-)$ | 0.008 | 0.010 | 0.008 | 0.020 | 0.010 |

${S}_{B}(-)$ | 0.086 | 0.058 | 0.086 | 0.086 | 0.319 |

**Table 5.**3D model initial and estimated dispersion coefficients based on the method of moments in m

^{2}/s.

Initial Dispersion Coefficient | Estimated Dispersion Coefficient (Method of Moments) | Difference |
---|---|---|

0.0 | 0.437 | 0.437 |

0.5 | 1.707 | 1.207 |

0.6 | 1.789 | 1.189 |

0.7 | 1.872 | 1.172 |

0.8 | 1.963 | 1.163 |

0.9 | 2.055 | 1.155 |

1.0 | 2.150 | 1.150 |

1.1 | 2.229 | 1.129 |

1.2 | 2.307 | 1.107 |

1.3 | 2.385 | 1.085 |

1.4 | 2.463 | 1.063 |

1.5 | 2.541 | 1.041 |

1.6 | 2.619 | 1.019 |

1.7 | 2.697 | 0.997 |

1.8 | 2.776 | 0.976 |

1.9 | 2.854 | 0.954 |

2.0 | 2.932 | 0.932 |

**Table 6.**Longitudinal dispersion coefficients. ${D}_{m}\text{}$are the coefficients from numerical modeling and field experiments, ${D}_{p}$ are the coefficients predicted by Equation (8).

Bathymetry Scenarios | ${\mathit{D}}_{\mathit{m}},{\mathbf{m}}^{2}/\mathbf{s}$ | ${\mathit{D}}_{\mathit{p}},{\mathbf{m}}^{2}/\mathbf{s}$ |
---|---|---|

3 | 2.110 | 2.025 |

6 | 2.194 | 2.270 |

9 | 2.572 | 2.589 |

Bellavista 1 (Tracer experiments) | 1.214 | 1.222 |

Bellavista 2 (Tracer experiments) | 0.305 | 1.032 |

Bellavista 3 (Tracer experiments) | 0.998 | 2.212 |

Bellavista 4 (Tracer experiments) | 0.854 | 0.862 |

Bellavista 5 (Tracer experiments) | 1.306 | 1.335 |

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**MDPI and ACS Style**

Fuentes-Aguilera, P.; Caamaño, D.; Alcayaga, H.; Tranmer, A.
The Influence of Pool-Riffle Morphological Features on River Mixing. *Water* **2020**, *12*, 1145.
https://doi.org/10.3390/w12041145

**AMA Style**

Fuentes-Aguilera P, Caamaño D, Alcayaga H, Tranmer A.
The Influence of Pool-Riffle Morphological Features on River Mixing. *Water*. 2020; 12(4):1145.
https://doi.org/10.3390/w12041145

**Chicago/Turabian Style**

Fuentes-Aguilera, Patricio, Diego Caamaño, Hernán Alcayaga, and Andrew Tranmer.
2020. "The Influence of Pool-Riffle Morphological Features on River Mixing" *Water* 12, no. 4: 1145.
https://doi.org/10.3390/w12041145