# Assessing Controls on the Displacement of Tracers in Gravel-Bed Rivers

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rationale

_{S}) can be estimated from the dimensions of the active layer of the streambed and the mean travel distance of bed sediments [17,52,53,54,55,56,57,58].

_{a}is the mean active channel width (in m), which is the portion of the channel that is active in bedload terms during transport episodes, h is the average depth of the active layer (in m), and p is the fractional porosity of the channel sediment. We can then re-arrange Equation (1) and express L as the dependent variable to give

_{s}), times the duration of the competent flow (t, in seconds).

_{84}is the 84th percentile of the surface grain-size distribution. Somehow, Equation (5) normalizes the specific stream power by the submerged weight of surface sediment, which arrives at an expression equivalent to the Shields parameter, but is based on flow discharge.

_{Di}is the transport distance of individual grains of diameter D

_{i}, L

_{D}

_{50}is the mean transport distance of the median grain-size (D

_{50}), and a and b are the empirical intercept and coefficient, respectively. Assuming that L ~ L

_{D}

_{50}, and combining Equations (6) and (7), the following expression for the mean travel distance of tracers (L

_{T}) can be derived.

_{T}).

_{W}is the fraction of the total channel width that is involved in the bedload. Similarly, it is reasonable to assume that active depth scales with the grain-size of the streambed’s surface [65]. Then, the active depth can be related to the D

_{84}, considering that the 84th percentile is an adequate descriptor of the overall roughness of the streambed-surface [66,67].

_{H}characterizes the vertical extent of the active layer in terms of the surface grain-size. Incorporating Equations (9) and (10) into Equation (8), the following expression is reached.

_{W}and γ

_{H}, which describes the vertical and horizontal extent of the active layer. It may be argued that these parameters are stage-dependent, so they may show some degree of co-linearity with the ω*/ω

_{c}* ratio. Similarly, they could also be influenced by the time duration of the competent flow [37]. Furthermore, channel dimensions may exert influence as well. For example, narrow channels likely have a narrower active width. However, wider channels may exhibit large patches of immobile bed sediment and, consequently, a larger proportion of inactive bed-surface than narrow channels for equivalent discharge conditions, which partially buffers this effect. Similarly, major protruding elements in the streambed have been reported to control the thickness of the active layer, with coarser channels tending to show thicker active layers for equivalent hydraulic conditions [65,68]. In contrast, coarser streambeds may be more armored or paved, which prevents scour, and, hence, decreases the horizontal and vertical extent of the active layer. Moreover, bed state and grain structures can exert control on grain displacements and bed mobility and influence the γ

_{W}and γ

_{H}parameters, which characterize the vertical and horizontal extent of the active layer. The influence of the bed state on bed mobility could be individually specified for each streambed through the critical stream power for incipient motion (ω

_{c}). With all this in mind, we postulate the following.

_{W}and γ

_{H}to other parameters is scarce [26,54,65]. In this paper, and, for the sake of simplicity, we assume that relationships between these parameters can be best fitted by power equations. We can then modify Equation (11) to the formula below.

#### 2.2. Data Set Compilation and Grouping

#### 2.3. Data Analysis

^{2}was inflated due to collinearity between the predictor variables. As a rule of thumb, when a regression model incorporates several explanatory variables, the value of VIF is acceptable if it is less than 10 [76]. Lastly, and with the aim of assessing the relative importance of each one of the independent variables, we used the method proposed by Lindeman et al. [77], which is often recommended for assigning shares of the relative weight of predictors to R

^{2}, while also accounting for the sequence of predictors appears in the model. Moreover, previous tracer works have shown significant correlations between mean tracer travel distance and excess stream power [17], accumulated stream power [25,26], and dimensionless impulse [28]. With the aim of assessing whether Equation (13) improves upon previous approaches, we compared the performance of Equation (13) to two regression models applied to the same data (Table 1).

_{c}) was estimated from the value of threshold discharge (Q

_{c}) provided by the different papers (Table 1). The comparison between the three regression models was based on the percentage of explained variance, evaluated for each regression model through R

^{2}. Subsequently, and in order to assess the robustness of each regression model, we used Jacknife resampling methods [78,79]. For this, we sequentially extracted one data point from the data set, fitted the three regression models with the remaining data, and estimated the particle travel length for the extracted data with each regression model. Then, we defined two additional scores: (i) the average ratio (r) between regression model estimates and field measured transport distances, in order to quantify the accuracy of each regression model, and (ii) the number of data plots in which r falls between 0.5 and 2, as a way to measure their precision.

_{c}is the critical value of ω for incipient sediment motion. We explored differences in the values of EEI according to channel morphology.

## 3. Results

#### 3.1. Regression Model

^{2}= 0.87, p-value < 0.05), and confirms that the proposed equation reproduces the compiled data with a high degree of adequacy. All of the variables are significant in explaining mean travel lengths at a 90% confidence level (p-values < 0.1), and they are all positively correlated with mean travel lengths (Table 2). The fitted model coefficients are shown in Table 2. Values of VIF are high in the case of width and tracer size, but all perform below 10, which indicate that multicollinearity can be dismissed.

^{2}is explained by channel width (W

_{T}) (Figure 1). The ω/ω

_{c}ratio and flow duration explain, together, around 25% of R

^{2}. D

_{84}and critical stream power account for ~20% of the explained variance. Lastly, the tracer size relative to the bed sediment is responsible for 16% of the explained variance.

^{2}= 0.87 in comparison to R

^{2}= 0.18 (Equation (14)) and R

^{2}= 0.03 (Equation (15)) (Figure 2, Table 3), even when channel morphology is accounted for.

#### 3.2. Influence of Channel Morphology and Experimental Conditions

_{84}tends to be finer in riffle-pools compared to step-pools and plane-beds (Figure 5A,B). Dimensionless critical stream power is also larger in riffle-pools than in step-pools (Figure 5C). Additionally, differences in competent flow duration between different morphologies, particularly between riffle-pool and step-pool channels, can also be observed (Figure 5D). Hence, all the parameters incorporated into Equation (13) show differences, according to channel-morphology. This may explain why data from different morphologies collapse into the same trend when Equation (13) is used.

_{84}.

## 4. Discussion

#### 4.1. Variability in Particle Travel Distances

_{84}, tracer size, the ω*/ω

_{c}* ratio, competent flow duration, and ω

_{c}*. Channel geometry (proxied by channel width) is the parameter explaining the largest differences in travel length (Figure 1), which is followed by hydraulic forcing. The bed structure (proxied by critical stream power) and tracer size appear to play a less determinant role compared to hydraulic constraints and channel dimensions. The channel shows a positive power scaling with travel length. On the one hand, channel width provides a measure of the hydraulic geometry of the study sections. Hence, the longest travel lengths are expected in larger channels due purely to their larger dimensions. On the other hand, the length-scale of the channel determines the average spacing between macro-bedforms [30], which, in turn, is typically argued to control travel length (e.g., Pyrce and Ashmore work [32]).

_{84}. The role of D

_{84}on tracer travel lengths is interesting because it is related to the depth of scour, and, at the same time, prevents entrainment from occurring. That said, a larger D

_{84}involves a thicker active layer (through scaling the depth caused by particle entrainment). Furthermore, D

_{84}is related to the length scale of the major grains on the streambed, and large protruding clasts act as obstacles that may stop and entrap sediment [63], which may increase the chances of a given tracer becoming buried. In principle, both situations may decrease the amount of tracer displacement, so travel lengths should show a negative relationship to D

_{84}. However, larger grains control the mobility of the bed-surface sediment in gravel and cobble streams, and a larger D

_{84}may involve a more stable streambed [66,67], which increases the chances of tracers showing pass-over behavior. In this regard, travel length also showed a positive scaling to ω

_{c}*, which is likely to be a relevant measure of bed-surface stability. Thus, the pass-over behavior of tracers, promoted by an increase in streambed stability with larger D

_{84}, appears to compensate for the potential influence of deeper tracer burial in thicker active layers, and may explain the observed positive scaling of the travel length with D

_{84}.

#### 4.2. Morphological Control of Travel Length

_{84}and larger ω

_{c}* values are expected. Lastly, step-pool streams drain, on average, into smaller drainage areas compared to other channel types, and are characterized by a flashier regime, with short storm flows (see the discussion by Comitti [17]), which likely have an effect on the reported shorter flow duration in our step-pool data. In summary, our analysis (see Section 3.1) outlined how all the parameters included in Equation (13) are significant in explaining differences in travel distance between different channels. Consequently, if these parameters are assumed to be a proxy of channel morphology, it can be concluded that Equation (13) actually incorporates the role of morphological controls into travel length. However, we are aware that this hypothesis needs further assessment from field data in new tracer studies which, unfortunately, are less and less available.

#### 4.3. Structural Controls on Travel Length

_{c}* as the predictor parameter in the regression model. This parameter was obtained for each selected site from the information provided in the original studies, since it was established by authors from field observations on incipient tracer motion. The analysis suggested that this parameter is significant in explaining ~10% of the variance in tracer data, which highlights the appreciable role of sediment structures in gravel transport.

_{84}, tracer size, and channel morphology. Nevertheless, when the different morphologies are considered separately, differences are observed between ‘unconstrained’ and ‘constrained’ tracer conditions in step-pool channels. This finding resembles the difference between structured and travelling bedload proposed by Piton et al. and Piton and Recking [90,91], for step-pool channels. No differences could be identified in the riffle-pool and plane-bed channels. Similar trends in tracer data, according to channel morphology and experimental conditions, were reported by Vázquez-Tarrío et al. [29].

#### 4.4. Methodological Uncertainties

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

b | Model coefficient defining how travel lengths scale to the tracer size |

c | Model coefficient defining how travel lengths scale to ω_{c}* |

d | Model coefficient defining how travel lengths scale to ω*/ω_{c}* |

D_{i} | Tracer size |

D_{50} | Median diameter of the surface grain-size distribution |

D_{84} | 84-th percentile of the surface grain-size distribution |

γ_{H} | Parameter multiplying D_{84} and characterizing the active layer depth |

e | Model coefficient defining how travel lengths scale to t |

f | Model coefficient defining how travel lengths scale to W_{T} |

g | Gravity acceleration |

h | Average depth of the active layer |

l | Model coefficient defining how travel lengths scale to D_{84} |

L | Average distance travelled by the bedload |

L_{Di} | Average transport distance of individual grains of diameter D_{i} |

L_{D}_{50} | Average transport distance of the median grain-size |

p | Fractional porosity of channel sediment |

Q | Water discharge |

Q_{c} | Threshold discharge for incipient sediment motion |

q_{s} | Average bedload rate of the transport episode |

Q_{s} | Event-based bedload volumes |

S | Channel slope |

t | Time duration of the competent flow |

W_{a} | Mean active channel width |

W_{T} | Total channel width |

γ_{W} | Fraction of the total channel width involved in the bedload |

ω | Specific stream power |

ω_{c} | Threshold value of ω for incipient sediment motion |

ω* | Dimensionless specific stream power |

ω_{c}* | Threshold value of ω* for incipient sediment motion |

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**Figure 2.**Relationships between measured tracer travel lengths and (

**A**) stream power, (

**B**) stream power times duration of competent flow duration, and (

**C**) Equation (13). SP: Step-Pool. RP: Riffle-Pool. PB: Plane-Bed. MT: Multi-Thread.

**Figure 3.**Assessment of the regression model robustness using jackknife resampling procedures: (

**A**) discrepancy between model estimates and field measurements, and (

**B**) percentage of data where model estimates are between 0.5 and 2 of field measurements. (*) Results of applying the simple regression model independently to each group of channel morphologies (plane-bed, riffle-pool, step-pool, and multi-thread).

**Figure 4.**Differences in EEI (see Section 2.3 in the text for a description of this index) according to channel morphology for the compiled data set.

**Figure 5.**Differences in (

**A**) channel width, (

**B**) D

_{84}, (

**C**) dimensionless critical stream power, and (

**D**) and flow duration, according to channel morphology for the compiled data set.

Data Set | W (m) | S | D_{50} (mm) | L (m) | Measuring Procedure | Observations | Recovery (%) | Q (m^{3}/s) | Q_{c} (m^{3}/s) [Estimation Method] | Seeding Procedure | Survey Duration | Stream Type | Source |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lainbach | 10.0 | 0.020 | 120 | 15–451 | No information | Mean distances estimated for all tracers and for moving tracers only | 17–100 | 3.2–165.0 | 3 [First tracer movements] | Tracers seeded on pools, steps, bars, and the toss side of large boulders | 4 years | Step-pool | [71]: Gintz et al. (1996) |

Spruce creek | 6.0 | 0.120 | 86 | 163.8–513.9 | Straight line distance between the initial location and the point of the final deposition | Mean distances estimated for moving tracers only | 83–92 | 1.2–2.0 | 1 [Discharge above 25% of clasts are mobile] | Tracers seeded along cross sections on the channel bed | 3.5 years | Step-pool | [24]: Lamarre and Roy (2008) |

Halfmoon Creek | 2.2 | 0.010 | 57 | 10.0–144.8 | Distances along and away from the channel centerline | Mean distances estimated for all tracers and for moving tracers only | 93–98 | 5.6–14.5 | 3.5 [Estimate based on critical shields stress] | A grid of 893 tracers in rows across the channel with 0.5 m between each tracer and each row | 4 years | Riffle-pool | [72]: Bradley and Tucker (2012) |

Bouinenc river | 24.0 | 0.020 | 20 | 299.0–775.0 | Distances measured along the axis of the main low flow channel | Mean distances estimated for all tracers and for moving tracers only | 65–88 | 33.4–41.1 | 2.5 [Estimate based on critical shields stress] | Tracers seeded along transverse lines crossing several morphological units | 3 years | Multithread channel | [73]: Liebault et al. (2012) |

Strimm Creek | 3.5–4.0 | 0.080-0.150 | 62–76 | 0.2–185.0 | Distances measured along the thalweg | Mean distances estimated for moving tracers only | 54–100 | 0.32–1.81 | 0.3–0.4 [Discharges able to mobilize clasts from all size classes] | Tracers seeded along transverse ribs on the streambed | 4 years | Step-pool/Plane- bed/Cascade | [74]: Dell’Agnese et al. (2015) |

East Creek | 2.3–2.8 | 0.018-0.020 | 49–55 | 0.3–35.7 | Distances measured along the thalweg | Mean distances estimated for moving tracers only | 77–88 | 0.9–4.7 | 0.5 [Discharge at which ¡ mobility is initiated for the median grain size] | Tracers seeded on the surface in rows spanning the entire width of the channel | 8 years | Riffle-pool/Plane-bed | [70]: Papangelakis and Hassan (2016) |

_{50}: Median size of the surface grain-size distribution. L: Mean travel length. Q: Event discharge. Q

_{c}: Threshold discharge for incipient sediment motion.

**Table 2.**Results of multiple regressions using stepwise procedures based on Equation (13) (see text for details).

Variable | Coefficient | Standard Error | t | p-value | VIF ^{1} |
---|---|---|---|---|---|

Intercept | 1170.146 | 2.686 | 3.47 | 0.001 ^{++} | |

1-log_{10}(D_{i}/D_{50}) | 4.859 | 1.516 | 3.204 | 0.003 ^{++} | 4.574 |

D_{84} | 3.239 | 0.455 | 7.12 | 2.68 × 10^{−8 ++} | 2.431 |

Width | 3.878 | 0.51 | 7.608 | 6.38 × 10^{−9 ++} | 6.239 |

ω*/ω_{c}* | 1.931 | 0.278 | 6.945 | 4.49 × 10^{−8 ++} | 1.908 |

Flow duration | 0.167 | 0.084 | 1.991 | 0.054 ^{+} | 1.908 |

ω_{c}* | 2.742 | 0.554 | 4.952 | 1.85 × 10^{−5 ++} | 4.002 |

^{2}= 0.87. Adjusted R

^{2}= 0.85. F-statistic: 38.86 on 6 and 35 degrees of freedom. p-value = 4.64 × 10

^{−4}.

^{1}Variance Inflation Factor (see Section 2.3 in the text for details). D

_{i}/D

_{50}: Ratio between tracer size (D

_{i}) and the median size of surface sediment (D

_{50}). D

_{84}: 84th percentile of the surface grain-size distribution. ω*: Dimensionless specific stream power. ω

_{c}*: Threshold value of ω* for incipient sediment motion.

^{++}: p-value statistically significant at a 95% confidence level.

^{+}: p-value statistically significant at a 90% confidence level.

Model | Intercept | b^{1} | c^{2} | d^{3} | e^{4} | f^{5} | l^{6} | R^{2} |
---|---|---|---|---|---|---|---|---|

Equation (13) (all) | 13406.86 | 4.859 | 2.742 | 1.931 | 0.167 | 2.878 | 3.239 | 0.87 |

(ω*−ω_{c}*) (all) | 0.81 | 0.82 | 0.18 | |||||

(ω*−ω_{c}*) (RP) | 0.21 | 0.81 | 0.12 | |||||

(ω*−ω_{c}*) (PB) | 0.18 | 0.57 | 0.43 | |||||

(ω*−ω_{c}*) (SP) | 0.54 | 1.04 | 0.46 | |||||

(ω*−ω_{c}*) × t (all) | 2.99 | 0.14 | 0.03 | |||||

(ω*−ω_{c}*) × t (RP) | 0.00 | 1.24 | 0.72 | |||||

(ω*−ω_{c}*) × t (PB) | 0.02 | 0.26 | 0.37 | |||||

(ω*−ω_{c}*) × t (SP) | 0.35 | 0.30 | 0.27 |

^{1}The model coefficient defining how mean travel lengths scale to tracer size.

^{2}The model coefficient defining how mean travel lengths scale to critical dimensionless stream power (ω

_{c}*).

^{3}The model coefficient defining how mean travel lengths scale to tracer size ω*/ω

_{c}* or (ω*−ω

_{c}*).

^{4}The model coefficient defining how mean travel lengths scale to the duration of the competent flow (t).

^{5}The model coefficient defining how mean travel lengths scale to the channel width.

^{6}The model coefficient defining how mean travel lengths scale to D

_{84}; RP: Riffle-Pool Channels. PB: Plane-Bed Channels. SP: Step-Pool Channels. (all): All data.

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## Share and Cite

**MDPI and ACS Style**

Vázquez-Tarrío, D.; Batalla, R.J.
Assessing Controls on the Displacement of Tracers in Gravel-Bed Rivers. *Water* **2019**, *11*, 1598.
https://doi.org/10.3390/w11081598

**AMA Style**

Vázquez-Tarrío D, Batalla RJ.
Assessing Controls on the Displacement of Tracers in Gravel-Bed Rivers. *Water*. 2019; 11(8):1598.
https://doi.org/10.3390/w11081598

**Chicago/Turabian Style**

Vázquez-Tarrío, Daniel, and Ramon J. Batalla.
2019. "Assessing Controls on the Displacement of Tracers in Gravel-Bed Rivers" *Water* 11, no. 8: 1598.
https://doi.org/10.3390/w11081598