# A New Framework to Model Hydraulic Bank Erosion Considering the Effects of Roots

^{1}

^{2}

^{3}

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## Abstract

**:**

^{−1}and a RAR of 1% to 2%. The results show that hydraulic bank erosion can be significantly decreased by the presence of roots under certain conditions and its contribution can be quantified considering different conditions of channel geometry, forest structure and discharge scenarios.

## 1. Introduction

## 2. Material and Methods

#### 2.1. Description of the Framework

^{−1}) at the streambank toe is modeled using the excess shear stress equation [9,26,27,29,58]:

^{3}N

^{−1}s

^{−1}), ${\tau}_{a}$ is the boundary shear stress applied by the flow (Pa) at the streambank assuming the cross section is trapezoidal, ${\tau}_{c,veg}$ is critical shear stress considering the mechanical effects of roots (Pa) and $a$ is a dimensionless empirically derived exponent. It is usually assumed that $a$ takes values close to 1 [27,59]. ${k}_{d}$ can be estimated as:

^{−3}), $g$ is the gravitational acceleration (9.81 m s

^{−2}), ${R}_{h}$ is the hydraulic radius (m), $S$ is mean channel slope (m m

^{−1}) and $\overline{b}$ is mean channel width (m). ${R}_{h}$ is derived using the Gauckler–Manning–Strickler equation, where Strickler’s roughness coefficient is derived empirically as a power function of ${D}_{50}$ (mm) for every cross section. To characterize the erodibility of streambank material, critical material dependent shear stress ${\tau}_{c}$ (Pa) is estimated using Shields criterion [61,62]. Rearranging the equation, critical shear stress can be formulated as:

^{−3}) and ${D}_{50}$ is median grain size (m). Reported values of θ scatter between 0.012 and 0.3 [62] and are divided into classes based on the particle size classification. θ hereby defines the threshold at which sediment particles are at incipient motion for noncohesive materials [3], but this approach has also been applied for cohesive materials in few cases [63]. As θ depends on the sediment diameter, it can be classified based on measured ${D}_{50}$ values. Depending on ${D}_{50}$, BankforNET uses a fitted normal distribution function to determine random values of θ (Figure 1) within the range of permissible θ values for each particle size class, where the upper and lower threshold are based on values found in the literature [3,38,62,64,65]. For example, if we have coarse gravel (with ${D}_{50}$ = 27 mm, as presented in Figure 1), the lower and upper permissible threshold of θ range from 0.044 to 0.052 based on this defined particle size class [64]. During the iteration process, 10,000 possible values for θ are randomly generated considering a normal probability distribution. The mean value is defined as a function of ${D}_{50}$ and the standard deviation of the normal distribution is defined based on the particle size class and the corresponding upper and lower threshold for θ. Subsequently, 10,000 possible ${k}_{d}$ values are computed resulting in a total of 10,000 ε values. The final modeled erosion rate then represents the mean cumulative erosion computed from all 10,000 iterations. Although we are not aware of any comprehensive data collection for the characterization of the distribution, the implementation of a normal probability distribution for a permissible and feasible range of θ is important to emphasize how the modeled results are influenced by the estimation of θ.

_{s}= 2.4 represents the soil volume of the plot that was used to calibrate the equation [42]. The work from Pasquale and Perona [42] studied the effects of roots on streambed erosion and found that local hydrodynamic bed shear stress conditions when exceeding some critical value gradually cause erosion. This can ultimately lead to uprooting and subsequent entrainment of vegetation. The change in local hydrodynamic bank shear stress conditions also causes streambank erosion when the critical value is exceeded. Even though the uprooting and entrainment process for roots situated on the bed or the streambank may be different, we assume that the effects of roots affecting critical shear stress are the same for both processes.

^{−2}) is then calculated as:

^{3}/s) for different return periods is estimated based on a modified and adapted empirical relation proposed by Kölla [73]:

^{2}). The function in brackets substitutes rainfall intensity in the equation from Kölla [73] for different scenarios (return periods) and was established based on observed data. The presented empirical equation was developed specifically for this model framework and was calibrated using observed precipitation and discharge data.

^{2}):

#### 2.2. Case Studies

#### 2.2.1. The Selwyn/Waikirikiri River Catchment

^{3}s

^{−1}occurred. During this event, the right streambank at an investigated cross section experienced 15.00 m of retreat. Based on a pre- and post-flood digital elevation model (DTM), the models were tested (Table 2). The duration of the modeled hydrograph was reduced to 38.9 h; during this time the river was morphologically active. BankforNET needs few input parameters which are provided by Stecca et al. [28], except for mean bank angle. Mean bank angle is assumed to be 85° based on the preflood survey terrain analysis. Based on the information provided by Stecca et al. [28], BankforNET was validated to estimate how reliably it can predict hydraulic bank erosion without considering the effects of roots as both streambanks were not covered by vegetation.

#### 2.2.2. The Thur River Catchment

^{2}at the observed cross section. Dense vegetation (mature trees) cover the streambanks on both sides. During the hydrological year 2010, impressive retreat of approximately 50 m on the right streambank (cut bank) occurred [76]. The input parameters (Table 3) for BankforNET are based on reported data [77,78,79]. The erosion events were selected based on discharge data provided by the Swiss Federal Office for the Environment (FOEN) [80], where the monthly maximum discharge was used for 12 events representing each month from the hydrological year 2010 (Table 4). Since no event lasted longer than one day, the discharge duration was modeled to be 24 h. For each erosion event, we assumed that the input parameters remained the same except for discharge and the adaption of channel width due to the erosion of the previous erosion event because no information on how the other parameters changed during the erosion events was found in the literature.

^{3}s

^{−1}), 5 years ($H{Q}_{5}$ = 827 m

^{3}s

^{−1}), 30 years ($H{Q}_{30}$ = 952 m

^{3}s

^{−1}), 100 years ($H{Q}_{100}$ = 1068 m

^{3}s

^{−1}) and 300 years ($H{Q}_{300}$ = 1158 m

^{3}s

^{−1}) were used to model erosion scenarios and to quantify relative cumulative erosion reduction for three RAR classes of 0.1%, 1% and 2%. The streambank height of the Thur River catchment was approximately 3.5–4.0 m and therefore, observed effects of roots were negligible under current conditions. However, it is possible that under different conditions (e.g., decreased discharge for longer time periods), roots grow deeper, and root distribution will reach higher values at the streambank toe. Under these hypothetical but realistic conditions, the magnitude of the effects of roots for five different return periods were calculated. While the observed effects of roots on reducing the susceptibility of hydraulic bank erosion were negligible under current conditions, it is possible that if the water level drops and remains low over longer time periods, roots penetrate to deeper depths to reach the water table [68]. Roots would then have a significant effect on reducing the susceptibility to hydraulic bank erosion.

#### 2.2.3. The Sulzigraben Catchment

^{2}. In 2012 and 2015, peak discharge caused noticeable hydraulic and geotechnical bank erosion triggering sediment mobilization as well as LW recruitment and transport [81]. No hydrographic recording station is installed in the Sulzigraben, but during the 2012 and 2015 events, precipitation intensities were reconstructed based on meteorological radar data. Precipitation intensities were approximately 60 mm h

^{−1}to 100 mm h

^{−1}for the 2012 event, and 100 mm h

^{−1}to 160 mm h

^{−1}for the 2015 event [81]. To estimate total streambank retreat, a DTM of the year 2012 with a spatial resolution of 0.5 m × 0.5 m was used. In 2019, a 2 m wide and 0.8 m deep profile was dug on the left streambank to measure the RAR of a sycamore maple (Acer pseudoplatanus L.). The tree stem was standing 2.5 m away from the streambank/water interface with a DBH of 36 cm. Area sectors were defined based on the vertical position of the roots resulting in 5 sectors in depths of 0–15 cm, 15–30 cm, 30–45 cm, 45–60 cm and 60–75 cm. For every sector, roots were counted and individual root diameters were measured. Considering the DTM of 2012, streambank height was approximately 60 cm corresponding to a measured RAR of 1% in 2019. In 2012, the distance of the tree stem to the streambank/water interface was 3.35 m with lower RAR values compared to the ones observed in 2019. The root distribution model was used to calculate changes in the vertical RAR distribution to consider that the streambank/water interface comes closer to the tree stem as the erosion progresses. Further, modeled vertical RAR distribution for white alder was scaled to match those of the investigated vertical RAR distribution of sycamore maple at the Sulzigraben catchment. Roots were present in the left streambank and no roots were present in the right streambank. Subsequently, hydraulic bank erosion for the left streambank was modeled considering the effects of roots and the right streambank was modeled without considering the effects of roots. Further, we assume that the tree present in 2019 was also present in 2012 and that no other trees were standing on or adjacent to the investigated cross section in 2012.

^{2}at the studied cross section. Discharge for both events were estimated using BankforNET considering the precipitation intensities of 2012 and 2015. Other input parameters (Table 5) are based on the field survey from 2019 and estimations using the DTM. Channel and streambank geometries were measured using measuring tape, a double meter stick and a TruePulse 200 laser rangefinder. Channel slope, mean streambank angle and bend radius were similar in 2012 and in 2019. The initial streambank width is based on the 2012 DTM for the first erosion event. For the second erosion event, modeled total erosion of the first event was added to the initial streambank width, increasing the width. Median sediment diameter was measured in the field using the line-by-number analysis [82]. Measured median sediment diameter were assumed to be the same for both erosion events.

#### 2.3. Sensitivity Analysis

## 3. Results

#### 3.1. Validation

#### 3.1.1. The Selwyn/Waikirikiri River Catchment

^{3}s

^{−1}and the rate of cumulative erosion increases nonlinearly with increasing discharge.

#### 3.1.2. The Thur River Catchment

#### 3.1.3. The Sulzigraben River Catchment

#### 3.2. Sensitivity Analysis

#### 3.3. Susceptibility to Hydraulic Bank Erosion Considering the Effects of Roots

^{−1}. The cumulative erosion reduction varies between 0% and 60% with a RAR of 0.1%, between 0% and 100% with a RAR of 1.0% and between 2% and 100% with a RAR of 2% under specific channel conditions. The distance of the tree stem to the streambank/water interface was placed at 1.00 m. Relative cumulative erosion reduction of 100% was reached by four different cross sections: At the four cross sections, total modeled absolute erosion reduction was at most 1.04 m and the median sediment diameter was exclusively ≥ 112 mm.

^{−1}are unlikely to exist in nature.

## 4. Discussion

^{−1}. For the Sulzigraben catchment, mean channel slope is 0.07 m m

^{−1}and varying this parameter +/− 10% produces a range of 0.063 to 0.077 m m

^{−1}. The variability of mean channel slope for the Thur River catchment lies between 0.00144 and 0.00176 m m

^{−1}, much smaller than for the other two catchments. As such, the relative deviation is the same, but the absolute deviation is greater for the Selwyn/Waikirikiri River catchment and the Sulzigraben catchment, resulting in a higher variability of the modeled results. Although several hydrologic model calibration and validation concepts exist [88], no descriptive guidelines for sensitivity analyses are available. A relative deviation was chosen over a fixed value (e.g., +/− 0.01 m m

^{−1}for channel slope) as we believe that measuring errors of +/− 10% magnitude can occur in the field. In addition, determining mean channel width, mean channel slope, mean streambank angle and mean bend radius can be done rather easily and accurately in the field, but the exact determination of ${D}_{50}$ is more time consuming and is spatially heterogeneous. Based on this, considerable effort is still needed to improve a more cost-effective method for the estimation of this parameter for practical applications. Further, median grain size is not the same on the surface of a streambank or riverbed as it is in depth. While the framework estimates erosion for static ${D}_{50}$ in time, ${D}_{50}$ is not static as it changes as soon as material is removed and mobilized sediment from upstream is deposited, as shown by Pasquale et al. [79] for the Thur River catchment. Their results show that the standard deviation of the observed ${D}_{50}$ at the surface is approximately 15 mm, whereas the standard deviation at depths of 40 cm is approximately 2 mm. To take this effect into account, a probability-density function adapting ${D}_{50}$ values during an erosion event could have a significant effect on the modeled results as ${D}_{50}$ will most likely change during erosion events. In addition, regarding the simplicity of the framework: measuring ${D}_{50}$ is rather time consuming. The inclusion of an empirically derived function to estimate ${D}_{50}$ as a function of mean channel slope and mean channel width [89] could be considered and used to compare measured and estimated ${D}_{50}$ values to find a suitable density function or to define permissible ${D}_{50}$ values.

^{−1}. Based on a GIS analysis using the ecological morphology data provided by the Swiss Federal Office for the Environment, these criteria correspond approximately to 42% of all Swiss torrents and streams. Considering the effects of roots (RAR in vertical direction) was validated at the Sulzigraben catchment. A sycamore maple with a DBH of 36 cm had a measured RAR of max. 2.3% at 30 cm depth and 1% at 60 cm depth—the point where erosion occurred and was modeled—standing 2.50 m away from the streambank/water interface. Considering a RAR of 1% at a radial distance of 2.50 m away from the tree stem, the ideal riparian forest conditions for the Sulzigraben catchment to decrease the susceptibility of hydraulic bank erosion can be reached with a forest stand of 510 trees per hectare and a mean DBH of 36 cm. With continuous erosion, RAR increases, subsequently reducing hydraulic bank erosion. As such, the reduction of hydraulic bank erosion is dependent on RAR, rooting depth, the difference and modification of critical and applied shear stress and streambank height. As long as channel incision is not greater than rooting depth, roots have a stabilizing (positive) effect. If no roots are present at the area affected by the flow, roots have no or very little influence on erosion reduction (see Figure 13).

## 5. Conclusions

^{−1}and RAR of 1% to 2%, reducing the susceptibility of hydraulic bank erosion up to 100%. Channel and forest managers could use the susceptibility matrix to identify areas where roots have significant effects on decreasing the susceptibility of hydraulic bank erosion. Their efforts should then be focused on areas where vegetation effects (i.e., root density) are high or variable to adapt forest densities or to perform targeted afforestation. Reducing the susceptibility of streambank erosion is only possible if roots are present at the area affected by the flow. Therefore, the matrix is only valid if streambank height is smaller than the maximum average rooting depth.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Example of the modeled Shields entrainment parameter θ distribution for coarse gravel (${D}_{50}$ = 27 mm). The red dotted line represents the fitted density line of the distribution and the y-axis (density) refers to the amount of modeled θ used for all 10,000 iterations to model ${\tau}_{c}$, ${k}_{d}$ and subsequent ε.

**Figure 2.**Modeled mean root area ratio (RAR) distribution of white alder (Alnus incana L.) in a modeled vertical profile at six lateral (radial) positions from the tree stem (d

_{stem}= 0.5 m, 1.0 m, 1.5 m, 2.0 m, 2.5 m and 3.0 m) with a DBH of 36 cm. Maximum modeled rooting depth was 74 cm. Note that the integral of the RAR distribution in vertical direction equals the mean RAR per total surface of the plot at six lateral (radial) positions.

**Figure 3.**Conceptual illustration of how BankforNET models hydraulic bank erosion considering RAR. The density of the RAR is a function of rooting depth and roots protecting the streambank affected by the flow. On the left is a conceptual illustration of root density (i.e., RAR) as a function of rooting depth. The red dot and arrow on the right represent the one-dimensional erosion at the bank toe that is modeled by BankforNET considering the vertical RAR distribution if roots are present.

**Figure 4.**Example of the modeled cumulative erosion for the Selwyn/Waikirikiri River catchment (where no roots were present). The blue line represents the triangular hydrograph, the red lines represent each erosion iteration and the black line represents the mean cumulative erosion (m). The two black vertical lines indicate when erosion is equal to zero (at discharge values ≤ 11 m

^{3}s

^{−1}for the Selwyn/Waikirikiri River catchment).

**Figure 5.**Cumulative erosion reduction (%) due to the mechanical effects of roots for return periods (RP) of 2, 5, 30, 100 and 300 years with RAR of 0.1%, 1% and 2% for the Thur River catchment.

**Figure 6.**Measured vertical RAR distribution for sycamore maple (Acer pseudoplatanus L., blue) and modeled vertical RAR root distribution for white alder (Alnus incana L.), scaled to represent the measured sycamore maple vertical RAR distribution (red). At the end of both erosion events, the distance of the tree stems to the streambank/water interface was 2.50 m and the DBH was set to 36 cm for both trees (

**top**). Modeled RAR for both erosion events at the Sulzigraben catchment considering the adaption of RAR as a function of erosion and the changing distance of the tree stem to the streambank/water interface for the left streambank (

**bottom**).

**Figure 7.**Sensitivity analysis using the data for the Selwyn/Waikirikiri River catchment. For every input parameter, the values varied +/− 10% from the original input value. The red line is the reference for the modeled erosion = 17.40 m using the original input values. Each boxplot represents the quantile of the cumulative erosion.

**Figure 8.**Sensitivity analysis representing event 12 for the Thur River catchment. For every input parameter, the value varied +/− 10% from the original input value. The red line is the reference for the modeled erosion = 9.18 m using the original input values. Each boxplot represents the quantile of the cumulative erosion.

**Figure 9.**Sensitivity analysis for the Sulzigraben catchment for the first erosion event in 2012. For every input parameter, the value varied +/− 10% from the original input value. The red line is the reference for the total modeled erosion = 1.10 m for the first event using the original input values without considering the effects of roots. The blue line is the reference for the total modeled erosion = 0.85 m for the first event using the original input values considering the effects of roots (left streambank). Each boxplot represents the quantile of the cumulative erosion.

**Figure 10.**Stabilizing effects of roots on hydraulic bank erosion in relationship to channel width. The points represent the cumulative erosion reduction (%) with RAR = 0.1% (green), RAR = 1% (blue) and RAR = 2% (red).

**Figure 11.**Stabilizing effects of roots on hydraulic bank erosion in relation to channel slope. The points represent the cumulative erosion reduction (%) with RAR = 0.1% (green), RAR = 1% (blue) and RAR = 2% (red).

**Figure 12.**Susceptibility matrix to hydraulic bank erosion highlighting the stabilizing effects of roots.

**Figure 13.**Streambank in the Zulg river catchment, Canton of Bern, Switzerland. In the lower area of the streambank, no roots are present. In this area, roots have no influence on reducing the susceptibility of hydraulic bank erosion. The black arrow indicates the flow direction. Source: Eric Gasser.

Model Name | Modeled Process | Effects of Vegetation | Reference |
---|---|---|---|

BSTEM & RipRoot | Geotechnical bank erosion, hydraulic bank erosion | Root reinforcement by adapting apparent cohesion (geotechnical bank erosion), adaption of critical shear stress (hydraulic bank erosion) based on values found in literature | [17,34,46,47,48] |

CONCEPTS & REMM | Geotechnical bank erosion, hydraulic bank erosion | Root reinforcement by adapting apparent cohesion (geotechnical bank erosion) | [45,49,50,51,52] |

SWAT | Hydraulic bank erosion, bed erosion | Adapting critical shear stress based on an empirical effect | [53,54,55] |

Geotechnical bank erosion | Root reinforcement by adapting apparent cohesion | [44] | |

SedNet | Hydraulic bank erosion | Consideration of vegetation cover (extent of vegetation cover) using a vegetation factor | [56,57] |

Hydraulic bank erosion | Consideration of vegetation cover (extent of vegetation cover) using a vegetation factor | [36] |

**Table 2.**Input parameters to perform BankforNET based on the analysis presented in Stecca et al. [28].

Input Parameters | Symbol | Dimension | Value |
---|---|---|---|

Discharge | Q | m^{3} s^{−1} | 130 |

Duration of flood event | t | h | 38.9 |

Mean channel width | W | m | 62 |

Mean channel slope | S | m m^{−1} | 0.007 |

Mean streambank angle | BA | ° | 85 |

Mean bend radius | R | m | 185 |

Median sediment diameter | D50 | mm | 27 |

Root area ratio | RAR | % | - |

**Table 3.**Input parameters for the cross section at the Thur River catchment in Canton of Thurgau, Switzerland. Discharge (Q) for the 12 scenarios (events) are presented in Table 4.

Input Parameters | Symbol | Dimension | Value |
---|---|---|---|

Discharge | Q | m^{3} s^{−1} | see Table 4 |

Duration of each flood event | t | h | 24 |

Mean channel width | W | m | 30 |

Mean channel slope | S | m m^{−1} | 0.0016 |

Mean streambank angle | BA | ° | 45 |

Mean bend radius | R | m | 100 |

Median sediment diameter | D50 | mm | 10 |

Root area ratio | RAR | % | 0.1/1/2 |

**Table 4.**Discharge (Q) scenarios (events) for the Thur River catchment in Canton of Thurgau, Switzerland.

Event | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Q | m³ s^{−1} | 122 | 206 | 362 | 174 | 120 | 170 | 61.9 | 340 | 501 | 390 | 492 | 628 |

**Table 5.**Input parameters for the cross section at the Sulzigraben catchment in the Canton of Bern, Switzerland for the two erosion events.

Input Parameters | Symbol | Dimension | Value |
---|---|---|---|

Discharge | Q | m^{3} s^{−1} | 11.4/18.9 |

Duration of flood event | t | h | 6.5 |

Mean channel width | W | m | 4/5 |

Mean channel slope | S | m m^{−1} | 0.07 |

Mean streambank angle | BA | ° | 29 |

Mean bend radius | R | m | ∞ (straight reach) |

Median sediment diameter | D50 | mm | 98 |

Root area ratio | RAR_{max} | % | 1 |

Input Parameters | Symbol | Dimension | Event | Number | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |||

Modeled streambank erosion (no roots) | E | m | 1.37 | 3.20 | 5.90 | 2.56 | 1.32 | 2.47 | 0.10 | 5.58 | 7.75 | 6.32 | 7.64 | 9.18 |

Standard deviation | - | m | 0.55 | 1.22 | 2.16 | 0.99 | 0.54 | 0.96 | 0.04 | 2.04 | 2.76 | 2.29 | 2.73 | 3.23 |

Modeled streambank erosion (RAR = 0.1%) | E | m | 1.32 | 3.14 | 5.84 | 2.50 | 1.28 | 2.41 | 0.08 | 5.51 | 7.68 | 6.25 | 7.57 | 9.11 |

Standard deviation | - | m | 0.54 | 1.20 | 2.13 | 0.97 | 0.52 | 0.94 | 0.03 | 2.02 | 2.74 | 2.27 | 2.71 | 3.21 |

Modeled streambank erosion (RAR = 1%) | E | m | 0.93 | 2.61 | 5.20 | 2.00 | 0.89 | 1.93 | 0.01 | 4.90 | 7.05 | 5.64 | 6.95 | 8.48 |

Standard deviation | - | m | 0.38 | 1.02 | 1.94 | 0.79 | 0.37 | 0.77 | 0.006 | 1.82 | 2.54 | 2.07 | 2.51 | 3.01 |

Modeled streambank erosion (RAR = 2%) | E | m | 0.57 | 2.06 | 4.58 | 1.50 | 0.53 | 1.44 | 0.00 | 4.26 | 6.37 | 4.98 | 6.27 | 7.78 |

Standard deviation | - | m | 0.25 | 0.82 | 1.72 | 0.61 | 0.23 | 0.58 | 0.00 | 1.61 | 2.33 | 1.86 | 2.29 | 2.80 |

**Table 7.**Cumulative erosion reduction for the Thur River catchment considering different return periods and root area ratios.

Return Period | RAR | Cumulative Erosion Reduction |
---|---|---|

(Year) | (%) | (%) |

2 | 0.1 | 1 |

2 | 1 | 8 |

2 | 2 | 17 |

5 | 0.1 | 1 |

5 | 1 | 7 |

5 | 2 | 14 |

30 | 0 | 1 |

30 | 1 | 6 |

30 | 2 | 13 |

100 | 0 | 1 |

100 | 1 | 5 |

100 | 2 | 11 |

300 | 0.1 | 1 |

300 | 1 | 5 |

300 | 2 | 10 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gasser, E.; Perona, P.; Dorren, L.; Phillips, C.; Hübl, J.; Schwarz, M. A New Framework to Model Hydraulic Bank Erosion Considering the Effects of Roots. *Water* **2020**, *12*, 893.
https://doi.org/10.3390/w12030893

**AMA Style**

Gasser E, Perona P, Dorren L, Phillips C, Hübl J, Schwarz M. A New Framework to Model Hydraulic Bank Erosion Considering the Effects of Roots. *Water*. 2020; 12(3):893.
https://doi.org/10.3390/w12030893

**Chicago/Turabian Style**

Gasser, Eric, Paolo Perona, Luuk Dorren, Chris Phillips, Johannes Hübl, and Massimiliano Schwarz. 2020. "A New Framework to Model Hydraulic Bank Erosion Considering the Effects of Roots" *Water* 12, no. 3: 893.
https://doi.org/10.3390/w12030893