# Constraints on Entrainment and Deposition Models in Avalanche Simulations from High-Resolution Radar Data

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## Abstract

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## 1. Introduction

## 2. Avalanche Model and Experimental Data

#### 2.1. Mathematical Model

#### 2.2. Experimental Avalanche Data

## 3. Simulation Setup and Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Sample Availability: The model is integrated as a module within OpenFOAM-v1812 and freely available under GNU General Public License in the OpenFOAM community repository https://develop.openfoam.com/Community/avalanche and additional tools and scripts are available at https://bitbucket.org/matti2/fasavagehutterfoam and as supplement in [32]. The GEODAR data can be accessed from the repository GEODAR data of snow avalanches at Vallé de la Sionne: Seasons 2010/2011, 2011/2012, 2012/2013 & 2014/2015 [Data set] hosted by Zenodo https://zenodo.org [42]. |

**Figure 1.**Velocity profile of an infinitely long avalanche with colour marking the time. (

**A**) Acceleration on an inclined plane. The simulation started from rest, the velocity profile tends to the Bagnold profile in steady state (black dotted line). (

**B**) Deceleration on a flat plane. The simulation started with the steady state Bagnold profile of an inclined plane from panel A, and leads to a gradual deposition starting from the bottom.

**Figure 2.**The deposition depth as a function of the depth-integrated momentum from granular simulations (not depth-averaged). The respective velocity profiles during deceleration on a flat plane for each given momentum $h{\overline{u}}_{x}$ are shown in orange. The dashed line shows the behaviour of the relation for the depth-integrated model derived in Equation (10).

**Figure 3.**Overview of the avalanche test site Vallée de la Sionne (VdlS) with both simulated avalanches (#0017 lower release area, #0019 above) from release areas Crêta Besse (CB1/2). The simulated mass balance summarizing entrainment (red) and deposition (blue) of the two consecutive avalanches is shown, together with the entrainable snow cover (green) that linearly increases from 0–1 m above 1200 m range until the mountain top. Black contour lines indicate the radar range (distance from GEODAR in the valley floor). The area in the black square is zoomed in the upper right corner showing the computational mesh and the deposition patterns of both avalanches with the dark blue area corresponding to the runout of avalanche #0019. Coordinates are given the Swiss coordinate system CH1903 (SRID 21781).

**Figure 4.**Moving target identification plot of avalanche #0017. The GEODAR data are shown in the background. Black lines give the front and tail position extracted from the simulations. Material and model parameters are optimized to this avalanche. Simulation no. 1 solely includes entrainment, simulation no. 2 includes entrainment and deposition, vertical lines correspond to the snapshots (

**A**–

**D**) in Figure 6. Simulation no. 3 is drawn as reference without entrainment and deposition. The slope angle along the line of steepest descend is shown in the right panel. A velocity legend is included for quick translation of front inclination to approach velocity. Black lines represent the front and tail position extracted from simulations (Section 3).

**Figure 5.**Moving target identification plot of avalanche #0019. The GEODAR data are shown in the background. Black lines give the front and tail position extracted from the simulations. No optimization of parameters has been conducted for this avalanche, but material and model parameters are taken from optimization to avalanche #0017 in Figure 4. Simulation no. 1 solely includes entrainment, simulation no. 2 includes entrainment and deposition, vertical lines correspond to the snapshots (

**E**–

**H**) in Figure 6. Simulation no. 3 is drawn as reference without entrainment and deposition. The slope angle along the line of steepest descend is shown in the right panel. A velocity legend is included for quick translation of front inclination to approach velocity. Black lines represent the front and tail position extracted from simulations (Section 3).

**Figure 6.**Time series of the simulation for avalanche #0017 (

**A**–

**D**) and #0019 (

**E**–

**H**) at 10, 30, 60 and $90\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. Concentric circles mark the line-of-sight distance from the bunker giving the radar range. The flowing avalanche is indicated with the depth-averaged velocity $\overline{\mathbf{u}}$. In the background, snow cover changes due to entrainment (red) and deposition (blue) are indicated with the same colour scale as the simulated mass balance in Figure 3.

**Figure 7.**The sum of the mobilized volume in the simulation for (

**A**) avalanche #0017 and (

**B**) avalanche #0019 for the three simulation setups no. 1, no. 2 and no. 3. The vertical markers A–D and E–H correspond to the snapshots of the simulations in Figure 6 and the markers in the MTI images (Figure 4 and Figure 5). Gray horizontal lines give volumes of initial release (lower line) and total volume (upper line), reported by Köhler et al. [29].

**Table 1.**Summary of simulation parameters for both consecutive avalanches #0017 and #0019. $\mu $, $\chi $ and ${e}_{\mathrm{b}}$ are determined with simulation no. 1 on avalanche #0017, ${u}_{\mathrm{dep}}$ resulted from simulation no. 2 on the same avalanche.

Simulation | $\mathit{\mu}$ | $\mathit{\chi}\phantom{\rule{0.166667em}{0ex}}\left({\mathbf{m}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathbf{s}}^{-2}\right)$ | ${\mathit{e}}_{\mathbf{b}}\phantom{\rule{0.166667em}{0ex}}({\mathbf{m}}^{2}/{\mathbf{s}}^{2})$ | ${\mathit{u}}_{\mathbf{dep}}\phantom{\rule{0.166667em}{0ex}}(\mathbf{m}/\mathbf{s})$ |
---|---|---|---|---|

No. 1 | $0.24$ | 6000 | 600 | – |

No. 2 | $0.24$ | 6000 | 600 | $1.5$ |

No. 3 | $0.24$ | 6000 | – | – |

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**MDPI and ACS Style**

Rauter, M.; Köhler, A.
Constraints on Entrainment and Deposition Models in Avalanche Simulations from High-Resolution Radar Data. *Geosciences* **2020**, *10*, 9.
https://doi.org/10.3390/geosciences10010009

**AMA Style**

Rauter M, Köhler A.
Constraints on Entrainment and Deposition Models in Avalanche Simulations from High-Resolution Radar Data. *Geosciences*. 2020; 10(1):9.
https://doi.org/10.3390/geosciences10010009

**Chicago/Turabian Style**

Rauter, Matthias, and Anselm Köhler.
2020. "Constraints on Entrainment and Deposition Models in Avalanche Simulations from High-Resolution Radar Data" *Geosciences* 10, no. 1: 9.
https://doi.org/10.3390/geosciences10010009