# Inferences on Mixed Snow Avalanches from Field Observations

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## Abstract

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## 1. Introduction

## 2. Summary of Observations

## 3. Inferences on Flow Regimes from Deposit Properties

#### 3.1. Correspondence between Flow Regimes and Deposit Types

#### 3.2. Granulometry and Deposit Density

#### 3.3. Where Do the Snow Clods Originate?

## 4. Dynamical Properties of the Fluidized Flow Regime

#### 4.1. Flow Depth and Density of Fluidized Flow

- We surmise that fluidized flow requires a minimum velocity, or else it reverts to dense flow or stops. This threshold velocity will depend on the slope angle, particle size and presumably the properties of the snow pack as well; we conjecture it to be in the range ${u}_{\mathrm{thr}}=15$–20 m s${}^{-1}$.
- For densities in the range ${\rho}_{f}=20$–100 kg m${}^{-3}$, the (time-averaged) impact pressure on a wide obstacle is ${p}_{\mathrm{imp}}>{\rho}_{f}{u}_{\mathrm{thr}}^{2}=5$–40 kPa.
- If the pressure had exceeded 10–20 kPa near the snow surface or 5–10 kPa more than 2 m above ground, the ski lift shed at Albristhorn or the cabins at Grand Moilles (Scex Rouge) would likely have been structurally damaged to some degree. At those locations, we can infer ${h}_{f}\lesssim 2$ m, ${\rho}_{f}\lesssim 50$ kg m${}^{-3}$ and ${u}_{f}\lesssim 20$ m s${}^{-1}$.
- Let ${l}_{d}$, ${h}_{d}$, and ${\rho}_{d}$ be the length, height, and density, respectively, of the Type 2 deposit, and denote the corresponding quantities of the fluidized flow shortly before deposition by ${l}_{f}$, ${h}_{f}$ and ${\rho}_{f}$. Then, the masses per unit width, ${m}_{d}\equiv {l}_{d}{h}_{d}{\rho}_{d}$ and ${m}_{f}\equiv {l}_{f}{h}_{f}{\rho}_{f}$, should be approximately equal.
- At Albristhorn, we use the following values characterizing the deposit along its centerline: ${\rho}_{d}=400$ kg m${}^{-3}$, ${l}_{d}=250$ m, ${\overline{h}}_{d}=0.2$ m, giving ${m}_{d}\approx 2\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{4}$ kg m${}^{-1}$. The corresponding values for the Scex Rouge avalanche are ${\rho}_{d}=500$ kg m${}^{-3}$, ${l}_{d}=300$–500 m, ${\overline{h}}_{d}=0.3$–0.4 m, thus ${m}_{d}\approx (4.5\u201310)\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}{10}^{4}$ kg m${}^{-1}$.
- We assume ${l}_{f}$ in the range 200–400 m at Albristhorn and 400–800 m at Scex Rouge, using the profiling radar measurements at Vallée de la Sionne [9] and pressure measurements at Ryggfonn [15,37] as reference points. Mass conservation demands ${h}_{f}={m}_{d}/\left({l}_{f}{\rho}_{f}\right)$, and with ${\rho}_{f}=20$–50 kg m${}^{-3}$ we obtain the ranges ${h}_{f}=1$–5 m at Albristhorn and ${h}_{f}=1.25$–12.5 m at Scex Rouge.

#### 4.2. Velocities of the Dense and Fluidized Flows

#### 4.3. Relative Mobility of the Three Flow Regimes

#### 4.4. Density Estimates for the Suspension Layer

#### 4.5. Impact Pressures and Densities

- ${C}_{d}$ decreases significantly with increasing Mach number for $\mathrm{Ma}<1$, but is almost constant for supersonic speeds.
- The drag coefficient increases by about a factor of 2 from $\mathrm{Kn}\ll 1$ to $\mathrm{Kn}=2$ and is about constant for $\mathrm{Kn}>5$.
- ${C}_{d}$ depends only mildly on the restitution and friction coefficients of inter-particle collisions.
- Except at very low Kn, ${C}_{d}$ for a cylinder in dilute granular flows is in the range 1.5–2.5 and thus roughly a factor 2 larger than in turbulent subsonic flows.

## 5. Possible Fluidization Mechanisms

#### 5.1. Aerodynamic Forces in the Head of the Suspension Layer

#### 5.2. Dispersive Pressure Due to Collisions between Snow Clods

#### 5.3. Fluidization by Compression of the Snow Pack?

## 6. Inferences on Entrainment and Mass Balance

#### 6.1. Entrainment by the Dense Flow

#### 6.2. Entrainment by the Fluidized Flow

#### 6.3. Entrainment by the Suspension Flow

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Energy Balance of a Suspension Flow on a Counter-Slope

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**Figure 1.**Vilan avalanche, snow pit in the Type 2 deposit. The avalanche eroded approximately 1 m of cold dry snow before depositing 40–50 cm of fine-grained, compressed snow with embedded spruce twigs and snow clods. The largest clods are sintered to the surface (at their original location behind the pit).

**Figure 2.**Aerial view of the run-out zone of the Albristhorn avalanche. The boundary of the Type 2 deposit (yellow line) is much less certain than that of the Type 1 deposit (red line), particularly in the southern parts (top of photo). The damaged ski lift shed is hidden behind trees in the top center part.

**Figure 3.**Cross-section and detail of a large snow clod in the run-out zone of the 10 February 2005 Salezertobel avalanche, Davos, Switzerland. The texture of the snow is made visible by spraying a mixture of ink and alcohol onto the smoothened surface, warming it carefully with a camping cooker and waiting for the ink to diffuse into the snow by capillary suction. Note the layered core (oblique stripes (

**left**) and the small snow clods sintered onto it (

**right**).

**Figure 4.**Large snow clod near the right edge of the avalanche artificially released from Gotschnawang, Klosters, Switzerland on 12 March 2006. Note that the approximately 10 cm deep furrow carved by the block abruptly swerves to the right at the end.

**Figure 5.**Definition sketch of the topographic quantities referred to in the topographical–statistical $\alpha $–$\beta $ model.

**Figure 6.**Schematic representation of the proposed fluidization mechanism by air escaping from the snow cover under compression. The degree of shading indicates the relative densities. The ambient air flow around and partly through the avalanche head is also schematically indicated.

**Table 1.**Approximate run-out angles ${\alpha}_{d}$, ${\alpha}_{f}$ of the dense (d) and fluidized (f) parts of the Albristhorn and Scex Rouge avalanches and twelve selected events from the experimental site Ryggfonn (western Norway). These values are compared to the predictions of the Norwegian topographical–statistical $\alpha $–$\beta $ model ($\beta $, ${\alpha}_{\mathrm{stat}}$).

Event | $\mathit{\beta}$ | ${\mathit{\alpha}}_{\mathbf{stat}}$ | ${\mathit{\alpha}}_{\mathit{d}}$ | ${\mathit{\alpha}}_{\mathit{f}}$ | ${\mathit{\alpha}}_{\mathit{d}}-{\mathit{\alpha}}_{\mathit{f}}$ |
---|---|---|---|---|---|

Albristhorn 1995 | 29.5° | 26.9° | 25.8° | 23.3° | 2.5° |

Scex Rouge 1995 | 28.1° | 25.6° | 25.5° | 20.9° | 4.6° |

Ryggfonn | |||||

10 January 1983 | 29.8° | 27.2° | 29.3° | 27.3° | 2.0° |

8 March 1983 | 29.3° | 29.3° | 0.0° | ||

13 February 1985 | 29.8° | 28.8° | 0.9° | ||

28 January 1987 | 29.3° | 26.4° | 2.9° | ||

11 April 1988 | 29.3° | 29.1° | 0.2° | ||

23 December 1988 | 29.1° | 29.0° | 0.1° | ||

7 March 1990 | 29.3° | 28.8° | 0.5° | ||

27 March 1993 | 28.8° | 27.3° | 1.5° | ||

24 January 1994 | 29.3° | 29.1° | 0.3° | ||

3 March 1995 | 29.4° | 28.5° | 0.9° | ||

8 February 1997 | 28.4° | 26.9° | 1.5° | ||

17 February 2000 | 27.1° | 23.1° | 4.0° |

**Table 2.**Estimates of volumetric particle concentration $\nu $, Knudsen number, Mach number and drag coefficient ${C}_{d}$ of a tree trunk in fluidized and suspension flow based on numerical results from [59]. For comparison, typical values of ${C}_{d}$ for a turbulent air flow and the experimental results of Hauksson et al. [58] for a free-surface flow with Froude number $\mathrm{Fr}=13$ around a cylinder are also listed (Kn, Ma: our estimates).

Flow Type | $\mathit{\nu}$ | Kn | Ma | ${\mathit{C}}_{\mathit{d}}$ |
---|---|---|---|---|

Fluidized flow, small particles | 0.03 | $<0.1$ | 0.2–0.7 | 1–1.5 |

Fluidized flow, large particles | 0.02 | 3–10 | 0.1–0.5 | 2–2.5 |

Suspension flow | 0.001 | 0.1 | 0.3 | 1–1.5 |

Storm | 0 | ${10}^{-6}$ | 0.1 | $\sim 1$ |

Dense granular flow | 0.55 | $\sim 0.01$ | $>1$ | 0.3–0.5 |

**Table 3.**Assumptions about the properties of the fluidized flows of the Vilan avalanche near the gully bend and the Albristhorn and Scex Rouge avalanches in the run-out zone (counter-slope, ↑) with Type 2 deposit, and the resulting mean erosion rates.

Vilan | Albristhorn | Scex Rouge | ||
---|---|---|---|---|

Location | Track | Run-out | Run-out | |

Slope angle | (°) | 20–25 | 5–30↑ | 0–10↑ |

Snow density | (kg m${}^{-3}$) | 100 | 215 | 200 |

Erosion depth | (m) | 0.6–1 | $<0.1$ | $<0.3$ |

Deposit density | (kg m${}^{-3}$) | 300 (?) | 420 | 520 |

Deposit depth | (m) | 0.2–0.5 | 0.1–0.3 | 0.2–0.5 |

Flow length ${l}_{f}$ | (m) | 30–50 | 100–200 | 200–400 |

Flow speed ${u}_{f}$ | (m s${}^{-1}$) | 15–20 | 20–40 | 20–30 |

Passage time | (s) | 1.5–3 | 2.5–10 | 7–20 |

Erosion rate | (kg m${}^{-2}$ s${}^{-1}$) | 20–70 | $<8$ | $<10$ |

© 2019 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/).

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**MDPI and ACS Style**

Issler, D.; Gauer, P.; Schaer, M.; Keller, S.
Inferences on Mixed Snow Avalanches from Field Observations. *Geosciences* **2020**, *10*, 2.
https://doi.org/10.3390/geosciences10010002

**AMA Style**

Issler D, Gauer P, Schaer M, Keller S.
Inferences on Mixed Snow Avalanches from Field Observations. *Geosciences*. 2020; 10(1):2.
https://doi.org/10.3390/geosciences10010002

**Chicago/Turabian Style**

Issler, Dieter, Peter Gauer, Mark Schaer, and Stefan Keller.
2020. "Inferences on Mixed Snow Avalanches from Field Observations" *Geosciences* 10, no. 1: 2.
https://doi.org/10.3390/geosciences10010002