# On a Continuum Model for Avalanche Flow and Its Simplified Variants

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## Abstract

**:**

## Editor’s foreword

## 1. Introduction

## 2. The Full Model

## 3. A Simplified Version of the Full Model

## 4. Comparison of Computations with the Full and Simplified Models

## 5. Improvement of the Simplified Model

## 6. The Simplest Model for Rough Estimations

#### 6.1. Solutions for the Coulomb and Voellmy Friction Laws

#### 6.2. Solution with the Stress-Limited Friction Law

## 7. Influence of Snow Entrainment on Avalanche Dynamics

## 8. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Issler, D. Comments on “On a Continuum Model for AvalancheFlow and Its Simplified Variants” by S. S. Grigorian and A. V. Ostroumov. Geosciences
**2020**. (Under review). [Google Scholar] - Grigorian, S.S. Mechanics of snow avalanches. In Snow Mechanics–Symposium–Mécanique de la Neige. (Proceedings of the Grindelwald Symposium); IAHS Publ. 114; Institute of Hydrology: Wallingford, UK, 1974; pp. 355–368. [Google Scholar]
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**1979**, 24, 110–111. [Google Scholar] - Grigorian, S.S.; Ostroumov, A.V. On Mechanics of Formation and Collapse of Rock Masses Deposits; Institute for Mechanics, Moscow State University: Moscow, Russia, 1975. (In Russian) [Google Scholar]
- Grigorian, S.S.; Ostroumov, A.V.; Stacheiko, S.F. A Simplified Method for Calculations of Slope Collapse Processes; Institute for Mechanics, Moscow State University: Moscow, Russia, 1979. (In Russian) [Google Scholar]

**Figure 1.**Schematic representation of an avalanche flowing in a rectangular channel of variable width $L\left(S\right)$, with S the coordinate along the centerline of the path and $\psi \left(S\right)$ the local slope angle.

**Figure 2.**Entrainment near the front of the avalanche. The interface between the snow cover (with original depth ${\delta}_{0}\left(S\right)$ and instantaneous depth $\delta (S,t)$) and the flowing avalanche is regarded as a shock front that propagates into the snow cover with normal velocity $\omega $.

**Figure 3.**Downslope evolution of the avalanche flow from the initial condition with fracture depth ${h}_{0}\left(S\right)$ from the crown at ${S}_{0}$ to the stauchwall at ${S}_{f}\left(0\right)$ to intermediate configurations $h(S,t)$ extending from ${S}_{0}$ to ${S}_{f}\left(t\right)$. The front propagates at the speed $w\left(t\right)$.

**Figure 4.**Initial conditions for calculations with the full model: The release depth was chosen either with a parabolic or a triangular longitudinal profile (

**left panel**), whereas the initial velocity was assumed uniform (

**right panel**).

**Figure 5.**Representative example (variant 1 in Table 1 and Table 2) of simulation results from the full model on an inclined plane (slope angle 30${}^{\circ}$). The flow started with a parabolic shape with maximum depth 1 m and a uniform initial velocity of 10 m s${}^{-1}$. The friction coefficients are $\mu =0.5$ and $k=0.02$, the density is $\rho =500\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ and the strength of the snow cover 10 kPa. (

**a**) Evolution of the longitudinal profile of the flow depth h, scaled by the instantaneous front positions, ${S}_{f}$ (in m), and maximum depth $H\left({S}_{f}\right)$, of the flow. (

**b**) Evolution of the relative position of the maximum flow depth, ${S}_{*}/{S}_{f}$, of the shape factor $\varkappa $, the front velocity w and the maximum flow depth H with front position ${S}_{f}$.

**Figure 7.**Integral curves corresponding to solutions of Equation (38) together with the hyperbolas $y=A/\left(BS\right)$, on which the integral curves attain their maximum, $y={\gamma}_{*}/S$ and $y=AS/(B{S}^{2}-1)$, on which the integral curves have their inflection point.

**Figure 8.**Examples of simulations with the full model, showing the distributions of new-snow depth and flow depth (full lines), and velocity (dashed lines) when the avalanche front reaches ${S}_{f}=500$ m. Here, and in Figure 9 and Figures S2.10–S2.14, the scale of the abscissa (S) is the same as for h and $\delta $ in the 10 m right behind the avalanche front, but strongly compressed for $0\phantom{\rule{0.166667em}{0ex}}\mathrm{m}<S<490\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$. (

**a**) Simulation with basic parameter set (no. 1 in Table 3). (

**b**) Same as (

**a**), except for the flow density increased from 300 to 500 $\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ (case 2 in Table 3).

**Figure 9.**Further examples of simulations with the full model (case 13–15 in Table 3), showing the distributions of new-snow depth and flow depth (full lines), and velocity (dashed lines) when the avalanche front reaches ${S}_{f}=500$ m. Note the change of horizontal scale at $S=490\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$. (

**a**) Slowly moving, non-eroding avalanche with dry-friction coefficient $\mu =0.55$. (

**b**) “Ignited” avalanche with $\mu =0.42$. (

**c**) With $\mu =0.40$, the avalanche behaves similarly as in the case $\mu =0.55$ (panel (a)).

**Table 1.**The set of numerical values of the problem parameters for tested variants ([1], Note 8).

Variant N | $\mathit{\mu}$ — | k — | ${\mathit{\tau}}_{*}$ kPa | $\mathit{\rho}$ $\mathbf{kg}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{-3}$ | $\mathit{\psi}$ ${}^{\circ}$ | ${\mathit{U}}_{0}$ $\mathbf{m}\phantom{\rule{0.166667em}{0ex}}{\mathbf{s}}^{-1}$ | ${\mathit{H}}_{0}$ m | Profile ${\mathit{h}}_{0}\left(\mathit{s}\right)$ |
---|---|---|---|---|---|---|---|---|

1 | 0.5 | 0.02 | 10 | 500 | 30 | 10 | 1 | parabola |

2 | 0.5 | 0.02 | 10 | 500 | 30 | 1 | 10 | parabola |

3 | 0.5 | 0.02 | 10 | 500 | 30 | 10 | 10 | triangle |

4 | 0.5 | 0.02 | 10 | 500 | 30 | 10 | 1 | triangle |

5 | 0.5 | 0.02 | 10 | 500 | 30 | 10 | 1 | parabola |

6 | 0.5 | 0.02 | 10 | 500 | 30 | 10 | 60 | triangle |

7 | 0.5 | 0.02 | 10 | 500 | 46 | 1 | 10 | parabola |

8 | 0.25 | 0.02 | 10 | 500 | 30 | 1 | 10 | parabola |

9 | 0.5 | 0.06 | 10 | 500 | 30 | 1 | 10 | parabola |

10 | 0.5 | 0.02 | 2.5 | 500 | 30 | 1 | 10 | parabola |

**Table 2.**The results of calculations with the full model (Section 2), the simplified model of Section 3 (formulas (23) and (24)), and the improved simplified model described in Section 5 (formulas (36), (39) and (40)) are presented in terms of the flow depth H and the front velocity w as functions of the front position ${S}_{f}$. For the simplified models, the value of the shape factor was always chosen as $\varkappa =0.36$ ([1], Note 9).

Variant | S_{f} | H(m) | w(ms^{−1)} | |||||
---|---|---|---|---|---|---|---|---|

N | (m) | Full Model | Formula (23) | Full Model | Formula (24) | Formula (36) | Formula (40) | Formula (39) |

1 | 100 | 1.000 | 1.00 | 1.00 | 1.00 | (10.64) | (7.8) | |

200 | 0.722 | 0.929 | 5.23 | 11.49 | 5.98 | 5.862 | 5.51 | |

600 | 0.323 | 0.309 | 3.37 | 25.61 | 3.21 | 3.203 | 3.18 | |

1000 | 0.200 | 0.185 | 2.60 | 34.35 | 2.60 | 2.469 | 2.47 | |

1400 | 0.416 | 0.132 | 2.22 | 41.27 | 2.09 | 2.080 | 2.08 | |

2 | 100 | 10.000 | 1.00 | 1.00 | 1.00 | — | (24.65) | |

200 | 4.733 | 9.290 | 18.48 | 25.64 | 23.45 | — | 17.43 | |

400 | 2.900 | 3.620 | 13.50 | 36.39 | 21.13 | 14.62 | 12.32 | |

600 | 1.973 | 3.090 | 8.46 | 39.83 | 10.96 | 10.78 | 10.07 | |

1000 | 1.607 | 1.850 | 7.60 | 45.95 | 8.01 | 7.78 | 7.80 | |

1400 | 1.272 | 1.320 | 6.64 | 51.35 | 6.68 | 6.67 | 6.59 | |

7 | 100 | 10.00 | 18.52 | 1.0 | 1.00 | 1.00 | — | (58.17) |

200 | 6.25 | 9.29 | 28.0 | 32.00 | 30.12 | — | 41.13 | |

400 | 4.00 | 4.63 | 28.5 | 51.00 | 30.52 | 34.49 | 29.10 | |

600 | 3.00 | 3.09 | 25.5 | 64.00 | 25.82 | 25.45 | 23.75 | |

1000 | 2.20 | 1.85 | 21.2 | 84.14 | 18.88 | 18.84 | 18.38 | |

1400 | 1.50 | 1.32 | 16.0 | 100.0 | 15.74 | 15.73 | 15.52 | |

2000 | 1.00 | 0.93 | 14.0 | 120.0 | 13.09 | 13.08 | 13.01 | |

2500 | 0.75 | 0.74 | 12.0 | 134.0 | 11.75 | 11.68 | 11.63 | |

8 | 100 | 10.00 | 18.52 | 1.0 | 1.00 | 1.00 | — | (50.72) |

200 | 4.8 | 9.29 | 21.5 | 25.64 | 21.6 | — | 35.86 | |

400 | 3.5 | 4.63 | 23.5 | 42.10 | 26.60 | 30.07 | 25.36 | |

600 | 2.9 | 3.09 | 21.6 | 53.67 | 22.51 | 22.18 | 20.69 | |

1000 | 2.0 | 1.85 | 17.0 | 71.63 | 16.47 | 16.42 | 16.03 | |

1400 | 1.7 | 1.54 | 14.7 | 78.83 | 14.89 | 14.88 | 14.63 | |

9 | 100 | 10.00 | 18.52 | 1.0 | 1.00 | 1.00 | — | (14.23) |

200 | 4.90 | 9.29 | 13.5 | 25.64 | 18.41 | 12.84 | 10.06 | |

400 | 2.70 | 4.63 | 5.9 | 36.39 | 7.92 | 7.49 | 7.11 | |

600 | 2.25 | 3.09 | 5.3 | 39.83 | 5.95 | 5.94 | 5.81 | |

1000 | 1.65 | 1.85 | 4.8 | 45.95 | 4.54 | 4.54 | 4.50 | |

2000 | 1.50 | 1.54 | 4.2 | 48.70 | 4.14 | 4.14 | 4.11 |

**Table 3.**Parameter values of numerical simulations with the full model for investigating the influence of entrainment on avalanche dynamics ([1], Notes 8 and 24). Figure numbers preceded by ‘S2’ refer to the Supplementary Materials Document 2.

No. | $\mathit{\mu}$ | k | ${\mathit{\tau}}_{*}$ | ${\mathit{p}}_{*}$ | C | $\mathit{\sigma}$ | $\mathit{\rho}$ | ${\mathit{\rho}}_{0}$ | Figure |
---|---|---|---|---|---|---|---|---|---|

— | — | kPa | kPa | — | — | $\mathbf{kg}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{-3}$ | $\mathbf{kg}\phantom{\rule{0.166667em}{0ex}}{\mathbf{m}}^{-3}$ | ||

1 | 0.25 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure 8a |

2 | 0.25 | 0.1 | 4.0 | 4.0 | 1 | 1 | 500 | 300 | Figure 8b |

3 | 0.25 | 0.1 | 4.0 | 4.0 | 0.7 | 1 | 300 | 300 | Figure S2.10.a |

4 | 0.25 | 0.1 | 4.0 | 4.0 | 0.1 | 1 | 300 | 300 | Figure S2.10.b |

5 | 0.25 | 0.1 | 1.5 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.10.c |

6 | 0.25 | 0.1 | 0.5 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.10.d |

7 | 0.25 | 0.1 | 4.0 | 6.5 | 1 | 1 | 300 | 300 | Figure S2.11.a |

8 | 0.25 | 0.1 | 4.0 | 4.5 | 1 | 1 | 300 | 300 | Figure S2.11.b |

9 | 0.25 | 0.1 | 4.0 | 0.5 | 1 | 1 | 300 | 300 | Figure S2.11.c |

10 | 0.25 | 0.1 | 4.0 | 4.0 | 1 | 1.3 | 300 | 300 | Figure S2.12.a |

11 | 0.30 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.12.b |

12 | 0.10 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.12.c |

13 | 0.55 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure 9a |

14 | 0.42 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure 9b |

15 | 0.40 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure 9c |

16 | 0.25 | 0.06 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.13.a |

17 | 0.25 | 0.01 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.13.b |

18 | 0.30 | 0.1 | 4.0 | 4.0 | 1 | 1 | 300 | 300 | Figure S2.14 |

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**MDPI and ACS Style**

Grigorian, S.S.; Ostroumov, A.V. On a Continuum Model for Avalanche Flow and Its Simplified Variants. *Geosciences* **2020**, *10*, 35.
https://doi.org/10.3390/geosciences10010035

**AMA Style**

Grigorian SS, Ostroumov AV. On a Continuum Model for Avalanche Flow and Its Simplified Variants. *Geosciences*. 2020; 10(1):35.
https://doi.org/10.3390/geosciences10010035

**Chicago/Turabian Style**

Grigorian, Samvel S., and Alexander V. Ostroumov. 2020. "On a Continuum Model for Avalanche Flow and Its Simplified Variants" *Geosciences* 10, no. 1: 35.
https://doi.org/10.3390/geosciences10010035