Snow avalanches cause fatalities and economic damage. Key to their mitigation is the understanding of snow avalanche dynamics. This study investigates the dynamic behavior of snow avalanches, using the material point method (MPM) and an elastoplastic constitutive law for porous cohesive materials. By virtue of the hybrid Eulerian–Lagrangian nature of the MPM, we can handle processes involving large deformations, collisions and fractures. Meanwhile, the elastoplastic model enables us to capture the mixed-mode failure of snow, including tensile, shear and compressive failure. Using the proposed numerical approach, distinct behaviors of snow avalanches, from fluid-like to solid-like, are examined with varied snow mechanical properties. In particular, four flow regimes reported from real observations are identified, namely, cold dense, warm shear, warm plug and sliding slab regimes. Moreover, notable surges and roll waves are observed peculiarly for flows in transition from cold dense to warm shear regimes. Each of the flow regimes shows unique flow characteristics in terms of the evolution of the avalanche front, the free-surface shape, and the vertical velocity profile. We further explore the influence of slope geometry on the behavior of snow avalanches, including the effect of slope angle and path length on the maximum flow velocity, the runout angle and the deposit height. Unified trends are obtained between the normalized maximum flow velocity and the scaled runout angle as well as the scaled deposit height, reflecting analogous rules with different geometry conditions of the slope. It is found that the maximum flow velocity is mainly controlled by the friction between the bed and the flow, the geometry of the slope, and the snow properties. We reveal the crucial effect of both flow and deposition behaviors on the runout angle. Furthermore, our MPM modeling is calibrated and tested with simulations of real snow avalanches. The evolution of the avalanche front position and velocity from the MPM modeling shows reasonable agreement with the measurement data from the literature. The MPM approach serves as a novel and promising tool to offer systematic and quantitative analysis for mitigation of gravitational hazards like snow avalanches.

Snow avalanches have long been threatening infrastructures and human
lives. Buildings, roads and railways can be severely damaged, causing
profound economic losses. Moreover, the fatalities induced by snow
avalanches are significant and are about 100 annually in the
European Alps during the last 4 decades

Snow can behave as a fluid or as a solid under different conditions,
leading to distinct behaviors of snow avalanches in reality

Popular classical numerical tools for modeling snow avalanches
primarily apply two-dimensional (2D) depth-averaged methods based on
shallow-water theory

This study applies the MPM in 2D (slope parallel and slope normal) to
explore the distinct behaviors of snow avalanches and the key
controlling factors of snow avalanche dynamics. To facilitate
efficient computation and capture important flow features along the
surface-normal direction, our 2D MPM modeling neglects variations
along the flow width. The MPM is a hybrid Eulerian–Lagrangian approach,
which uses Lagrangian particles to track mass, momentum and
deformation gradient, and adopts an Eulerian background mesh to solve and
update the motion of the particles. By virtue of the hybrid
Eulerian–Lagrangian nature of the MPM, processes with large deformations,
fractures, collisions and impacts can be well simulated

Assuming a continuous material, the MPM discretizes it into Lagrangian
particles (material points) to trace mass, momentum and deformation
gradient and adopts Eulerian grids to solve the motion of the
particles and update their states. In particular, the particle motion
is governed by mass and momentum conservation as follows:

The MPM relies on a continuum description and requires a constitutive
model for the considered material. The Cauchy stress

The elastoplastic model in this study is borrowed from

When the

A flow rule needs to be adopted when plastic behavior
occurs. Referring to

Model setup for MPM modeling of snow avalanches on an ideal slope.

Parameters adopted in the MPM simulations of snow avalanches on ideal slopes.

To examine the behavior of snow avalanches under well-controlled
conditions, the setup with a rectangular snow sample and an ideal slope
is adopted as illustrated in Fig.

In each group of our MPM simulations, four typical flow regimes are
captured with the changing mechanical properties of
snow. Figure

The four flow regimes in Fig.

Snow properties adopted in the MPM modeling of the flows in the four typical flow regimes.

All the demonstrated flows in the four typical regimes share identical
initial and boundary conditions except for the snow properties. From
the simulations, it is clear that higher

Figure

Evolution of

When the flows are on the slope (

Evolution of free-surface shape

Figure

Figures

Evolution of the maximum velocity with the normalized deposit height for varying slope angles

Evolution of the maximum velocity with the normalized deposit height for different horizontal lengths

Unified relation between the scaled maximum velocity and the scaled deposit height.

Slope angle is a key factor in evaluating the trigger, flow and
deposition of snow avalanches

The normalization of

The runout angle

Evolution of the maximum velocity with

Evolution of the maximum velocity with

From Figs.

Unified relation between the scaled maximum velocity and the scaled

Figure 10 demonstrates a unified trend with the dimensionless velocity

To testify the capability of the MPM modeling in capturing key dynamic
features (i.e., front velocity and position) of snow avalanches, five
reported real avalanches with different complex terrains are
simulated. All the cases are modeled in 2D, neglecting the variation
along the flow width direction. The adopted geometry of the terrains
is borrowed from the corresponding literatures. As no detailed snow properties of
the avalanches were measured and reported, the applied snow properties
in the MPM refer to the description of the snow type and snow
condition. In particular, three of the avalanches mainly consisted of
dry, loose and new snow, while the other two were chiefly composed of
wind-packed and settled old snow. Correspondingly, two groups of snow
properties are adopted as summarized in
Table

Adopted parameters in the five MPM simulations of snow avalanches on real terrains.

Case I and II in Figs.

Front velocity distribution along the flow path for Case I:
Weissfluh north ridge, 12 March 1982, a1 (Davos, Switzerland). Drop height

Front velocity distribution along the flow path for Case II:
Weissfluh north ridge, 12 March 1982, a2 (Davos, Switzerland). Drop height

Figure

Front velocity distribution along the flow path for Case III:
Himmelegg, 14 February 1990 (Arlberg, Austria). Drop height

Front velocity distribution along the flow path for Case IV:
Ryggfonn, 2 May 2006 (Stryn, Norway). Drop height

Front velocity distribution along the flow path for Case V:
Vallée de la Sion (VdlS), 31 January 2003 (Sion, Switzerland). Drop height

Case IV in Fig.

Figure

This study explores the dynamics of snow avalanches with the material
point method (MPM) and an elastoplastic constitutive model. By virtue
of the capability of the MPM in simulating processes with large
deformations, fractures and collisions and coexistence of solid- and
fluid-like behaviors, a wide range of distinct snow avalanches with
diverse flow behaviors have been investigated. The reported four flow
regimes for dense snow avalanches from real observations have all been
captured from our MPM simulations, including cold shear, warm shear,
warm plug and slab sliding regimes. Moreover, in transition from cold
shear to warm shear flow regimes, flows with surges and small granules
are observed. The evolution of the avalanche front, the free-surface
shape and the vertical velocity profile shows distinct
characteristics for the different flow regimes, underpinning the
identification of flow regimes. In addition to the flow surface and the
shear behavior presented in this study, other features of the flow may
also be used to pinpoint the flow regimes, such as snow temperature
and liquid water content

We have systematically examined the effects of snow properties, slope
angle and path length on the flow and deposition behaviors of snow
avalanches, including the maximum flow velocity on the slope, the
runout angle and the avalanche deposit height. It is found that snow
friction and cohesion are closely related to the behavior of snow
avalanches. Low snow friction and cohesion give fluid-like behavior
and highly sheared flows, while high snow friction and cohesion lead
to solid-like flow with limited shearing. Both slope angle and path
length have a positive correlation with the maximum flow velocity on
the slope, while their effects on the deposit height are
trivial. Furthermore, unified trends have been obtained with
normalization of the maximum flow velocity, the deposit height and the
runout angle, revealing analogous physical rules under the different
conditions. Key controlling factors of

The MPM modeling has been calibrated and tested through simulations of real snow avalanches on irregular terrains. The calculated avalanche front position and velocity from the MPM show reasonable agreement with the measurement data from the literature. The behavior of dense snow avalanches has been well recovered by the MPM. A discrepancy was observed particularly for avalanches which developed a powder cloud above the dense core, as the powder cloud has not been modeled here.

The presented research focuses on examining the flow regimes and flow
dynamics of snow avalanches with idealized conditions, which is a
preliminary study serving as the basis for investigating more
realistic and complex snow avalanches. The 2D ideal slope with a
constant inclination could be further changed to other shapes to be
more realistic, such as a parabolic track. Although the 2D setups were
used to efficiently conduct the systematic study including more than
1000 cases, it is fully possible to explore interesting cases with 3D
MPM simulations

The constitutive model adopted in this study perfectly satisfies the
second law of thermodynamics. Following the derivation in

Evolution of potential and kinetic energy of the flows in the
four typical flow regimes.

The evolution of kinetic and potential energy of the flows in the four
typical flow regimes (i.e., cold dense, warm shear, warm plug, sliding
slab) is demonstrated in Fig.

Figure

Evolution of dissipated energy of the flows in the four typical flow regimes.

Energy dissipation inside the flow and through the boundary bed in the flows with the different flow regimes.

The energy dissipation is contributed to by (1) the internal force of the
material and (2) the external force on the material from the
boundary slope. As illustrated in Fig.

All the relevant data are available on Zenodo at

Videos of the avalanches presented in Fig. 2 are available on Zenodo at

The supplement related to this article is available online at:

JG designed the study and obtained the funding. XL conducted the study, performed the simulations and wrote the paper under the supervision of JG. BS commented on the paper and provided guidance on the flow regime transitions. CJ developed the MPM tool, which has been used in this study.

The authors declare that they have no conflict of interests.

The first author acknowledges Lars Blatny for his support with the language editing of this paper.

This research has been supported by the Swiss National Science Foundation (grant no. PCEFP2_181227).

This paper was edited by Florent Dominé and reviewed by Frédéric Dufour and two anonymous referees.