The failure of a weak snow layer buried below cohesive slab layers is a necessary, but insufficient, condition for the release of a dry-snow slab avalanche. The size of the crack in the weak layer must also exceed a critical length to propagate across a slope. In contrast to pioneering shear-based approaches, recent developments account for weak layer collapse and allow for better explaining typical observations of remote triggering from low-angle terrain. However, these new models predict a critical length for crack propagation that is almost independent of slope angle, a rather surprising and counterintuitive result. Based on discrete element simulations we propose a new analytical expression for the critical crack length. This new model reconciles past approaches by considering for the first time the complex interplay between slab elasticity and the mechanical behavior of the weak layer including its structural collapse. The crack begins to propagate when the stress induced by slab loading and deformation at the crack tip exceeds the limit given by the failure envelope of the weak layer. The model can reproduce crack propagation on low-angle terrain and the decrease in critical length with increasing slope angle as modeled in numerical experiments. The good agreement of our new model with extensive field data and the ease of implementation in the snow cover model SNOWPACK opens a promising prospect for improving avalanche forecasting.

Snow slab avalanches range among the most prominent natural hazards in snow-covered mountainous regions throughout the world. The winter 2014/15 served as a cruel reminder of the destructive power of this ubiquitous natural hazard with 132 fatalities, just for the European Alps. The ability to reliably forecast avalanche danger is therefore of vital importance and requires a sound understanding of avalanche release processes.

Avalanches are the result of numerous factors and processes interacting over
a large range of temporal and spatial scales

^{©}Grant Gunderson).

Information on snow cover stratigraphy, especially the presence and
characteristics of WLs and the overlying slab, is thus essential for
avalanche forecasting. Traditionally, such information is obtained through
manual snow cover observations, such as snow profiles and stability tests

Due to the very complex nature of crack propagation in multilayered elastic
systems under mixed-mode loading, theoretical and analytical approaches are
not yet conceivable

To evaluate the critical crack length based on the anticrack model, the
WL specific fracture energy is required. Presently, it can be estimated using
three existing methods: (i) through PTV or finite element analysis of the PST

Clearly, the various methods to estimate the critical crack length all have
their respective shortcomings, and a unified approach which incorporates all
relevant processes is thus far missing. To overcome these limitations and
take into account all the important physical ingredients, we propose to
evaluate the critical crack length for different snowpack stratigraphies
using discrete element simulations. Similar to the field experiments, in the simulations we
gradually create a crack in the WL with a saw until rapid propagation occurs
(Fig.

Successive snapshots

We model crack propagation in
a slab–WL system using the discrete element method (DEM). DEM is well suited to represent large
deformations as well as the evolution of the microstructure of materials in a
dynamic context

The simulated system (Fig. 2a) is 2-D and composed of a fixed substratum, a
WL of thickness

We used the cohesive contact law detailed in

The applied loading represents a typical experimental setup of a PST

Failure criterion FC

Critical length

The data set consists of 93 PST experiments which were presented in

The Young's modulus of the slab

In the simulations, the crack of length

We performed a series of systematic simulations to investigate the influence
of snow cover parameters on

The discrete element simulations revealed that the maximum shear stress at
the crack tip can be decomposed into two terms related to slab tension
(

The tension term alone is unable to predict stress concentrations and thus
crack propagation on flat terrain (

Ratio between the shear strength

From Eq. (

The agreement between Eq. (

Comparison between measured and modeled critical crack lengths using
the anticrack model

The predictions of Eq. (

We compare how well our new analytical expression (Eq.

The decrease of

For low-angle terrain, the anticrack model and our new formulation yield
similar results. However, this is where the similarities end. Indeed, overall
the anticrack model overestimates

By comparing the anticrack model to the 93 PST measurements
(Fig.

We showed that the critical crack length

Effect of the slab elastic modulus on the slope angle dependency of
the critical crack length (Eq.

However, if slab depth

Finally, geometrical effects significantly influence how the critical crack
length depends on slope angle. Figure

Performing DEM simulations allowed us to investigate crack propagation in
weak snow layers without relying on the same strong assumptions concerning
the weak layer as previous research

In a recent study,

The main limitation of our model is the uniform character of the slab. In
this paper, the multilayered character of the slab was not accounted for, for
clarity reasons since the phenomenon is already very complex. However, the
elastic moduli of the slab layers have a very important influence on slab
deformation and thus on the critical crack length

Concerning the weak layer, the schematic microstructure considered in this
study is sufficient to capture the main features of the failure envelope

Another important aspect is the relevance of our new model with regards to
slab avalanche release. We showed that our model was able to reproduce crack
propagation at the scale of the PST. However, at the slope scale, 3-D
effects, slope-transverse propagation, terrain, and snowpack variability

The snow cover model SNOWPACK

On 3 March 2015 we performed several PSTs on three WLs at the location of the
automatic weather station. The SNOWPACK simulation for that specific day
clearly shows local minima in the calculated critical crack length for these
three WLs (Fig.

We proposed a new analytical expression to assess the conditions
for the onset of crack propagation in weak snowpack layers. The formulation
was developed based on discrete element simulations; it accounts for crucial
physical processes involved in crack propagation in snow, namely the complex
mechanical behavior of the WL and the mixed stress states in the slab
induced by slab tension and bending resulting from WL collapse. A critical
parameter in the formulation is the length scale

The analytical expression for the critical crack length reproduced field data
obtained with 93 PST experiments. In contrast, the anticrack
model

Finally, our new expression was implemented in the snow cover model SNOWPACK to evaluate the critical crack length for all snow layers throughout the entire season. While validation is still required, this opens promising perspectives to improve avalanche forecasting by combining traditional stability indices with a new metric to evaluate crack propagation propensity.

The critical crack length model is implemented in the SNOWPACK model, which
is available under the GNU Lesser General Public Licence Version 3 and can be
retrieved at

The authors declare that they have no conflict of interest.

We are grateful to all SLF colleagues who assisted in field data collection. We thank Benjamin Reuter for the SMP-derived specific fracture energy data and for insightful discussions and comments on the paper. We acknowledge the constructive comments of two anonymous reviewers as well as Ned Bair who helped us to improve our paper. Johan Gaume has been supported by the Ambizione grant of the Swiss National Science Foundation (PZ00P2_161329). Edited by: E. Larour Reviewed by: E. H. Bair and two anonymous referees