| 25 Feb 2026
Parameterization of the snow fracture energy to model the onset of crack propagation in snowpack models
Abstract. Snowpack models are widely used to complement field observations in avalanche risk forecasting. They help estimate key indicators related to dry-snow slab avalanche triggering processes, such as failure initiation and crack propagation. In recent decades, several models have been developed to predict crack propagation propensity, typically characterized by the critical crack length (the cut length beyond which an initial crack self-propagates), measurable in the field using the Propagation Saw Test (PST). However, these models often depend on poorly constrained parameters, particularly, the weak layer fracture energy wf. In this work, we relate the fracture energy to snow properties that can be measured directly or simulated by detailed snowpack models. To this end, we first exploit microstructural knowledge and data by computing the min-cut on about 300 three-dimensional snow microstructure images. The min-cut represents the smallest ice interface that would need to be fractured to separate two opposing sides of the structure, thus providing a quantitative proxy of fracture energy at the microscale. We fitted a relation between the min-cut and snow properties that can be measured or simulated, namely density and grain morphology. The min-cut is then linearly related to the fracture energy wf using measurements from Richter et al. (2019), which track weak layers across multiple seasons and sites using PST and observed profiles. We retrieved wf by inverting the state-of-the-art slab model WEAC (Weißgraeber and Rosendahl, 2023), based on manual measurements and snowpack simulations from the Crocus model. After calibration, wf is evaluated in two slab models of different complexity for their ability to reproduce observed critical crack lengths, using both observed data and Crocus outputs. Although the correlation between min-cut and wf remains moderate (i.e. R of 0.5 at most), likely due to measurement and modeling uncertainties, the parameterization demonstrate a clear added-value over the use of a constant wf in reproducing realistic critical crack lengths (i.e. R = 0.59 and RMSE of 12.5 cm against R = 0.39 and RMSE = 14.5 cm). Moreover, the proposed parameterization performs well for identifying and monitoring weak layers over the course of the season. Its consistent performance across slab models and strong results using Crocus outputs highlight its potential for operational dry-snow slab avalanche hazard monitoring.
Dear authors,
The topic is highly relevant for avalanche forecasting and aims to bridge microstructural physics with operational snowpack models. In particular, the analysis of the CT dataset is valuable and extensive, contributing to an improved understanding of how microstructural properties relate to macroscopic properties.
However, the manuscript currently suffers from conceptual inconsistencies, unclear physical interpretation, and methodological gaps that limit the strength of its conclusions.
1. Critical energy release rate vs. critical crack length
From a fracture mechanics perspective, it remains unclear what is meant by the term “critical crack length.” One could interpret it as the finite crack size required for the onset of crack propagation within a coupled criterion framework (Leguillon, https://doi.org/10.1016/S0997-7538(01)01184-6), but this is not the case here. Renaming this quantity to “critical cut length” would help avoid this ambiguity.
Furthermore, the manuscript neglects the second condition required for fracture initiation, namely that the stress must exceed the strength of the weak layer (Leguillon). The context of coupled criteria is therefore missing.
“The stability of a snowpack and the critical length above which a crack spontaneously propagates depend on both the fracture energy of the weak layer w_f and on the mechanical behavior of the overlying slab”. What critical length is ment here, the finite length of a coupled criterion which needs to be exceeded (like everyone with a mechanical background reads) or the “critical crack length” of a PST (like snow practitioners reads it) ?
“The fracture energy itself is known to be a function of the snow layer density and microstructure”. This has never been stated in the mentioned reference Adam et al., https://doi.org/10.1038/s41467-024-51491-7
The magnitude of the “critical crack length” derived from a PST does not provide direct information about crack propagation propensity. It is a geometric measure rather than a material property. For the same weak layer, different critical cut lengths can be obtained depending on slope inclination, slab stratification, and cutting direction (see Adam et al., https://doi.org/10.1038/s41467-024-51491-7).
"A small a_c means that the weak layer below the slab is prone to crack propagation.” This statement is incorrect. A small “critical crack length” is not a direct indicator that the weak layer is prone to crack propagation, but a configuration-dependent system response of the slab–weak layer–geometry setup as you partly mention: “The critical crack length ac thus depends on both how the slab deforms when it loses support from the weak layer, and the specific fracture energy of the weak layer “. Same applies for “Finally, we examine the potential of predicted critical crack lengths to discriminate between weak layers and more stable layers within a profile.”
The “critical crack length" has so many geometric and material dependencies which scale differently that its nearly impossible to interpret it. E.g. a longer “crack length” can be related to a stiffer slab due to a crust and even though the weak layer might be potentially dangerous. Vice versa how should be the small “crack length” at lower slab height be interpreted in Figure 9 without knowledge? Those interactions makes the interpretation highly complex, particularly for operational applications or practitioners.
The profiles in Figure 9 mainly reflect the effect of slab stiffness, which causes the “critical crack length” to increase more or less linearly with depth. The observed discontinuities are introduced by the min-cut parameterization associated with different grain types. If the min-cut parameterization were plotted with the corresponding calculated critical energy release rates, this would contain essentially the same information but would be easier to interpret. In this context, the level of “critical cut length” is unnecessary. These values are absolute and configuration-dependent, and therefore not intrinsic material properties. As such, they are not suitable quantities to describe or predict fracture initiation or propagation.
The terminology “crack propagation propensity” is vague. The critical energy release rate describes fracture initiation (or “onset,” as stated in the title). If crack propagation itself is to be addressed, a steady-state framework (e.g., Rosendahl et al., https://doi.org/10.5194/nhess-25-1975-2025) or dynamic fracture processes (e.g., Bergfeld et al., https://doi.org/10.5194/nhess-23-293-2023) must be considered. From a fracture mechanics standpoint, the PST-derived “critical crack length” is not a valid indicator of crack propagation and has historically been misinterpreted in the snow science community.
Since PST measurements are partly conducted on inclined slopes, the relative contributions of mode I and mode II should also be clarified. A brief statement quantifying that the loading is predominantly mode I (e.g., ~95%) and that the analysis is therefore restricted to mode I-driven initiation would improve clarity.
2. Missing physical justification of the min-cut → fracture energy link
No direct experimental validation is provided to support a correlation between min-cut and critical energy release rate.
The argumentation of the parameterization follows an indirect route via the “critical crack length.” The comparison with a constant energy release rate primarily shows that, within the PST dataset, the inferred energy release rate is not constant but varies, likely with density. Since density is also a primary input to the min-cut formulation, the resulting improvement in modelled “critical crack length” does not constitute a robust justification for a physically meaningful parameterization between min-cut and fracture energy.
Rather, this highlights the need for a controlled experimental campaign combining fracture mechanical field tests with direct CT measurements across different grain types.
The correlation on which the parameterization is based remains only moderate.
It is unclear whether the PST dataset alone is suitable to validate such a relationship across different grain types. It should first be demonstrated that grain types can be statistically correlated with fracture energy derived from the PST dataset itself. If this is not possible, the dataset may not be appropriate for validating the proposed parameterization.
In addition, it is unclear whether the PST dataset contains a sufficient diversity of grain types to support such validation.
The assumption of a fixed weak-layer thickness in the PST analysis is not explained, as this information is available in the original dataset. Similarly, the weak-layer elasticity is parameterized via density relationships derived for general snow types rather than specifically for weak layers. It is not stated which parametrisation is chosen (AC, CT, SMP) from Gerling 2017. This would improve clarity. Furthermore, density-based parameterizations reflect an apparent stiffness rather than the theoretical Young’s modulus. Consequently, the experimental method used to derive such parameterizations should be consistent with the actual loading conditions, particularly with respect to strain rate and time-dependent behavior. For example, formulations such as Gerling (2017) may yield Young’s modulus values several times higher than the effective modulus relevant for PST-type configurations (see Schöttner et al., 2025, Fig. 13a). In addition, more appropriate parameterizations exist for specific weak layer grain types (e.g., Schöttner et al., 2025). A clear separation between the derivation of weak layer and slab elastic moduli is necessary.
Since the weak-layer Young’s modulus strongly influences the calculated energy release rate (see sensitivity study Adam et al., https://doi.org/10.1038/s41467-024-51491-7), it is important to assess whether the observed correlation between min-cut and fracture energy isn’t in fact primarily a density correlation. Given the multiple sources of uncertainty in deriving fracture energy from PSTs, this should be explicitly tested. For example, a direct comparison between weak-layer density and min-cut may already yield similar or even stronger correlations.
The manuscript itself acknowledges that density is the main variable controlling min-cut and that measured densities exhibit strong scatter: ” Since the density of the weak layer is the main variable used to estimate the min-cut in the parameterization Eq. 5,” and “we assume that the min-cut can be related to a macroscopic quantity, in particular the snow density.” A sensitivity analysis quantifying this effect would therefore be highly valuable.
While the min-cut may indeed be a promising proxy for fracture energy, the current approach does not convincingly demonstrate this. It would be more straightforward to directly test correlations between fracture energy and other available microstructural descriptors (e.g., from CROCUS or grain-type parameterizations) rather than introducing an additional layer of assumptions via min-cut.
3. Further comments
The general approach of estimating the resistance of each snow layer against crack propagation using snowpack models such as Crocus, which resolve microstructural properties, is very promising.
However:
a) The methodological justification of the min-cut parameterization for fracture energy is currently insufficient.
b) The use of the geometrical measure “critical crack length” introduces conceptual ambiguity and reinforces a historically misinterpreted quantity in the snow science community.
A more appropriate framework would directly assess, for each layer, how the available energy release compares to the intrinsic fracture toughness of that layer. This ultimately calls for a physically consistent approach in which fracture toughness is determined through controlled fracture experiments across different grain types combined with direct CT measurements.