Overcoming the stauchwall: Viscoelastic stress redistribution and the start of full-depth gliding snow avalanches

Authors


Abstract

[1] When a full-depth tensile crack opens in the mountain snowcover, internal forces are transferred from the fracture crown to the stauchwall. The stauchwall is located at the lower limit of a gliding zone and must carry the weight of the snowcover. The stauchwall can fail, leading to full-depth snow avalanches, or, it can withstand the stress redistribution. The snowcover often finds a new static equilibrium, despite the initial crack. We present a model describing how the snowcover reacts to the sudden transfer of the forces from the crown to the stauchwall. Our goal is to find the conditions for failure and the start of full-depth avalanches. The model balances the inertial forces of the gliding snowcover with the viscoelastic response of the stauchwall. We compute stresses, strain-rates and deformations during the stress redistribution and show that a new equilibrium state is not found directly, but depends on the viscoelastic properties of the snow, which are density and temperature dependent. During the stress redistribution the stauchwall encounters stresses and strain-rates that can be much higher than at the final equilibrium state. Because of the excess strain-rates, the stauchwall can fail in brittle compression before reaching the new equilibrium. Snow viscosity and the length of the gliding snow region are the two critical parameters governing the transition from stable snowpack gliding to avalanche flow. The model reveals why the formation of gliding snow avalanches is height invariant and how technical measures to prevent snowpack glide can be optimized to improve avalanche mitigation.

1. Introduction

[2] Full-depth cracks can form in the tensile stress regions of the mountain snowcover (Figure 1). The cracks initiate when the snowcover glides on terrain with reduced basal friction, such as grassy slopes or smooth rock surfaces [McClung, 1981; Lackinger, 1987; Conway, 1998; Clarke and McClung, 1999; McClung and Schaerer, 2006]. The gliding zones can also be created when liquid water (meltwater, rain) accumulates at the snow-ground interface [McClung and Clarke, 1987; Conway and Raymond, 1993; Stimberis and Rubin, 2011]. The fracture widens as the gliding snowcover pulls the crack apart, exposing the ground. The cracks are an indication of the loss of static equilibrium: full-depth glide avalanches can release immediately when the snowcover opens or delayed action avalanches start hours, even days, after the initial break [Lackinger, 1987; Conway and Raymond, 1993]. Glide avalanches are a concern for operational avalanche forecasting because they are difficult to predict and endanger highways, railroads, housing and ski runs [Stimberis and Rubin, 2011; Peitzsch et al., 2012].

Figure 1.

(left) A tensile crack in the snowcover near Davos, Switzerland (Photo. R. Meister, 2012, SLF). The gliding zone and stauchwall are visible. The cracks are termed “fishmouths” because of their downward looking opening. (right) A gliding snow avalanche near Crans Montana, Canton Wallis, Switzerland (Photo. F. Meyer, 2008). A long tensile crack opened along the mountain crest. The stauchwall on the left withstood the stress redistribution while the stauchwall on the right failed, initiating a gliding snow avalanche. In both cases the stable snowcover in the gliding zone has developed undulations: these folds are a precursor to eventual failure by buckling.

[3] Holding the sliding mass in place is a stable snowpack zone fixed to the ground: the stauchwall (Figure 1) (in German, the verb “stauchen” signifies a compressive loading with deformation, hence the name in snow science, “stauchwall”). Mechanically, the sudden opening of a tensile crack will cause an immediate redistribution of the internal forces within the snowcover; forces are transferred from the crown to a compression zone [Lackinger, 1987; Conway and Raymond, 1993]. The gliding snowcover is then free to accelerate by gravity and depending on the basal friction of the gliding zone and the strength of the stauchwall, either a new static equilibrium is found, or the stauchwall fails in compression (Figure 2). The stauchwall is an indicator of material failure. The interplay between gliding and non-gliding snowpack zones plays a fundamental role in understanding the start of gliding snow avalanches. The important role of the stauchwall is often observed when these compressive, non-gliding zones are removed by natural (avalanches) or technical means (explosives, snow clearing along steep embankments), releasing secondary avalanches and snow slides.

Figure 2.

A rigid avalanche slab of height h and length l releases spontaneously on a slope of angle θ at time t = 0. The slab accelerates with inline image and deforms the stauchwall over the length ls. The stauchwall deforms elastically (reversible) and viscously (irreversible) and will break if the strain rate inline image. We consider two cases: (a) a detachment zone with constant sliding friction μ; (b) the detachment zone consists of a length l0 with no friction and a length lμ with friction.

[4] In this paper we investigate under what conditions the stauchwall fails, leading to spontaneous, full-depth gliding snow avalanches. This problem can be reversely stated: we find the snowpack conditions under which the stauchwall can successfully accommodate the lost tensile force at the crown to find a new stable (however tenuous) equilibrium state. That is, under what conditions can a tensile crack open and there is no avalanche. We assume that the gliding friction remains constant and the lost resistance must be taken up entirely by the stauchwall. This assumption is justified by the fact that smooth gliding zones are required to create tensile zones that can fail: it is unlikely that these gliding zones can take up the lost tensile force. An interesting physical interaction results: the gliding snowcover accelerates, but the acceleration is controlled by the viscoelastic resistance of the stauchwall, which, in turn, depends on the magnitude of the acceleration. For a particular density and temperature, the viscosity of snow controls the force on the stauchwall and therefore if the stauchwall fails. We mathematically express the feedback between inertial forces and viscoelastic material response with a system of two ordinary, but strongly coupled, differential equations. One equation describes the motion of the slab; the other the viscoelastic response of the compression zone. The movement is defined by the compressive stressσ and the associated strain rate inline image at the stauchwall. The ( inline image) phase space of this system is of interest, since it defines not only the new equilibrium, but also how the system reaches this equilibrium and, more importantly, whether the path to equilibrium transgresses the compressive failure criteria of snow.

2. Stauchwall Model

[5] We model the mechanical system depicted in Figure 2a: A compact snow slab of height h and length l begins to move as the tensile region (the crown) releases. At initiation we consider the slab to be a block moving as a rigid body. Coulomb friction at the ground (parameter μ) prevents the slab from accelerating fully. At the lower end of the slab, the stauchwall experiences an axial compression and therefore an increase in stress σ. Denoting the downslope velocity of the slab u and the acceleration inline image, the momentum balance of the slab is:

display math

where m is the total mass of the slab with unit width, m = ρhl, with ρthe density. For simplicity, we assume a homogenous density from the bottom to top of the snowcover. This is a realistic assumption especially for full-depth, wet snow gliding snowpacks [McClung and Schaerer, 2006]. The angle of the slope θ determines the slope parallel component of the gravitational acceleration gx = g sin(θ), with g the gravitational acceleration. The slope normal component gz = g cos(θ) defines the normal stress N acting on the ground N = ρgzh and together with the gliding Coulomb coefficient μ the frictional shear stress at the gliding surface S = μN. Theoretically, the tensile force that is no longer carried by the crown could be taken up by an increase in frictional force at the ground, retaining the slab in static equilibrium. In this case, the lost tensile force is balanced by an increase in basal shear stress. We assume this does not occur; the gliding friction coefficient is constant and therefore the tensile force results in an increase in stress σ at the stauchwall. For this to happen, the friction on the ground must vary between two regions: (1) the detachment region l where the sliding friction is lower than the tangent of the slope angle and (2) the stauchwall, which is fixed rigidly to the ground. The difference in surface friction could be caused by a variation in surface roughness; for example, changes in surface properties e. g. small scale terrain undulations [Conway, 1998], meltwater accumulations [McClung and Clarke, 1987], vegetation or natural or man-made obstacles. Mathematically, these changes in surface properties divide the snowcover into the gliding region with lengthl and the stauchwall region with length ls. The length ls is the distance the snowcover requires to transfer the excess force to the ground. Observations indicate that this region is typically several meters long. After the release of a slab, the overrun remnants of a stauchwall are often visible as snowcover patches that remain fixed to the ground (Figure 1).

[6] A slight variation of this failure scenario is to assume that there is a snowcover region of length l0 with no friction μ = 0 (Figure 2b). We imagine this region to be a zone of meltwater accumulation. A tensile crack opens and this unstable snowcover exerts a slope parallel pressure σ0 on the snowcover below:

display math

However, the snowcover has some sliding friction μ ≠ 0, and, before the tensile crack opens, is in equilibrium. It has length lu. It may have some residual strength τmax,

display math

which depends on the sliding friction μ. Thus, this zone can support the additional load. Note that μ > gx/gy = tan(θ), since the region lμ was in static equilibrium before the crack opened. The magnitude of the reserve strength depends on the length of the gliding zone lμ. When τmax = σ0, a critical situation is attained: the initial snowcover region with length l0 can overcome the basal shear resistance of the zone with length lμ and this zone becomes activated; that is, it is also no longer in static equilibrium, and additionally stresses the stauchwall. This occurs when

display math

These considerations help determine the length of the detachment zone and reveal why even small unstable zones can lead to avalanching on slippery slopes. In this failure scenario, the length l of the snowcover acting on the stauchwall is now the sum of the two regions, the initial region l0 and the failure of the gliding interface lμ

display math

This procedure can be repeated with more than one zone with length lμ until the stauchwall is reached. However, for avalanches to occur, the stauchwall must fail. There must be a material failure of the stauchwall, not only a sliding failure at the basal interface. This is why, in the end, we must also consider the snow properties of the stauchwall. The procedure outlined above helps to understand the role of the frictional variability at the ground snowcover interface.

[7] Snow is a viscoelastic material (see overview works of Mellor [1974], Voytkovskiy [1977], and Salm [1982], or more recent investigations of von Moos [2001], von Moos et al. [2003], Scapozza and Bartelt [2003], and Scapozza [2004]). The total compressive strain rate inline image can therefore be split into elastic (reversible) and viscous (irreversible) parts. Triaxial tests with snow show that at strain rates inline image ≈ 10−2, snow will fail in brittle compression [Scapozza and Bartelt, 2003; Scapozza, 2004]. We compute the relationship between strain rate inline image and stress σ below this failure limit using a Burger's viscoelastic model [Mellor, 1974; Salm, 1982]. This general model was first proposed by Salm [1974] to model the viscoelastic response of snow under stress loading and later applied by von Moos [2001]to model the viscoelastic behavior of snow under different strain rates in triaxial tests. Burger's model divides the stress response of snow into Maxwell and Kelvin spring (elastic) - dashpot (viscous) elements in series. The governing differential equation relating stressσ and strain rate inline image is [Mellor, 1974]:

display math

which contains the four material constants Em, Ek, ηm and ηk, the Maxwell (subscript m) and Kelvin (subscript k) elasticity and viscosity, respectively. Values for different snow densities can be found in von Moos et al. [2003]. At the instant of release the slab is motionless, but as it displaces, it axially deforms the stauchwall. The axial deformation of the stauchwall and the motion of the rigid slab must fulfill the continuity constraint: inline image and inline image. This relation couples the gliding velocity u with the axial deformation of the stauchwall. Also note that the second derivative of the strain is related to the slab acceleration, indicating that the inertial forces have a corresponding constitutive response. Equation 6 becomes

display math

This is a trivial substitution but it reveals the coupling between the acceleration inline image and the viscoelastic response of the stauchwall during avalanche release. The stauchwall model consists of two coupled ordinary differential equations (equations (1) and (7)) with primary unknowns slab velocity u(t) and stauchwall stress σ(t). As a weak basal interface is necessary to initiate the tensile failure, the slab may have an initial velocity that causes the initial fissure. We consider t = 0 to be the time the fissure has completely developed. In the following, we solve the two equations numerically with the boundary conditions: u(t = 0) = 0 and inline image. Of interest is the presence of both the inertial forces inline image and therefore the rate of change of strain rate inline image. These terms are typically neglected in snowpack stability investigations [Bader and Salm, 1990; Stoffel and Bartelt, 2003].

3. Results

[8] A mathematical feature of the coupled solution of equations (1) and (7) is the existence of a steady equilibrium point (ϵsσs), see Figure 3. This point represents the refound equilibrium of the snowcover. A stability analysis (and the numerical solutions) reveal that this point is a focus point: all solution trajectories approach the new equilibrium in a counter clockwise spiral (Figures 3 and 4). This first result indicates two salient features of the viscoelastic stress redistribution and the stauchwall:

Figure 3.

(a) Influence of stauchwall density on strain rate. The larger the density the smaller the strain rate and fewer stress oscillations are required to find equilibrium. (b) Strain rate as a function of time. (c) Influence of gliding friction μ on strain rate. Without gliding friction μ = 0, the maximum strain rate increases significantly. There is little difference in strain rate between the μ = 0.2 and μ = 0.4 cases. The example calculations model a l = 20 m slab on a θ = 35° slope. The stauchwall length is ls = 2 m. The viscoelastic model parameters are for ρ = 210 kg/m3 snow: Em = 1.0e8 Pa, Ek = 2.0e6 Pa, ηm = 1.5e9 Pa s, ηk = 1.0e6 Pa s; for ρ = 250 kg/m3 snow: Em = 1.5e8 Pa, Ek = 1.5e7 Pa, ηm = 1.4e9 Pa s, ηk = 2.5e6 Pa s.

Figure 4.

(a) Influence of slope angle on critical strain rate, l = 30 m, ls = 2 m, ρ = 250 kg/m3. (b) Influence of slab length on critical strain rate, θ = 35°, ls = 2 m. (c) Strain rate as a function of time for different detachment zone lengths.

[9] 1. The new snowcover equilibrium is not found immediately, but requires several stress and strain rate oscillations with decreasing amplitude. The counter clockwise spiral stipulates that the stauchwall will experience strain rates and stresses greater than the final, refound equilibrium. If large enough, these excess strain rates can cause brittle, compressive failure of the stauchwall.

[10] 2. For a wide range of snow densities (150 kg/m3ρ ≤ 400 kg/m3), the new snowcover equilibrium is found relatively slowly, compared to the time required for a stress wave to reach the stauchwall, which depends on the speed of sound in the snow-ice matrix. When the tensile crack opens at the crown, a tensile pressure wave will be transmitted to the stauchwall at an approximate speed of 1000 m/s (ice). For a slab length ofl = 20 m, 0.02 s are required to propagate the disturbance to (and past) the stauchwall whereas the new visco-elastic snowcover equilibrium will be reached within 0.5 s to 1.0 s.

[11] In general a stauchwall zone with mean lower density (ρ ≤ 250 kg/m3) will experience larger deformation rates and therefore are “weaker”, in the sense that they will be loaded more closely to the critical brittle strain rate, approximately10−2 1/s (Figure 3a). Such critical strain rates were found by [Scapozza, 2004] in triaxial tests for a wide range of snow densities. The value depends on the magnitude of the sliding friction coefficient μ (Figure 3c). Varying the sliding friction coefficient reveals another significant feature of the relationship between strain rate and ground conditions: even detachment regions with high sliding friction can induce deformation rates near the critical strain rate. For example, in Figure 3c there is no difference between the maximum calculated strain rate for the μ = 0.2 and μ= 0.4 cases. This result suggests that gliding snow avalanches can form even on rough gliding surfaces. The two most important components are the strength (density) of the stauchwall and the length of the detachment zone. Full-depth, gliding snow avalanches are therefore best mitigated first by hindering the creation of a gap (by increasing the friction of the gliding zone, parameterμ), but if this is not possible, by reducing the length l of the gliding zone, creating “artificial” stauchwalls (berms). Another result concerns the absence of the snow cover height in the model equations. This occurs because of our assumption of depth invariance in the x-direction; the height of the stauchwall is the same as the snowcover height in the detachment zone. If this is the case, our results indicate that gliding snow avalanches are mechanically height invariant: the response of the stauchwall to the stress redistribution does not depend on the height, rather the frictional properties of the substrate. Gliding snow avalanches can occur for all snowcover heights. Of course, the damage potential very much depends onh. The likelihood of an avalanche releasing depends on the length l of the detachment zone and the friction μof this zone. Once these are given, the start of full-depth avalanches depend on the properties of the stauchwall (ls and ρ).

4. Discussion and Conclusions

[12] The problem of stauchwall failure is only one part of the more general problem of avalanche release. Previous studies of snowcover stability that included viscous deformations determined the increase in strain rate, caused by stress concentrations at the ends of weak snowpack layers [Bader and Salm, 1990; Stoffel and Bartelt, 2003]. The approach considered herein deviates from this analysis as inertial forces of an unstable snowcover are considered. We do not begin with a statically stable snowcover and seek the mechanical conditions for failure. Rather, we begin with an unstable snowcover with glide crack and seek the role of the stauchwall in preventing avalanches from starting. This is not a static creep analysis but a question of the dynamic stability of a visco-elastic system subjected to an initial perturbation.

[13] We find that the material properties and length of the stauchwall play a decisive role in preventing gliding avalanches after the glide crack initially opens. The stauchwall can withstand the viscoelastic transfer of stress from the crown. However, the mechanical response of the stauchwall is indirect; a new equilibrium is obtained after several viscoelastic stress strain-rate oscillations, which leads to a time delay. Mathematically, the new equilibrium is a spiral shaped attractor, indicating that high strain rates are encountered on the way to equilibrium, which may lead to brittle, compressive failure. The model results indicate that the critical material parameter is the Maxwell viscosity. This viscosity increases (exponentially) with density and decreases with increasing temperature [Scapozza and Bartelt, 2003]. The occurrence of gliding snow avalanches, especially during warming periods, highlights the important role of understanding the snow viscosity which governs the competition between the density increase (snowcover settlement and smaller viscous deformation rates) and the temperature rise (higher viscous deformation rates). This model result is in agreement with gliding avalanche observations which reveal that air temperature [Peitzsch et al., 2012] and the shear resistance between the snow and ground [Lackinger, 1987] appear to be the most important variables in full-depth avalanche occurrence. It should be possible in the near future to develop a simple mechanical model capturing the mechanical result of this competition. However, the primary unknown in the analysis remains the length of the gliding zone and the properties of the ground-snow interface [Conway and Raymond, 1993]. Therefore, an operational forecasting model is probably not feasible in the near future.

[14] Avalanche safety experts are often confronted with the problem of how to mitigate the danger of a spontaneous avalanche release once a tensile crack has opened. Attempts to use explosives placed near the crack opening often do not have the desired effect of forcing the release of the slab. Our analysis reveals that blasting the stauchwall might be more effective. The bulging of the slab makes it easy to identify the stauchwall in clear weather. The effectiveness of this approach will depend on the frictional property of the glide surface, which changes during the course of the day. Mobilizing the mass of the slab at the right moment (when the basal friction is lowest) in combination with the removal of the stauchwall might mitigate the problem slide. However, the stauchwall is often located near ski-lift pylons, roads and other infrastructure which makes the application of explosives impossible. It is certainly not advisable to dig out the stauchwall from below (or just above the road).

[15] We emphasize that we have not solved the follow-up problem of creep failure. Once the stauchwall has reached a new equilibrium the snowcover continues to glide and deform. The snow slab upslope the stauchwall buckles and folds (Figure 1), indicating considerable gliding and storage of gravitational work into deformation energy. The secondary equilibrium we describe must not be stable, and that raises some questions. First, how long will it be stable before it fails? Secondly, and more specifically, how wide and how fast can the tensile crack grow before failure? This secondary failure process is governed by different aspects of snow mechanics. During the spontaneous failure of the stauchwall and the viscoelastic stress redistribution, we need not to consider the evolution of temperature or anelastic (long term time-dependent) stresses. The inertial forces determine the deformation rates and the immediate response of the stauchwall. The secondary, time delayed failure is under constant stress (assuming the weight of the slab does not change, no inertial forces) and therefore initial deformations, buttressing or inhomogeneous material properties leading to imperfect load distributions become significant. These imperfections are induced during the glide and perhaps during the redistribution of viscoelastic stress. Our analysis reveals that the snowcover in the detachment zone must endure tensile and compressive strain oscillations. These will not act at the center of mass of the slab, as our simple model assumes, but will cause folds, that would not only excite the onset and growth of the next unstable phase of gliding, but also provide the necessary force to lift the slab above the stauchwall.

Acknowledgments:

[16] The authors thank many members of the SLF for their helpful discussions during the writing of this article: P. Bebi (gliding snow avalanches in forests), S. Margreth (stauchwall mechanics and mitigation) and C. Pielmeier, T. Stucki, R. Meister and S. Harvey (Swiss avalanche warning). We especially appreciate the helpful review comments of Karl Birkland and another unknown reviewer that improved the paper greatly.

[17] The Editor thanks Karl Birkeland and an anonymous reviewer for assisting in the evaluation of this paper.

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