Snow avalanche origin

The problem of the stability of snow on mountain slopes and avalanche origin is key in avalanche science. During statistical simulation, the model of the stress state of a snow slab on mountain slopes based on the equations of the momentless theory of thin elastic (viscous) shells was used (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2002a). The zone of the avalanche origin or instability is determined by the critical thickness H_cr of a snow slab:

where c is cohesion, f is the coefficient of internal friction, ρ is the snow density, and is the local slope angle.

The output characteristics of the two-dimensional model are normal stresses S₁₁, S₂₂ (along and across a slope) and shear stresses S₁₂ in a plane of a slab. Thus, it is possible to determine an avalanche origin not only from a maximum tensile stress, which is usually observed on the upper contour (fracture line), but also by more complicated criteria, for example, the intensity S_i of shear stresses (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2002a), which achieves a maximum along longitudinal edges of the instability zone (flank shear fractures):

From the solution of the two-dimensional problem, it is possible to estimate a configuration of an instability zone and a volume of snow, which is potentially unstable. However, this volume practically always exceeds the actual volume of snow originally involved in the motion, because the action of maximal stresses is spread to only a part of the instability zone (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2002a).

The statistical simulation of snow instability on a slope was carried out for avalanche path No. 22 (Khibiny, Russia). The stress state and stability of the snow slab before an avalanche release was estimated. The avalanche occurred on 9 February 1987, as a result of mortar firing, and had a volume of 31 × 10³ m³. The distribution of the snow slab thickness within a starting zone was determined from field measurements on 40 stakes and was approximated by a cubic polynomial, as shown in Figure 1. Storm-accumulated snow is noticeable in the left upper corner of the starting zone. Data on the strength characteristics of snow are lacking, so typical ranges of values were assumed: 0.4 ≤ f ≤ 0.5; 300 ≤ c ≤ 700 Pa. The strength of the snow slab on rupture and shear, and accordingly the limiting value S_i ^* of the shear stress intensity in a plane of a slab, were also random. The assumed value was 1 × 10⁴ ≤ S_i ^* ≤ 2 × 10⁴ Pa. The assumed ranges for strength parameters approximately correspond to experimental data for storm-packed snow with a density of about 300 kgm^–3 (Reference VoitkovskiyVoitkovskiy, 1977).

Fig. 1. Distribution of snow slab thickness; contour of H level is 0.1m. Avalanche site No. 22 (Khibiny).

A uniform distribution of strength characteristics within the assumed ranges was supposed. One hundred values of random variables c; f; S_i ^*were generated, and then the stress distributions within the avalanche starting zone are obtained. It is necessary to underline that, for a given problem, as well as for two other problems considered in this paper, the number of generations has an illustrative character.

The fragment of the starting zone, corresponding to the left upper corner of Figure 1, is shown in Figure 2. There the typical distribution of S_i is given. It is seen that the contours of the maximum stress S_i form a near-triangular shape. Maximum tensile stresses act at the crown, and maximum shear stresses and transverse normal stresses on the sides. The thick solid line marks the fracture line of the avalanche. Good agreement between the model calculations and field data is noticeable.

Fig. 2. Contours of shear stress intensity S_i.

The maximum values S_i for each realization were compared to a random value S_i ^*, and the random difference ΔS_i = S_i ^* = S_i was determined. The histogram and distribution function of ΔS_i are shown in Figure 3. The distribution has a negative skewness and excess. By definition, ΔS_i > 0 corresponds to stability and ΔS_i < 0 to instability of the slab. According to the distribution function in Figure 3, the probability of stability and instability of the snow slab is equal to 0.11 and 0.89, respectively. The probability of instability of the snow slab is rather great, which also proves to be true of the avalanche release.

Fig. 3. Histogram and distribution function of ΔS_i.

However, as the boundary values of ranges of change of snow-strength characteristics are not exact, the influence of the first moments of distributions (mathematical expectation and variance) on stability of the snow slab was investigated. In Figure 4, the curves of probability of stability and instability of the slab are shown to depend on the variance of S^* _i ; DS^* _i : The mean value of S^* _i ; ξ_S, was held constant and set to 1.5 × 10⁴ Pa. This corresponds to expansion or narrowing of the range of changing of S^*i . It is apparent that the influence of the variance of S^* _i on stability of the snow slab is rather weak. At the same time, the dependence of the probability of snow-slab stability on the mean value of cohesion ξ_c is noticeably stronger (Dc is constant, which corresponds to a shift of the range of changing of c) (see Fig. 5). The curves corresponding to stability and instability of the slab draw together with increasing ξ_c, and at ξ_c ≈ 700 Pa the probabilities of avalanche release and snow-slab stability are nearly identical and equal to 0.5. Thus, control of the process of the avalanche origin is determined by mean values of strength parameters of a snow slab, whereas the influence of variance of these parameters is secondary. This conclusion is important for modelling and should be proved by the subsequent calculations.

Fig. 4. Probability of snow slab instability (solid line) and stability (dashed line) υs dispersion of S_i.

Fig. 5. Probability of snow slab instability (solid line) and stability (dashed line) υs mean value ξ_c of snow cohesion.

Snow avalanche dynamics

The following avalanche problem is connected with describing the process of an avalanche motion on a slope. Here, the deterministic one-dimensional hydraulic model, which takes into account an entrainment of new snow masses into the motion, is applied (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2001). Three parameters of the model, namely, coefficients μ; k of dry and turbulent friction, respectively, and the coefficient m_e of the snow mass entrainment into the motion, are considered as random variables. The statistical simulation of the avalanche-motion process is carried out for the well-known avalanche path “Domestic” (Elbrus region, Russia), where there is a series of field observations of run-out distances (38 events). The uniform distribution of random model parameters within ranges 0.15 ≤ μ ≤ 0.525; 0.005 ≤ k ≤ 0.2, and 0.002 ≤ m_e ≤ 0.01 was assumed. The mean values of μ and k were assumed based on several back model calculations of actual avalanches. The boundaries of ranges cannot be prescribed precisely, and naturally contain some indeterminacy, but they correspond to rather rare avalanches (short and long run-out distances). The boundaries of m_e correspond approximately to one-quarter and total erosion of snow cover. The remaining model parameters and input data were deterministic and set to an average value. The density ρ of the avalanche body was equal to 200 kgm^-3, the snow-cover thickness in a transit and deposition zone H₀ = 1.5m, and the initial volume (per unit of width) of snow involved in the motion, V₀ = 50 m².

Two hundred values of random model parameters were generated. As a result of statistical simulation, the 200-term series of the following output model characteristics were obtained: the run-out distances of avalanches, the maximum thickness of depositions, the time of motion, and the volume and length of an avalanche deposit body.

The statistics (the mathematical expectation ξ, the sample standard deviation σ, the coefficient of skewness _a, the coefficient of excess _e, the coefficient of variation C_v) of the empirical and model run-out distance S^* are shown in Table 1. The mean values of distributions practically coincide; also, the coefficients of skewness are close. The scatter of model values is less noticeable. Moreover, the model distribution has a more acute peak. A comparison of the model and empirical cumulative distribution functions F(S^*) of the run-out distance is depicted in Figure 6. According to the Kolmogorov–Smirnov test, the maximum value of the module of the difference is used as a measure of deviation-model (m) and empirical (e) distributions (Reference VentselVentsel, 1969):

Fig. 6. Model and empirical distribution function of avalanche run-out. Avalanche site “Domestic”.

Then the value λ = dN¹/² is determined, where N is the number of observations, and under the relevant table the probability p(λ) is found, which means that the discrepancy between model and empirical distributions will be not less than the actually observed one. In the case under consideration, d ≈ 0.12; N = 38; λ ≈ 0.7 and ρ(λ) ≈ 0.7.This probability is not small; therefore, it is possible to consider a hypothesis about the goodness of model and empirical distributions compatible with data from field observations. The hypothesis is also not rejected according to Pearson’s chisquare test. At the same time, some discrepancy is found in zones of rather rare values, which is due to both ignorance of precise distribution laws of model parameters, and restricted series of field observations.

The histogram and distribution function of the maximum thickness of avalanche depositions are shown in Figure 7. The distribution is characterized by positive asymmetry (γa = 0.63) and almost by lack of excess (_γe = -0.11): The mean value is ξ_Hmax = 5.5m. This value is in accordance with field observations, which, unfortunately, are unit.

Fig. 7. Histogram and distribution function of H_max. Avalanche site “Domestic”. Slushflow dynamics

The two-layer deterministic hydraulic model of a slushflow is a basis for developing a probabilistic model of a slushflow and statistical simulation (Reference Bozhinskiy and NazarovBozhinskiy and Nazarov, 1998).

Series of numerical experiments using the model have shown that the coefficient μ of dry friction of slush and the intensity Q of a water inflow on the “tail” strongly influence dynamic characteristics, run-out distance, thickness and volume of slushflow depositions. The influence of the other model parameters is weaker. In this connection, the following model parameters were assumed to be deterministic: the drag coefficients k_s; k_w; k_sw of the turbulent flow of slush and water over the snow-cover surface and over the slush– water interface, respectively, and the intensity of water percolation from the lower to the upper layer. Morphometric parameters of the slope were also considered as prescribed.

It was supposed that the inflow Q of water on the “tail” of the flow damps in time (t) within 600 s under the square law

The parameters μ and Q₀ were considered as random. The uniform distribution law was accepted. The statistical simulation of slushflow dynamics was carried out for basin No. 166, which is located on the western slope of Khibiny in the basin of Bear brook. Tracks of slushflow, released presumably in 1950/51, 1968/69 and reliably in 1977 and 1984, are traced on terrain. The slushflow in 1984 left the most numerous and reliable tracks, making it possible to produce a survey of boundaries and to determine some parameters of slushflow. The flow has passed 3.35 km up the channel. The height of the leading front reached >10m. The basic mass of wet snow was deposited by the extended “tongue”, about 250 m long and up to 40 m wide. The total volume of snow deposits was of the order of 3.5 × 10⁴ m³. The run-out distance of slushflow in 1984 was not maximal. From the character of depositions on boards and at the bottom valley, the age of new growth and damages on tree trunks, it appears that in the beginning of the 1950s, slushflows advanced 900 m further (Reference Bozhinskiy, Nazarov and SapunovBozhinskiy and others, 2002c).

The data from field observations and morphometric parameters of the slope were used during statistical simulation of slushflow dynamics. From a relation of water and snow masses in slushflow it was assumed Q_0min = 5 ms^- ₁, Q_0max = 15ms^-1. The minimum value μ_min of the coefficient of dry friction was estimated according to the possibility of model slushflow reaching the visual boundaries on the terrain, and was equal to 0.1. The maximum value μ_max was assumed equal to 0.2. The values of the deterministic model parameters were set to: snow porosity P = 0.5; k_s = k_w = 0.03; k_sw = 0.02; = 0.05. The snow-cover thickness in the channel was H_c = 1m; the density of involved snow was equal to 450 kgm-³. The value of the coefficient of snow entrainment was assumed such that the snow accumulated in the channel was completely involved in the motion according to field observations.

Fifty values of random variables μ; Q₀ were generated, and then, using the two-layer model of slushflow, 50 numerical experiments were carried out. Thus, the 50-term series of the following output model characteristics were obtained: the run-out distance S; the length l_s of a zone of slushflow depositions; the length l_sw of the lower water layer; the maximal thickness H_s of snow depositions; the maximal depth H_w of the lower water layer; the volume (per unit width) V_s of snow depositions and the total volume of water V_w; the time of motion T. The outcomes of the statistical simulation of slush flow dynamics are shown inTable 2 and Figure 8. It is necessary to emphasize that the distributions of all output model characteristics, excluding V_w, have appeared essentially non-uniform, though the input distributions of μ and Q₀ were uniform. This outcome is due to non-linearity of the operator of the slushflow dynamic model. In addition, it is possible to note that the majority of distributions are characterized by rather weak asymmetry. At the same time, the coefficients of excess of distributions noticeably differ (from 1.24 to -1.33).

Fig. 8. Histogram and distribution function of slushflow run-out. Bear brook, Khibiny.

The distribution of the run-out distance of slushflow is characterized by positive excess. Figure 8 testifies to a concentration of stop points of the leading front of slushflow within a slope-length interval of 2700–3200m. For gravitational avalanche-type flows, as a rule, this is due to morphometric peculiarities of a profile of a channel in this interval (a flattening zone of a slope). According to the distribution function obtained, the run-out distance of slushflow in 1984 approximately corresponds to a probability of 0.05. The distant run-outs are rather rare; their realization requires a combination of considerable water inflow on the “tail” and low coefficients of dry friction.