Introduction

The dynamic characteristics and parameters of deposit of an avalanche body can be determined by applying mathematical models. As a rule, two-parameter dynamic models including the coefficients of dry and turbulent friction are used (Reference Bozhinskiy and LosevBozhinskiy and Losev, 1987; Reference Eglit and HestnesEglit and Revol, 1998). Recalculations for avalanche sites, where field data for avalanche events are available, allow the ranges of model parameters to be estimated (Reference Buser and FrutigerBuser and Frutiger, 1980; Reference Martinelli, Lang and MearsMartinelli and others, 1980). Hydraulic models were applied to five European avalanche sites to determine the sensitivity of the models to input parameters (Reference Barbolini, Gruber, Keylock, Naaim and SaviBarbolini and others, 2000). It was shown that the dry friction coefficient is the predominant influence on avalanche run-out. However, the number of such avalanche sites is limited. Moreover, the calculated values of model parameters are often not regional, and their use for neighboring avalanche sites in the same region is not always justified. Another approach is statistical simulation of the avalanche process (Reference Barbolini and SaviBarbolini and Savi, 2001; Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2001; Reference Ancey , Gervasoni and MeunierAncey and others, 2004; Reference BozhinskiyBozhinskiy, 2004; Reference Meunier and AnceyMeunier and Ancey, 2004). The model parameters are assumed random, with average values corresponding to recalculated estimations, based on field data. However, the laws and ranges of model parameter distributions are poorly known, so a more detailed investigation of the influence of model parameters on avalanche dynamics is needed. The earlier investigation was incomplete (Reference BozhinskiyBozhinskiy, 2007): only general tendencies of change of output model characteristics (run-out, thickness of avalanche deposits, velocity and height of leading front) were illustrated by varying the model parameters. The purpose of this work is to research in depth the influence of the interaction of model parameters on output model characteristics, using a three-parameter dynamic model.

Model

The avalanche dynamics model used for research is characterized by three non-dimensional parameters: the coefficients μ and k of dry and turbulent friction, respectively, and the coefficient m _e of new-snow-mass entrainment into the motion (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2001). The system of model equations has the following form. The mass conservation equation is

and the momentum conservation equation is

where H and U are the depth and velocity of the avalanche body, respectively, is the local slope angle, q _b is the rate of specific snow volume (per unit area of the bottom), g is the gravitational acceleration, s is the coordinate along the slope and t is time. The parentheses with attached lower index designate partial derivatives with respect to the variable, indicated by this index. The avalanche density ρ is assumed constant.

The specific volume of snow masses entrained into the motion is assumed to be proportional to the avalanche velocity

According to this phenomenological law of entrainment, the leading edge of the avalanche spreads over undisturbed snow cover, and additional snow mass is entrained by gradual shearing of the layer along the entire avalanche body. The other law characterizes the entrainment of snow peeled off by the avalanche at the leading edge. When the law (Equation (3)) is used, the dynamic characteristics of an avalanche body near the leading edge change evenly (Reference Bozhinskiy and LosevBozhinskiy and Losev, 1987). Recently, different laws of snow entrainment have been analyzed, and it was shown that the dynamic characteristics of an avalanche body only weakly depend on the form of the entrainment law (Reference Eglit and DemidovEglit and Demidov, 2005).

Finally, the snow-cover thickness h during the avalanche motion on the slope diminishes according to

The deterministic model (Equations (1–4)) was used to obtain dynamic characteristics and parameters of avalanche deposits. It was also applied during statistical simulation of an avalanche process, when the initial data and the model parameters were considered as random (Reference Bozhinskiy, Nazarov and ChernoussBozhinskiy and others, 2001). The model acted as an operator. The output model characteristics were also found to be random, and the model captured the probabilistic contents.

Results and Discussion

The ‘Domestic’ avalanche site in Elbrus region, Caucasus, Russia, was chosen as the model avalanche range. The snow-cover distribution along the slope was assumed uniform and equal to a mean observed value of 1.5 m. Equations (1–4) were integrated numerically using the ‘large-particle’ numerical method, known in continuum mechanics (Reference BelotserkovskyBelotserkovsky, 1984).

The fixed (^*) avalanche run-out S ^*, according to the model, can be achieved with various combinations of model parameters. When m _e = const., we have a curve on the plane k, μ, which illustrates possible combinations of parameters of dry and turbulent friction ensuring achievement of the given run-out S ^* by model avalanche. For series values of S ^* we receive, by definition, the family of uncrossed curves (Fig. 1); the value of m _e = 0.005 corresponds to entrainment about a half of thickness of the snow cover on a slope. However, the situation changes when the third model parameter, m _e, is taken into account. On the plane k, μ, the family of curves (m _e is a parameter) was constructed (Fig. 2). The curves were found to be crossed at the single point (Fig. 2). The minimum value, m _e = 0.001, corresponds to entrainment into an avalanche flow of no more than 0.1 of the initial thickness of a snow cover on a slope, and the maximum value, m _e = 0.009, corresponds to the complete entrainment of a snow cover into an avalanche flow. The existence of the cross-point for the m _e-parameter family of curves means that there is a pair of values k, μ for which the avalanche run-out does not depend on new-snow-mass entrainment. It is known that the entrainment of new snow masses into an avalanche flow results in both increasing active force and additional resistance to motion (Reference Bozhinskiy and LosevBozhinskiy and Losev, 1987). The obtained value of k, corresponding to the cross-point, is small, resulting in suppression of turbulent resistance. At the same time, the active force and the dry friction force grow proportionally to increased mass of an avalanche body. Therefore, an independence of run-out on entrained masses takes place. Thus, when for determination of run-out the values of k, μ appropriating to the cross-point are used, the model appears as two-parametric, because the influence of snow-cover entrainment at such a combination of model parameters is practically excluded. However, if it is necessary to estimate thickness or volume of an avalanche body, the prescribing of a concrete value of the coefficient of entrainment m _e is essential.

Fig. 1.

Family of curves (run-out S ^* is a parameter) on the plane k, μ; m _e = const.

Fig. 2.

Family of curves (m _e is a parameter) on the plane k, μ; S ^* = const.

A series of values of S ^* was further analyzed. Within the range of 400 m varying S ^* (maximum and average avalanche run-outs), the values of k, appropriate to cross-points, were practically identical. The effect of ‘rapprochement’ of curves, corresponding to minimum and maximum values of the coefficient of entrainment, with decreasing k is also illustrated in Figure 3. It can be seen that when k = 0.0011 the curves merge, and the single dependence of run-out on the coefficient of dry friction appears. The result obtained means that, for determination of run-out, it is possible to assume the value of the turbulent friction coefficient k is constant and to vary only the coefficient of dry friction μ. The curve illustrating the dependence of avalanche run-out on the coefficient of dry friction is shown in Figure 4. The approximation of this dependence appears to be close to linear. Similar dependence of the observed run-outs on μ was obtained by Reference Naaim, Naaim-Bouvet, Faug and BouchetNaaim and others (2004). A weak influence of k within the range 0.01 < k < 0.025 on a family of S (μ) was found using the rigid-body model of avalanche dynamics with two friction coefficients (Reference Ancey , Gervasoni and MeunierAncey and others, 2004).

Fig. 3.

Run-out vs dry friction coefficient for the ‘Domestic’ site; k is a parameter. Closed symbols, solid lines: minimum m _e; open symbols, dashed lines: maximum m _e.

Fig. 4.

Run-out vs dry friction coefficient for the ‘Domestic’ site; k corresponds to cross-point.

The result obtained is important for the statistical simulation of avalanche dynamics, because it is possible to assume only one random model parameter μ. The constant parameter of turbulent friction k ≈ 0.004 and random parameter μ during statistical simulation were used in the two-parameter dynamic model (Reference Barbolini and SaviBarbolini and Savi, 2001).

Similar analysis of the influence of dynamic model parameters on the output characteristics of avalanches was carried out for other, arbitrarily chosen, avalanche sites: Nos 2, 22 and 43 (Khibiny mountains, Kola Peninsula, Russia), Cheget-Kara (Elbrus region) and an idealized avalanche site with an exponential longitudinal profile. The results are shown in Table 1, where the value of the coefficient of turbulent friction k corresponding to the crosspoint of curves is given.

Table 1.

Values of the coefficient of turbulent friction corresponding to the cross-point

The values of parameter k at the cross-point are very small for all the avalanche sites, testifying to the relatively weak turbulent mixing of snow into the modeled avalanche flow. However, this does not mean that the turbulent resistance can be neglected, because, for example, for the ‘Domestic’ avalanche site, the difference of run-outs when k = 0 and k = 0.0011 is within the 100–200m range. For every site, the dependences of run-out on the coefficient of dry friction are constructed (Fig. 5). These dependences were approximated by linear functions S = A+Bμ. The values (in meters) of the coefficients of equations are given in Table 2; R is the coefficient of correlation.

Fig. 5.

Run-out vs dry friction coefficient for six avalanche sites.

Table 2.

Values of the coefficients of equations

The obtained dependences are valid within the range of change of S, including the maximum and average run-outs. For small values of S, the model coefficient of turbulent friction, appropriate to the cross-point, increases slightly. This is related to the fact that such avalanches stop on relatively steep segments of a slope, and it is practically impossible to create resistance to movement of an avalanche on these segments due to dry friction only. Therefore, turbulent friction, which is weak on flatter segments, should grow. Then the drag and stop of an avalanche body is possible. The example of dependence of k at the cross-point for ranges of change of S, including relatively short run-outs, is given in Figure 6. The significant range of S within which k ≈ const. is noticeable.

Fig. 6.

Turbulent friction coefficient corresponding to cross-point vs run-out for avalanche site No. 22, Khibiny mountains.

Available field data of run-outs for the avalanche sites ‘Domestic’ (39 events), No. 22 (30 events) and No. 43 (24 events) and the received linear dependences S(μ) allow distributions of the model parameter μ to be constructed. The histogram and distribution function of μ are given for the ‘Domestic’ avalanche site in Figure 7. Characteristic statistics of distributions are given in Table 3.

Table 3.

Statistics of distributions of μ. M is the mathematical expectation, σ is the root-mean-square deviation, λ _a, λ _e and C _v are the coefficients of skewness, kurtosis and variation, respectively, and max and min are the maximum and minimum values, respectively

Fig. 7.

Histogram and distribution function of dry friction coefficient for ‘Domestic’ avalanche site.

The distribution of μ for the ‘Domestic’ and No. 22 avalanche sites is close to normal, though a positive kurtosis and skewness are available. Note that as the dependences of μ(S) are linear, then Mμ = a+bM _s and σ_μ = | b |σ_s , where a and b are the coefficients of linear dependence. As the coefficients of skewness and kurtosis are proportional to the third and fourth central moments, respectively, it is obvious that λ _a μ = –λ _a s and λ _e μ = λ _e s. It follows that the coefficient b is negative and the third moment is proportional to the cube, and the fourth moment to the fourth degree, of b (Reference VentselVentsel, 1969). Therefore, the mode and the tail of distribution of run-out shift to the right and left, respectively; the distribution of dry friction coefficient is the reverse (Fig. 7). For avalanche site No. 43, the distribution is close to uniform, there is almost no skewness, but a large negative kurtosis testifies the flatness of distribution.

Conclusions

The obtained dependences S(μ) should be considered as the property of an avalanche site. Application of the three-parameter avalanche dynamic model allows the value of the turbulent friction coefficient to be estimated and the dependence of run-out on the coefficient of dry friction to be constructed. Knowledge of the longitudinal profile and the distribution of a snow-cover thickness at the avalanche site is only needed to determine the value of the turbulent friction coefficient. This means that the model can be used for poorly investigated avalanche sites. Further development can provide a simplified statistical simulation of snow avalanche dynamics, since it is possible to use only one random model parameter (the coefficient of dry friction) to determine an avalanche run-out. It may be possible to construct distributions of the dry friction coefficient on the basis of available field data and statistical simulation for well-investigated avalanche sites.