# Models for Long Run-out Avalanches

## Introduction

Avalanches are one of nature's most ferocious phenomena. Triggered by anything from earthquakes to catastrophic rock failure, they strike fear and awe into the hearts of humans everywhere. Avalanches are interesting to scientists not just for their destructive power, but also because they seem to defy basic Newtonian mechanics. Like a skier atop a slope, rocks sitting atop a mountain have a certain amount of gravitational potential energy that is converted into velocity as the rocks slide down the mountain. The rocks are slowed, however, by the frictional force from rubbing against the mountain. What is bizarre, is that the rocks in an avalanche sometimes travel much farther than would be expected given their potential energy and the friction of the mountain. This phenomenon is known as long run-out.

## Coulomb Frictional Force

The Coulomb frictional force is the force created by two surfaces rubbing against each other, commonly known as the frictional force. In the flowing of granular mass, such as in an avalanche, the Coulomb frictional force generates the most resistance. Studies have shown that the Coulomb frictional force due to gravity dominates most inter-granular stresses, and is thus the primary source of energy loss . When one assumes no cohesion among the rocks of the avalanche, the Coulomb frictional equation becomes τ = σ * tan(θ); where τ is the coefficient of the frictional force, σ is the stress between the constituent rocks, and θ is the angle of descent . The potential energy of the avalanche is characterized by U = MGH; where H is the height of the rock fall, G is the acceleration of gravity, and M is the mass of the avalanche.

Conservation of energy requires MGH = τFn*L = τ*MG*L; where L is the distanced traveled by the avalanche. This basic equation does not accurately describe observations in two ways. First, the ratio of H/L is calculated to be on the order of 0.5 for avalanches. However, ratios as low as 0.1 and possibly 0.06 have been observed. Moreover, this equation implies that H/L should be unaffected by the mass (or volume) of the avalanche, but observations have shown that H/L ∝ 1/log(Volume) .

## Models for Long Run-Out

For many years, researchers have investigated possible explanations for the long run-out of avalanches. The main hypotheses proposed have been:

1. Mobilization via air cushion. In this hypothesis, an avalanche will either override and trap gas beneath the rock flow or vaporize groundwater found in the path of the avalanche. The rock flow will then ride on this "cushion" of gas, which decreases the coefficient of friction (τ), thus accounting for long run-out. This model postulates gas under high pressure at the base of the avalanche, which would cause a grading of the rock as the gas escaped (smaller particles nearer the surface of the rock flow). In observations of avalanche profiles, however, most avalanches exhibit reverse grading (smaller particles nearer the bottom) which is opposite to what this hypothesis would predict. Air is also inefficient for fluidising, and would likely escape through the debris . Moreover, similar long run-outs have been observed on the moon where there is no atmosphere .

2. Rock melting. In this hypothesis, the extreme pressure of the avalanche causes the rocks nearest the mountain to liquidize. The rock flow then rides on the layer of melted rock, which decreases the coefficient of friction. Rock melting might occur in the thickest of avalanches, but this theory has not been borne out by observations as very few examples of melted rock found within avalanches have been reported .

3. Mechanical Fluidization. In this hypothesis, the high rate of shearing found among the rocks of the avalanche leads to a spontaneous reduction in the frictional force. The large energy input to granular material such as rocks causes high impulsive contact pressure between the rocks, which causes them to separate and decreases the internal resistance of the avalanche. Observation of sand flow have shown that the apparent coefficient of friction τ ∝ (λD)2S2; where S is the rate of shear, λ is the linear density of the sand, and D is the grain diameter . However, it has yet to be shown experimentally that such high rates of shearing within a rock flow actually lead to a reduction in the frictional force .

4. Acoustic Fluidization. In this hypothesis, the acoustic-frequency vibrations at the base of the avalanche cause a reduction in the frictional force. This theory has been experimentally verified. In fact, the avalanches studied in the Haloran and Silurian hills are best fit by the acoustic fluidization model . However, this theory cannot explain why the run-out length is affected by the volume of the avalanche.

5. Soil Entrainment. In this hypothesis, a rock slide is initiated by a rock fall. The rock fall then flows over a deposit of loose saturated material, such as mud or clay. The front end overrides the soil, and the pressure from the avalanche liquefies it. The body of the avalanche then flows atop the layer of liquefied soil, thus reducing the coefficient of friction. This can be used to explain the long run-out of avalanches. This model can also be used to explain the increase in run-out distance with volume. Avalanches with a larger volume cover more area, and thus are more likely to encounter thicker, more saturated deposits of soil. This model is also supported by direct field evidence.

## Field Evidence for the Entrainment Model

Direct experiments with avalanches are almost impossible to perform because of the dangers they entail. Thus most of the data from avalanches are collected after they occur. While attributes like initial volume and the velocity of the avalanche are difficult to determine, data such as the avalanche composition and final volume can be reliable measured. When the avalanche at Rainbow Creek at Mount Baker of 2006 was studied, it was determined that the avalanche had an unusually long run out. Profile samples of the avalanche taken from 11 dispersed sites contained 8-21% fine materials, such as clay and silt that might have been entrained from the valley floor . These findings are consistent with the soil entrainment model. Likewise, when the Eagle Pass Slide of 1999 was profiled, it was found to contain 5-15% fine materials, such as silt .

The Eagle Pass Slide and Nomash River Slide (both 1999) were both studied in depth. The distance both slides traveled were recorded, and their velocities were estimated using in locations where the avalanches followed a curved path using vortex equation for flow through bends: v = (R*g*tanβ)1/2; where R is the radius of curvature and β is the angle between the lines defining the flow of avalanche .

The avalanches were then simulated using the Voellmy model. The Voellmy model was initially used for snow avalanches, but the model offered such accurate simulations of rock avalanches that it is now used for rock avalanches as well. It is a two parameter model that combines the Coulomb frictional force and Chezy formula to determine the coefficient of friction.

τ = γhf*cos(θ)+γv2

Where γ is the density of the avalanche, h is the depth of the avalanche, v is the velocity of the avalanche, and f and χ are the two parameters of the model . Due to the high non-linearity of the equations of motion of an avalanche, the values of f and χ cannot be calculated, but they can be found through the process of trial and error when trying to model observed avalanches. The entrained soil decreases the overall frictional force by lower the parameter f. Using this model, researchers were able to reproduce the avalanches extremely well. When the model was run again with no soil entrainment, the slides produced were much smaller than their physical counterparts .

Also, computer simulations of 2-D particle have supported the soil entrainment model. Using an expression for friction that is very similar to the Voellmy model:

τ = (ρsf) νsgh*cos(θ)-P

where ρs would correspond to the rock density, ρf would correspond to the soil density, νs is the flow density, θ is the angle of declination, and P is the interganular pressure . Simulations using this model have shown that lower values of friction in a large slide can be explained via material riding on a thin layer of highly agitated particles at low concentration, and that this is the a stable state which granular flows tend toward .

## Conclusion

There is no reason yet to believe that all long run-out avalanches are caused by the same mechanism. In fact, it might be the case that several of the hypotheses are at work in each avalanche. However, the overwhelming majority of direct observations and simulations are consistent with the soil entrainment model. As of now, it seems to be the best explanation for why avalanches can be excessively mobile.

© 2007 D. Silverstein. The author grants permission to copy, distribute and display the work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

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