1. Introduction
Avalanche motion is complex, varying from the pure, sliding motion of large blocks to thehighly turbulent motion of fully developed powder avalanches. No single existing theory canhandle this full spectrum of motion. Instead, several simplified approaches offer reasonably good predictions of run-out distance, impact forces, debris location, and debris distributionunder most conditions even though the theoretical basis for such predictions falls far short ofmatching actual avalanche conditions. To assure realistic results from such approaches, it isnecessary to calibrate the computation schemes against actual events.
In this study, data from a number of avalanches were used to estimate the frictioncoefficients needed for two avalanche-dynamics computation procedures—the computerprogram AVALNCH, and Voellmy’s avalanche-dynamics equations. Program AVALNCH(Reference LangLang and others, 1979[a], [b]; Reference LangLang and Martinelli, 1979), is a numerical, finite-differencecomputer program based on the Navier-Stokes equations. The Reference VoellmyVoellmy (1955) equationsbased on open channel flow, were developed over 20 years ago and have since been used andslightly modified by subsequent workers (Reference SchaererSchaerer, [1975]; Reference MearsMears, 1976; Reference Leaf and MartinelliLeaf and Martinelli, 1977; Reference BuserBuser and Frutiger, 1980).
Table I. Summary of the physical parameters and field data for nineteen case studies
2. Case studies using program AVALNCH
For each of the 18 avalanches in Table I one or more computer runs were made usingprogram AVALNCH to see what coefficients of internal friction v and of surface friction fgave the best approximation of the observed run-out distance and debris distribution. Forthese calculations run-out distance is considered to be the slope distance from the place in thelongitudinal profile where slope angle ≤18°-2° to the end of the debris. When good fielddata were available for leading-edge velocity and/or flow heights, these were also comparedto the computations.
The results of this analysis are summarized in Table II, The cell numbers referred to inthis table and elsewhere in the paper are units of slope distance on the longitudinal profile ofthe avalanche path. Cells were 20 m long on all paths except Breckenridge, Floral Park, andRed Lady Basin where they were 10 m.
Table II. Summary of case studies using program AVALNCH
The eight soft-slab avalanches (Group 1, Table II) all ran in fresh, soft, dry snow. Thefirst four of these ran the same day in adjacent paths on the same ridge. Battleship avalanche,9 km south-east of these four paths, also ran on the same day. Slab thickness was about 1 mfor all eight of the soft-slab avalanches, yet three of them (Blikes “a”, Chapman Gulch, andBlikes “b”) had run-out distances greater than 800 m. In general, friction coefficientsv = 0.4 to 0.5 m2/s and f = 0.4 to 0.5 gave good estimates of run-out distances but tended toshow debris concentrated more toward the end of the run-out zone than was actually observed.
The two hard-slab avalanches (Group 3, Table II) also showed consistent results. Coeffi-cients of v = 0.7 to 0.8 m2/s and f = 0.7 to 0.8 represent conditions well. Again, run-outdistances were closely approximated by the computations, but debris tended to be confinedmore toward the end of the run-out zone than it should have been, and maximum debrisdepths were definitely underestimated.
The three design-size avalanches (Group 2, Table II), were all large for their paths. Forthis reason, they are important for developing design criteria for avalanche zoning and fordesigning structural controls in the lower track and run-out zone. These avalanches alsofurnished useful data on some unusual conditions. For example, the Bird Ridge avalanchestarted in cold, dry, wind-toughened snow, encountered damp or wet snow in the track, andran out 600 m over the unfrozen tidal water of Turnagain Arm. The Dam avalanche, whichwas very large for its path, traveled down a gully, through a dense stand of mature coniferoustrees, and crossed a stream channel before piling debris 9 m deep on the highway 60 m up a 15° adverse grade. Program AVALNGH handled all these conditions well, especially for theDam avalanche where actual conditions were duplicated very accurately using v = 0.4 m2/sand f = 0.5.
In each of the last five avalanches in Table II there were some exceptional field dataavailable for comparison with computed results. At Floral Park, snow plastered on the up-hillside of trees in the track and the run-out zone gave good evidence of flow heights. At Red LadyBasin the avalanche was a slow-moving (estimated < 10 m/s), wet slab that ran over wet grassbut sleftped on a steep (21°) slope. The Gothic Mountain No. 4 avalanche ran through rockpinnacles and boulders in the starting zone. At West Guadalupe the avalanche fell over a cliffin the lower track and came to rest 320 m up an adverse grade of 9.6°. Battleship came down amoderately steep track (26°), crossed a narrow stream, and ran up a steep (23°), adversegrade for a slope distance of 140 m.
Program AVALNGH duplicated run-out distance well for all the avalanches in Group 4(Table II) except Battleship and West Guadalupe. In these two cases the program under-estimated run-out distance by 5 and 12% respectively—probably because of the steep,adverse grade in the run-out zone. The measured flow heights of about 2 m in the track andrun-out zone at Floral Park were estimated by program AVALNGH to be only 0.3-0.6 m.This underestimation is the result of the partial fluidization of fast-moving, soft-slab snow. This type of snow has a greater flow height than the height of the dense flow regime (corematerial) predicted by program AVALNGH because the program assumes incompressibility. Table III For the slow-moving, wet-slab avalanche at Red Lady Basin, two sets of coefficients, v = 0.55m2/s, f — 0.55 and v = 0.85 m2/s, f = 0.4, give acceptable results. In the first case, theleading-edge velocity was computed to be ≤10 m/s, but the f value seems too high for wetsnow sliding over grass. In the second case, leading-edge velocity was ≤8 m/s , and the f valueseems more reasonable. Table III is a summary of v and f values suggested for use in programAVALNCH for a variety of snow and terrain conditions. These values should be used withcaution until more case studies can be made and more experience can be gained in the use ofthe program.
Table III. Internal friction v and surface friction f coefficients for usewith program AVALNGH
3. Case studies using Voellmy’s approach
Run-out distances were calculated using Voellmy’s approach (Reference VoellmyVoellmy, 1955; Reference Leaf and MartinelliLeaf and Martinelli, 1977) for most of the avalanches that ran on non-channeled (or open) tracks andfor one-channeled avalanche. The procedure outlined by Reference MearsMears (1976) was used for thechanneled path. The following equations were used for the paths with open tracks:
Where V is the maximum velocity in the starting zone; ξ is the coefficient of turbulent friction(different values can be used in different parts of the path); h' is flow height in the startingzone; Ψ is slope angle in the starting zone; μ is the coefficient of kinetic or sliding friction(can be varied for different parts of the path); s is run-out distance from the break in gradeat the left of the run-out zone to the end of the debris; g is gravitational acceleration (takenas to m/s2);β is the slope angle of the run-out zone (minus values indicate adverse grade).Subscripts 1 and 2 refer to sections of the track and LT to the lower track.
3.1. Non-channeled avalanches
For avalanches with non-channeled tracks, Equation (1) was used to compute velocity inthe starting zone. Equation (2) give velocity and flow height for uniform sections of the trackworking down from the starting zone. Equation (3) was used for run-out distance based on thelower-track velocity and flow height from Equation (2). The same ξ and μvalues used tocompute velocity were also used to compute run-out distance. (Average ξ and μ’s were usedto facilitate comparisons of coefficients for the various events. Some workers feel μ in therun-out zone should be larger than in the track.) Flow height in the lower track (h LT) ascomputed by the second of Equation (2) was used in most cases. For the Nicholson Lakeavalanche, the measured flow height in the lower track was also used. Although the measuredheight in the lower track was three times that computed using the second of Equation (2),this made no significant difference to the run-out distance. In most cases a range of p valueswas used, including the Schaerer ([1975]) convention of μ=5/V (where V is in m/s). Severalcombinations of ξ and μ could often be found to satisfy the field observations.
For the two open-slope avalanches that ran in soft, fresh snow (Saddle and SnodgrassNo. 2), ξ values of 1 000 to 1 200 m/s2, combined with p values of 0.10 to 0.15, gave good fitsto the field data (Table IV), At Snodgrass No. 2, ξ = 700 m/s2 and μ = 0.10 were alsosatisfactory. The values that gave the best fit at Saddle (ξ = 1 000; μ = 0.10) overestimatedrun-out on Snodgrass No. 2 by almost a factor of two even though the two avalanches actuallystarted from the same fracture line. This most likely reflects the more uneven nature of theterrain at Snodgrass No. 2.
The two hard-slab avalanches (Breckenridge and Pallavicini) fit the Voellmy approachbetter than any of the others. Both conform well to ξ ≈ 700 m/s2 andμ. = 5/V.
The Floral Park avalanche was also very tractable with ξ ≈ 700 m/s2 and μ = 5/V,provided the measured value of 1.24 m was used for flow height in the starting zone h'. Thish’ value, when adjusted to the lower track by the second of Equation (2), closely duplicatedthe measured flow height of 2 m in the lower track and run-out zone. The one apparentlyinconsistent value computed for this avalanche is the high value of about 0.3 for μ. This value gave a good fit for the hard-slab cases, but is much higher than the best-fit values for the othersoft-slab cases. A lower μ value was expected based on the field description of the snow;however, flow through the trees on this slope probably contributed to the higher value.
The steep run-out zone (21°) of the Red Lady Basin avalanche offers some problems. Any μ value smaller than 0.38 does not permit the avalanche to stop. Values that high, however,seem unreasonable for wet snow moving over smooth, wet grass. Using different μ’s for the starting zone and run-out zone may provide better results in this case. Good results were obtained with values of 700 < ξ < 800 m/s2 for the entire path, 0.15 < μ < 0.25 for thestarting zone, and 0.4 for the run-out zone. Although this gives a good estimate of run-outdistance, the lower track velocity of about 15 m/s is half again more than the field estimate of10 m/s or less.
A more logical approach for estimating values for this type of avalanche is to assume the motion was sliding rather than flowing. This assumption enables one to disregard frictionwork caused by internal flow deformation. For sliding motion, the net slope-parallel force Facting on the block (Fig. 1) is F = μmg cos β—mg sin β, Since the block is decelerating in the run-out zone, the net force is acting up the slope. The friction work done by the block movingthrough distance s is Fs. From the work-energy theorem, disregarding flow work, dissipation of kinetic energy is equal to the friction work over the distance s, thus 1/2mV02=(μmg cos β—mg sin β)s. Hence
Fig. 1. Sliding block run-out model
where V 0 is velocity at the up-hill end of the section of the avalanche path in question, s is travel distance (along the slope), g is the acceleration due to gravity (9.8 m/s2), µ is thecoefficient of sliding friction, and β the slope angle in the run-out zone.
Substituting an assumed velocity of 10 m/s and the measured run-out distance s of 260 min Equation (4) gives a μ value of 0.405. The above approach may be a more realisticcalculating procedure for low-velocity avalanches that move primarily as sliding blocks.
Two of the design avalanches (Nicholson Lake and Bird Ridge) had very steep startingzones, steep lower tracks, flat run-out zones, and average starting zone-slab thicknesses of 1.5 to 2 m. Both had very large snow-dust clouds and Nicholson Lake had air blast in the far end of its run-out zone. The midwinter slide at Nicholson Lake, where h' = 1.5 m, required ξ = 1 800 m/s2 and μ = 0.15; ξ = 1 800 m/s2 and μ = 5/V or ξ = 1 200 m/s2 and μ = 0.1to match field run-out conditions.
The spring avalanche at Bird Ridge, which had damp snow in the lowest 300 m of itspath, required ξ > 2 000 m/s2 with μ = 0. 1 and h' = 1.5 m or ξ ≈ 1 500 m/s2 with μ = 0.1and h' = 2 m to produce a 600 m run-out distance.
Sensitivity testing at ξ = 1 800 m/s2 and μ = 0.15 for the Nicholson Lake avalanche showed ξ for the run-out zone and flow height in the lower track hLT make little difference in run-out distance when other things are held constant. Small changes in p and β however,make great differences in run-out distance. For example, a change in β from 5° to —2°shortens run-out distance by 500 m. At ξ = 1 400 m/s2 and h' = 1.5 m a change in μ from 0.15 to 0.1 lengthened the predicted run-out distance by 150 m.
3.2. Channeled avalanche
Chapman Gulch was the one channeled avalanche analyzed by the Voellmy approach (asmodified by Reference MearsMears, 1976). Starting-zone velocities were estimated from Equation (1). The combinations of ξ and μ values needed to produce starting-zone velocities of 15, 25, and 30 m/s are given in Table V. From these velocities, the length of the starting zone, and the volume of snow released, three estimates of the discharge of snow from the starting zone were computed. Assuming continuity of discharge down the track, values for μ and ξ in the track (Table V) are obtained by the simultaneous solution of the equations for velocity and dis-charge. Based on the computed track velocities, the measured run-out distance s (870 m on a 12° slope), and assumed values of ξ (500, 1 000, and 2 000 m/s2), p values for the run-outzone can be computed (Table V). The cluster of μ values around tan β (0.212 56) sharply emphasizes the physically unacceptable fact that as μ approaches tan β, run-out distance ascomputed by Equation (3) becomes independent of velocity.
3.3. Summary
In four of the nine cases analyzed by the Voellmy method a turbulent coefficient ξ between700 and 800 m/s2 with kinetic friction, μ = 5/V gave good duplication of observed run-outTable V distances (Table IV). These cases include the two hard-slab avalanches, a soft slab that ranthrough an open stand of mature trees, and a slow-moving, wet slab. Two of the designavalanches required ξ values between 1 200 and 2 000 m/s2 even with μ values as low as 0.10 to 0.15. The long-running, soft slabs of modest size required ξ between 1 200 and 1 600 m/s2for μ of 0.15 or ξ of 700 to 1 000 m/s2 for μ of 0.10. Had μ been set at 0.2 to 0.25 for therun-out zone, as is often done in practice, even higher ξ values would have been required.
Table V. SUMMARY OF COEFFICIENTS NEEDED TO FIT FIELD DATA AT CHAPMAN GULCH
The extreme sensitivity of the run-out equation to slope angle β and µ in the run-out zonecreates uncertainty in the use of this approach, especially as p approaches tan β.
Fig. 2. Slushman avalanche flowing through the upper part of the run-out zone. (Photograph bv R. G. Oakberg, Montana State University.)
4. Additional dynamics data
Direct computations were also made of the coefficient of sliding friction μ and the leading-edge velocity V in the lower part of the run-out zone from movies of the Slushman avalanchein the Bridger Range of southern Montana. This hard-slab avalanche ran in March 1978. It consisted of relatively dry, dense snow that encountered wet snow in the run-out zone. Debrismovement in the areas under study showed negligible internal agitation.
Five trees along the edge of the avalanche path were used as markers (Fig. 2). The distances and slope angles between marker trees were measured in the field. Travel time forthe leading edge of the avalanche was determined by identifying the movie frames when theleading edge passed the markers and estimating the lapsed time based on filming speed. Because of the parallax problem when viewing the film, several observers made independentestimates of when the avalanche passed the markers and an average was used to establishtravel time between markers (Table VI). Velocities are based on the measured distances divided by these average travel times.
Table VI. Travel time and velocity between marker trees,Slushman avalanche
The reduction in velocity, starting at about tree three, is of interest. Although there is onlya small reduction in slope angle, velocity drops by about 40%. On film this reduction invelocity appears to take place about where the avalanche flow becomes laterally confined in amore channeled part of the run-out zone (Fig. 2). Ordinarily, in open-channel flow a lateralconstriction would be expected to produce an increase in velocity as a reflection of greater flowheight. In the case of sliding motion, however, there is no velocity gradient, hence no increasein velocity when flow height increases. In addition, the constriction produces more friction,thus reducing velocity.
The coefficient of sliding friction μ can be calculated by equating the change in kineticenergy over some given reach of length s to the frictional work done within this same reach. Therefore, the difference of kinetic energy equals the sliding work or
where V 0 is the velocity at the beginning of the control reach, V 1 is the velocity at the end ofthe reach, and m is mass. Other terms were defined in Section 4.1. Solution of Equation (5) for the friction coefficient gives
over three measured reaches (1–2 to 2–3; 2–3 to 3–4; and 3–4 to 4–5), μ equals 0.35, 0.43,and 0.32 respectively (Table VII).
Table VII. Changes in velocity and coefficients of friction,Slushman avalanche
These friction values are close to those estimated by the Schaerer convention ([1975]) oassuming μ = 5/V for 10m/s < V < 50 m/s. For example 5/14.93 = 0.335 and 5/14.44 = 0.346. These two also are close to the assumed value of 0.3 used by Reference LaChapelle and LangLaChapelle and Lang (1980) in their analysis of this same avalanche.
5. Conclusions
With the proper selection of friction coefficients, program AVALNCH gives good estimatesof run-out distance for a wide variety of snow and terrain conditions. There were not enoughdata to get a good evaluation of the program’s ability to predict leading-edge velocity and flowheight in the track, or debris distribution in the run-out zone. There appears to be a tendency,however, to under-predict flow heights, debris dispersal, and run-out distance in cases ofsteep, adverse grade in the run-out zone. The failure to predict proper debris dispersal may bedue, in part, to snow entrainment during flow, which program AVALNCH can handleprovided sufficient data are available. The rather narrow range of values for the frictioncoefficients that suffice for most snow conditions, together with the wide range possible todescribe unusual conditions combined with the simplicity of the input data, make the programeasy to use and invites experimentation.
Voellmy turbulence coefficients ξ of 1 500 m/s2 or greater, and kinetic friction μ values of 0.15 or lower, were needed to duplicate some of the field data. If μ. in the run-out zone is setat 0.2 to 0.25, as is common practice with some workers, even higher ξ values are needed to match observed conditions. These are much higher £ values than were suggested by Voellmy(1955), but are close to those given by Schaerer ([1975]) for flow over a deep, dense snowcover. The extreme sensitivity of the run-out equation to kinetic friction μ and slope angle βspecifically when μ approaches tan β, greatly limits the usefulness of this equation.
The lack of unique solutions for velocity and run-out distance is a major shortcoming ofboth the Voellmy approach and program AVALNCH. Both techniques also requireexperience and judgment in the selection of proper coefficients for the conditions encountered. Additional case studies are badly needed to improve objectivity.
