At present, few measurements of avalanche velocity, dynamic pressures, densities, flow dimensions or length of travel (runout distance) have been made. Such data are difficult to obtain for obvious reasons, but are gradually being accumulated through efforts of the U. S. Forest Service, the National Research Council of Canada, and the Center of Snow Studies in France. Because of this paucity of data, the present discussion does not advocate a particular model of avalanche flow. Rather, it summarizes existing information and presents some basic ideas about the forces operating within avalanche as related to snow type. Further research and data collection is necessary for understanding of the basic principals because any theoretical treatment of the phenomenon, regardless of its complexity, can produce misleading results if the analytical manipulations are based on unrealistic assumptions.
Driving Force and Motion Resistance
After fracture and release of a slab the snow accelerates downslope as an avalanche. The initially rigid slab is broken up into a sliding, tumbling, and bounding movement of fragments. If it travels far enough or is broken up into fine enough particles the motion develops into a flow. Two opposing sets of forces operate upon or within the flowing avalanche: (1) the driving force, F, is the component of its weight parallel to the slope, and (2) the resisting force, R, is composed of several forces which oppose F and prevent the avalanche from accelerating indefinitely (fig. 1).
Figure 1. The driving force, F, and the resisting forces, R, act on a
moving avalanche to determine its acceleration and maximum velocity.
The relative importance of the various resisting forces depend on the
type of avalanche.
Regardless of details about the forces, the avalanche will accelerate as long as F exceeds R because there will be a net force (and a resulting acceleration) down the slope. The condition (F>R) is satisfied in the upper portion of the avalanche path, or starting zone. Here acceleration occurs because terrain is steep and F is large. Additional unstable snow on the steep slopes of the starting zone can be dislodged and incorporated into the avalanche causing the flowing mass to increase. For avalanche dynamic analysis the starting zone is usually defined as the area within the avalanche path where slopes exceed roughly 30 degrees. This usually exceeds the area of the initial slab fracture.
Constant velocity is achieved when F equals R. This condition of maximum or terminal velocity occurs in the transition between the starting zone and the bottom of the avalanche path. This transition area is usually referred to as the track. In fact, even in the track velocity continually changes in response to changes in gradient, entrainment of new snow, deposition, and topographic irregularities.
Deceleration occurs in the runout zone where R exceeds F. Here the gradient is reduced and as a result the driving force, F, is also reduced. Furthermore, the underlying snow pack is stable, the kinetic energy of flow is dissipated, and a deposit of avalanche debris is formed. The distribution of the avalanche deposit defines the limit of the runout zone and probably depends more on released snow type and avalanche characteristics than on ground roughness in the runout zone.
The above discussion applies to all avalanches regardless of type or size. They all begin in a starting zone and stop in a runout zone. In any given path the type and amount of snow released and incorporated into the avalanche determines the balance between forces, the velocity, the dimensions, the amount of snow entrained, and the location of the runout zone. Thus a given avalanche path is the location of avalanches of many sizes and, as with other natural phenomena, there exists an inverse relationship between the size and the probability of the event.
Figure 2. Parts of an avalanche path as related to the balance between forces and slope inclination.
Resisting Forces and Snow Type
The following are probably the most important factors contributing to frictional resistance within an avalanche. Although they are separated here for purposes of discussion, they probably interact with one another within an avalanche.
R1: Sliding friction between the avalanche and the underlying snow or ground.
R2: Internal dynamic shear resistance due to collisions and momentum exchange between particles and blocks of snow.
R3: Turbulent friction within the snow/air suspension.
R4: Shear between the avalanche and the surrounding air.
R5: Fluid-dynamic drag at the front of the avalanche.
The net force available to accelerate the avalanche results from the summation of all the forces acting upon it or
where F is the downslope component of the avalanche weight. The relative importance of the individual resistance terms varies within the avalanche and depends on the type of avalanching snow. The type of avalanching snow, in turn, depends on the characteristics of the snow pack prior to avalanche release, on the type of snow encountered in the track, and on avalanche path topography. In a dry snow avalanche the density and mechanical strength of the released slab probably determines the importance of the various resistances.
In hard slabs the snow is bonded together strongly, Thus, shortly after release the majority of the avalanche mass is apparently composed of relatively large snow blocks (probably 10 to 100 cm in length). These blocks slide, roll, bound, and collide with one another but because of their large size and high free fall velocity in air they never become suspended well above ground level by turbulence. Instead the mass moves as a cascade of discrete snow blocks, and may never become a true "flow." Resistance to movement is primarily from forces R1 and R2, Forces R3, R4, and R5 probably contribute little to avalanche resistance, and purely fluid-dynamic models of avalanche motion probably do not accurately describe this particular type of avalanche.
After a soft slab release, disintegration of the slab and air entrainment are very rapid and as the avalanche develops much of the mass is suspended well above ground level. As a result of the increased mean distance between snow particles in the flowing snow/air suspension, flow height increases, velocity increases, the avalanche assumes the form of a fluid, and forces R1 and R2 diminish in importance. Thus fluid-dynamic resistances (R3, R4, R5) limit velocity. In some cases a large mass of the flowing snow is whirled into suspension and held well above ground level by turbulence as a powder avalanche. The denser mass, comprised of large snow particles, remains closer to the ground and is called a flowing avalanche. Powder avalanches can sometimes attain very high velocities (over 60 m/sec) and may travel very long distances in the low-gradient runout zone at the base of an avalanche path. Most dry soft slab releases develop into mixed flowing and powder avalanche forms.
Figure 3. An avalanche resulting from a released hard slab is
comprised of large snow fragments.
Figure 4. An avalanche resulting from a released soft slab is
comprised of a "fluidized" suspension of snow and air.
Avalanches resulting from the release of wet snow, either point failures or slab fractures, usually disintegrate quickly into a pasty mass or slurry which moves at relatively low velocities and follows gullies faithfully. Because there is very little air entrainment in wet snow avalanches the flow does not attain great heights. Resisting forces R1 and R2 are much more important than R3, R4, and R5. However, in spite of relatively low velocities, wet snow avalanches can be very damaging upon impact because of their high densities.
Table 1 lists typical velocities for various types of avalanches. The values are opinions of researchers in Europe and North America, and are only best estimates.
Table 1. Typical avalanche velocities (m/sec)
||10 to 20
||20 to 35
||10 to 35
||35 to 60
||25 to 60
||60 to 90
Avalanches can produce very large dynamic forces on objects; thus knowledge about these forces becomes an important practical design criterion for objects located within avalanche paths.
High velocity, low density dry snow avalanches (velocities > 20m/sec; densities < 200 kg/m3) may flow over and around objects, engulfing them in much the same way as a true fluid. This produces a fluid-dynamic stagnation pressure (a force per unit area). The stagnation pressure is calculated simply as
P = 1/2 pV2 (1)
where P is the pressure, p is the avalanche density, and V is avalanche velocity. Total force on the object consists of drag and uplift forces which act, respectively, parallel to the flow direction, and perpendicular to the flow direction, upward on the object. These forces are calculated by multiplying the stagnation pressure by the exposed surface area of the object and by a coefficient of drag of lift. Appropriate values for these coefficients may be found in standard texts on fluid mechanics.
Denser, slower-moving avalanches will generally not engulf an object. Instead, some of the mass is brought to rest against the object and some is deflected. In these cases the total pressure is calculated simply as
P = pV2 (2)
Thus the reference pressure level is exactly twice that of the faster moving, less dense avalanches.
Because important differences exist between the mechanics of impact of different types of avalanches, one must decide which type is likely to reach the area of interest and constitute the design case.
Figure 5. Impact of a dry snow or powder avalanche may produce
both drag and lift forces on an object.
Figure 6. Impact of a dense, wet snow avalanche.
Avalanche Dynamic Equations
The theoretical basis of avalanche dynamics most commonly used in analysis in Europe and North America was first derived by Voellmy (1955). Little additional research has been done on the flow of snow since that time, even in the Swiss Alps where severe land-use problems in avalanche paths exist. In the United States, research on this subject has been postponed until recently when we have begun to experience increasing pressures to develop in potential avalanche zones. Consequently, the technical or engineering approaches to avalanche dynamic analysis has closely followed Voellmy's fluid mechanics model of avalanche movement.
As discussed earlier, not all avalanches move as fluids or can be adequately modeled through fluid mechanics, however, such an approach is preferable to purely subjective ones. In time more data will be collected, measurements will improve, and perhaps new experiments will be carried out which will affirm, modify, or negate the present model. In the meantime, realizing that the results of the use of the present approach are only approximate, land-use decisions will and must be made based on what we consider to be the state of the art in avalanche analysis. This is a normal procedure in science and engineering, and if allowed to operate, it will gradually provide improved models upon which more confident estimates of the extent of the avalanche hazard can be made.
Voellmy's equation for the maximum velocity an avalanche will reach on a uniform track inclined at an angle o is
where h' is the flow height, e is the coefficient of turbulent friction, and u is a coefficient of sliding friction. For avalanches confined to a channel the flow height, h', is replaced by the hydraulic radius, R.1/ This equation has been found to be reasonably reliable in cases in which it was applied to hilly developed dry flowing avalanches.
Voellmy's equation is highly dependent on chosen values of e, u and h', but only rough guidelines can be given for selection of proper values. E probably varies with the roughness of the ground surface in much the same way as empirical river friction coefficients.
The dynamic friction coefficient u , probably varies between 0. 1 and 0.3 depending on avalanche velocity. At higher velocities the lower values of u approaching 0.1 should be used. The flow height, h', depends on the amount and type of snow released, consequently proper selection of this value requires knowledge about expectable snowpack conditions in an area. For soft slab releases it is not justifiable to assume the flow height is equal to the height of the released slab because turbulent motion tends to quickly disaggregate and fluidize the slab. Equation (3) probably does not apply to hard slab avalanches in which the blocks in the flow are large, or to wet snow avalanches which slide at velocities of less than 10 m/sec.
Voellmy ( 19 5 5) also gives the following equation for the distance, S, an avalanche will travel in its decelerating phase in the runout zone. This is written
where is the slope of the runout zone and h' is the flow height. Note that this equation is also very sensitive to e, h', u, and also to V, thus it is advisable to use as many other indicators of avalanche runout distance, such as debris distribution, as possible when calculating potential avalanche effects.
Determination of the runout distance, S, of large, infrequent avalanches in an area is of primary importance in land- use planning near avalanche paths. Thus, proper selection of "reasonable" values of the various coefficients of equations (3) and (4) is an important practical problem.
Present research on avalanche dynamics by the U. S. Forest Service attempts, in part, to place reasonable constraints on the various coefficients through observations of large avalanches.
1/ The hydraulic radius is equal to A/P where A is the cross-sectional area of the avalanche and P is the wetted perimeter of that area.
Leaf, Charles F., and M. Martinelli, Jr. 1977 Avalanche Dynamics: Engineering applications for land use planning. USDA For. Serv. Res. Pap. RM- 183, 51p. Rocky Mt. For. and Range Exp. Stn., Port Collins, Colo. 8052 1.
Mears, Arthur I. 1976. Guidelines and methods for detailed snow avalanche hazard investigations in Colorado. Colo. Geol. Surv. Bull. 38, 125p. Denver.
Voellmy, A. 1955. Uber die Zerstorungskraft von Lawinen. Schweiz. Bauzeitung, Jahrg. 73, S. 159-165, 212-217, 246- 249, 280-285. (In English as: On the destructive force of avalanches. 63p. Alta Avalanche Study Center, Transl. 2, 1964. On file at Rocky Mt. For. and Range Exp. Stn., Fort Collins, Colo.)