Introduction

Flowing avalanches (see Fig. 1) are those with a dense core of granular flowing snow at the base, usually with flow depth of one to several meters. Calculations of avalanche run-up for flowing avalanches are very important for the design of defenses to slow down or stop the avalanche debris core, and the problem is strongly related to calculations of run-out distances using dynamic methods.

In this paper, a practical method is introduced for calculating run-up and run-out, using avalanche-dynamics principles. The model is compared to a related model used in Switzerland which is designated as the present, standard Swiss guidelines (Reference Dent,Salm and others, 1993) tor avalanche-dynamics calculations in run-out zones. The results illustrate important differences in the mathematical and philosophical principles used in estimating stopping distances of avalanches from dynamics principles. The differences are illustrated using field-documented examples of avalanche run-up and run-out and theoretical comparisons using identical model parameters.

Fig. 1. Flowing avalanche enveloped in a turbulent dust cloud engaged in run-up on the side of a mountain.

The focus of our paper is a simple, practical method far run-up or run-out prediction similar to that proposed in the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990). Thus, our work is intended to apply to only one, small but important, part of the avalanche-dynamics problem, Specifically, we provide a simple theoretical framework from which it is possible to calculate run-up or run-out for avalanches containing dry snow, provided that the friction coefficients (including internal friction), flow depth of material approaching the run-up or run-out slope, and slope geometry are known.

We do not address here the important question of haw to estimate incoming speeds and friction coefficients. We however, provide our prescription of how friction coefficients enter the run-up (or run-oat) dynamics problem, since our view differs from the somewhat traditional one expressed in the Swiss guidelines. Due to the complexity of the avalanche-dynamics problem and the simplifications introduced for one-dimensional motion, it probably is not yet possible to say definitely which view of this important question is most nearly correct. Our comparisons with field examples do not completely resolve this issue: high uncertainty remains with respect to friction-parameter input.

Flow and Boundary Conditions

We consider first a model for run-up and run-out calculations based upon solution of the dynamic-equilibrium equations and continuity proposed by Reference Hungr, and McClung,Hungr and McClung (1987) and modeling at slope transitions proposed by Reference Takahashi,Takahashi and Yoshida (1979) and Reference Takahashi, and H,Takahashi (1991) for the related problem of debris-flow dynamics. The model is a depth-averaged, one-dimensional formulation with constant densities assumed within, ρ, and at the top of the core ρ_t, of the flowing avalanche.

In order to develop a practical dynamics model relevant to run-up (and run-out) problems, it is very important to begin from a realistic physical picture of the avalanche-flow conditions. Since, in reality, the processes involved are very complex, it would be very difficult to model them in great detail. One must, then, rely on a proper interpretation of the trends in the processes involved to arrive at a model with the correct form of boundary conditions. The boundary conditions will be crucial for determining the dynamic behavior of snow avalanches.

Direct observations of the flow conditions inside avalanches are very rare and the information available is not very precise. We are concerned here with physical modeling of the conditions at the top and bottom of the flowing snow as well as with the internal friction between snow particles. Observations, measurements of flow conditions and avalanche impact pressures indicate that the core of flowing avalanches is a dense granular material (Reference McClung, and Schaerer,McClung and Schaerer, 1985, Reference McClung, and Schaerer,1993; Reference Gubler,, Hiller,, Klausegger, and Suter,Gubler and others, 1986; Reference Salm,Salm, 1993). By “dense”, we mean that the concentration of solid material (volume fraction filled by solid material) is high enough that the probability of collisions between particles is high during all stages of the flow. However, during run-out and run-up there will also be considerable rubbing friction between the particles as they become close enough for enduring particle contact. In the final stage, locking takes place and static friction is approached. The volume fraction filled by solids has not been precisely measured in run-up and run-out problems. However, hack calculations by Reference McClung, and Schaerer,McClung and Schaerer (1985) yielded estimates of 30–50% solids concentration within the How for measurement in the track (or transport zone) before significant deceleration and run-out deposition took place. These measurements were taken just above the ground surface but are not precise enough to give estimates relevant to the crucial boundary condition at the bottom. In general, we expect the volume concentration of solids to increase from die top to the bottom of a flowing avalanche.

If the estimates of solids concentration from impact data are reasonable, very important implications arise with respect to run-up and run-out problems. In the final stages of motion, we may expect the solids concentration to be even higher than the 30–50% range measured in the track. Reference Gubler,, Hiller,, Klausegger, and Suter,Gubler and others (1986) estimated from precise radar measurements that the flowing material shows concentrated shearing deformation at the bottom with very little internal shearing in the main body of the flow (Reference Salm,Salm, 1993). This information is consistent with the picture that the relevant volume concentration of solids is high for run-up and run-out problems, with perhaps a slightly reduced value within the zone of active shearing at the bottom.

Observations of actual dry deposits from large avalanches show that particle size decreases rapidly with depth into the deposit. Particle sizes at the surface are typically in the range 1–10 cm, but deeper in the deposit particles are of much smaller sizes down to the millimeter Scale. These observations show that there have been intense interactions (collisions) between particles and with the sliding surface at some stages of their travel down the mountain. Again, the indication is that the flow must be considered dense: the collision mean free path is smaller than the distance between the particles. In addition, since the density of the fluid (air) is at least 10⁻²–10⁻³ that of the particles, these two trends (low fluid density and high volume concentration of particles) mean that we expect momentum transfer to be due to particle and rubbing friction between particles, with the fluid ignored in the mechanical description. This is particularly likely to be true in run-up and run-out problems dealing with the final stages of motion, Reference McClung,McClung (1990) has discussed this physical condition extensively.

For run-up and run-out, then, we consider the motion of a dense core of material with lower-density material suspended in turbulent eddies around the upper surface of the core. Reference Takahashi, and H,Takahashi (1991) took the bold step of uncoupling turbulent fluid processes and particle-particle-collision and rubbing-friction effects to enable the basal shear resistance to be written as the sum af two terms. In what follows, we adopt Takabashi’s strategy initially, but in the final formulation of our model we do not retain the uncoupled form to express the motion resistance at the upper and lower boundaries. The assumption that the basal shear resistance can be written as a sum (Equation (1) below) has been traditional in avalanche-dynamics formulations since Reference Takahashi, and H,Voellmy (1955) and Reference Salm,Salm (1966). Since the Swiss guidelines essentially contain this assumption, it is retained here initially, but it is not contained in our model. At the bottom of the flow, then, we take the downslope resistance, τ_b, to motion as the sum of two terms:

where T is resistance due to particle interaction with the sliding surface and adjacent particles within the flow, and τ^′ turbulent friction generated by any turbulence within the interstitial air at the base of the flow. For the dense flow at the base of the flow, however, we assume that τ^′ ≪ T. This is because the high density (volume fraction) of particles at the base of the flow will prevent turbulence from forming within the interstitial air. Furthermore, we assume (Reference Dent,Dent, 1986, Reference Dent,1993; Reference McClung,McClung, 1990) that bottom friction (due to collisions and rubbing) is coupled to the normal stress provided by the overburden of material by a dynamic coefficient of friction, μ, which may in general depend on rate of shearing and volume fraction as well as roughness and condition of the sliding surface. Basal drag will always be dominant in run-out and run-up, with drag at the top of the flow often being negligible. We then adopt a dynamic Coulomb-type relation

for our definition of basal resistance. In Equation (2), ρ is depth-averaged density, h is mean flow depth along the run-up (or run-out) slope and ψ is slope angle. We later derive an approximate expression for h and replace it by the expected mean deposition depth in the run-out zone. Reference Dent,Dent (1993) shows examples of how μ may vary with speed, particle-restitution coefficient and inter-particle friction for dense, rapidly sheared granular materials.

At the top of the core, we envision resistance to motion being represented by turbulent drag due to air/snow dust by representing the upper surface of the flow. If one adopts Takahashi’s strategy, an equation analogous to Equation (2) can be written for the top of the flow, but in this case Τ is negligible and τ^′ is of primary importance. The upper surface of the core presents a rough surface which generates friction as the avalanche is transported along the incline with speed, υ. The upper surface of the core is envisioned as consisting of a rough surface with saltating particles at the top immersed in a snow-dust and air mixture suspended in turbulent eddies. The drag at the top of the flow is represented as:

where ρ_i is density of the snow-dust-air mixture and is about 10 kg m⁻³ (Reference McClung,McClung, 1990) and C _D is a drag turbulent flow over a rough upper periphery of the core. Following Reference McClung,McClung (1990), we estimate that the value of C _D is about 0.01, Fran Schlicting (1972). In at least some problems of run-up and run-our, we believe that the contribution of Equation (3) to stopping dynamics may often be very small (if not negligible). For completeness, however, we will retain Equation (3) to represent drag at the top of the flow instead of prescribing a free surface there.

For Equation (2), ρ is the mean density of material in the tore of the avalanche. Since the mixture there may consist of snow particles (a mixture of ice and air) and air, we may write

for the mixture, where C is the volume fraction filled by snow particles, ps is the density of snow particles, and ρ_A is the air density. With C in the range 0.30–0.50, as estimated by Reference McClung, and Schaerer,McClung and Schaerer (1985), and with ρs in the range 200–500 kg m⁻³ (Reference McClung, and Schaerer,McClung and Schaerer, 1993) and ρ_A = 1 kg m⁻³ (density of air), ρ is in the range of about 100–300kg m⁻³, or at least ten times ρ_t, the density at the top of the flow. At the bottom of the flow, due to destructive collisions, it is possible that the particles consist of millimeter-size individual ice grains (917 kg m⁻³), and the flow density may be as high as 450 kg m⁻³.

In the interior of the flow, it may be necessary to account far internal friction between snow particles in run-up and run-out problems. Reference Salm,Salm (1993) points out that internal shearing is only possible if the internal friction angle Φ is less than the slope angle in first approximation. Since run-up and run-out problems are, from field observations, those in the last stages of motion in which at least sons of the material is “locked” or not undergoing rapid shearing, we must have a mechanism to account far the internal resistance in the flow mechanics. Reference Salm,Salm (1966, Reference Salm,1993) proposes that allowance be made for passive snow pressure analogous to passive earth pressure for the compressive states of stress expected in run-up and run-out problems. The longitudinal passive snow pressure (per unit width) may be represented as (Reference Craig,Craig, 1988):

where (ϕ Is internal friction angle)

In Equation (6), cohesion is ignored, and h0 and ψ₀ represent flow depth and slope angle on entering the runout zone, respectively. For ϕ< ψ₀, Equation (5) reduces to the fluid analogy introduced by Reference Hungr, and McClung,Hungr and McClung (1987) for which k _p = 1. For ψ = 0° Equation (6) reduces to the expression given by Reference Salm,Salm (1993). There is considerable uncertainty with respect to the role of passive snow pressure in avalanche dynamics. Since the constitutive equation for flowing snow is unknown, our formulation takes the simplest possible form by specifying a constant internal friction angle. Below we discuss some preliminary suggestions for a rough determination of ϕ based upon Reference Salm,Salm (1993) discussion and our own observations. Reference Salm,Salin and others (1990) and Reference Salm,Salm (1993) estimate for run-out zones based upon avalanche deposits that ϕ is about 25°. This estimate (ϕ = 25°) is recommended in the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990), and we shall also retain this estimate far run-up and run-out problems. With ϕ = 25° and ϕ₀ = 0°, Equation (6) gives k _p = 2.5. Equation (6) with ϕ = 25° predicts that locking in the flow will begin at slope angles of about 25°. Thus passive snow pressure will begin to become a factor in rim-up and run-out problems fer slope angles less than 25°, if ϕ is taken as 25°. This seems to be reasonable as a rough estimate, as speeds in large, dry avalanches are usually beginning a decelerating phase for such slope angles (Reference Gubler,, Hiller,, Klausegger, and Suter,Gubler and others, 1986), and the flowing snow should be entering a compressive phase.

Evidence from avalanche deposits (described above) shows that particle sizes decrease rapidly below the surface of deposits to produce a uniform distribution of particles of small size. This indicates that shearing deformations and intense destructive collisions of particles have taken place through the depth of the flow at some stages of motion. Reference Salm,Salm (1993) points out that ϕ has never been estimated in flowing snow. Static values for ϕ are estimated by Reference McClung,McClung (1987) in the range 50–60°; such values may be reasonable for deposited snow, but here we provide estimates during run-up and run-out for which we believe the static estimates (50–60°) would be too high. Physically, k _p then depends on flow conditions and state of stress, but for our practical work we estimate it by choosing a reasonable value for ϕ and then incorporate geometric effects using Equations (5) and (6). Field estimates of impact pressures in the transport zone (track) by Reference McClung, and Schaerer,McClung and Schaerer (1985) show that pressures in large, dry avalanches have a highly fluctuating component for a slope angle of 31°. These measurements indicate that the material is not in a completely locked state until lower slope angles are encountered in the deposition zone. These observations make it reasonable that passive snow pressure begins to take effect as the mass approaches the deposition zone consistent with a value far ϕ near 25°. We recommend adopting an angle, ϕ, with an approximate value intermediate between 0° (no friction] and 50–60° (static value from in situ tests) to describe flowing snow in the final stages of run-up or run-out. Accordingly, we have chosen ϕ= 25°, consistent with the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990).

Leading-Edge Theoretical Model

For run-up and run-out, we adopt a one-dimensional model to calculate the position of the leading edge of the avalanche, with quantities averaged through the depth of the flowing snow measured perpendicular to the surface over which the avalanche is flowing. We consider the flow as incompressible so that the density of material in the body of the flowing snow is assumed constant. We also assume that the discharge per unit width, Q (m²s⁻¹), is constant throughout run-up or run-out. The assumption of constant Q is a very rough approximation. Consistent with the background material above, the resistive forces are a dynamic Coulomb friction force at the bottom of the avalanche (Equation (2)) and a turbulent resistive force at the top and around the upper periphery of the avalanche body (Equation (3)). Note that our boundary-condition formulations do not require the assumption implied by Equation (1) of decoupling the stress terms, even though this assumption is rooted in mast avalanche-dynamics models since Reference Voellmy,Voellmy (1955).

Considering the problem of run-up, following Reference Hungr, and McClung,Hungr and McClung (1987) and Reference Takahashi, and H,Takahashi (1991), we express Newton’s second law (see Figure 2 for the geometry) as:

Fig. 2. Schematic of flowing-avalanche run-up on an adverse slope; (a) lumped-mass or centre-of-mass model, (b) leading-edge model.

The quantities are expressed as force per unit width of the flow, and terms T ₁–T ₂ are driving and resistive farces affecting motion, h is mean flow depth along the run-out slope, and x is distance along the run-out slope measured from zero at the beginning of the run-out slope. Terms T ₁–T ₂ are given below (force per unit width); of these, T ₃ and T ₄, are always resistive perms.

• T ₁ Driving force

  

• T ₂ Momentum flux

  
  approaching the run-out or run-up slope, where ψ₀, υ₀ and h ₀ are initial slope angle, speed and flow depth, respectively

• T ₃ Dynamic Coulomb resistive force (at the bottom)

  

• T ₄ Turbulent resistive farce (at the top)

  

• T ₅ Passive snow-pressure force term

  

Equation (7) is derived far a very general momentum principle for constant supply to the flow. The momentum flux term T ₂ is calculated as the incoming mass flux (per unit width) times the incoming velocity, with the velocity taken as the mean value through the depth of flow. It represents the discharge (or supply) of momentum to the run-out or run-up slope. See Appendix Β far an explanation of this term.

By analogy to fluid dynamics, we assume supercritical at the beginning of the run-out slope so that conditions upstream are not influenced. Typical speeds and flow depths for avalanches indicate this is a good assumption. The other terms on the righthand side of Equation (7) come from the sum of all the forces acting an the flawing avalanche mass, including gravity, friction at the top and bottom and internal, passive snow pressure.

These formula differ from Reference Hungr, and McClung,Hungr and McClungs (1987) for terms T ₄ and T ₅. In Τ ₄, we assume here that turbulent resistive force acts only at the top and upper surface, not the bottom, of the avalanche. In T ₅, we have included friction between the snow particles (for ϕ ≥ ψ₀) as expressed in Equation (6). In order to solve for the run-out or run-up distance, we salve Equation (7) for the distance, x_R , when ν = 0 with the initial condition that t = 0 when x = 0. In our formulation, slope angles (ψ, ψ₀) are taken positive if the slope is downward from horizontal and negative if upward (adverse slopes). In Equation (7), ψ is taken positive on the approach slope and negative on run-up slope (or for adverse slopes in run-out). In addition, we must invoke the depth-averaged, integrated continuity equation so that mass is conserved:

The derivation of Equation (8) requires the assumption of constant mean dynamic flow depth, h , in the run-up (or run-out) zone.

Rearrangement of Equation (7) gives:

where

with

On the run-up or run-out slope we seek a numerical solution of Equations (8) and (9) for ν as a function of time. The run-up or run-out distance is then given by the integral of ν from t = 0 until t = t _R, the time taken on the run-up slope until the front of the avalanche stops. It is of interest to compare the two resistance terms G ₀ t and D ₀ υ ² t for a run-up slope. The ratio of these terms is: G ₀/D ₀υ² where, on the adverse run-up slope, G ₀ is represented by

i.e. ψ is negative.

Typical values for D ₀ result from (Reference McClung, and Schaerer,McClung, 1990)

and in run-up problems. If ψ = 30°, μ = 0.3, and ν = 20 m s⁻¹, the ratio becomes G ₀/D ₀υ² = 112. For our model, therefore, in some cases of practical interest, the term D ₀υ² t may be ignored in run-up (and run-out) problems, and drag may be attributed to dynamic Coulomb drag at the base of the avalanche. When the term D ₀υ² t is ignored, an analytical solution is available and the run-out distance is

and the run-up height is

Following Reference Salm,Salm (1993), we derive an approximate expression for the mean deposition depth on the run-up (or run-out) slope. Neglecting the turbulent term, the solution for the speed along the deceleration slope is:

For an approximate mean flow depth in the run-up (or run-out) zone, we assume from conservation of mass, following Reference Salm,Salm (1993), a constant supply of mass per unit width entering into the run-up (or run-out) zone until the tip comes to rest. The balance for the flux of material (per unit width) is:

where h is instantaneous flow depth during run-out.

Equation (15) is written for one-dimensional motion, and if the approximations dh/dx ≪ C 1 and constant density are assumed, simplification results. Integration of Equation (15) all along the deposition zone, with h= h taken as constant, gives the mean deposition-zone depth (Reference Salm,Salm, 1993):

In Equation (16), quantities with a bar ( v ) are averaged along the length of the run-up or run-out zone. Integration of Equation (14) (assuming neglect of turbulent drag) gives:

where V is given by Equation (10), With neglect of passive snow pressure and the slope-angle momentum correction

, Equation (17) predicts that h is about 1.5h ₀.

When slope-angle momentum correction is important (such as in run-up), Equation (17) predicts that h can be as high as 3h ₀. Field measurements show that this approximate range of values (1.5/h ₀ to 3h ₀) is reasonable. The approximations and one-dimensional character of Equation (17) mean that only rough estimates are provided. In problems for which turbulent drag is important, the estimate in Equation (17) could be revised by numerical integration to derive the speed estimate analogous to Equation (14) which is given for neglect of turbulent drag. However, since Equation (17) is only a rough estimate, and since we expect turbulent drag to be small in many run-up and run-out problems, this added sophistication is probably unnecessary. Following Reference Salm,Salm (1993), for practical problems we now replace h in our theoretical model with h given by Equation (17). This assumption is equivalent to taking the mean deposition depth as our estimate of the mean flow depth in the run-out Or run-up zone.

Equation (17) provides only an approximate estimate (e.g, Reference Salm,Salm, 1993): dynamic dependence on flow-depth changes and turbulence during run-out are ignored and constant discharge is assumed. In reality, input discharge is expected to decrease from a certain time after the passage of the avalanche front. Since deceleration and deposition processes relate to non-equilibrium states, an expression like Equation (17), derived from idealized simplifications, is only a rough estimate. Avalanche-flow height variations and deposition depth often display two- and three-dimensional effects, not included in Equation (17).

Lumped-Mass Models

Models containing the assumption that motion may be described by the centre of mass of the avalanche have been the most popular for run-up and run-out problems. The models of Reference Voellmy,Voellmy (1955), Salm (1979), Reference Perla,, Cheng, and McClung,Perla and others (1980) and Reference Salin,, Burkard, and Gubler,Salin and others (1990) are mathematically equivalent to motion description as if the avalanche mass were concentrated at a point at the centre of mass. The model of Reference McClung,McClung (1990) is also a centre-of-mass model but is intended only for speed estimates, not run-out estimates. Lumped-mass models have two disadvantages in run-up and run-our problems:

• 1) Since motion description is with respect to the centre of mass instead of the tip of the avalanche, such models predict shorter run-up or run-out (Reference Hungr, and McClung,Hungr and McClung, 1987; Reference Chu,, Hill., McClung,, Ngun, and Sherkat,Chu and others, 1995) if the same friction coefficients are used.

• 2) It is not possible to include passive snow pressure explicitly in the run-up or run-out formulation since only external forces can be specified in a point-mass model.

Salm (1979), Reference Perla,, Cheng, and McClung,perla and others (1980) and Reference McClung,McClung (1990) have derived the differential equation for the lumped-mass models analogous to Equation (9):

In Equation (18), ds is an element of path length along the incline. For the simple geometry of Figure 2, we assume, following Reference Perla,, Cheng, and McClung,Perla and others (1980), that only the slope-parallel component of velocity is transferred on to the run-up or run-out slope:

when ψ₀ is greater than ψ. The solution to Equation (18) is then (e,g. Salm, 1979; Reference Perla,, Cheng, and McClung,Perla and others, 1980):

where V ₀ ² = G ₀/D ₀. Equation (19) is mathematically equivalent to speed estimates in the Swiss guidelines (Salm, 1993) except that the Swiss guidelines do not account for the momentum loss at slope transitions when the slope angle decreases (it is assumed that (cos(ψ₀ – ψ → 1)). Also, in the guidelines, the parameter D ₀ is envisioned as accounting for turbulent drag at the base of the flow instead of at the top of the flow as we envision it. Appendix A explains the relation between D ₀ h and its alternative representation in the Swiss guidelines (g/ξ) for our model and the Swiss guidelines. With the substitutions, D ₀=g/ξ h (Swiss guidelines) instead of

and cos(ψ₀ – ψ) → 1, the speed equations of the Swiss guidelines are recovered.

The solution to Equation (18) for the run-out distance is:

For the run-up height, we get (ψ is taken negative)

With the substitutions above, cos(ψ₀ – ψ) → 1 and D ₀=g/ξ h Equation (20) is identical to that given for run-out distance in the guidelines (Reference Salm,Salm, 1993). Using L’Hospital’s rule, in the limit as D ₀ → 0 (neglect of turbulent drag), Equation (20) reduces to

For run-up, we take ψ negative in G ₀, and the run-up height is:

Equations (22) and (23) may be compared to Equations (12) and (13) for the leading front model. If passive snow pressure is neglected, Equations (12) and (13) predict exactly twice the run-out or run-up. This comparison shows a fundamental difference between the models: the leading-edge model always predicts longer run-up or run-out if the same incoming speed and friction coefficients are used, because it simulates the motion of the tip of the avalanche (instead of the centre of mass) and because it contains passive snow pressure explicitly. Reference Hungr, and McClung,Hungr and McClung (1987) have emphasised these differences previously. In addition to a different momentum formulation, the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990; Reference Salm,Salm, 1993) contain no provision for momentum Loss as slope angle decreases. Experimental data from small-scale experiments on granular flows show that the correction,

, is important in run-up (Reference Chu,, Hill., McClung,, Ngun, and Sherkat,Chu and others, 1995).

To compare solutions for the leading-edge model and the centre-of-mass approach for cases in which turbulent drag is important, the numerical solution to Equations (7), (8) and (17) may be compared to Equations (20) and (21). To complete the model for the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990; Reference Salm,Salm, 1993) it is necessary to introduce the mean run-out zone deposition depth analogous to Equation (17) for our model. Reference Salm,Salm (1993) gives the expression from the Swiss guidelines from an engineering hydraulics argument:

where λ = tan²(48° + ϕ/2). This can be related to our expression for passive snow pressure by substitution: λ —> k _p cos ψ₀ with ψ₀ = 0°. In other words, cos ψ₀ is explicitly taken as I in the Swiss guidelines instead of accounting for the multiplier cos ψ₀ as is usual in passive-pressure theory (Reference Craig,Craig, 1988). For comparison between the leading-edge and lumped-mass models in the next section, we shall specify a value for ϕ to achieve the same assumptions.

For run-up estimates, the guidelines (Reference Salm,Salin and others, 1990) simply specify:

for an approximate estimate of run-up from analogy to engineering hydraulics.

Model Comparisons and Field Examples

In order to calculate run-out or run-up from our model or the Swiss guidelines, one must have estimates of υ₀, h ₀, μ, ϕ, and D ₀ (our model) or ξ (Swiss guidelines). In addition, the geometry must be known (ψ₀ and ψ). Field examples in which all of these parameters arc known accurately are virtually non-existent; one must always rely on theoretical estimates for the parameters to some extent. In this section, we compare the models theoretically by using the same geometry and parameter estimater with υ₀ varying, and we also attempt comparison for field examples in which at least the incoming speed, υ₀, is measured and path geometry is known.

Summary

We have presented a method for calculating run-up and run-out in the deceleration phase of avalanche motion. Our method applies only to the important but restricted case in which initial conditions can be estimated: incoming speed, incoming flow depth, friction coefficients and internal friction, Our method is similar in spirit to that proposed in the Swiss guidelines (Reference Salin,, Burkard, and Gubler,Salin and others, 1990), but the modelling details differ. Some differences include:

• 1) Instead of a centre-of-mass model, as in the Swiss guidelines, we calculate the stopping position of the tip of the avalanche by simultaneous solution of the momentum and continuity equations. The implication is that, given the same friction conditions and flow depth, our model gives longer (and more conservative) run-out. Similar findings arc reported by Reference Salm,Salm (1993) for another model which takes into account momentum and continuity: longer run-out is predicted.

• 2) We estimate mean deposition depth (or mean run-out zone flaw depth) from supply as proposed by Reference Salm,Salm (1993). This (approximate) estimate gives reasonable flow depths which are used in the calculations. According to Reference Salm,Salm (1993) and H. Gubler (personal communication), the deposition depths (Equation (24)) used in the calculations for the Swiss guidelines from engineering-hydraulics analogy are unrealistically high. The application of these high flow depths in the guidelines causes increases in run-out to partially make up for difference (1).

• 3) In our model, we are able to input passive snow pressure directly, and in doing so we have also accounted for slope-angle dependence. The Swiss guidelines is a center-of-mass (or lumped-mass) model, into which only external forces may be input: there is no way of accounting for passive snow pressure explicitly. Instead, the Swiss guidelines input passive snow pressure implicitly through flow-depth dependence using an engineering-hydraulics argument. Reference Salm,Salm (1993) constructed a model as an alternative to the Swiss guidelines which did include passive snow pressure explicitly.

• 4) We have introduced slope-angle corrections into our model to account for momentum losses due to slope-angle decreases in run-out or run-up (adverse-slope) problems. This effect will be most important in run-up at sharply increasing slope-angle changes. Granular-flow experiments (Reference Chu,, Hill., McClung,, Ngun, and Sherkat,Chu and others, 1995) in a flume show that this effect is important at sharp dope-angle transitions. For the run-up example from Battleship, Colorado, the equations adapted From the Swiss guidelines (Equations (21) and (24)) produce reasonable results for two reasons:

  • a) momentum loss at the slope-angle transition was not accounted for;

  • b) unrealistically high deposition depths are predicted.

  These two effects increase the run-up to compensate for the fact that the centre-of-mass model will normally produce lower run-up.

Our limited experience with the run-up Equation (25) proposed in the Swiss guidelines suggests that it may underestimate run-up. This belief is based on model comparison as well as the example from the Battleship avalanche, Colorado. In a previous study (Reference Chu,, Hill., McClung,, Ngun, and Sherkat,Chu and others, 1995) for granular flows in a flume, it was shown that during run-up there is a tendency for material to he deposited at a sharp slope-angle transition, with maximum run-up occurring by material overriding that trapped at the corner. The implication is that run-up in very sharp slope-angle transitions may exceed even that for the leading-edge run-up equations we have proposed (Reference Chu,, Hill., McClung,, Ngun, and Sherkat,Chu and others, 1995). Therefore, it is possible that in some field examples Equation (25) may underestimate run-up even more than in the Battleship example. Our model does not contain a mechanics formulation sophisticated enough to be applied when the slope-angle transition approaches 90°. Equation (25) from the Swiss guidelines contains no explicit slope-angle dependence. It must be kept in mind that the formulation we have proposed here is one-dimensional. Neither it nor that for the Swiss guidelines is designed to handle the two-dimensional and three-dimensional features which can be important in run-out and run-up for real avalanches. A disadvantage of our formulation compared to the Swiss guidelines is that, in general, no analytic solutions are available when turbulent drag is important. Numerical solutions are easy to obtain, however, and it is likely that in the future numerical solutions will be required when, two- or three-dimensional aspects of the problem are incorporated.

We believe that our formulation as applied in this paper is more suitable for run-up than run-out, Surely, the basal friction conditions will change as the avalanche mass decelerates and stops, which implies that the friction coefficient μ may need to be varied along the run-out zone (Reference McClung,McClung, 1990). Since run-up slope distances are shorter than run-out distances, basal friction variations during deposition may not be as serious in run-up. If theoretical estimates of friction-parameter variations along the run-out zone were known, it would be easy to incorporate them in our numerical solution scheme.

Another reason our formulation is more suitable for run-up than run-out is our assumption of constant discharge (or supply) in the derivation of the momentum Equation (7). This assumption is closer to being satisfied for run-up or for large avalanches. If the discharge decreases with time, the momentum flux (term T2) would have to be reduced with time in Equation (7). Such modification would, however, result in prediction of shorter run-out distances. Therefore, our assumption of constant supply (or discharge) instead of decreasing discharge with time is likely to produce conservative estimates of run-out, which is usually an advantage in engineering calculations. See Appendix B for details.

Aside from the modelling differences described above, our view about the values of the friction coefficients μ and ξ (or D ₀ h ) is different from that of Reference Salm,Salin and others (1990) and Reference Salm,Salm (1993). We believe that basal friction in run-out and run-up can be accounted for almost entirely by the dynamic Coulomb friction coefficient μ, with ξ (or playing only a role as turbulent friction at the upper surface of the flowing mass. Our view is conditioned by the results of Reference Dent,Dent (1986, Reference Dent,1993) showing how basal shear and normal forces can be coupled in a dense, rapidly deforming granular material such as we envision for avalanching material in run-out and run-up problems. The model of Reference McClung,McClung (1990) for estimating speeds all along the incline is also based on the concept that shear and normal forces are entirely coupled at the lower basal boundary. Since the basal friction conditions in dry-flowing avalanches have never been measured, we must rely at this point on physical principles and models for information about basal friction mechanisms. This uncertainty should not detract from the model we have presented. If turbulent drag does play a significant role in basal friction (e.g. Reference Salm,Salm, 1993), our model can still be applied, and indeed we have presented example calculations in our paper to represent this alternative view. Regardless of which of these two pictures is most nearly correct, large-scale roughness features will also play a role in basal friction (Reference Salm,Salm, 1993), and no theoretical model exists for proper inclusion of this effect.

Reference Salm,Salm (1993) showed that the run-out estimates of the Swiss guidelines are very sensitive to initial flow depth. His calculations for the Ariefa avalanche in Switzerland, for example, showed an increase of 200 m run-out with an initial flow-depth increase from 0.8 to 1.0 m. This effect is found to some degree in our model as well, but it is highly dependent on the choice of friction coefficients: μ and D ₀ h (or ξ) taken for model comparisons. From Figures 3 and 4, we show that given an initial approach speed the equations of the Swiss guidelines are essentially independent of initial flow depth when turbulent drag is small (Fig. 4) (we believe that turbulent drag should be small in run-up and run-out). When turbulent drag is small, our model shows increasing run-out distance with flow depth (Fig. 4), but the sensitivity is not large. This flow-depth dependence results because we are able to input passive snow pressure explicitly. Our analysis, together with Salm’s example, field observations and examples such as Battleship, Colorado, reinforces our belief that turbulent drag is small in run-out and run-up. Run-out and run-up should depend on initial flow depth, but the sensitivity implied in the models with respect to flow-depth dependence, when turbulent drag is taken to be significant, seems too great in our experience.