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 runup 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 runout distances using dynamic methods.
In this paper, a practical method is introduced for calculating runup and runout, using avalanchedynamics principles. The model is compared to a related model used in Switzerland which is designated as the present, standard Swiss guidelines (Salm and others, 1993) tor avalanchedynamics calculations in runout 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 fielddocumented examples of avalanche runup and runout and theoretical comparisons using identical model parameters.
Fig. 1. Flowing avalanche enveloped in a turbulent dust cloud engaged in runup on the side of a mountain.
The focus of our paper is a simple, practical method far runup or runout prediction similar to that proposed in the Swiss guidelines (Salin and others, 1990). Thus, our work is intended to apply to only one, small but important, part of the avalanchedynamics problem, Specifically, we provide a simple theoretical framework from which it is possible to calculate runup or runout for avalanches containing dry snow, provided that the friction coefficients (including internal friction), flow depth of material approaching the runup or runout 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 runup (or runoat) dynamics problem, since our view differs from the somewhat traditional one expressed in the Swiss guidelines. Due to the complexity of the avalanchedynamics problem and the simplifications introduced for onedimensional 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 frictionparameter input.
Flow and Boundary Conditions
We consider first a model for runup and runout calculations based upon solution of the dynamicequilibrium equations and continuity proposed by Hungr and McClung (1987) and modeling at slope transitions proposed by Takahashi and Yoshida (1979) and Takahashi (1991) for the related problem of debrisflow dynamics. The model is a depthaveraged, onedimensional 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 runup (and runout) problems, it is very important to begin from a realistic physical picture of the avalancheflow 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 (McClung and Schaerer, 1985, 1993; Gubler and others, 1986; 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 runout and runup 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 runup and runout problems. However, hack calculations by 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 runout 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 runup and runout 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. 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 (Salm, 1993). This information is consistent with the picture that the relevant volume concentration of solids is high for runup and runout 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 runup and runout problems dealing with the final stages of motion, McClung (1990) has discussed this physical condition extensively.
For runup and runout, then, we consider the motion of a dense core of material with lowerdensity material suspended in turbulent eddies around the upper surface of the core. Takahashi (1991) took the bold step of uncoupling turbulent fluid processes and particleparticlecollision and rubbingfriction 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 avalanchedynamics formulations since Voellmy (1955) and 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 (Dent, 1986, 1993; 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 runout and runup, with drag at the top of the flow often being negligible. We then adopt a dynamic Coulombtype relation
for our definition of basal resistance. In Equation (2),
ρ
is depthaveraged density,
h
is mean flow depth along the runup (or runout) 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 runout zone. Dent (1993) shows examples of how μ may vary with speed, particlerestitution coefficient and interparticle 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 snowdust 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 snowdustair mixture and is about 10 kg m^{−3} (McClung, 1990) and C
_{D} is a drag turbulent flow over a rough upper periphery of the core. Following McClung (1990), we estimate that the value of C
_{D} is about 0.01, Fran Schlicting (1972). In at least some problems of runup and runour, 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 McClung and Schaerer (1985), and with ρs in the range 200–500 kg m^{−3} (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 millimetersize 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 runup and runout problems. 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 runup and runout 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. Salm (1966, 1993) proposes that allowance be made for passive snow pressure analogous to passive earth pressure for the compressive states of stress expected in runup and runout problems. The longitudinal passive snow pressure (per unit width) may be represented as (Craig, 1988):
where (ϕ Is internal friction angle)
In Equation (6), cohesion is ignored, and h0 and ψ_{0} represent flow depth and slope angle on entering the runout zone, respectively. For ϕ< ψ_{0}, Equation (5) reduces to the fluid analogy introduced by Hungr and McClung (1987) for which k
_{p} = 1. For ψ = 0° Equation (6) reduces to the expression given by 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 Salm (1993) discussion and our own observations. Salin and others (1990) and Salm (1993) estimate for runout zones based upon avalanche deposits that ϕ is about 25°. This estimate (ϕ = 25°) is recommended in the Swiss guidelines (Salin and others, 1990), and we shall also retain this estimate far runup and runout problems. With ϕ = 25° and ϕ_{0} = 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 rimup and runout 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 (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. Salm (1993) points out that ϕ has never been estimated in flowing snow. Static values for ϕ are estimated by McClung (1987) in the range 50–60°; such values may be reasonable for deposited snow, but here we provide estimates during runup and runout 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 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 runup or runout. Accordingly, we have chosen ϕ= 25°, consistent with the Swiss guidelines (Salin and others, 1990).
LeadingEdge Theoretical Model
For runup and runout, we adopt a onedimensional 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 (m^{2}s^{−1}), is constant throughout runup or runout. 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 boundarycondition formulations do not require the assumption implied by Equation (1) of decoupling the stress terms, even though this assumption is rooted in mast avalanchedynamics models since Voellmy (1955).
Considering the problem of runup, following Hungr and McClung (1987) and Takahashi (1991), we express Newton’s second law (see Figure 2 for the geometry) as:
Fig. 2. Schematic of flowingavalanche runup on an adverse slope; (a) lumpedmass or centreofmass model, (b) leadingedge 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 runout slope, and x is distance along the runout slope measured from zero at the beginning of the runout 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 runout or runup slope, where ψ_{0}, υ_{0} 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 snowpressure 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 runout or runup slope. See Appendix Β far an explanation of this term.
By analogy to fluid dynamics, we assume supercritical at the beginning of the runout 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 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 ≥ ψ_{0}) as expressed in Equation (6). In order to solve for the runout or runup 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, ψ_{0}) 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 runup slope (or for adverse slopes in runout). In addition, we must invoke the depthaveraged, 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 runup (or runout) zone.
Rearrangement of Equation (7) gives:
where
with
On the runup or runout slope we seek a numerical solution of Equations (8) and (9) for ν as a function of time. The runup or runout distance is then given by the integral of ν from t = 0 until t = t
_{R}, the time taken on the runup 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 runup slope. The ratio of these terms is: G
_{0}/D
_{0}υ^{2} where, on the adverse runup slope, G
_{0} is represented by
i.e. ψ is negative.
Typical values for D
_{0} result from (McClung, 1990)
and in runup problems. If ψ = 30°, μ = 0.3, and ν = 20 m s
^{−1}, the ratio becomes
G
_{0}/
D
_{0}υ
^{2} = 112. For our model, therefore, in some cases of practical interest, the term
D
_{0}υ
^{2}
t may be ignored in runup (and runout) problems, and drag may be attributed to dynamic Coulomb drag at the base of the avalanche. When the term
D
_{0}υ
^{2}
t is ignored, an analytical solution is available and the runout distance is
and the runup height is
Following Salm (1993), we derive an approximate expression for the mean deposition depth on the runup (or runout) slope. Neglecting the turbulent term, the solution for the speed along the deceleration slope is:
For an approximate mean flow depth in the runup (or runout) zone, we assume from conservation of mass, following Salm (1993), a constant supply of mass per unit width entering into the runup (or runout) 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 runout.
Equation (15) is written for onedimensional 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 depositionzone depth (Salm, 1993):
In Equation (16), quantities with a bar (
v
) are averaged along the length of the runup or runout 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 slopeangle momentum correction
,
Equation (17) predicts that
h
is about 1.5
h
_{0}.
When slopeangle momentum correction is important (such as in runup), 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 onedimensional 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 runup and runout problems, this added sophistication is probably unnecessary. Following 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 runout Or runup zone.
Equation (17) provides only an approximate estimate (e.g, Salm, 1993): dynamic dependence on flowdepth changes and turbulence during runout 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 nonequilibrium states, an expression like Equation (17), derived from idealized simplifications, is only a rough estimate. Avalancheflow height variations and deposition depth often display two and threedimensional effects, not included in Equation (17).
LumpedMass Models
Models containing the assumption that motion may be described by the centre of mass of the avalanche have been the most popular for runup and runout problems. The models of Voellmy (1955), Salm (1979), Perla and others (1980) and 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 McClung (1990) is also a centreofmass model but is intended only for speed estimates, not runout estimates. Lumpedmass models have two disadvantages in runup and runour problems:

1) Since motion description is with respect to the centre of mass instead of the tip of the avalanche, such models predict shorter runup or runout (Hungr and McClung, 1987; Chu and others, 1995) if the same friction coefficients are used.

2) It is not possible to include passive snow pressure explicitly in the runup or runout formulation since only external forces can be specified in a pointmass model.
Salm (1979), perla and others (1980) and McClung (1990) have derived the differential equation for the lumpedmass 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 Perla and others (1980), that only the slopeparallel component of velocity is transferred on to the runup or runout slope:
when ψ
_{0} is greater than ψ. The solution to
Equation (18) is then (e,g. Salm, 1979;
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(ψ_{0} – 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(ψ_{0} – ψ) → 1, the speed equations of the Swiss guidelines are recovered.
The solution to Equation (18) for the runout distance is:
For the runup height, we get (ψ is taken negative)
With the substitutions above, cos(ψ_{0} – ψ) → 1 and D
_{0}=g/ξ
h
Equation (20) is identical to that given for runout distance in the guidelines (Salm, 1993). Using L’Hospital’s rule, in the limit as D
_{0} → 0 (neglect of turbulent drag), Equation (20) reduces to
For runup, we take ψ negative in G
_{0}, and the runup 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 runout or runup. This comparison shows a fundamental difference between the models: the leadingedge model always predicts longer runup or runout 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. Hungr and McClung (1987) have emphasised these differences previously. In addition to a different momentum formulation, the Swiss guidelines (Salin and others, 1990; Salm, 1993) contain no provision for momentum Loss as slope angle decreases. Experimental data from smallscale experiments on granular flows show that the correction,
, is important in runup (
Chu and others, 1995).
To compare solutions for the leadingedge model and the centreofmass 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 (Salin and others, 1990; Salm, 1993) it is necessary to introduce the mean runout zone deposition depth analogous to Equation (17) for our model. Salm (1993) gives the expression from the Swiss guidelines from an engineering hydraulics argument:
where λ = tan^{2}(48° + ϕ/2). This can be related to our expression for passive snow pressure by substitution: λ —> k
_{p} cos ψ_{0} with ψ_{0} = 0°. In other words, cos ψ_{0} is explicitly taken as I in the Swiss guidelines instead of accounting for the multiplier cos ψ_{0} as is usual in passivepressure theory (Craig, 1988). For comparison between the leadingedge and lumpedmass models in the next section, we shall specify a value for ϕ to achieve the same assumptions.
For runup estimates, the guidelines (Salin and others, 1990) simply specify:
for an approximate estimate of runup from analogy to engineering hydraulics.
Model Comparisons and Field Examples
In order to calculate runout or runup from our model or the Swiss guidelines, one must have estimates of υ_{0}, h
_{0}, μ, ϕ, and D
_{0} (our model) or ξ (Swiss guidelines). In addition, the geometry must be known (ψ_{0} and ψ). Field examples in which all of these parameters arc known accurately are virtually nonexistent; 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 υ_{0} varying, and we also attempt comparison for field examples in which at least the incoming speed, υ_{0}, is measured and path geometry is known.
i) Model comparison
Figure 3 shows model comparisons for runout with the same input parameters: ψ_{0} = 25°, ψ = 8°, ϕ = 25° with input flow depth (0.5–2.5 m) and approach speed (15–35 m s^{−1}) varied. For this example, we have used friction parameters specified by Salm (1993), μ = 0.155, ξ = 1000 m ^{−2}, and equivalent to suggestions in the Swiss guidelines for large avalanches. The results show that the predictions are similar, with the leadingedge model usually (not always) producing longer (more conservative) runout fur initial flow depths ≥ 1 m. In Figure: 3b the deposition (flow) depths arc shown as input to the calculations of Figure 3a. The runout distance predictions for the Swiss guidelines are significantly boosted by assuming very high deposition depths. In the guidelines, as in our model, the dynamic flow depth is taken constant during runout and is equated to mean deposition depth. Salm (1993) showed that deposition depths predicted by the Swiss guidelines may be higher than the real ones by a factor of about 3. Salm (1993) has noted that the Swiss guidelines use unrealistically high flow depths. If the same flaw depths were assumed for both models for the calculations in Figure 3, the guidelines would predict much shorter runout than the leadingedge model because the guidelines arc essentially a centreofmass model (see discussion above). Another difference is that the leadingedge model accounts for momentum loss at the slope transition whereas the guidelines do not. This is not usually very important in runout problems, but it is important in runup (see Chu and others (1995) and the discussion below) where the dopeangle transition is abrupt. For the calculations in Figure 3, turbulent drag is important and influences the results.
Fig. 3. (a) Runout predictions for the leadingedge model (—) and the Swiss guidelines (centreofmass model) (O) as a function of approachflow depth (h_{0}) and approach speed (υ_{0}). Calculations are shown for h_{0} = 0.5, 1.0, 1.5, 2.0 and 2.5 m.(b) Deposition flow depths for the calculations in (a); ψ_{0} = 25°, ψ = 8°, ξ = 1000 m s^{−2}, μ = 0.155, ϕ = 25°.
Figure 4 shows runnut comparison for different friction parameters than in Figure 3: ψ_{0} = 25°, ψ = 8°, μ = 0.3, ξ= 19.620 m s^{−2}, ϕ = 25°. See Appendix A for an explanation of the choice of ξ. In this example, turbulent drag will be small, as we envision for runout or runup problems. For the results in Figure 4, the calculations far the Swiss guidelines are essentially independent of initial flow depth when initial speed is specified, because for the guidelines flow depth enters only through the turbulent drag term (which is small in this example). For the results in Figure 4, the Swiss guidelines predict about half the runout of the leadingedge model. For the leadingedge model, flowdepth dependence enters explicitly from the passive snow pressure.
Fig. 4. (a) Runout predictions for the leadingedge model (—) and the Swiss guidelines (centreofmass model) (O) as function of approachflow depth (h_{0}) and approach speed (υ_{0}). Calculations are shown for h_{0} = 0.5, 1.0, 1.5, 2.0 and 2.5 m. Note that, for the friction parameters used, the Swiss guidelines runout predictions are independent of initial flow depth if υ_{0} is specified. (b) Deposition flow depths for the calculations in (a); ψ_{0} = 25°, ψ = 8°, ξ = 19.620 m s^{−2}, μ = 0.3, ϕ = 25°.
Figure 5 shows a comparison of three models for runup calculations with μ = 0.155, ξ = 1000 m s^{−2}, ϕ = 25°, ψ_{0} = 10° and ψ = −30°, Included are: (1) the leadingedge model; (2) Equation (21) for Swiss guidelines runout adapted for an adverse slope (ψ negative), with cos(ψ_{0} – ψ) − 1 with
h
predicted by Equation (24); and (a) Runup height as a function of approach speed and approachflow depth. —, leadingedge model; O; Swiss guidelines (Equation (25)); and □, Swiss guidelines runout equations for an adverse slope (Equation (21) with cos(ψ_{0} – ψ) − 1, D
_{0}=g/ξ
h
, and Equation (24)). Initial flow depths used are h_{0} = 0.5, 1.0, 1.5, 2.0 and 2.5 m. For all calculations: ψ_{0} = 10°, ψ = −30°, μ = 0.155, ξ = 1000 m s^{−2}, ϕ = 25°. (b) Deposition flow depths for the calculations in (a). —, leadingedge model; O, Swiss guidelines (Equation (24)). (3) Equation (25) as the recommendation for the Swiss guidelines. Since the guidelines contain assumptions of very high flow depths and there is no account of momentum loss at the slope angle transition, runup is boosted to partially make up for higher energy loss in the centreofmass formulation contained in the guidelines. Figure 5 shows that the Swiss guidelines predict much lower runup (Equation (25)). When runout Equations (21) and (24) from the guidelines are used on the adverse slope, the predictions are comparable to the leadingedge model. This similarity results because cos (ψ0 – ψ) is assumed to be 1 and because very high flow depths are used in the guidelines. The final results from Equations (21) and (24) are essentially independent of initial flow depth for a given initial speed. Again, if the same friction coefficients and flow depths were used in the models (Equations (21) and (24) and Equations (7), (8) and (17)), the leadingedge model would predict about twice the runup (Hungand McClung, 1987).
Figure 6 gives calculations of Q(m^{2}s^{−1}): discharge per unit width as a function of incoming speed and mean deposition depth (m) for three incoming speeds: 15, 25 and 35 m s^{−1}. For the leadingedge model, Equation (17) is used, and for the Swiss guidelines Equation (24). Salm (1993) and H. Gubler (personal communication) have argued that the guidelines result in use of deposition depths (taken as a flew depth in dynamics calculations) which are too high; based on Figure 6, we agree. Again, when turbulent drag is important, runout predicted by the guidelines is dependent on flow depth, and if very high flow depths are used runout is increased in a manner that seems physically unrealistic by this mechanism.
ii) Field examples
Field example 1: Battleship, Colorado, U.S.A. (February 1987)
Figure 7a gives the geometry of an example of avalanche runup from the Battleship avalanche (February 1987) in Colorado, U.S.A. Speed estimates are available from timing the avalanche between known points on the path. In particular, it is estimated that the avalanche had a speed of about. 45m s^{−1} on entering the runup segment. Total runout along the slope was about 93 m with runup height being about 58m. The average slope angle along runup was −31°; average deposit depth was 1–5 m. We estimated the basal friction coefficient, μ, from the centreofmass model (McClung, 1990) by assuming an initial flow depth of 2–3 m, with ρ_{t}
ρ=0.1 and C
_{D} = 0.01 to give D
_{0}
h
=0.0005 With D
_{0}
h
=0.0005 (m^{−1}) we found an approximate value for μ by selecting the value for which the two speed estimates (60 m s^{−1} reducing to 45 m s^{−1}) are matched for the terrain in the track in between the two estimates. The initial flow depth was estimated by knowledge of the release volume (2 × 10^{5} m^{3}), the measured time for the volume to flow through the lower track and the confined path geometry. The fracture depth averaged about 1 m, but we estimated the flow depth to be 2–3 m an entering the runup slope. This gave the estimated friction coefficients of μ = 0.32; D
_{0}
h= 0.0005 (m^{−1}) or ξ= 19 620 m s^{−2} for use in the runup calculations. We assume ϕ = 25° to give k
_{p} = 1.87. Model calculations gave: μ = 0.32, D
_{0} = 0.0005/2.28 = 0.0002 (m^{−1});
h
=4.5 m (h
_{0} = 2m); runup = 57 m. For the same parameters, except h
_{0} = 3 m. we get
h
=6.8 m runup = 58 m, In this example, the agreement is very close to the observed runup, and we believe it is fortuitous because of all the uncertainties in parameter estimates and incoming speeds.
Fig. 6. Discharge per unit width (Q in m s^{−2}) as a function of mean deposition depth for υ_{0} = 15, 25 and 35 m s^{−1}. — leadingedge model; O, Swiss guidelines, ψ_{0} = 25°, ψ = 8°, Q = υ_{0}h_{0}, ϕ = 25°. Initial flow depth used are h_{0} = 0.5, 1.0, 1.5, 2.0 and 2.5 m.
Fig. 7. (a) Slope profile, speed estimates and runup for the Battleship avalanche, February 1987. (b) Slope profile, speed estimates and runout for Mount Pizzac avalanche, January 1994.
If we assume Equation (25) to represent runup for the Swiss guidelines, we estimate (υ_{0} = 45 ms^{−1}; ϕ = 25° (λ = 2.5), h
_{0}= 2 m) runup = 43 m. For the same parameters, except h
_{0} = 3 m, we get nutup = 44 m.
From the runout, equations for the Swiss guidelines (Equations (21) and (24)) art applied, with ψ_{0} = ψ, μ = 0.32, ξ = 19 620 m s^{−2}, h
_{0} = 2 er 3m and λ = 2.5; we get runup = 67 m for both
h
=22.6 m and
h
=23.6 m(h
_{0}=3 m)
If we use friction coefficients recommended in the guidelines and Salm (1993) in Equations (21) and (24), μ = 0.155, ξ = 1000 m s^{2}, λ _= 2.5 (ϕ = 25°) and h
_{0} = 2 m, we get runup = 77 m,
h
=22.6 m For both sets of friction coefficients, Equations (21) and (24) overestimate runup because very high flow (meandeposition) depths are used in the calculations (much higher than observed) and because all the momentum is taken to be conserved on the runup slope, instead of just the slopeparallel component. Granularflow experiments by Chu and others (1995) show the importance of taking only the slopeparallel component in runup problems.
Field example 2: Mount Pizzac, Italy (January 1994)
Figure 7b shows geometry for the slope profile above and in the runout zone far Mount Pizzac. The avalanche was small with a volume of 2000 m^{3} and was triggered by explosives (Natale and others, 1994). Average slope angle for 267 m above the last measurement point; was 29°. The estimated speed at the final measurement point was 2223 m s^{−1}, and the speed approaching the calibration zone 267 m above the last measurement point was 25 m s^{−1}, we assume C
_{D} = 0.0 1, and ρ_{t}/ρ = 0.1, to give ξ = 19 620 m s^{−2}, D
_{0}
h
=0.0005 m as in previous examples. With this assumption about turbulent drag, if the speed drops from 25 to 22ms 1 over the 267m interval upslope of the runout calculation point, we estimate from our model that μ = 0.6, and this estimate along with ξ = 19 620 m s^{−2} is used in the runout calculations. To estimate passive pressure, we assume ϕ = 25°, and since ψ_{0} = 29°, i.e. greater than 25°. we assume k
_{p} = 1 (from Equation (6)). Measured flow depth approaching the runout zone was 1 mt and we used this value for ho in oar calculations.
With the above inputs, the calculations from our model yield:
h
=1.5 m (maximum flow depth measured was 2.5 m), runout 190m (if υ_{0} = 22 m s^{−1}) or 207m (if υ_{0} = 23 ms^{−1}) beyond the last measurement point. Measured runout distance was 170 m. For this example, the calculated mean (or deposition) flow depth is less than the measured maximum depth. The runout distance is overestimated but the differences are well within the range of uncertainty of the model parameter inputs. In Appendix B, we offer a partial explanation for the overestimate: for such a small avalanche the supply (or discharge) may decrease with time, resulting in shorter runout. This example is the only one we have found for which there are reasonably accurate measurements of speeds, flow depths and geometry to enable all model parameter inputs to be measured.
This avalanche is very small and does nor fit the Swiss guidelines for extreme avalanches. According to the guidelines, an extreme avalanche would not stop on a slope of 19°. Our model is more general; it can be used to predict runout or runup for even small dryflowing avalanches, provided input parameters can be estimated and the basic assumptions are satisfied as well.
Field examples 3 and 4: Aulta, Switzerland (8 and 10 February 1984)
Salm and Gubler (1985) and Gubler and others (1986) documented speeds, initial fracture depths, runout and path geometry far two avalanches on the Aulta avalanche path. We have used our model to predict the runout. For bath avalanches we assume ϕ = 25°, and h
_{0} = 1 m (close to estimated fracture depths). For both avalanches, we assume ξ = 19 620 m s^{−2}, as in previous comparisons, and have estimated the Mction coefficient μ by using speeddeceleration information and path geometry in our model prior to the point at which we begin our runout calculations. Both examples from Aulta have high uncertainty because neither runout Is well documented. In addition, the avalanche path has a gully in the final stages which opens up in the runout area. Caution should therefore be used when taking these examples to support our model.
Figure 8a shows path geometry, speed estimates and runout position for the 8 February avalanche (Gubler and others, 1986). The speed drops from about 55 to 45 m s ^{−1} over a horizontal reach distance of about 480 m. From these data and the assumptions above, we estimate μ = 0.4. For the runout (runup) calculations, we take ψ_{0} = 0°; υ_{0} = 26m s^{−1}. When input in our model, these estimates gave a runout distance of 110m which compares favourably with the measured estimate of about 90 m.
Fig. 8. Slope profile, speed estimates and runout for the Aulta avalanche, 8 February 1984. (b) Slope profile, speed estimates and runout for the Aulta avalanche, 10 February 1984.
Figure 8b shows path geometry, speed estimates and runout: position far the 10 February avalanche. The speed drops from about 52 to 35 m s^{−1} over a horizontal distance of 250 m on a slope with mean angle of 18.4°. With the assumption ξ = 19 620 m s^{−2}, from our model, we estimate μ = 0.7, using ϕ = 25°, and h
_{0} = 1 m. For the runout calculations, we take ψ_{0} = 18.4°, ψ = 12°. These assumptions give:
h
=1.5 m, runout = 244 m. The measured runout distance is about 260 m (Salm and Gubler, 1985; Gubler and others, 1986).
Summary
We have presented a method for calculating runup and runout 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 (Salin and others, 1990), but the modelling details differ. Some differences include:

1) Instead of a centreofmass 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) runout. Similar findings arc reported by Salm (1993) for another model which takes into account momentum and continuity: longer runout is predicted.

2) We estimate mean deposition depth (or mean runout zone flaw depth) from supply as proposed by Salm (1993). This (approximate) estimate gives reasonable flow depths which are used in the calculations. According to Salm (1993) and H. Gubler (personal communication), the deposition depths (Equation (24)) used in the calculations for the Swiss guidelines from engineeringhydraulics analogy are unrealistically high. The application of these high flow depths in the guidelines causes increases in runout 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 slopeangle dependence. The Swiss guidelines is a centerofmass (or lumpedmass) 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 flowdepth dependence using an engineeringhydraulics argument. Salm (1993) constructed a model as an alternative to the Swiss guidelines which did include passive snow pressure explicitly.

4) We have introduced slopeangle corrections into our model to account for momentum losses due to slopeangle decreases in runout or runup (adverseslope) problems. This effect will be most important in runup at sharply increasing slopeangle changes. Granularflow experiments (Chu and others, 1995) in a flume show that this effect is important at sharp dopeangle transitions. For the runup example from Battleship, Colorado, the equations adapted From the Swiss guidelines (Equations (21) and (24)) produce reasonable results for two reasons:
These two effects increase the runup to compensate for the fact that the centreofmass model will normally produce lower runup.
Our limited experience with the runup Equation (25) proposed in the Swiss guidelines suggests that it may underestimate runup. This belief is based on model comparison as well as the example from the Battleship avalanche, Colorado. In a previous study (Chu and others, 1995) for granular flows in a flume, it was shown that during runup there is a tendency for material to he deposited at a sharp slopeangle transition, with maximum runup occurring by material overriding that trapped at the corner. The implication is that runup in very sharp slopeangle transitions may exceed even that for the leadingedge runup equations we have proposed (Chu and others, 1995). Therefore, it is possible that in some field examples Equation (25) may underestimate runup even more than in the Battleship example. Our model does not contain a mechanics formulation sophisticated enough to be applied when the slopeangle transition approaches 90°. Equation (25) from the Swiss guidelines contains no explicit slopeangle dependence. It must be kept in mind that the formulation we have proposed here is onedimensional. Neither it nor that for the Swiss guidelines is designed to handle the twodimensional and threedimensional features which can be important in runout and runup 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 threedimensional aspects of the problem are incorporated.
We believe that our formulation as applied in this paper is more suitable for runup than runout, 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 runout zone (McClung, 1990). Since runup slope distances are shorter than runout distances, basal friction variations during deposition may not be as serious in runup. If theoretical estimates of frictionparameter variations along the runout zone were known, it would be easy to incorporate them in our numerical solution scheme.
Another reason our formulation is more suitable for runup than runout is our assumption of constant discharge (or supply) in the derivation of the momentum Equation (7). This assumption is closer to being satisfied for runup 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 runout distances. Therefore, our assumption of constant supply (or discharge) instead of decreasing discharge with time is likely to produce conservative estimates of runout, 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 Salin and others (1990) and Salm (1993). We believe that basal friction in runout and runup 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 Dent (1986, 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 runout and runup problems. The model of 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 dryflowing 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. 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, largescale roughness features will also play a role in basal friction (Salm, 1993), and no theoretical model exists for proper inclusion of this effect.
Salm (1993) showed that the runout 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 runout with an initial flowdepth 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 runup and runout). When turbulent drag is small, our model shows increasing runout distance with flow depth (Fig. 4), but the sensitivity is not large. This flowdepth 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 runout and runup. Runout and runup should depend on initial flow depth, but the sensitivity implied in the models with respect to flowdepth dependence, when turbulent drag is taken to be significant, seems too great in our experience.
Acknowledgements
We are grateful for the ideas of Dr B. Salm, Swiss Federal Institute for Snow and Avalanche Research. This research was sponsored by the Natural Sciences and Engineering Research Council of Canada (NSERC). We thank NSERC for support of this and other avalanche research in western Canada. Special thanks are owed to L, Nettuno, University of Pavia, Italy, and Dr H. Gubler, Davos, Switzerland, for supplying field examples.
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Appendix A Estimate of Turbulent Drag Coefficient
In our formulation, the quantity D
_{0}
h
is represented by:
Solving for ξ gives:
If ρ/ρ_{t}≃0.1 and if C
_{D} ≃ 0.01, calculation gives: ξ= 19 620 m s^{−2}. This result contains the assumption that turbulent drag acts at the top (upper surface) of the flow. If ρ/ρ_{t}=1 (turbulent drag is assumed to act at the base of the flow only), then ξ = 1962 m s^{−2} (or 982 m s^{−2} if C
_{D} = 0.02, nearly equivalent to Salm’s estimate of 1000 m s^{−2}). Both our model and the prescription given by Salm (1993) contain the assumption that C
_{D} ≈ 0.01 to 0.02, which is equivalent to a coefficient (Schlicting, 1972) for turbulent flow over a rough boundary. The important difference between these results stems from our belief that far flowing avalanches, particularly in runout and runup problems, turbulent drag is negligible at the base of the flow. We believe that the high volume fraction filled by solid materials (up to 50%) will prevent turbulence from forming in the interstitial air between the particles and that momentum will be transferred by particle collisions and rubbing friction. Due to the large difference in density between the particles and the air (ratio 100 to 1000), we believe that the presence of the air at the base of the flow will play a negligible role in the mechanics of runup and runout problems.
Appendix Β Momentum Principle Applied to Avalanche Dynamics
Following Daugherty and others (1985), we consider an avalanche mass contained at time t in a control volume which is fixed in position. At time t + Δt, the avalanche mass has moved to a new position along the runout or runup slope. At time t = 0, the tip of the avalanche mass is just coincident with the start of the runout or runup slope (see Fig. 9).
Definitions of important quantities are:
Fig. 9. Schematic of fixed control volume for momentum principle at runout (or runup) slope.
At time t, by definition, (m
V)_{t} = (m′ V′)_{t}. At time t + Δt, (shaded portion of Figure 9), the momentum of the entire avalanche mass is the sum of the momentum within the control volume at t + Δt, plus the momentum that has flowed out of the control volume during Δt, minus the momentum that has flowed into the control volume (unshaded area of Figure 9) during Δt. This balance may be written:
Therefore, the change of momentum of the avalanche mass is:
Combining these expressions and substituting into Newton’s second law, we get:
where Σ F is the vector sum of all forces acting on the avalanche mass including gravity, drag at the top and bottom and passive snow pressure.
Equation (B3) is a perfectly general momentum principle (Daugherty and others, 1985). It applies for compressible or incompressible flow, and steady or unsteady flow. For steady flow, which we assume in this paper, the momentum balance becomes:
because the last term of Equation B(3) is zero. Equation B(4) stares that, for steady flow, the sum of the forces on the avalanche mass is equal to the net rate of outflow of momentum across the original (fixed) control surface. The second term in Equation B(4) represents the momentum rate or momentum flux into the control volume from oncoming avalanche material reaching the beginning of the runout slope (control volume): it is a physically necessary force which cannot be neglected.
For our model, we have assumed an incompressible flow in one dimension and we have implicitly taken the velocity as the mean value over the face of the control volume. The rate of momentum transfer into the control volume (momentum flux) is given by (ρ
vdA
v
_{0} where dA is the element of area on the face of the control volume. The rate of momentum transfer across the entire area (per unit width) is then ρ
v
_{0}
^{2}
h
_{0}: the second term in Equation B(4) (see Fig. 2). For a runup (or adverse) slope, however, we assure that only the component of momentum parallel to the slope is effective in pushing the avalanche mass forward (the force component perpendicular to the controlvolume face). With respect to Figure 2, this gives
term
T
_{2} in
Equation (7). All the other terms in
Equation (7) came from the lefthand side of
Equation B(4): gravity, forces at the top and bottom and passive snow pressure.
From Equation B(4), if the momentum flux into the control volume (discharge or supply) decreases with time, as might happen some time after the passage of the avalanche front, the momentum rate out of the control volume also reduces with time. For our model, such a situation would imply reduced runout distance. Therefore, our assumption of constant discharge is a conservative one from the perspective of runout: our calculated runout distances may exceed the real ones. It is important to remember that other uncertain features of the avalanchedynamics problem may mask this effect.