Snow Avalanches - Environmental Geomechanics - page 50

tan θ<μ , there is no real solution: the frictional force equation [2.3] outweighs the
downward component of the gravitational force. It is therefore considered that the flow
slows down. The point of the path for which tan θ = μ is called the characteristic point
(point P ). It plays an important role in avalanche dynamics since it separates flowing
and stopping phases. In the stopping zone, we deduce from the momentum equation
that the velocity decreases as follows:

d u 2
d x

1
2

+ u 2 g
ξh

= g cos θ (tan θ − μ ) . [2.5]

The runout distance is easily inferred from equation [2.5] by assuming that at a point
x =0, the avalanche velocity is u p . In practice, the origin point is point P but attention
must be paid in the fact that, according to equation [2.4], the velocity at point P should
be vanishing; a specific procedure has been developed to avoid this shortcoming (see
Salm et al. [SAL 90]). Neglecting the slope variations in the stopping zone, we find:

ln 1+

u P
ξh cos θ ( μ − tan θ )

x a = ξ 2 g

. [2.6]

This kind of model enables us to easily calculate the runout distance, the maximum
velocities reached by the avalanche on various segments of the path, the flow depth
(by assuming that the mass flow rate is constant and given by the initial flow rate just
after the release), and the impact pressure.

Ancey and Meunier [ANC 04c] performed a back analysis on 15 well-documented
avalanches by inferring the bulk frictional force from avalanche velocity. To that end,
they used equation [2.2]: if one has a record yielding the body velocity as a function
of the position along the path, then it is possible to directly deduce the frictional
force components and its relationship with the velocity u to a multiplicative factor m .
Plotting the resulting force per unit mass in a phase space ( u,F/m ) can give an idea
of the dependence of the frictional force on the mean velocity and normal component.

For most events, the frictional force was found to be weakly dependent on velocity
or to fluctuate around a mean value during the entire course of the avalanche.
Figure 2.7 shows a typical example provided by the avalanche at the Arraba site
(Italy) on 21 December 1997. This figure reports the variation in the frictional force
per unit mass with velocity (solid line) and the downward component of the driving
force per unit mass g sin θ (dashed line). In the inset, we have plotted the measured
velocities (dots) together with the interpolation curve (Legendre polynomials) used
in the computations. On the same plot, we have drawn the velocity variations as
if the avalanche was in a purely Coulomb regime (dashed line): assuming that the
frictional force is in the Coulomb form F = fmg cos θ , where f is the bulk friction
coefficient, we numerically solved the equation of motion (equation [2.2], in which
F/m is replaced with the expression of F above). As shown in Figure 2.7, in the early
phases (between points A and B), the frictional force gently decreased with increasing