Snow Avalanches - Environmental Geomechanics - page 55

where g denotes the reduced gravity g = g Δ ρ/ρ a and Δ ρ = ρ − ρ a is the buoyant
density.

Usually a smaller Ri value implies predominance of inertia effects over the
restoring action of gravity, thus greater instability and therefore a higher entrainment
rate; it is then expected that the entrainment rate is a decreasing function of the
Richardson number. Although the details of the mixing mechanisms are very complex,
a striking result of recent research is that their overall effects can be described in quite
a simple way [FER 91]. To express the volume balance equation, the commonest
assumption is to state that the volume variations come from the entrainment of the
ambient fluid into the cloud and that the inflow rate is proportional to the exposed
surface area and a characteristic velocity u e . This leads to the equation:

d d t

= E v Su e , [2.13]

where E v is the bulk entrainment coefficient. According to the flow conditions,
different expressions of E v have been drawn from experiments. Interestingly enough,
the value of E v has been expressed very differently depending on whether the current
is steady or unsteady. There is, however, no clear physical reason justifying this
partitioning. Indeed, for most experiments, the currents were gradually accelerating
and mixing still occurred as a result of the development of Kelvin-Helmotz billows,
thus very similarly to the steady case. This prompted Ancey [ANC 04a] to propose a
new expression of the entrainment coefficient for clouds, which holds for both steady
and slightly unsteady conditions: Ancey [ANC 04a] related E v (or α v ) as a function
of Ri (instead of θ as done by previous authors).

The cloud mass can vary as a result of the entrainment of the surrounding
fluid and/or the entrainment of particles from the bed. The former process is easily
accounted for: during a short time increment δt , the cloud volume V is increased by a
quantity δV mainly as a result of the air entrainment, thus the corresponding increase
in the cloud mass is ρ a δV . The latter process is less well known. In close analogy with
sediment erosion in rivers and turbidity currents, [FUK 90] assumed that particles are
continuously entrained from the bed when the drag force exerted by the cloud on the
bed exceeds a critical value. This implies that the particle entrainment rate is controlled
by the surface of the bed in contact with the cloud and the mismatch between the drag
force and the threshold of motion. Here, since in extreme conditions the upper layers
of the snowcover made up of new snow of weak cohesion can be easily entrained,
it is reasonable to think that all the recent layer ahead of the cloud is incorporated
into the cloud: when the front has traveled a distance U f δt , where U f is the front
velocity, the top layer of depth h n and density ρ s is entirely entrained into the cloud
(see Figure 2.9). The resulting mass variation (per unit width) is written: ρ s U f h n δt .
At the same time, particles settle with a velocity v s . During the time step δt , all the
particles contained in the volume Lv s δt deposit. Finally, by taking the limit δt →

0,