Snow Avalanches - Environmental Geomechanics - page 56

we can express the mass balance equation as follows:

d d t d d t

= ρ a + ρ s U f h n − φρ s Lv s ,

where m = ρV is the cloud mass. Usually the settling velocity v s is very low
compared to the mean forward velocity of the front so that it is possible to ignore the
third term in the right-hand side of the equation above. We then obtain the following
simplified equation:

dΔ ρV
d t

= ρ s U f h n . [2.14]

The cloud undergoes the driving action of gravity and the resisting forces due to
the ambient fluid and the bottom drag. The driving force per unit volume is ρg sin θ .
Usually, the bottom frictional force is written in a Chézy form: C D ρU 2 L where
C D ( Re ) is the Chézy friction factor, dependent on the overall Reynolds number.
Most of the time, the bottom drag effect plays a minor role in the accelerating and
steady-flow phases but becomes significant in the decelerating phase [HOG 01]. Since
we have set aside a number of additional effects (particle sedimentation, turbulent
kinetic energy), it seems reasonable to also discard this frictional force. The action
of the ambient fluid can be broken into two terms: a term analogous to a static
pressure (Archimede's theorem), equal to ρ a Vg , and a dynamic pressure. As a first
approximation, the latter term can be evaluated by considering the ambient fluid
as an inviscid fluid in an irrotational flow. On the basis of this approximation, it
can be shown that the force exerted by the surrounding fluid on the half cylinder is
ρ a Vχ d U/ d t , where

χ = k, [2.15]

is called the added mass coefficient . Since at the same time volume V varies and the
relative motion of the half cylinder is parallel to its axis of symmetry, we finally take:
ρ a χ d( UV ) / d t . Note that this parameter could be ignored for light interstitial fluids
(e.g. air), whereas it has a significant influence for heavy interstitial fluids (basically
water). Thus the momentum balance equation can be written as:

d( ρ + χρ a ) VU
d t

=Δ ρgV sin θ. [2.16]

Analytical solutions can be found for equations [2.10, 2.11, 2.13] in the case of a
Boussinesq flow ( ρ/ρ a → 1); for the other cases, numerical methods must be used.

In the Boussinesq limit, since the final analytical solution is complicated, we only
provide an asymptotic expression at early and late times. To simplify the analytical
expressions, without loss of generality, here we take: U 0 =0and x 0 =0and we
assume that the erodible snowcover thickness h n

and density ρ s are constant. The