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International Snow Science Workshop, Davos 2009, Proceedings
519
International Snow Science Workshop Davos 2009
Avalanche Simulation with SAMOS-AT
Peter Sampl1 and Matthias Granig2
1 AVL List GmbH, Graz, Austria
2 Stabstelle Lawinensimulation, Schwaz, Austria
ABSTRACT: The simulation software SAMOS for dry snow avalanches has been used by the Aus-
trian Service for Torrent and Avalanche Control since 1999. The software was based on a two-
dimensional, depth-averaged model with Coulombian and turbulent bottom friction for the dense flow
part and on a three-dimensional mixture-model for air and ice-particles, which form the powder snow
part. The mixture model involved separate mass-balances for the air and ice-particles and one single
balance for the momentum. Additional balances for the turbulent energy and dissipation rate were
considered. The amount of powder snow generated from the dense flowing snow was predicted as-
suming an analogy of momentum- and mass-transfer above the dense-flow-surface. A triangular grid,
moving with the flowing mass over the terrain surface, was used for the numerical solution of the
dense-flow equations in Lagrangean formulation, while a three-dimensional, non-moving (Eulerian)
grid, adapted to the terrain surface, was employed to solve the equations for the air-and-ice mixture.
An improved version, SAMOS-AT, has been completed in 2007. The dense flow friction model was
changed, so that the turbulent friction part depends also on the flow-depth. The mixture-model for the
powder snow part was replaced by a full two-phase-model for air and ice-particles, with separate mass
and momentum balances. A smoothed-particle hydrodynamics method is used now to solve the
dense-flow-equations. The improved model is described in this paper.
KEYWORDS: Avalanche dynamics, simulation, dense flow avalanches, powder avalanches.
1 INTRODUCTION
SAMOS is a simulation-software for dry
snow avalanches. It describes both the dense-
flow-layer (DFL) and the powder-snow-layer
(PSL) of an avalanche, as well as the interaction
between them. The software has been devel-
oped for the Austrian Service for Torrent and
Avalanche Control (WLV), which has used a first
version since 1999 (Zwinger et al., 2003; Sampl
and Zwinger, 2004). The use of the software
revealed several problems: too large DFL-width
in channelled terrain and often too large run-out
distances of the DFL and also the PSL in terms
of the 5 kPa zone. An improved version,
SAMOS-AT, has been developed and com-
pleted in 2007. The main changes are different
bottom friction models for the DFL, the treatment
of the PSL as two-phase-flow and a different
numerical solution scheme for the DFL. Fur-
thermore, the usage of the software has been
substantially simplified.
2 SAMOS-AT AVALANCHE MODEL
In SAMOS, dry snow avalanches are
thought to consist of air and ice particles in vari-
able concentrations. With the volume-fraction c
and intrinsic density
ρ
ice of the ice particles and
the air density
ρ
air, the mixture density
ρ
is given
by
airice
cc
ρρρ
)1( −+= . (1)
A vertical three-layer-structure (Figure 1) is
assumed in the avalanche, the layers being
identified by different ranges of the mixture den-
sity: a dense flow layer (DFL) with 200 to 300
kg⋅m-3 at the bottom, a powder snow layer (PSL)
with 1 to 10 kg⋅m-3 at the top and a transition
layer in between.
Powder snow layer
(PSL)
Transition layer
Dense flow layer
(DFL)
Sliding surface
(DHM)
Powder snow layer
(PSL)
Transition layer
Dense flow layer
(DFL)
Sliding surface
(DHM)
Figure 1. Vertical layer-structure of a dry snow
avalanche as assumed in the SAMOS-AT
model.
The transition layer and the PSL are as-
sumed to develop above the DFL with time, de-
pending on the flow conditions. The transition
layer is not spatially resolved in the model, but
reduced to an interface which connects DFL and
PSL. The DFL is modelled as shallow flow in
two dimensions on the terrain surface (pre-
scribed by a digital height model (DHM)), while
______________________
Corresponding author address: Peter Sampl,
AVL List GmbH, Hans-List-Platz 1, 8020 Graz,
Austria;
tel: +43 316 787 439; fax: +43 316 787 1923;
email: peter.sampl@avl.com
International Snow Science Workshop, Davos 2009, Proceedings
520
International Snow Science Workshop Davos 2009
the PSL is modelled as three-dimensional flow
above the DFL, including the air masses a few
hundred meters above ground.
2.1 Dense Flow Layer (DFL) Model
The DFL model is obtained by dividing the
DFL into a large number of small mass elements
and writing the integral mass- and momentum-
balances for each of it. Each element contains a
mass me, touches the bottom surface at an area
Ae and extends up to the DFL surface. The bal-
ances result in a set of ordinary differential
equations, which are coupled by the lateral
forces between the elements. The DFL is as-
sumed to be shallow: the ratio of the character-
istic DFL depth H (normal to the terrain) to the
DFL length (and width) L is taken to be small. All
terms of order (H/L)1.5 or smaller are neglected
(following the work of Savage and Hutter, 1989;
derivation see Zwinger et al., 2003). A constant
bulk density
ρ
is assumed for the entire DFL. A
smoothed-particle hydrodynamics (SPH)
method (Monaghan, 1988) is used, so that the
flow depth h at each point
x
on the terrain sur-
face is expressed as weighted sum over all ele-
ment with centers at position
e
x
:
=
e
ee
xxWmxh ),(
1
)(
ρ
(2)
The SPH-kernel function
),(
e
xxW
(see 3.1)
has a dimension of m-2. The bottom area Ae of
an element follows as
)(
e
e
e
xh
m
A
ρ
=
(3)
The mass and momentum balances for each
element are formulated in a local coordinate
system with direction 1 along the elements cur-
rent flow direction
u
(defined as average veloc-
ity in the element), direction 3 normal to the ter-
rain and direction 2 normal to both. For brevity,
the element indices e are dropped below.
1
,eu
3
e
2
e
A
h
1
,
eu
3
e
2
e
A
h
Figure 2. A DFL mass element with bottom sur-
face A, average depth h and average velocity
u
and the local coordinate system
321
,, eee
.
Snow entrainment is considered at the DFL
front and specified by qent, the entrainable mass
per unit surface area. The front width wf within
an element, measured normal to the flow direc-
tion, is zero for elements not at the front. If the
mass transferred to the PSL per unit surface
area is termed js and the time t, the mass bal-
ance for an element can be written as
Ajuqw
dt
dm
sentf −= . (4)
The momentum balance in direction 3 yields
an equation for the bottom pressure in the ele-
ment
A
m
ugp
b
)(
2
13
)(
κ
−−= (5)
with gi the vector of gravitational accel-
eration and
κ
1 the terrain curvature in flow direc-
tion. The momentum balances for directions i
=1,2 can be written
)(
)(
)(
)( 2
)(
res
i
i
b
b
i
i
i
i
F
u
u
A
h
p
K
x
Amg
dt
mud
+−
∂
∂
−=
τ
(6)
with
τ
(b) the bottom shear stress and Fi
(res)
the resistance force resulting from obstacles (an
additional resistance force to break entrained
snow from the ground can be considered op-
tionally; see Sailer et al, 2008). The second term
on the right hand side describes the force result-
ing from the lateral normal stresses, which are
expressed as product of the bottom pressure
with the normal stress coefficients K(i). A linear
drop of the stresses from the bottom to zero at
the DFL surface is assumed, hence the average
over the depth is half the bottom value. The
shallowness assumption allows to neglect lateral
shear stresses. The resistance force due to ob-
stacles like trees in a forest or fields of boulders,
defined by a characteristic obstacle height hres,
diameter dres, average lateral distance sres and
shape resistance coefficient cw is computed as
u
u
s
A
hhd
u
cF
i
res
resre sw
res
i
2
2
)(
),min(
2
ρ
−=
(7)
For a granular material with internal Mohr-
Coulombian friction, as assumed in SAMOS, the
coefficients K(i) depend on the flow state and the
friction parameters (Zwinger et al., 2003). The
normal stress is larger than p(b), if the material is
compressed, and smaller, if it expands. This
results in a net energy loss in each compres-
sion-expansion cycle. In practice, this led to a
considerable reduction of runout-distances in
SAMOS when finer DHMs were used, since
these resolve more surface-ripples, each caus-
ing a compression-expansion-cycle. Hence K(i) ≡
1 is assumed in SAMOS-AT, as for a Newtonian
fluid, and the retarding effect of ripples is con-
sidered in the bottom friction parameters.
International Snow Science Workshop, Davos 2009, Proceedings
521
International Snow Science Workshop Davos 2009
The bottom friction in SAMOS-AT is com-
puted as
2
2
0
0
)(
0
)(
ln
1
+
1
+
+
++=
D
DD
ss
s
bb
B
R
h
u
RR
R
p
κ
ρ
µττ
(8)
In the above relation,
τ
0 is a yield stress
(which is applied in small-avalanche-simulations
only; see Sailer et al., 2008),
µ
is the Coulom-
bian bed friction coefficient, Rs a “fluidization fac-
tor”, RD the surface roughness, and Rs0,
κ
D and
BD empirical constants. The values used for
µ
,
Rs0,
κ
D, RD and BD are 0.155, 0.222, 0.43, 0.1 m,
and 4.13, respectively.
The fluidization factor Rs is defined as the ra-
tio of dispersive stresses to the effective bottom
pressure:
)(
2
b
s
p
u
R
ρ
= (9)
The Rs-term serves to increase the Coulom-
bian bed friction at low fluidization, so as to stop
slow avalanche parts already in terrain steeper
than atan(
µ
) and to prevent spreading when
mass is close to stopping.
The last term on the right hand side in (8) is
inferred from classical boundary layer theory for
turbulent fluid flow and the logarithmic law-of-
the-wall for rough plates (see e.g. in Gersten
and Herwig, 1992). These laws may be obtained
by assuming turbulent eddies (i.e. larger clusters
of non-coherent ice particles, or coherent snow
clods), whose size is proportional to the distance
from the terrain surface. The difference to the
turbulent friction term as in the classical Voellmy
model is the (moderate) reduction of friction with
increasing flow depth h.
2.2 Transition Layer Model
The transition layer is dominated by salta-
tion, particle-collisions, sedimentation and turbu-
lent suspension due to the aerodynamic forces
acting onto the particles at the DFL surface. The
interaction of particle-suspension and the turbu-
lence in the air is intense and complex. A
strongly simplified model thus has been devel-
oped for the initial version of SAMOS and has
been kept with a few modifications.
The DFL surface with the transition layer is
considered to be a “rough wall” for the PSL (this
wall is of course moving with the DFL-speed).
With
∆
u the magnitude of the velocity difference
between the DFL surface and the PSL at a ref-
erence height y above, equal to the assumed
transition layer depth, the wall shear stress in
turbulent flows at rough walls is, again from
classical boundary layer theory, given by
+
∆
=
P
P
air
w
B
R
y
u
ln
1
2
κ
ρ
τ
(10)
Rp is the roughness length and
κ
(again
0.43) and Bp (8.5) constants. The suspension
mass flux per DFL surface unit js is assumed to
be proportional to the momentum flux, which is
equal to
τ
w. Since
τ
w is proportional to
∆
u and js
is to the particle concentration difference
∆
c,
between DFL and PSL at y, the mass flux is re-
scaled accordingly and computed as
∆
∆
−=
p
dsusp
ws d
cc
u
c
j
τ
(11)
The correction term in brackets is introduced
to account for dissimilarities between momen-
tum and particle transport. It shall reflect that
particles with larger diameter dp and smaller
form-drag-coefficient cd are transported to a
lesser extent. csusp is an empirical correction con-
stant with the dimension of a length. No size
distribution is considered for the particle diame-
ter in the model. The roughness seen by the
PSL is computed as
33
2
0g
c
g
u
cRR
ice
w
ressaltP
ρ
τ
++=
(12)
where R0 is the geometric roughness of the
DFL surface without saltation. The second term
models the increase of roughness due to salta-
tion (with empirical constant csalt) and the third
the increase due to the suspended particles
themselves (with yet another empirical constant
cres). Furthermore, it is assumed that suspension
starts only at a critical saltation height of the par-
ticles at the DFL surface. The saltation height,
relative to the particle diameter, is proportional
to a Froude number Frs, computed with the DFL
velocity and the condition for suspension is for-
mulated as
crits
p
sFr
dg
u
Fr ,
3
2
>=
(13)
This condition blocks suspension specifically
when the DFL comes to rest. Values typically
used for csusp, cd, Frs,c rit, R0, csalt and cres are 0.01,
3, 400, 0.1 m, 10-4 and 10, respectively. The
particle diameter dp is assumed in the range 0.5
to 1 mm.
2.3 Powder Snow Layer (PSL) Model
The PSL model in SAMOS-AT treats air and
ice particles as separate phases, i.e. separate
velocities are computed for both (in contrast to
International Snow Science Workshop, Davos 2009, Proceedings
522
International Snow Science Workshop Davos 2009
the initial SAMOS version, which assumed zero
slip velocity). The two phase approach allows for
sedimentation of the particles and a separation
of particle trajectories and air-streamlines, which
may be significant in highly curved terrain.
The well known Navier-Stokes equations
with turbulence-averaging are solved to com-
pute the flow of the air phase. The compressibil-
ity of air is considered via the ideal-gas-law
TRp air
ρ
=
(14)
with the static air pressure p, the air tem-
perature T , which is assumed to be constant
constant (273 K), and Rair the specific gas con-
stant of air. In the following equations, all vari-
ables refer to the air, if not indicated otherwise,
and all dependent variables are turbulence-
averaged. The volume of the ice particles is ig-
nored due to their small volume fraction. A
global, static (Eulerian) coordinate system (co-
ordinates xi) with directions i =1,2,3 is used and
Einsteins summation convention applies to the
equations below. The air mass balance reads
0
)( =
∂
∂
+
∂
∂
i
i
x
u
t
ρ
ρ
(15)
The momentum balance considers the parti-
cle drag fi and the drag fi
(res) resulting from ob-
stacles not representable via the geometric
boundary conditions (e.g. forest):
)(
)(
)(
res
ii
i
j
j
i
eff
j
i
i
j
ji
i
ff
x
u
x
u
x
x
p
g
x
uu
t
u
++
∂
∂
+
∂
∂
∂
∂
+
∂
∂
−=
∂
∂
+
∂
∂
η
ρ
ρ
ρ
(16)
The effective viscosity
η
eff consists of the
laminar viscosity
η
lam and the apparent turbulent
viscosity, resulting from turbulent mixing. The
widely used k-
ε
turbulence model (Launder et
al., 1972) is employed to capture the turbulent
flow effects. It requires the solution of additional
transport equations for the turbulent fluctuation
energy k and the turbulent dissipation rate
ε
. For
brevity, these equations are not reproduced
here. Turbulence generation and dissipation due
to the particles are assumed to cancel. With the
k-
ε
-model constant C
µ
(0.09), the effective vis-
cosity reads
ε
ρηη
µ
2
k
C
lame ff += (17)
The resistance force per volume due to for-
est is written analogously to (7) as
u
u
u
c
s
d
f
i
w
res
res
res
i
2
2
)(
ρ
= (18)
The particles are treated as separate rigid
bodies. The drag force exerted by the surround-
ing air onto a particle is formulated as (suffix p
indicates particle values)
p
ipp
p
dip u
ud
u
cF
∆
∆
∆
=,
2
2
,42
π
ρ
(19)
with the drag coefficient c
d
and the particle
diameter d
p
, introduced in 2.2. The slip velocity
ipiiip
uuuu
,,
)( −
′
+=∆ (20)
must consider the turbulent fluctuation part
u’
i
of the air velocity at the particle position. W ith
the k-
ε
–model, a fluctuation velocity can be de-
termined stochastically by picking a random vec-
tor according to a Gaussian distribution with
mean value zero and a standard deviation of the
vector magnitude of
ku
i
3
2
2
=
′
(21)
The same random velocity is kept as long as
the particle travels with the same turbulent eddy,
which is of size L
turb
and has a life time t
turb
, de-
termined from k and
ε
: This time interval t
t
ends
when either the eddy decays or it is traversed.
∆
=
==
turb
p
turb
t
turbturb
t
u
L
t
k
t
k
CL
,min
5.1
75.0
εε
µ
(22)
The particle drag force in (16) hence can be
formulated as limit of a sum over all particles
within a volume
∆
V around the considered posi-
tion
∆
→∆
∆
−=
V
p
ipVi
F
V
f
,0
1
lim
(23)
and the momentum balance for a particle
with mass
6/
3
πρ
picep dm =
reads
ipip
ip
pgmF
dt
du
m+= ,
,
(24)
Particles that hit obstacles are removed. The
hit-probability within a time interval
∆
t is
res
p
s
tu
res
res
hit
s
d
P
∆
−−=
11 (25)
A particle is removed, if a random number in
the range [0,1), queried each time interval
∆
t, is
smaller than P
hit
. W hen a particle hits the DFL
surface, it is always reflected. If it hits the ter-
rain, it is reflected only if the velocity is larger
than a threshold (of 3 ms-1) and the angle be-
International Snow Science Workshop, Davos 2009, Proceedings
523
International Snow Science Workshop Davos 2009
tween its velocity vector and the surface normal
is smaller than 30°. Otherwise it is deposited.
3 NUMERICAL METHODS
3.1 DFL Numerics
The DFL momentum equation (6) is solved
for a large number of mass elements (typically
about 2000 kg each) explicitly in time (usually
steps of 0.1 s). A regular DHM grid with a reso-
lution of typically 5 m is used to represent the
terrain and to define the SPH-kernel-function W
in (2).
Figure 3. DHM-grid (detail), seen from above,
mass element centers (dots), element velocities
(light vectors) and smoothed velocity field at grid
points (bold vectors).
To obtain W, a bilinear interpolation is ap-
plied first to distribute element values to the
nodes of the grid cells containing the element-
centers, and the same bilinear interpolation is
used afterwards to get a smoothed value at the
element center from the values at the surround-
ing grid nodes. SPH methods have the advan-
tage of small numerical diffusion, while they may
show stability problems due to the forces be-
tween the elements. In the DFL the latter are
small (of order H/L), however. Furthermore, an
artificial viscosity is applied to the difference be-
tween element velocity and the smoothed veloc-
ity at the element center.
3.2 PSL Numerics
A fully implicit Finite Volume method is used
to solve the balances of the air flow in a SIMPLE
scheme (Patankar, 1990; for implementation
details, see FIRE Manual, 2009). The momen-
tum balances for the ice-particles are integrated
explicitly in time. Of course, not all particles can
be followed separately. Instead, they are
grouped in parcels (of about 100 kg each) and
the equations are solved for one exemplaric par-
ticle in each parcel. The numerical PSL-mesh is
created by extruding a regular grid (resolution
typically 15 m) in direction of the average terrain
normal. About 20 cell layers with increasing
depth are added, with the depth of the first ap-
proximately 4 m.
Figure 4. 3D-grid for simulation of PSL adapted
to the terrain surface.
4 APPLICATION RESULTS
Model validation, application and compari-
sons to the initial SAMOS version are described
in a separate article (Granig et al., 2009).
5 REFERENCES
FIRE CFD Solver Manual v2009, 2009. AVL List
GmbH, Graz.
Gersten, K. and Herwig, H. 1992.
Strömungsmechanik, Vieweg, Braunschweig,
Wiesbaden.
Granig, M., Sampl, P., Tollinger, C., Jörg, Ph., 2009.
Experiences in avalanche assessment with the
powder snow avalanche model SamosAT.
Accepted for the International Snow Science
Workshop 2009, Davos, Switzerland.
Launder, B.E. and Spalding, D.B., 1972.
Mathematical models of turbulence. Academic
Press, London.
Monaghan, J.J., 1988. An introduction to SPH.
Computer Physics Communications, vol. 48, pp.
88-96.
Patankar, S.V., 1980. Numerical heat transfer and
fluid flow. McGraw-Hill, New York.
Sailer, R., Fellin, W., Fromm, R., et al. 2008. Snow
avalanche mass-balance calculation and
simulation-model. Annals of Glaciology, Volume
48, 183-192.
Sampl, P., Zwinger, T., 2004. Avalanche simulation
with SAMOS. Annals of Glaciology, Volume 38,
393-398.
Savage, S.B. and Hutter, K., 1989. The motion of a
finite mass of granular material down a rough
incline. J. Fluid Mech. 199, 177-215.
Zwinger, T., Kluwick, A., Sampl, P., 2003. Simulation
of Dry-Snow Avalanche Flow over Natural
Terrain. In: Hutter, K., Kirchner, N. (Editors.),
Dynamic Response of Granular and Porous
Materials under Large and Catastrophic
Deformations, Lecture Notes in Applied and
Computational Mechanics. Volume. 11, Springer,
Heidelberg, p. 161-194.