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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.