A new model for snow avalanche dynamics based on

Bingham fluids

E. Boveta,∗,1, B. Chiaiaa,∗,1, L. Preziosib,2

aPolitecnico di Torino - Department of Structural and Geotechnical Engineering, ItalybPolitecnico di Torino - Department of Mathematics, Italy

Abstract

The purpose of this paper is to describe the snow avalanche dynamics empha-sizing the phenomenon of entrainment, the shape variation and the velocityprofile thank to the peculiar features of the Non-Newtonian fluids, in par-ticular those showing shear thinning and Bingham constitutive behaviour.The model is general in its present form, and could eventually be extendedto specific granular flow. Two different approaches are proposed to simulatethe avalanches numerically: the determination of the relations to transformthe avalanche domain in a simple shape domain that doesn’t change in timeand the level set method, suitable for the free boundary problems. Finally,the characteristics of the variation of the interface between avalanche andsnowcover under a similarity hypothesis is put forward. A further refinementof those methodologies, coupled with experimental data, will eventually allowto validate the model proposed.

Key words: Snow avalanches, Bingham fluid, Shear thinning, Level set

∗Corresponding author: bernardino.chiaia@polito.itEmail addresses: eloise.bovet@studenti.polito.it (E. Bovet),

bernardino.chiaia@polito.it (B. Chiaia), luigi.preziosi@polito.it (L. Preziosi)1Fax: +39 01156448992Fax: +39 0115647599

Preprint submitted to Journal of Mechanics and Physics of Solid August 7, 2008

1. Introduction

1.1. Non-newtonian fluids: shear thinning and Bingham behavior

In this paper, particular non-Newtonian fluids in which the shear stressis a non linear function of the shear strain rate are used. In particular, thepseudoplastic or shear thinning fluids and the Bingham one are analysed.The constitutive behavior of the shear thinning fluids, as Fig. (1) shows, ischaracterized by a progressively decreasing slope µa = τ/γ (called apparentviscosity) of the shear stress as a function of the strain rate. For high valuesof γ it reaches a constant value µ∞.The viscoplastics materials or Bingham ones have a little if not zero de-

shear strain rate

shea

r st

ress

Shear thinningBinghamNewtonianBiviscous material

τ0

Figure 1: Newtonian and non-Newtonian links.

formation, till a determined yield stress value. Above this threshold value,they behave as fluids. Some examples are paint, for the first time studied in1916 by E.C. Bingham, (thus the name Bingham fluids), oil (Farina,1997),(Farina et al.,1997b), (Farina et al.,1997a), as well as materials of commonuse as toothpaste or ketchup (Macosko,1994).

Bingham, in his original paper, didn’t consider the strain rate below athreshold value (Fig. 1), that is, the material was completely rigid for valuesτ < τ0: {

γ = 0 for τ < τ0

τ = ηγ + τ0 for τ ≥ τ0(1)

This model is extended to the case in which the (solid) material has anelastic behavior for τ < τ0. In this case, the link (1) can be rewritten by the

2

following relations (Fig. 1):

{τ = Gγ for τ < τ0

τ = ηγ + τ0 for τ ≥ τ0(2)

where G is the shear modulus and γ is the shear strain.The three-dimensional formulation of the system (2) is deduced by in-

troducing the second 3 invariant IIτ of the tensor:

τ = GB for IIτ < τ 20

τ =

[η +

τ0

|II2D|1/2

]2D for IIτ ≥ τ 2

0

(3)

where B = FF T is the Cauchy-Green strain tensor, F is the deformationgradient tensor and 2D = (∇v + (∇v)′). Other models were proposed byHerschel-Bulkley (Macosko,1994), to take in account high shear rate inter-vals:

τ =

2ηD for II1/22D ≤ γc

2

[τ0

|II2D|1/2 + m|II2D|(n−1)/2

]D for II

1/22D > γc

(4)

and by Papanastasiou (Macosko,1994) who, using an exponential function,allows the use of only one equation for the whole flux:

τ =

{η +

τ0[1− exp(−aγ)]

γ

}γ (5)

The latest two models have a viscoplastic constitutive equation that re-sults advantageous in numerical simulations. Using the modifications putforward by Papanastasiou, the 3D formulation of the Herschel-Bulkley equa-tion can be rewritten as

τ =

{m|II2D|(n−1)/2 +

τ0(1− exp(−a|II2D|1/2)]

|II2D|1/2

}2D (6)

where m, n, a are calibration parameters.

3IIA =12[(tr A)2 − tr A2] where tr A is the trace of A

3

In the model developed in this paper, the links (3) and (6) adapted tothe two-dimensions for the Bingham and shear thinning behaviour are used.

A viscoplastic fluid behaves like a very viscous material, almost like asolid, for low stress values. In a very small interval, that can be modelledas a single yield stress, its viscosity falls down suddenly. Above the yieldstress the material behaves like a low viscosity liquid, in the limit case like aNewtonian one.

An important aspect of such “plastic” behavior consists in the fact that,since the stress is not constant in the whole body, some portions can flowwhile others still behave like solids. This property, in particular, will be usedin our avalanche model.

To emphasize the difference among the different fluids and in particularthe fact that a Bingham fluid can be considered a limit situation of the shearthinning link, the velocity profiles of different flows in a channel are reportedFig. (2).

(a) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Velocity

Flo

w d

epth

x=0

(b) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Velocity

Flo

w d

epth

x=0.5

(c) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Velocity

Flo

w d

epth

x=1

(d) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Velocity

Flo

w d

epth

x=1.9

Figure 2: Velocity profile comparison of fluids in a channel having η =1/ [1 + α · abs (∂u/∂y + ∂v/∂x)]: (a) Newtonian (α = 0), (b)-(c) shear thinning (α = 5and 10), (d) quite similar to Bingham’s fluid (α = 19).

4

1.2. Entrainment modellingRecent observations recognized that the entrainment of snow strongly

influences the dynamics of avalanches. It was estimated that the major-ity of medium/large avalanches can increase their mass by a factor 2 or 3(Gauer et al.,2004), while the small avalanches of the Italian Monte Pizzac(Sovilla,2004) even reach a factor 9. For this reason, it is understandablethat the entrainment may significantly affect the avalanche behaviour and inparticular its velocity, flow height, runout distance and impact pressure onobstacles.

The term entrainment includes two aspects (Gauer et al.,2004). The firstone, called erosion, concerns the breaking up of the snowcover into particles.It is strictly linked to the erosion rate, that is the velocity, measured per-pendicularly to the ground, at which the surface of the untouched snowcoveris lowered because of the erosion. The second one, the entrainment itself,consists on the incorporation of the eroded snow in the flow. To this conceptis connected the entrainment rate (kg m−2 s−1), e.g. the snow mass per unittime and unit area incorporated in the avalanche.

Four mechanisms of erosion are classified (Gauer et al.,2004): (i) impacterosion: the avalanche particles, impacting on those of the snow pack, com-press and/or displace them, (ii) abrasion: snow particles indent the snowcover and slide parallel to it, (iii) fluidization: the avalanche front generatesa pore pressure gradient in the snow pack neutralizing the gravity and thecohesive forces and (iv) ploughing: the avalanche front push the snow aheadof it.

For what concerns the entrainment, three mechanisms (Sovilla,2004) aredefined. The frontal entrainment or ploughing is characterized by a dry,low-density and cohesionless snow entrained by the avalanche that slidesover a more resistant and older layer of snow. Since it lasts a very shorttime, mathematically it can be treated as a jump. The step entrainmenttakes places when low-strength snow layers are sandwiched between ice/snowcrusts. In fact, when a crust is broken, quickly a lot of snow enters theavalanche, even not in the front. Finally, the basal erosion occurs when thesnow pack is constituted by high shear strength layers. The avalanche scrapesmass from the sliding surface along its body, proportionally to the shearingforce exerted on the basal surface.

Recently, some dense-flow models describing the entrainment have beendeveloped (for instance (Naaim et al.,2003), for an exhaustive report on thestate of the art see (Barbolini et al.,2003)), but they are not used practically,

5

due to the lack of experimental data necessary for the validation. In literature(Eglit et al.,2005) entrainment rate is considered proportional to (i) the flowvelocity, with a coefficient depending on the snow properties and on thedifference between the density of the snow cover and that of the avalanche,(ii) the square of the velocity or (iii) the flow height, and consequently to theoverall avalanche load. Hence, since near the avalanche front the maximumvalues of velocity and flow depth are encountered, the entrainment rate ishigher in the frontal part, as the measurements confirm. In fact, in the basalerosion the entrainment rate is lower (up to 10 kg/m2s) than in the caseof ploughing and step entrainment (up to 350 kg/m2s) (Sovilla et al.,2006).However the basal erosion lasts more time, with the consequence than theeroded masses are comparable.

In the model presented in this paper, a new approach is presented todescribe entrainment.

2. Model definition

Let’s consider a slope as an inclined plane4 with an inclination θ describedby the x coordinate (Fig. 3). Let’s H(x) be the snow cover thickness mea-sured orthogonally to the ground along the y direction. At the initial timet = 0 a snow mass having the front in x = 0 begins to slip. Let’s the point ofcoordinate x = (x, y)′ have velocity v = (u, v)′. The air-snow interface is de-scribed by the material interface s2(x, t) = 0, while the interface against theunderlying motionless snow cover not eroded, by the function s1(x, t) = 0.Let’s consider, besides, that the avalanche and the snow cover have the samedensity ρ and are incompressible. The first hypothesis is in agreement, forinstance, with (Sovilla,2004) who considers that the flowing snow, the en-trained one and the snow cover have the same density. The hypothesis ofincompressibility is common in almost all the existent models, even if someexperimental measures show that the avalanche flowing slightly changes itsdensity (Coussot et al.,1999).

The system composed by the snow cover and the flowing mass is consid-ered as a shear thinning (SH) fluid or a Bingham one (B). In the second case,the avalanche is supposed to be in a fluid phase (in which τ > τ0), while the

4This hypothesis is easily removable and generalizable for slopes of arbitrary shapethrough a change in curvilinear coordinates, where one describes the ground topography(local tangent), while the second one is perpendicular to the soil (local normal).

6

Figure 3: Coordinates system of the model.

layer of non eroded snow results to be in the solid phase (τ < τ0). This rep-resents a substantial difference from the Dent and Lang’s biviscous modifiedBingham model (Dent et al.,1980), (Dent et al.,1983) in which the avalancheitself is considered as a combination between two linear viscous fluids, wherethe lower one has a viscosity higher than the upper one, as justified by thevelocity profiles observed in laboratory experiences (Nishimura et al.,1989).

Let’s note that this problem is simplified since the the avalanche hasalways the same width. Consequently, the chute experiments, for instance,can be correctly simulated, while to describe real avalanches the model shouldbe refined, considering e.g. that an avalanche is higher in a gully than in anopen slope.

Under the incompressibility hypothesis, the Navier–Stokes equations be-come:

∇ · v = 0 (7)

ρ

(∂v

∂t+ v · ∇v

)= ∇ · T + ρg (8)

where v is the velocity, g is the gravitational acceleration and T is the stresstensor given by the following expression:

T = −pI + Z(∇v + (∇v)′) (9)

7

where p is the pressure and Z is

Z =

ZB =

(η0 +

τ0

|II2D|1/2

)(B) case

ZSH =

{m|II2D|(n−1)/2 +

τ0(1− exp(−a|II2D|1/2)]

|II2D|1/2

}(SH) case

(10)where η0, τ0, m, n, and a are constants defining the Bingham and shearthinning constitutive laws. Adapting the three-dimensional formulation tothe two-dimensional situation, the second invariant of 2D is

II2D = 4∂u

∂x

∂v

∂y−

(∂u

∂y+

∂v

∂x

)2

(11)

.Consequently, the momentum conservation equation becomes:

ρ

(∂v

∂t+ v · ∇v

)= ∇ · [Z (∇v + (∇v)′)]−∇p + ρg (12)

where Z = ZSH in the (SH) case and Z = ZB in the (B) one.Let’s note that if in the (SH) situation Eq. (12) is available for the whole do-main, that is ∀IIτ , in the (B) case Eq. (12) is valid only inside the avalanche,that is where IIτ ≥ τ 2

0 . Outside, the snow cover is described by

τ = GB for IIτ < τ 20 (13)

2.1. Considerations about the interfaces

Since the differential problem is a parabolic one, two conditions are re-quested on the interface: a kinematic one and a dynamic one.

2.1.1. Avalanche/air interface s2(x, t) = 0

Let’s s2(x, t) = g(x, t) − y = 0 be the equation describing the interfacebetween the two different materials, snow avalanche and air. By deriving inthe time this expression, the equation called advection equation is deduced:

∂g

∂t− v + u

∂g

∂x= 0 (14)

In this way it’s possible to deduce the interface evolution by knowingthe initial shape s2(x, t = 0) = 0. Hence, contrarily to some models in

8

literature (Beghin et al.,1991), we have supposed that avalanche doesn’t growup keeping a fixed shape along the whole path, but that it modifies itselfwith time. As a matter of fact, the distribution of the mass in the avalanchebody can influence significantly the dynamics,, with the consequence that therunout distances, heights and velocities are different. Besides, the descriptionof the flow depth allows to make calculations for the design, e.g. of a damor an house (Egli,1999) along the avalanche path.

To define the boundary conditions, let’s note that the velocity continuityis necessary, that is the avalanche velocity must be equal to that of the air.

By the scalar product of the above equality by the normal n =1

c2

(∂g

∂x,−1)′,

where c2 = 1/√

(∂g/∂x)2 + 1 is the normalisation coefficient, the followingequation is obtained:

vair · n = vavalanche · n (15)

This expression indicates the non-penetrability between air and avalanche.Moreover, the continuity of the normal stress is valid, that is:

Tair· n = T

avalanche· n (16)

However, since Tair

is negligible, Eq. (16) can be reduced to:

Tavalanche

· n = 0 (17)

Eq. (17) is expressed in the following system, available on the interfaces2(x, t) = 0:

[2Z

∂u

∂x− p

]∂g

∂x− Z

(∂u

∂y+

∂v

∂x

)= 0

[Z

(∂u

∂y+

∂v

∂x

)]∂g

∂x− 2Z

∂v

∂y+ p = 0

(18)

where Z = ZSH in the (SH) case and Z = ZB in the (B) one, noting thatthe expression for the avalanche, and not that for snowcover, is taken intoaccount.

2.1.2. The snow/avalanche interface s1(x, t) = 0

The interface s1(x, t) = l(x, t) − y = 0 identifies the bottom limit ofthe avalanche, that divides the snow cover (y < l(x, t)) from the mass inmovement (y > l(x, t)), by considering the erosion/entrainment and depositphenomena. The first constraint is linked to the definition of erodible snow:

l(x, t) ≤ H(x) (19)

9

Besides, since the shear stress on the surface s1 is equal to the thresholdvalue τ0,

t′s1 · Tns1 = τ0 (20)

where ts1 =1

c1

(1, ∂l/∂x)′ and ns1 =1

c1

(∂l/∂x,−1)′ (where c1 =√

(∂l/∂x)2 + 1),

are respectively the tangential and normal vector at the interface. Hence:

1[(∂l

∂x

)2

+ 1

]{

2∂l

∂x

(∂u

∂x− ∂v

∂y

)+

[(∂l

∂x

)2

− 1

](∂u

∂y+

∂v

∂x

)}=

τ0

Z

(21)where Z = ZSH or Z = ZB in the (SH) case or in the (B) one, respectively.

It is finally necessary to assign the condition for describing the interfaceevolution y = l(x, t). Let’s note that the advection equation (14) is not ap-plicable, because the interface is a non-material one. Therefore let’s considerthat this line moves in time because the zone in which τ = τ0 is changed.Hence, let’s assume

dl

dt∝ (

τ0 − τ |y=l(x,t)

)(22)

In the one dimension case, that expression is reduced, by using the classicalPapanastasiou’s expression, in the following way:

dl

dt∝ (τ0 − (τ0 + ηγ + τ0[1− exp(−aγ)]))

∝ (ηγ + τ0[1− exp(−aγ)]))(23)

3. Numerical simulations

3.1. Front-tracking strategy

A first approach to implement the model is based on the fact that theshape of the avalanche is almost self-similar. For this reason, using a la-grangian coordinate system moving with the avalanche, instead of an eule-rian one could be advantageous. Therefore, to simulate the evolution of themass in movement without regenerating the mesh in each temporal step, theavalanche volume is transformed in a simple domain.

10

Let’s consider the situation in which the domain of the avalanche alone(coordinates x, y) is transformed in a rectangular domain of unit height (co-ordinates ξ, η) through the following:

η =y − l(x, t)

g(x, t)− l(x, t)ξ = x

⇔{

y = η[g(ξ, t)− l(ξ, t)] + l(ξ, t)x = ξ

(24)

In this way, the interface s2(x, t) = 0 is the upper side of the rectangle, thatis η = 1, while s1(x, t) = 0 is the lower side, that is η = 0.

To determine the differential operators in the coordinates (ξ, η) it is neces-sary to calculate the Christoffel’s symbols, defined by the following formula:

Γlih =

1

2glj (gij,k + gjk,i − gik,j) (25)

where gij = ei · e′j is the metric tensor, obtained by the product between thevectors of the basis in the reference frame (ξ, η), and gij is its inverse matrix.The notation gij,k means that the derivative of gij by the k−th component(where k = 1 is the derivative by ξ and k = 2 by η) is carried out. Using asimplified notation for h and l, even if they depend on ξ e t, we obtain:

Γ111 = Γ12,1 = Γ1

21 = Γ122 = Γ2

22 = 0

Γ211 =

1

h

(η∂2h

∂ξ2+

∂2l

∂ξ2

)

Γ212 = Γ2

21 =1

h

∂h

∂ξ

(26)

It is possible now to calculate the differentials terms included in theNavier-Stokes’ equation:

∇ · v = v1,1 + v2

,2 +1

h

∂h

∂ξv1 (27)

∇p =

(∂p

∂ξ− 1

h

∂p

∂ξ

(η∂h

∂ξ+

∂l

∂ξ

),1

h

∂p

∂ξ

)(28)

and the components of the velocity laplacian:

(∇ · ∇v)l=1 =1

h

∂h

∂ξv1,1 + hv1,11 − η

∂3h

∂ξ3− ∂3l

∂ξ3+ hv1,22 − ∂2h

∂ξ

∂ηv2

∂2

∂ξ2v2,2

e1

(29)

11

(∇ · ∇v)l=2 =1

h

[∂h

∂ξv1,2 + hv1,21 − ∂2h

∂ξ2v2 − ∂h

∂ξv2,1 +

1

h

∂h

∂ξv2,2 + v2,22+ . . .

. . . +1

h

(η∂2h

∂ξ2+

∂2l

∂ξ2

)[v2,2 − 1

h

(η∂2h

∂ξ2+

∂2l

∂ξ2

)] +

2

h

∂h

∂ξ(v1,2 − 1

h

∂h

∂ξv2)

]e2

(30)In order to transform the system made by the avalanche and snow do-

main (coordinates x, y) in a rectangular domain having the unitary height(coordinates x, y), it is sufficient to replace in the previous case l(x, t) = 0and h(x, t) = g(x, t):

{y =

y

g(x, t)x = x

⇔{

y = yg(x, t)x = x

(31)

3.2. The level set method

An alternative approach to simulate avalanche dynamics consists in usingthe level set method, suitable for free boundary problems (COMSOL,2005),(Bovet et al.,2007).

To test this method, for the sake of simplicity, only the avalanche, isconsidered, neglecting the snow cover and its entrainment.

To this aim, let’s consider the system constituted by air and avalanche asa domain composed by two fluids, modelled by the Navier-Stokes equationsand having different densities and viscosities:

ρ∂u

∂t+ ρ(u · ∇)u = ∇ · [−pI + η(∇u + (∇u)T )] + F

−∇ · u = 0(32)

where ρ is the density, u = (u, v)T is the velocity, p is the pressure, I is theidentity, η is the viscosity and F takes into account the gravitational andfriction forces (both a Coulomb force and a viscous one).

The method describes the evolution of the interface between the twofluids, tracing an isopotential curve of the level set function Φ. The interfaceis described by Φ = 0, the more dense and more viscous fluid is placed inthe domain where Φ > 0 and the less dense and less viscous one is situated

12

in the zone characterized by Φ < 0. The function Φ is transported by theadvection equation:

∂Φ

∂t+ u · ∇Φ = 0 (33)

It is important to underline that the density and the viscosity have to bedescribed through the level set function, since they move jointly to the func-tion Φ. Thanks to the Heaviside function, they can be defined in the wholedomain according to:

ρ = ρ1 + H(Φ)(ρ2 − ρ1) (34)

η = η1 + H(Φ)(η2 − η1) (35)

where ρ1 (η1) and ρ2 (η2) are, respectively, the density (viscosity) of the airand of the avalanche, and H(Φ) is the Heaviside function:

H(Φ) =

{0 if Φ < 01 if Φ > 0

(36)

In the simulations reported in this section, carried out using the ComsolMultiphysics tool, for the sake of simplicity a newtonian fluid (η2=costant)is used. However, it is possible to consider the avalanche itself has a shearthinning fluid, giving an appropriate viscosity law (η2 depending on γ) as in(Dent et al.,1983), or calculated experimentally by (Kern et al.,2004b). It isimportant to underline that the results obtained are still qualitative, becausethe model should be refined and calibrated with experimental data.

The first case considered, describes an avalanche through a Newtonianfluid using typical values for the density (300kg/m3) and for the viscosity(30 kg s−1m−1, (Dent et al.,1983)). By the analysis of the density (Fig. 4)we note that the mass of the avalanche is transported towards the front,as in nature occurs, and thanks to the introduction of friction forces theavalanche stops itself. Moreover the velocity representation (Fig. 5) showsthat the front of the avalanche is faster than the tail, as confirmed by severalexperimental observations.

Furthermore, the interaction between an avalanche and a rigid structureis represented. Fig. 6 shows the effects that a dam has in containing thesnow: part of the avalanche crosses the obstacle whereas the other part isstopped by the dam.

13

⇓

⇓

⇓

⇓

Figure 4: Snapshots showing the density (avalanche is the red zone, air is the blue one)at time t=0, 1, 2, 4 and 12 s.

14

Figure 5: Snapshot shows the velocity field (surface) and the interface between the twophases (solid red line) at time 1.2 s.

4. Discussion and further development

4.1. Formulation under self-similarity hypothesis

To study how the volume changes with the variation of the lowest interfacel(x, t), we develop a simple model in which the avalanche evolves maintainingthe same shape, increasing the volume because of the snow entrainment. Asimilar hypothesis was used by Hopfinger, Tochon-Danguy, Beghin, Britterand Rastello (Rastello,2002), (Beghin et al.,1991), i.e. the avalanche keepsself-similar during its evolution.

Hence let’s consider that an initial configuration (Fig. 7), in which theabscissa varies in 0 ≤ x ≤ L0, can be expanded until the abscissa varies in0 ≤ x ≤ L(t). A similar reasoning is available also for the vertical coordinate,that is: {

x = αxy = αy

(37)

and for the two-dimensional volume A:

A = α2A0 (38)

where α =L(t)

L0

and A0 is the initial area.

The variation of this two-dimensional volume in time can be calculatedby deriving Eq. (38):

dA

dt=

2A0

L20

L(t)dL

dt(39)

15

⇓

⇓

⇓

Figure 6: Simulation 3: Snapshots showing the density at time t=0, 1, 2.5 and 5 s.

16

Figure 7: Self-similar growth of the avalanche: the Lagrangian coordinate system has theorigin fixed on the avalanche tail

or by using the density definition ρ0 =M

A, related to the fact that mass

variation is due to entrainment of the snow in the area Ldl

dtdt, where l(t) is

the interface between the motionless snow and the avalanche.It is therefore possibile to deduce that:

dA

dt=

1

ρ0

dM

dt= L(t)

dl

dt(40)

Equating Eq. (39) to Eq. (40), we obtain:

dL

dt=

L20

2A0

dl

dt(41)

Therefore, Eqs. (40), (41) indicate that the avalanche length L and themass M vary proportionally to the entrainment velocity.To deduce an analytical expression for L and M , a further hypothesis is madebased on the classification of the different possible entrainment mechanisms,describred in section 1.2. Since each mechanism is characterised by differentsnow masses, the entrainment velocity assumes different values. For thesake of simplicity, we assume that between the time t0 and t1 the frontalentrainment occurs with a constant rate (∂l/∂t = rP ), then between t1 andt2 the step entrainment gives ∂l/∂t = rS and finally between t2 and t3, thebasal erosion gives ∂l/∂t = rB.

Under such hypothesis, it is possible to calculate the mass value at each

17

time. The final time tf is of particular interest:

M(tf ) = M0 + ρ0 ·{

L(t0) · t3(rP + rS + rB) +L2

0

2A0

·[t232

(r2P + r2

S + r2B)+

. . . + t3 · rP · [−rP t0 + (rS + rB)(t1 − t0)] + t3 · rS · [−rSt1 + rB(t2 − t1)]+. . . + t3 · rB · (−rBt2)]}

(42)Let’s note than even if the basal entrainment rate is negligible in comparisonwith the ploughing and step entrainment, the amount of eroded snow iscomparable (Sovilla et al.,2006). For this reason each contribute plays animportant role.

4.2. Validation of the model

One of the most important innovation of this model is that it allows todescribe the velocity profile, which is not done by the majority of the modelsthat consider a depth-averaged velocity or give a constant law for the velocitydependence on the depth (Harbitz,1998).

To validate the capability of the model to describe correctly the ve-locity profile, the result of the simulations should be confronted with theexperimental data available. These are, for instance, the velocity profilesobtained in the snow chute at the Weissfluhjoch near Davos, Switzerland,(Kern et al.,2004a), (Kern et al.,2004b) or, at full scale, in the Valle de laSionne test site, both recorded by the Swiss Federal Institute of Snow andAvalanche Research SLF.

In the above tests also the entrainment was analysed. Investigationsabout the mass balance and the local entrainment were carried out by pho-togrammetric analysis, video recording (Sovilla et al.,2002) and the FMCW(Frequency Modulated Continuous Wave) radar data (Sovilla et al.,2006),(Sovilla et al.,2008).

An alternative for validation is represented by the experiments made witha mass of granular material released in a channel. An automatic proce-dure shows the evolution of the surface that separates the material at restfrom that in movement (Barbolini et al.,2002), and thus the behaviour of theavalanche/snowcover interface can be validated.

The model presented here can be directly implemented to be confrontedwith the chute data. In particular, to compare the velocity profile of theWeissfluhjoch, in which there is no erosion, the level set technique presented

18

in sect. 3.2 can be used. Moreover, using the level set method, several con-stitutive equations could be tested to obtain the correct velocity profile.

On the contrary, we argue that, to compare the simulations with the re-sults of the Valle de la Sionne test site, a 3D-formulation or a width average ofthe Navier-Stokes equations are necessary to take in account the confinementand the expansion of the flow due to the real topography of the mountainsites.

5. Conclusions

The developed model describes the snow avalanches using non-newtonianfluids with shear thinning and Bingham constitutive behaviour. The modelallows to describe the shape variation of the avalanche as a function of thetime, the phenomenon of snow entrainment and the velocity profile, whichrepresents an improvement of others models in the literature. The first aspectis of fundamental importance to describe correctly the interaction betweenthe avalanches and the obstacles located along their paths, since the modeldescribes the height of the flow as well the velocity and the pressure at everypoint. The second aspect, hence, plays a decisive role to describe correctlythe complete avalanche dynamics, like the prediction of the runout distanceand the determination of the deposition depth (Sovilla et al.,2007) whichis of crucial importance for a proper zonation strategy of risk areas. Thethird aspect, the most innovative one, allows a detailed analysis of the snowrheology.

In the future, the model should be improved, by the application of thetechniques analysed in this article (front tracking and level set), and validatedand calibrated on the basis of experimental data available of both channelchute and real site avalanches. For this goal, the model will be refined bydescribing the width variation or, better, with a complete 3D-formulation.

Finally, since the model is general in its present form, it could eventuallybe extended to specific granular flow.

Acknowledgements

The authors wish to acknowledge financial support by the EuropeanUnion, the Regione Autonoma Valle d’Aosta (Italy), the Ministero del La-voro e della Previdenza Sociale (Italy) and the Fondazione CRT-ProgettoAlfieri (Italy).

19

References

[Barbolini et al.,2002] Barbolini, M., Biancardi, A., Cappabianca,F., Natale,L., 2005, Laboratory study of erosion processes in snow avalanche. ColdRegions Science and Technology 43, 1-9

[Barbolini et al.,2003] Barbolini, M., Cappabianca, F., Issler,D., Gauer, P.,Eglit, M., 2003,WP5: Model development and validationErosion anddeposition processes in snow avalanche dynamics: report on the state ofthe art, SATSIE PROJECT

[Beghin et al.,1991] Beghin, P., Olagne, X., 1991, Experimental and theo-retical study of the dynamics of powder snow avalanches, Cold RegionsScience and Technology 19, 317-326

[Bovet et al.,2007] Bovet, E., Preziosi, L., Chiaia, B., Barpi, F., 2007, Thelevel set method applied to avalanches, Proceedings of the COMSOLUsers Conference 2007 Grenoble

[COMSOL,2005] COMSOL, 2005, Comsol Multiphysics: rising bubble mod-eled with the Level Set Method

[Coussot et al.,1999] Coussot, P., Ancey, C., 1999, Rhophisique des pates etdes suspensions, EDP Sciences, 227-240

[Dent et al.,1980] Dent, J.D., Lang, T.E., 1980, Modeling of snow flow 26,131-140

[Dent et al.,1983] Dent, J.D., Lang, T.E., 1983, A biviscous modified Bing-ham model of snow avalanche motion, Annals of Glaciology 4, 42-46

[Egli,1999] Egli,T.,1999, Richtlinie Objektschutz gegen Naturgefahren,Gebaudeversicherungsanstalt des Kantons St. Galle

[Eglit et al.,2005] Eglit, M.E., Demidov, K.S., 2005, Mathematical modelingof snow entrainment in avalanche motion, Cold Regions Science andTechnology 43, 10-23

[Farina,1997] Farina, A., Waxy crude oils: some aspects of their dynamics,Mathematical models and methods in applied sciences 4, vol.7, 435-455

20

[Farina et al.,1997a] Farina, A., Preziosi, L., 1997, Flow of waxy crude oils,Progress in Industrial Mathematics, 306-313

[Farina et al.,1997b] Farina, A., Fasano, A., 1997, Flow characteristics ofwaxy crude oils in laboratory experimental loops, Mathl. Comput. Mod-elling 5, vol.25, 75-86

[Gauer et al.,2004] Gauer, P., Issler, D., 2004, Possible erosion mechanismin snow avalanches, Annals of Glaciology 38, 384-392

[Harbitz,1998] Harbitz, C., 1998, A survey of computational models for snowavalanche motion, Norwegian Geotechnical Institute, Oslo, NGI Report581220-1

[Kern et al.,2004a] Kern, M., F. Tiefenbacher, 2004, Experimental devicesto determine snow avalanche basal friction and velocity profiles, ColdRegions Science and Technology 38, 17-30

[Kern et al.,2004b] Kern, M., Tiefenbacher, F., McElwaine, J.N., 2004, Therheology of snow in large chute flows, Cold Regions Science and Tech-nology 39, 181-192

[Macosko,1994] Macosko, CH. W., 1994, Rheology: principles, measure-ments and applications, Wiley, 1-56081-579-5

[Naaim et al.,2003] Naaim,M., Faug, T., Naaim-Bouvet, F., 2003, Dry gran-ular flow modelling including erosion and deposition, Surveys in Geo-physics 24, 569-585

[Nishimura et al.,1989] Nishimura, K., Maeno, N., 1989, Contribution of vis-cous forces to avalanche dynamics, Annals of Glaciology 13, 202-206

[Rastello,2002] Rastello, M.C., 2002, Etude de la dynamique des avalanchesde neige en aerosol, Centre d’Etude du Machinisme Agricole du GenieRural et Forestier (CEMAGREF), Saint Martin d’Heres (France)

[Sovilla et al.,2002] Sovilla, B., Bartelt, P., 2001, Observation and modellingof snow avalanche entrainment, Natural Hazards and Earth System Sci-ences 2, 169-179

[Sovilla,2004] Sovilla, B., 2004, Field experiments and numerical modellingof mass entrainment

21

[Sovilla et al.,2006] Sovilla, B., Burlando, P., Bartelt, P., 2006, Field experi-ments and numerical modeling of mass entrainment in snow avalanches,Journal of Geophysical Research 111, F03007 1-16

[Sovilla et al.,2007] Sovilla, B., Margreth, S., Bartelt, P., 2007, On snowentraiment in avalanche dynamics calculations 47, 69-79)

[Sovilla et al.,2008] Sovilla, B., Schaer, M., Rammer, L., 2008, Measure-ments and analysis of full-scale avalanche impact pressure at the Vallede la Sionne test Site, Cold Regions Science and Technology 51, 122-137

22