A ONE-DIMENSIONAL MODEL OF DENSE SNOW AVALANCHES
USING MASS AND MOMENTUM BALANCES
by
Richard Michael Oremus
A Thesis
Presented to
The Faculty of Humboldt State University
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Environmental Systems: Mathematical Modeling
May, 2006
A ONE-DIMENSIONAL MODEL OF DENSE SNOW AVALANCHES
USING MASS AND MOMENTUM BALANCES
by
Richard Michael Oremus
Approved by the Master's Thesis Committee:
Dr. Ken Owens, Major Professor
Date
Dr. Elizabeth Burroughs, Committee Member
Date
Dr. Margaret Lang, Committee Member
Date
Dr. Sharon Brown, Graduate Coordinator
Date
Donna E. Schafer, Dean for Research and Graduate Studies
Date
ABSTRACT
A ONE-DIMENSIONAL MODEL OF DENSE SNOW AVALANCHES
USING MASS AND MOMENTUM BALANCES
by Richard Michael Oremus
Masters of Science in Environmental Systems: Mathematical Modeling
A one-dimensional dense-snow avalanche model has been developed to predict
avalanche runout distance, flow velocity and flow height over general terrain. The model
is designed to compute the motion and deformation of the dense-snow avalanche from
initiation to runout. By applying the principals of conservation of mass and momentum,
a system of hyperbolic partial differential equations has been developed that completely
describes the flow, which is assumed to be fluid-like and follow the basic laws of fluid
mechanics. The model includes sliding friction, internal viscous friction, and
gravitational forces, which are general enough to recover the basic properties of snow
avalanches. The model is numerically solved using conservative variables and the LaxFriedrichs finite difference scheme. It is shown that the dynamics of the avalanche are
strongly dependent on the choice of the flow friction parameters.
iii
ACKNOWLEDGEMENTS
There are a number of people I would like to thank for their support, direction,
and mentoring through the process of producing this thesis. First and foremost, I wish to
thank my advisor Dr. Kenneth J. Owens, Jr. for his assistance and inspiration at all stages
of my work. Without him, I would not have been able to complete such a challenge, and
would not have produced anything nearly as complete.
I would also like to thank the other members of my committee for their valuable
guidance. Specifically, I would like to thank Dr. Beth Burroughs of the Mathematics
Department and Dr. Margaret Lang of the Engineering Department at Humboldt State
University. Their help in this arduous process has provided valuable direction and
encouragement.
Other thanks go out to the all the staff member of the Mathematics Department
and Graduate Students that have provided me with valuable advice along the way. Also,
special thanks for the support of the Research and Graduate Studies office for the grant
that helped me pursue this task.
iv
TABLE OF CONTENTS
ABSTRACT....................................................................................................................... iii
ACKNOWLEDGEMENTS............................................................................................... iv
TABLE OF CONTENTS.................................................................................................... v
LIST OF FIGURES ........................................................................................................... vi
1.
INTRODUCTION ...................................................................................................... 1
2.
LITERATURE REVIEW ........................................................................................... 4
3.
THE MODEL............................................................................................................ 12
3.1 Volume Conservation ............................................................................................ 14
3.2 Momentum Balance ............................................................................................... 18
3.3 Conservative System.............................................................................................. 29
4.
EULERIAN APPROACH TO FLUID FLOW......................................................... 31
5.
FINITE DIFFERENCE METHODS ........................................................................ 33
5.1 The Lax-Friedrichs Method ................................................................................... 33
5.2 Numerical Flux ...................................................................................................... 35
5.3 Implementing the Lax-Friedrichs Method ............................................................. 36
6.
STABILITY OF THE LAX-FRIEDRICHS METHOD ........................................... 38
7.
MODEL SIMULATIONS ........................................................................................ 43
7.1 Simulation Design................................................................................................... 43
8.
DEMONSTRATION ................................................................................................ 53
9.
DISCUSSION ........................................................................................................... 59
9.1 Model Application ................................................................................................. 60
9.2 Future Model Improvements.................................................................................. 60
10.
REFERENCES ........................................................................................................ 63
v
LIST OF FIGURES
Figure
Page
1
Parts of an avalanche path as determined by areas where there is a shift in the
balance between stress and strain. .............................................................................. 2
2
Geometry of a finite volume of snow moving down an inclined plane showing
the assumptions of the depth-averaged velocity profile. .......................................... 13
3
Transport of snow through a channel of constant width showing mass
conservation. ............................................................................................................. 15
4
Free body diagram showing the forces acting on a slab of snow with dimensions
height-h, width-w and length Δx . ............................................................................ 22
5
The result of shear stress applied on a small element of snow is deformation......... 25
6
Hydrostatic pressure experienced by a small element of snow at a depth (h-y)
below the free surface of a snow avalanche.............................................................. 26
7
Viscous force is dependant on the depth of the snow below the free surface. ......... 27
8
Definition of the mesh and cell divisions and notation for the Eulerian
numerical scheme for a snow pile............................................................................. 31
9
Each unknown U nj +1 is computed from prior neighboring values accoding to
this stencil. ................................................................................................................ 35
10
Change in the amount of snow in an interval of width q centered at xn equals
the flow in minus the flow out. ................................................................................. 36
11
Computational region for the Lax-Friedrichs finite difference method.................... 37
12
Development of a parabolic cap-shaped snow mass released from rest down a
general sloped avalanche path. Fig. 12a shows the initial shape.After one
second the mass is in the transition zone; the height of the mass has decreased
as the avalanche begins to elongate (Fig. 12b.) The mass three-seconds after
release is shown in Fig. 12c. Here the mass is starting to decelerate. Fig. 12d
shows the mass six seconds after release when it has come to rest at the end
of the avalanche path and has re-accumulated in a pile............................................ 45
r
vi
LIST OF FIGURES (continued)
Figure
Page
13
Velocity at the tail and head of a finite volume of snow as it moves down the
general sloped avalanche path shown in Figure 12. ................................................. 46
14
Front and rear edges of an avalanche over time with friction parameters set to
μ = 0.49 and λ active / λ passive (a) 0 / 0.5, (b) 2 / 3 and (c) 4 / 5. ............................... 48
15
Predicted runout distance of the avalanche vs. the sliding friction parameter.
The head of avalanche is taken from numerical simulations. Flow simulations
are shown for the diamond values in Figure 16........................................................ 49
16
The motion and deformation of the snow from initiation to rest with specific
sliding friction coefficients of (a) μ =0.29, (b) μ =0.49 and (c) μ =0.69.
Simulations were conducted with internal viscous parameters λ active=0.5,
λ passive = 2.5. Flow heights have been exaggerated by a factor of 4. ...................... 51
17
Effect of sliding friction on the velocity of the head and tail of the avalanche for
internal viscous friction parameters λ active = 0.5, λ passive = 2.5. ............................... 52
18
Design of a demonstration involving the motion of a finite volume of sand
down an inclined plane. The total volume of sand released from rest is
4/1000 meters3. ......................................................................................................... 54
19
Avalanche chute constructed of two tracks of length 3.5 meters and 2 meters
adjoined with a smooth transition zone. The 35 cm wide track is lined with
shelf paper to reduce the roughness of the sliding surface. ...................................... 55
20
Computer representation of the numerical avalanche. Flow heights have been
exaggerated by a factor of 3...................................................................................... 56
21
Comparison between observed runout distances (squares) and numerically
simulated runout distances (circles) at various angles of inclination. ...................... 57
22
Comparison between observed runout lengths (squares) and numerically
simulated runout lengths (diamonds) at various angles of inclination. .................... 58
vii
1. INTRODUCTION
Avalanches are a significant mountain hazard. Their occurrences are very
unpredictable and constitute a threat to both human life and property. Once in motion,
snow avalanches are a powerful force, capable of snapping off full-grown trees,
damaging roads and destroying buildings. In practice any snow slide big enough to carry
a person is important. Like landslides and rock falls, snow slides usually begin on steep
slopes and can travel long distances before they come to rest. For this reason engineers
and land-use planners use avalanche dynamics models to predict the run-out distances,
flow velocities and the destructive force of snow avalanches. With this information
decisions can be made to protect investments and human life in mountainous regions
where the increased popularity of winter sports puts snow enthusiasts in extreme danger.
Avalanches can occur wherever there is snow lying on ground of sufficient angle.
As winter progresses more and more snow is deposited in successive layers. These layers
may have dissimilar physical properties. An avalanche occurs when there is a failure in
the stability of the snow in these layers. Thousands of snow avalanches occur each year.
Most occur in remote mountain areas and do not affect human populations, so as a result,
go unnoticed. Avalanches become a hazard when humans and their actions expose them
to these events. Thus, the purpose of an avalanche model is to reduce the risks that are
posed by avalanches.
1
2
There are many different types of avalanches, each of which is characterized by
the type of release and the characteristics of the flow once in motion (Ancey, 2002).
Some varieties of snow avalanches are more dangerous than others. Here we will focus
on the most destructive force of an avalanche, the dense flowing core. The density of the
flowing core ranges from 150 kg/m3 to 500 kg/m3, which is half that of liquid water.
Once in motion the avalanche is capable of reaching velocities greater than 60 miles per
hour and flow heights of several meters in depth (Ancey, 2002).
Avalanche dynamics models can be used to study and quantify velocities, flow
heights, impact pressures and runout distances of avalanches. This thesis develops a
dynamics model that can predict the velocity, flow heights and momentum along a
complete avalanche path from initiation to runout.
Generally, the terrain can be divided into three segments: acceleration zone, steady flow
zone and deceleration zone (see Figure 1). All avalanches begin in a starting
(acceleration) zone and stop in a runout (deceleration) zone.
Figure 1: Parts of an avalanche path as determined by areas where there is a shift in the
balance between stress and strain.
3
Mear (2005) presents some of the general characteristic of an avalanche flow.
Avalanches seldom occur on slopes that have an angle less than 30 degrees and hardly
ever on slopes below 25 degrees. Above 45-50 degrees small avalanches are common,
but do not accumulate sufficient depth to be dangerous. The most dangerous avalanches
begin on slopes with angles between 30 degrees and 45 degrees. They are so dangerous
because they can reach speeds greater than 60 mph within 5 seconds after release. Thus,
they move quickly and can engulf everything in their path with enough force to snap fullgrown trees.
An avalanche begins when the stress of gravity pulling the snow downhill exceeds the
strength of the snow pack. Once in motion less force is required to sustain the downhill
movement of the snow. The initiation of the avalanche depends on many factors
including the type of snow crystals, age and density of the snow, and depth of a weak
layer in the snow pack. Because of the complexity of the initiation of the avalanche the
model in this thesis will not attempt to predict the occurrence of an avalanche. The goal
of our model will be to capture the movement and deformation of the avalanche once in
motion.
2. LITERATURE REVIEW
There are many existing models used to describe the movement of snow and snow
mechanics, ranging from very simple theoretical models to complicated computational
models. Typical approaches employ analytical models, statistical models and, most
recently, numerical models. To formulate our approach, a study was conducted of the
current literature. Using this research we have extracted the essential features of a
transport model and developed a unique model to describe the motion of the dense
flowing core of an avalanche.
Traditionally, simple models have been used to describe avalanche events. These
models are based on point-wise and piecewise analytical solutions of governing
differential equations describing momentum conservation laws. Simple models have
been used for more than 80 years to get crude estimations of important avalanche features
such as velocity, pressure and runout distance. Although they are very simplistic, they
can provide valuable results (Ancey, 2002). The Swiss professor Logotala (1924) was
one of the first to develop a model to compute the velocity of avalanches down a
predetermined path. Since then, Voellmy, Salm, and Gubler (VSG) have proposed many
extensions to it that have made it more accurate. In the VSG model, the flowing
avalanche is treated as a sliding block. This model can be easily used to compute the
runout distance, the maximum velocity and impact pressure of an avalanche. However, it
4
5
has been found that this model is too restrictive regarding assumptions on the avalanche
path and is unrealistic (Mear, 2005). This model divides the avalanche path into three
segments: the release zone, the avalanche track, and the runout zone, the last of which is
the region where the avalanche slows to a stop. Each of these segments is assumed to
have a constant slope, constant width and uniform flow. These assumptions are not
sufficient to model all avalanche terrain and cannot model a system as dynamic as snow
flow (Mear, 2005).
Statistical approaches have also been taken to describe the runout distance of an
avalanche. Bovis and Mears (1976), Lied and Bakkehǿi (1980), and McClund and Lied
(1987) have introduced an entirely different approach to the problem. These researchers
found that avalanche runout distances can be predicted from topographic features for a
specific avalanche path. They have developed statistical approaches relating the slope
angle of the avalanche path to the runout distance of an avalanche. These types of
models have major advantages over some of the more complicated models discussed
below, since they are simple, easy to use and are derived from the local history of
avalanches in the region of interest. When a good record of past avalanches is available,
the statistical models are very accurate. There are two types of statistical models. They
are referred to as the “alpha-beta” (α − β ) model and the “runout ratio”
model (ΔX / Xβ ) . These two models rely on a correlation between runout distances and
some topographic features. They assume the longitudinal profile of the avalanche path
governs avalanche dynamics. Although results of the statistical methods can be
comforting, since they are based on real data, they have their limitations (Mear, 2005).
6
Each mountain will have its own unique relationship between its terrain and the runout of
past avalanches so that the model may only be applicable to the terrain where that data
was obtained. Furthermore, the statistical models only give limited information since
they do not provide a means to determine the velocity, flow depth or impact pressures of
an avalanche. The model that is the focus of our research is not statistically based
because of the limitations of such a model. Although the model will not be statistically
based these models may be used in the future to verify the accuracy of our model.
Hydraulic models have been explored as a means to capture the dynamic behavior
of the avalanche flow. The models are similar to models used to describe the flow of
water and other fluids (Mear, 2005). These models are very useful since they can provide
information about the velocity, flow depth and impact pressure of an avalanche over the
entire avalanche path.
Avalanches are extremely complex. This complexity has led to the development
of several approaches based on different scales and is discussed by Ancey (2002). The
smallest scale, the size of snow particles, leads to very complicated models. The largest
scale, corresponding to the entire flow, leads to simple models such as the VSG model.
At an intermediate scale are depth-averaged models. These models provide a means of
capturing the dynamics of the avalanche flow yet are not as complicated as the smallest
scale. Our model will use depth-averaged values to describe the physical factors of the
dense flowing core.
Voellmy (1955) was one of the first to develop a quasi-one-dimensional depthaverage continuum model (the Voellmy-fluid model). He used principles of conservation
7
of mass and momentum to write the governing differential equations describing densesnow avalanche movement. In this model, the avalanche is seen to move as a plug with
velocity that is constant over the depth of the flow. No shear is allowed through the
depth of the snow; it is concentrated at the base of the avalanche. This model is widely
used, not only in Switzerland but also all over the world, and gives correct and reliable
results for small-scale avalanches. However, the model produces unsatisfactory results
for the predictions of the runout distance of large-scale avalanches (Bartelt 1999).
Eglit et al. (1960) from Moscow State University made modifications to the
Voellmy-fluid model. Their model differs from the Swiss model in that there is an upper
limit placed on the dry sliding friction parameter. Physically, this means that the friction
force cannot increase indefinitely (i.e. as the flow depth increases). The Russian model
also does not make a distinction between active and passive parts of the flow. This
modification was introduced to resolve the unsatisfactory results obtained from the Swiss
model. Salm (1993), however, maintains that snow under tensile strain (active) or
compressive strain (passive) internal friction does arise and this distinction needs to be
made.
The Norwegian NIS model, described by Norem and others (1987, 1989), is also
based on a modification of the Voellmy-fluid model. The main difference between the
NIS and Voellmy-fluid model is that the NIS model contains no plug-flow regime and the
velocity profile through the depth of the snow is not constant. This modification results
in the top layer of snow traveling faster than the bottom layer. Also, the viscous drag
term is a function of the velocity at the base of the avalanche, which differs significantly
8
from the mean flow velocity of the avalanche. In our model it is not necessary to
consider different velocities through the depth of the snow since the velocity should be
relatively uniform everywhere except at the lowest layer due to the sliding friction.
Savage and Hutter (1988) described the motion of a finite mass of granular
material down a rough incline. Their model deals with a dry cohesionless granular
material. Using principles of conservation of mass and momentum a system of partial
differential equations was developed that resembles the nonlinear shallow-water
equations. The model accurately describes the motion of the front and rear edges of a
finite mass of gravel released from rest. For the calculation of the flow other than at the
leading and trailing edges an artificial viscosity term was added to dampen the numerical
ripples that tended to develop under some conditions. However, the general features of
the motion of the gravel mass were well predicted by the numerical solutions of the
model. We would like to develop a model that does not require an artificial viscosity
term to dampen numerical ripples.
Salm (1993) developed a simple quasi one-dimensional model for flowing
avalanches. In this model the force that causes shearing in the avalanche movement is
concentrated at the base of the snow pack and is proportional to the hydrostatic pressure.
The effect of this term is a stronger resistance force for deep snow pack. This is because
of the increasing mass, which increases the flow resistance. This change in the viscous
friction causes a large increase in the height in the front part of the avalanche and, thus, a
steep deposition front, which matches with observations. Also, a resistive force
proportional to the square of the velocity is used to account for dry friction at the base of
9
the flow. The model successfully predicts runout distance with smaller deposition depths
and smaller velocity gradients in the runout zone. The use of hydrostatic pressure to
model the effect of shear produces good results. We would like to include this
description of shear in our model, but will describe shear by a depth-averaged value since
it is different at all layers of the snow pack. At the surface of the avalanche, the shear is
less than at the base of the avalanche due to the increased weight of the snow that is
above it. The use of a depth-averaged value for shear will capture the difference of the
shear through the layers of the snow pack.
Turnbull et al. (2002) use a one-dimensional depth-averaged model to compute
the mass balance of a mixed flowing snow avalanche. Their model is comprised of three
basic components: the dense flowing avalanche, the powder cloud and a turbulent wake.
The dynamics of the mixed avalanche are strongly dependent on the interaction between
these components, the snow cover and properties of ambient air. The numerical results of
the model are successful in predicting runout lengths from three known avalanche tracks
for which well-documented data exist: Aulta, Galtüir and Vallée de la Sionne. The model
that is the focus of our research is concerned with the development and motion of only
the dense flowing core and so the Turnbull model is beyond the scope of this research.
The French research group Groupement De Recherche Milieux Divisés (2003)
presented a collective of results on steady uniform granular flows. The work contains a
collection of results of initial avalanche formation. For an avalanche to occur, the driving
force must overcome some “static” threshold (or yield stress) in order to enable a dense
granular flow. Once the flow is in motion, it can be sustained by driving forces that are
10
lower than this “static” threshold, which results in a hysteretic behavior. In our model we
will not tackle the problem of predicting an avalanche event but rather focus on the
avalanche in motion. Thus, we will not need to consider such complicated hysteretic
behavior. Included also is a study of the three scales of an avalanche: the microscopic
scale at which the contact between individual grains is established, the grain level at
which the different forces act and the scale of the flow itself (of the geometry). It has
been observed that the different length scales only modify the effective-friction
coefficients. In our model we will restrict our view of the avalanche to the flow of the
avalanche itself. Thus, we will be able to neglect grain-grain interactions through the
depth of the snow pile.
This study of avalanche models considers those existing models that describe the
runout distance and motion of an avalanche. The approaches taken to describe the
motion of the dense flowing core of the avalanche make important assumptions about the
characteristics of the flow. Our model uses a different approach to describe the physical
processes that cause motion of the dense flowing core. We developed our model by
using a depth-averaged value for the velocity, shear strain term through the depth of the
flow, and sliding friction at the lowest layer. The strain is shown to capture the presence
of the internal friction that is present in the flow. This strain is not constant through the
depth of the flow and is lower at the surface than at the bottom of the snow pile. The
relationship between the forces pulling the snow downhill and the resisting forces of
sliding friction and internal friction are shown to be sufficient to describe the motion of
the avalanche. The model does not need to prevent numerical ripples in an ad-hoc way
11
by including a numerical viscosity term. It captures the characteristics of an avalanche
using only simple physical descriptions of motion. It provides a simple tool to predict the
flow of an avalanche by numerically solving the depth-averaged equations of motion.
3. THE MODEL
It has been observed that snow avalanches behave in an almost fluid-like manner
(Savage, 1989). It is also known that fluid motion is governed by a set of fundamental
physical laws including conservation of mass, conservation of momentum and Newton’s
laws of motion. All of the laws governing the motion of a fluid can be stated for
avalanches. With these governing laws we will be able to develop a system of equations
that will completely describe the motion of snow down an avalanche path. The
development of these equations is based on several important assumptions:
•
Snow is only transported through convection.
•
Flowing snow is a fluid continuum of constant density.
•
There exists a clearly defined initial snow geometry.
•
The flow height can be represented by the average flow height across the
avalanche.
•
The width of the avalanche is constant.
•
Flow velocity and height are both time and space dependent.
In this model the fluid-like flow will be treated as a continuum. Since sliding of
snow is permitted, the model will need to include a sliding friction force that will be
constant and assumed to act only at the lowest layer of the flow. Additionally, it is
12
13
assumed the snow travels in a direction parallel to the surface with a velocity gradient
that is dependent in both time and space.
Depth-averaged equations for the conservation of mass and momentum balances
will be used to describe the evolution of the snow profile geometry and transport. The
depth averaging used in the momentum balance is allowable since the velocity is
assumed to be the same everywhere except at the lowest layer due to the sliding friction.
Eventually the momentum balance equation will contain terms involving external forces
of gravity, internal drag on the system and sliding friction. Mathematically the depthaveraged velocity u (t , x) is defined by
h
1
u (t , x ) = ∫ u (t , x, y )dy ,
h0
where u (t , x, y ) is the velocity at time t, y is the height and x is the downhill location.
(See Figure 2)
Figure 2: Geometry of a finite volume of snow moving down an inclined plane showing
the assumptions of the depth-averaged velocity profile.
14
In Section 3.1 the governing equations will be derived from a physical point of
view. After creating the depth-averaged mass and momentum balances equations a finite
difference method will be presented for solving them. Afterwards several computer
simulations will be presented showing the dependence of the proposed model on the
coefficient of sliding friction and the internal viscous friction parameters. Finally, an
avalanche demonstration will be discussed.
3.1 Volume Conservation
The goal of this section is to describe the dense flowing core by a partial
differential equation relating the cross-sectional area and the velocity of the flow. This
core is a granular flow of snow masses but resembles the flow of a liquid. In the model it
is assumed that the liquid-like avalanche flow is incompressible. In addition, the
possibility of the transfer of mass into or out of the dense flowing core of the avalanche
will not be considered. This means that mass entrainment from the snow cover or
dissipation into the turbulent wake will be neglected. With these assumptions, the mass
of the snow will remain constant as the flow moves down the mountain profile.
In the model, it is assumed that the motion of the dense flowing core will be well
behaved and flow smoothly. This “smooth” behavior occurs in laminar flows, such as the
flow of a very thick maple syrup. In addition, it is assumed that the snow travels down a
channel whose lateral width is constant. Now, since it is assumed that during the flow the
15
avalanche volume will be conserved, changes in the flow height reflect movement of
snow. A simple visualization of this concept is provided by the one-dimensional flow of
a fixed volume of snow as shown in Figure 3. The volume of snow to be considered is
the volume shown in Figure 3 at time t, with height ht and downhill velocity u(t,x). At
time t the volume of this plug of snow is given by ht ⋅ w ⋅ Δxt . Now if the velocity u(t,x) is
not constant over the mountain profile the downhill face of the volume may move at a
faster speed then the uphill face, causing a deformation of the volume as it slides down
the hill. Thus, at a short time later t + Δt , the snow plug will have moved slightly
downhill and changed shape.
∆xt
∆x
∆xt+∆t
ht+Δt
(t)
(t+Δt)
Figure 3: Transport of snow through a channel of constant width showing mass
conservation.
16
After a short time, the new height and thickness of the plug of snow will be equal
to the previous height and thickness, respectively, with adjustments to account for the
change in time and change in location. As the plug of snow moves, the back and front
faces can move at different velocities given by u (t , x ) and u (t , x + Δxt ) , respectively.
Then after a short time Δt the new location of the two faces will be given by
x + u (t , x )Δt
(x + Δxt ) + u (t , x + Δxt )Δt .
The new thickness of the plug of snow after it has changed shape will be given by the
difference of the position of the two faces as
Δxt + Δt = Δxt + (u (t , x + Δxt ) − u (t , x ))Δt .
This equation can be rewritten as follows to obtain a partial differential equation
describing how the thickness of the plug of snow changes over time
Δxt + Δt = Δxt +
(u (t , x + Δxt ) − u (t , x ))
Δ x t + Δt ≅ Δ x t +
Δxt
ΔtΔxt
∂u
Δxt Δt .
∂x
Thus, after the snow has moved down the channel its new volume at time t + Δt as
shown in Figure 3 will be given by ht + Δt
plug has moved and
⋅ w ⋅ Δxt + dt , where Δx is the distance the snow
17
ht + Δt = ht +
∂h
∂h
Δt + Δx
∂t
∂x
Δ x t + Δt = Δ x t +
∂u
Δx t Δ t .
∂x
(1)
(2)
Now since the mass of the plug of snow is assumed to be constant the two
volumes can be equated to obtain
ht w Δ x t = ht + Δ t w Δ x t + Δ t
∂h
∂h ⎞⎛
∂u
⎞
⎛
= w⎜ ht + Δt + Δx ⎟⎜ Δxt +
Δxt Δt ⎟
∂t
∂x ⎠⎝
∂x
⎝
⎠
= wht Δxt + wht
+w
∂u
∂h
∂h ∂u
Δxt Δt + w ΔtΔxt + w
Δxt Δt 2
∂x
∂t
∂t ∂x
∂h
∂h ∂u
ΔxΔxt + w
Δxt ΔxΔt .
∂x
∂x ∂x
(3)
Neglecting terms of order O (Δx n Δt m ), where n + m ≥ 3 , and canceling the ht wΔxt
term that appears on both sides of the equation, a simpler form for the conservation of
mass equation is obtained
∂h
∂h
⎛ ∂u
⎞
0 = w⎜ h Δxt Δt + ΔtΔxt + Δxt Δx ⎟ .
∂t
∂x
⎝ ∂x
⎠
(4)
The width of the avalanche is assumed to be constant and non-zero, thus it can be
omitted. Similarly dividing by the non-zero term Δxt Δt in Equation (4) the following is
obtained
∂h
∂u ∂h Δx
+h
+
= 0.
∂t
∂x ∂x Δt
(5)
18
In this expression Δx is the amount the snow moved in the time interval Δt .
Taking the limit as Δt → 0 , we obtain
Δx
→ u . With this substitution of u the expression
Δt
can be simplified to obtain
∂h
∂u
∂h
+h
+u
= 0.
∂x
∂t
∂x
(6)
Now rewrite the last two terms on the left side of Equation (6) using the chain
rule. This construction gives a partial differential equation dependent on space and time
that describes the movement of a finite volume of snow flowing down an inclined plane.
Notice the equation is reduced to a statement about only the height of the snow mass
∂h ∂(hu )
+
=0.
∂t
∂x
(7)
3.2 Momentum Balance
The development of the momentum balance equation for the incompressible flow
of snow is obtained with the same finite volume approach used in the development of the
mass conservation equation. The momentum of a plug of snow down an inclined plane is
defined as P=mu where m is the mass and u is the velocity. It is known that the rate of
change of momentum is equal to the forces acting on the avalanche as a system. These
forces will account for loss and gain of momentum as the plug of snow slides down the
hillside.
19
To obtain the momentum balance equation for a plug of snow moving down an
inclined plane, again refer to Figure 3. The momentum of the fixed volume of snow at
time t is given by Pt = ρ u t ht Δxt w where u t represents local velocity, ht the local height,
Δxt the local thickness, w the width of the plug of snow and ρ is the density of the snow.
The momentum of the fixed volume of snow at time t should be equal to the momentum
at time t + Δt with adjustments to account for external forces. After the snow has moved
down the channel its new momentum will be given by Pt +Δt = ρ ut +Δt ht +Δt Δxt +Δt w , where
ht + Δt = ht +
∂h
∂h
Δt + Δx
∂x
∂t
(8)
∂u
Δx t Δ t
∂x
(9)
Δ x t + Δt = Δ x t +
u t + Δt = u t +
∂u
∂u
Δx +
Δt.
∂x
∂t
(10)
Now since the rate of change of momentum of the plug of snow is equal to the net force
acting on the plug of snow, we write
Pt + Δt − Pt
= Fnet
Δt
u t + Δt ht + Δt Δxt + Δt wρ − u t ht Δxt wρ
= Fnet
Δt
((u t +
∂u
∂u
∂h
∂h
∂u
Δx +
Δt )(ht + Δt + Δx)(Δxt +
Δxt Δt ) − u t ht Δxt ) wρ
∂x
∂t
∂t
∂x
∂x
= Fnet
Δt
20
(
)
After multiplying this expression out, neglecting terms of O Δx n Δt m ,
where n + m ≥ 3 , and canceling the common term u t ht Δxt that appear this equation can
be rewritten in a simpler form given by
(uh
∂h
∂h
∂u
∂u
∂u
Δxt Δt + u ΔtΔxt + u (ΔxΔxt ) + h (ΔxΔxt ) + h ΔtΔxt ) wρ
∂t
∂x
∂t
∂x
∂x
= Fnet .
Δt
Canceling out the common term Δt and dividing by the non-zero term Δxwρ leads to
uh
Fnet
∂u
∂u ⎛ Δx ⎞
∂h ⎛ Δx ⎞
∂h
∂u
=
.
+u ⎜ ⎟+h ⎜ ⎟+h
+u
∂t Δxt wρ
∂x ⎝ Δt ⎠
∂x ⎝ Δt ⎠
∂t
∂x
Let F net be the net force per unit length. Then Fnet = Fnet Δxt , so that
uh
∂u
∂h
∂h ⎛ Δx ⎞
∂u ⎛ Δx ⎞
∂u Fnet
+u
+u ⎜ ⎟+h ⎜ ⎟+h
=
.
∂x
∂t
∂x ⎝ Δt ⎠
∂x ⎝ Δt ⎠
∂t wρ
Again note that the lim
Δt →0
Δx
= u . Taking this limit yields
Δt
uh
∂u
∂h
∂h
∂u
∂u Fnet
+u
+ u (u ) + h (u ) + h
=
∂x
∂t
∂x
∂x
∂t wρ
∂u
∂h ⎞ ⎛ ∂u
∂h ⎞ F
⎛
+ u2 ⎟ + ⎜h
+ u ⎟ = net .
⎜ 2uh
∂x
∂x ⎠ ⎝ ∂t
∂t ⎠ wρ
⎝
Collecting terms that contain derivatives of x and those that contain derivatives of
t, and applying the chain rule the equation is rewritten in Equation (11). This
construction gives a partial differential equation dependent on space and time that
describes the momentum of a finite volume of snow flowing down an inclined plane
21
( )
F
∂ (hu ) ∂ hu 2
+
= net .
∂t
∂x
wρ
(11)
It is known, from Newton’s laws, that the sources for the variation of momentum
in a physical system are the net forces acting on it. The forces present in this system are
the external volume forces due to gravity, Fg, and internal forces of drag, Fd. The latter
of these will be based on the internal property of the snow that resists deformations.
With this in mind, Fnet in Equation (11) can be replaced with Fg - Fd , where Fg and Fd
are the external downhill forces due to gravity and internal drag, respectively, measured
per unit length. The momentum balance equation then becomes
( )
Fg − Fd
∂ (hu ) ∂ hu 2
+
=
.
∂x
∂t
ρw
(12)
Eventually F g will contain terms accounting for gravitational acceleration and
sliding friction and F d will account for internal drag that will slow the movement and
deformation of the snow.
To develop a physical formulation for the forces of gravity term, consider the
free-body diagram for the plug of snow with volume hwΔx shown in Figure 4. For this
one-dimensional formulation, there are two types of external forces that can affect the
downhill momentum of the system.
22
Figure 4: Free body diagram showing the forces acting on a slab of snow with
dimensions height-h, width-w and length Δx .
The first external force is gravitational acceleration that is tangent to the mountain profile
defined by ρwhgΔx sin (θ ) , and the second is sliding fiction f k assumed to act only on the
bottom layer of the avalanche. With this formulation, the external forces due to gravity
per unit length can be written as
Fg =
Fg
Δx
= ρ [g sin (θ ) − gμ cos(θ )] hw ,
(13)
where the sliding friction has been defined using the normal force as indicated in
Figure 4 and is written as
fk = μ * n
f k = μ * (ρ gwhΔx cos(θ )) .
Regardless of the details about the measure of the slope angle and size of the
friction coefficients, the avalanche will accelerate as long as ρgwhΔx sin (θ ) exceeds f k
because there will be a net force (and a resulting acceleration) down the slope. This
condition is satisfied in the upper portion of the avalanche path, or starting zone. Here
23
acceleration occurs because the terrain is steep and ρgwhΔx sin (θ ) is large. Constant
velocity is attained when ρgwhΔx sin (θ ) = f k . This happens when the snow slide reaches
the maximum or terminal velocity. This takes place on the avalanche path in the
transition zone between the starting zone and the bottom of the avalanche path.
Deceleration of the avalanche flow occurs in the runout zone where
f k > ρgwhΔx sin (θ ) . Here the gradient of the mountain profile is smallest so that the
forces retarding the motion are greater than forces pulling the snow down the hill. The
exact location of the runout zone depends mostly on ground roughness and slope angle.
Friction is the most dominant force that slows down the movement of the snow slide on a
slope. Equation (13) allows a very simple measure of the minimum sliding friction
coefficient that is required to slow the movement of the slide to a standstill. For the
avalanche to come to rest, the sum of the external forces due to gravity must be less then
zero. This leads to the following condition that must be fulfilled for external forces to act
as a retarding force
sin (θ ) − μ cos(θ ) < 0 ,
sin (θ ) < μ cos(θ ) ,
sin (θ )
<μ,
cos(θ )
tan (θ ) < tan (β ) .
The sliding friction μ can be expressed as μ = tan (β ) , for some angle β .
Looking closely at Equation (14), if θ < β than the forces of gravity driving the
(14)
24
avalanche down the incline will be smaller then the forces of friction retarding the flow.
In this event, the avalanche will come to rest. The size of β will vary and depends on
the ground surface. The following are contributors to the size of β :
ƒ
Rough, boulder covered or timbered slope;
ƒ
“Average” unconfined slope;
ƒ
Smooth slope (on compact, old snow);
ƒ
“Average” gully.
Proper selection of the value of β will require knowledge about the expected
snow pack conditions in the area and roughness of the terrain.
Next the derivation of the dissipative viscosity term that causes a loss of
momentum in the flow of the avalanche is presented. To model the force of viscous
friction that causes internal drag in the system, we have followed the approach taken by
Salm (1993). This force is based on an internal property of the snow that resists
deformations in its shape. This type of deformation is common in thick viscous liquids.
To derive the dissipative viscosity term, the relationship between shear stress and shear
strain as shown in Figure 5 is used.
25
∆y
∆y
Figure 5: The result of shear stress applied on a small element of snow is deformation.
It is known that shear stress is the applied force F divided by the area of
application wΔx. Shear stress is related to the shear strain experienced by a fluid and is
given by
shear stress = η
∂u
,
∂x
where u is the depth-averaged velocity, x is the downhill direction and η is the viscosity
(Weisstein). The shear stress expresses the tendency of the snow to be “pulled apart”
(sheared) by a differential force, with a viscosity η acting as a resistance to the shear.
The shearing force on a unit volume V (as shown in Figure 5) in the downhill direction is
given by the relationship
ΔFviscous
∂ ⎡ ∂u ⎤
= ⎢η ⎥ ,
ΔV
∂x ⎣ ∂x ⎦
26
where u is the depth-averaged velocity, η is the dynamic viscosity, and ΔFviscous is the
force required to shear a unit volume ΔV in the downhill direction (Weisstein). Hereη
is a measure of how resistive the snow is to flow.
In the model, it is assumed that dynamic viscosity η is proportional to hydrostatic
pressure (Salm, 1993). Hydrostatic pressure is the force that is exerted on the snow that
is below the surface of the snow pile, and is related to the depth. For a surface inclined at
an angle θ the hydrostatic pressure is given by
P = ρg cos(θ )(h − y ) ,
(15)
where ρ is the density of the snow, g is the force of gravity, y is the location in the snow
pack above the hillside, h is the depth of the snow pack so that (h − y ) is the depth of the
snow below the free surface as shown in Figure 6. Here g cos(θ ) describes the force of
gravity that is perpendicular to the hillside.
Figure 6: Hydrostatic pressure experienced by a small element of snow at a depth (h − y )
below the free surface of a snow avalanche.
27
Thus, the dynamic viscosity at a depth (h − y ) is given by η = λρg cos(θ )(h − y ) where
λ is the friction coefficient, θ is the slope angle of the mountain profile, as shown in
Figure 4, and g is the acceleration due to gravity ( 9.8
m
). With this definition, viscosity
s2
is dependent on the depth below the surface and thus different at all layers. Using this
expression for the dynamic viscosity, the viscous force per unit volume in the downhill
direction at a depth (h − y ) is given by
ΔFviscous ( y ) ∂ ⎡
∂u ⎤
λρg cos(θ )(h − y ) ⎥.
=
⎢
ΔV
∂x ⎣
∂x ⎦
(16)
Consider now that the volume of snow has a height Δy , width w and length Δx as
shown in Figure 7. Then the viscous force in the downhill direction can be written as
ΔFviscous ( y ) ∂ ⎡
∂u ⎤
(
)
g
h
y
λρ
cos
θ
(
)
=
−
.
w ⋅ Δy ⋅ Δx ∂x ⎢⎣
∂x ⎥⎦
Figure 7: Viscous force is dependant on the depth of the snow below the free surface.
28
Solving for the viscous force term per unit length we obtain
ΔFviscous ( y ) =
ΔFviscous ( y ) ∂ ⎡
∂u ⎤
λρg cos(θ )(h − y ) ⎥ wΔy .
=
⎢
Δx
∂x ⎣
∂x ⎦
(17)
Integrating over the depth of the snow pack the total viscous force can be determined.
Recalling that λρwg cos(θ )
∫
∫
h
0
h
0
∂u
is independent of y the integration yields
∂x
∂ ⎡
∂u ⎤
λρwg cos(θ )(h − y ) ⎥dy
⎢
0 ∂x
∂x ⎦
⎣
(18)
∂ h⎡
∂u ⎤
λρ
wg
cos(
θ
)(
h
y
)
dy
−
∂x ∫ 0 ⎢⎣
∂ x ⎥⎦
(19)
ΔFviscous ( y ) = ∫
Δ Fviscous ( y ) =
Fviscous =
h
∂ ⎡ λρ wg cos( θ ) h 2 ∂ u ⎤
⎢
⎥
∂x ⎣
∂x ⎦
2
Fviscous
∂
=
∂x
⎡ λ ρ wh 2 ⎤
⎢ 2 ⎥
⎣
⎦
(20)
(21)
where the coefficient of friction λ is determined by the acceleration and is given by
⎧
g cos(θ )λ a
⎛ ∂u ⎞ ⎪
λ ⎜ x, ⎟ = ⎨
⎝ ∂x ⎠ ⎪ g cos(θ )
λb
⎩
∂u
≥0
∂x
∂u
for
< 0.
∂x
for
The friction parameter λ represents the active and passive parts of the flow. This
accounts for the fact that different amounts of internal friction may be present in the
system depending on the motion of the avalanche (Bartelt, 1999). It depends on whether
the snow avalanche is being pulled apart in the active case (in the release zone) or the
avalanche is being compressed in the passive case (in the run-out zone).
29
Viscous friction is a measure of how resistive the snow is to flow when force is
applied. This friction describes the inner snow movement and changes as the snow
moves down the hill since it is related to hydrostatic pressure. It transfers the momentum
through the snow pile as it accelerates and decelerates. Thus, we can replace Fd in
Equation (12) with Fviscous to describe how internal forces of drag cause the snow to
resist motion. This captures the effect that at different depths of the snow pack different
amounts of internal friction may be present because of the weight of snow above.
3.3 Conservative System
By applying conservation of volume (mass) and momentum balance laws, a
system of hyperbolic partial differential equations have been developed to describe the
flow of the dense snow avalanche down a general path
∂h ∂ (hu )
+
=0
∂t
∂x
( )
Fg − Fviscous
∂ (hu ) ∂ hu 2
.
+
=
ρw
∂t
∂x
(22)
Using Equation (21), Equation (22) can be rewritten as
2
2
∂ (hu ) ∂ (hu + 12 λ h ) Fg
+
=
.
∂t
∂x
ρw
(23)
At this point, the system can be rewritten in matrix form by introducing the vector of
r
r
conservative variables, U , and the vector of gravitational forces, S , as
30
r
r
∂U ∂F r
+
=S,
∂t
∂x
(24)
where
r ⎡h ⎤
U =⎢ ⎥ ,
⎣hu ⎦
⎤
r ⎡
hu
⎥
F =⎢
⎢ hu 2 + 1 λ h 2 ⎥
2
⎣
⎦
,
r ⎡0 ⎤
S=⎢ ⎥
⎣f⎦
f = h[g sin θ − gμ cos θ ] ,
⎧
g cos(θ )λ a
⎛ ∂u ⎞ ⎪
λ ⎜ x, ⎟ = ⎨
⎝ ∂x ⎠ ⎪ g cos(θ )
λb
⎩
∂u
≥0
∂x
∂u
< 0.
for
∂x
for
4. EULERIAN APPROACH TO FLUID FLOW
The Eulerian scheme makes use of a fixed grid of points that extend both
upstream and downstream of the moving snow pile (Figure 8). This scheme is
convenient since the geometry of the snow pile will constantly be changing shape from
initiation of the avalanche until runout. In the release stage the snow pile will spread out
as it moves down the mountain profile with the nose moving faster and the tail moving
slower than the center-of-mass velocity. Once the avalanche reaches the run out zone the
opposite will occur and the snow pile will come to a stop in the shape of a mound.
Figure 8: Definition of the mesh and cell divisions and notation for the Eulerian
numerical scheme for a snow pile.
To formulate an approach, the mountain snow profile is divided into a number of
cells as shown in the depth profile of Figure 8. We now set up indices i that correspond
to the cell centers and j that correspond to the cell boundaries. In this scheme, the heights
31
32
and momentums will be defined at the cells centers. The cell centers will represent the
average values of height and momentum across the entire cell. In this formulation, the
height and momentum will be zero on large portions of the avalanche path where there is
no snow present, particularly in front of the head of the avalanche.
5. FINITE DIFFERENCE METHODS
The goal in this section is to present a finite-difference method that is specifically
tailored to a first-order, non-homogeneous conservative law partial differential equation
of the form
r
r r
r
∂U ∂F U
+
= S (t , x ) .
∂t
∂x
( )
(25)
The advantages of finite-difference methods are that their implementation is
straightforward, they are particularly easy to code, and they can be used to approximate
the solution of Equation (25) on a rectangular grid in the xt-plane (DuChatuau, 1989).
The Lax-Friedrichs method is tailored for application to this conservation law difference
equation in that it preserves volume. Thus, it can be used to march through time to
determine the development of an avalanche from initiation to runout.
5.1 The Lax-Friedrichs Method
The Lax-Friedrichs method is a stable, two-level finite difference method. To
implement it, we need only to reach backward in time one step to be able to determine the
solution of Equation (25) for the next time step. It will require two arrays, one to store
the solution at time tj and one to store the solution at the next time step tj+1. Since the
Lax-Friedrichs method involves two time levels, the values at the initial time must be
33
34
supplied. The starting values at time t0 must be supplied for both the initial mass
distribution h(0, x) and momentum balance distribution g (0, x ) . Clearly, we should
initialize g (0, x) = 0 since the mass will begin from rest. DuChateau (1989) gives a
complete discussion of the Lax-Freidrichs method. The following encapsulates his
derivation. The method can be obtained beginning with the first order conservation law
( )
r
r r
r
∂U ∂F U
+
= S (t , x ) .
∂x
∂t
Let q denote the size of the x grid spacing and k the size of the t grid spacing.
r r
∂F U
with the centered difference of the left and right neighbor values
Approximating
∂x
( )
r
∂U
with the forward time step the conservation law can be expressed as
and
∂t
( ) ( )
r rj
r rj
r
r
r
U nj +1 − U nj F U n +1 − F U n −1
+
= S nj ,
k
2q
(26)
where the superscripts refer to time and the subscripts to space. Observe that if we
let σ =
k
; Equation (26) can be rearranged to read
q
r
r rj
r j +1 r j σ r r j
j
[
]
(
)
(
)
=
−
F
U
−
F
U
+
k
S
n.
n +1
n −1
Un Un 2
r
By replacing the U nj term with the average of its left and right neighbors’ values
the stable Lax-Friedrichs method for the conservative law Equation (25) is obtained
r
r rj
r j +1 1 ⎛ r j r j ⎞ σ r r j
j
U n = 2 ⎜⎝U n+1 + U n−1⎟⎠ − 2 [F (U n+1 ) − F (U n−1 )] + kS n .
(27)
35
r
Looking at Equation (27) it is seen that to determine the values of U nj +1 the values
r
of U at the previous time j at its left and right neighbors’ n+1 and n-1 must be known.
r
Each of the computational grid points U nj +1 can be associated with the computational
stencil shown in Figure 9.
r
Figure 9: Each unknown U nj +1 is computed from prior neighboring values according to
this stencil.
5.2 Numerical Flux
r r
The F (U ) terms of Equation (27) represent abstract snow fluxes that are shown in
r
r r
Figure 10 where Fn = F (U n −1 ) . Looking at Figure 10 it is seen that the conservation
difference law, Equation (27), states that the amount of snow in an interval is equal to the
amount of snow that was in that interval previously adjusted for the flow in and out of the
interval across the boundaries and for abstract sources that in this model include
gravitational forces, Fg , given in Equation (13).
36
q
n-1
n
n+1
Figure 10: Change in the amount of snow in an interval of width q centered at xn equals
the flow in minus the flow out.
For small flow heights, numerical complications in the calculation of this flux can
occur. To avoid these complications, a minimum cut-off value was set on the flow height
h. As soon as h < 10-10 meters the height is set to zero. Thus, the flux is not computed.
The disadvantage to this method is that the conservation of mass law is not exactly
fulfilled. However in all simulations the mass errors experienced were very small.
5.3 Implementing the Lax-Friedrichs Method
When implementing the Lax-Friedrichs method some care must be taken at the
boundary points of the computation domain. Since each calculation requires neighboring
nodes from the previous time step, the computational domain shrinks by 2 as we march
through each time step and advance the numerical solution of the initial-value problem.
This can cause numerical computational problems. To maintain a constant domain we
modify the solution domain as follows. For each time the boundary points
37
r
r
r
r
U (t , x min ) = U min and U (t ,x max ) = U max will be defined to be the value the previous time
r
step. The definition at U min is justifiable since the initial snow geometry will begin far
r
enough down the mountain profile that U min will always be zero. Extending the
computation domain far enough into the runout zone so that the avalanche will not reach
r
the computation boundary can solve the problem at the other boundary point, U max .
Figure 11 shows the initial computation region of the Lax-Friedrichs method and the
extended computation region using the above-mentioned scheme.
Figure 11: Computational region for the Lax-Friedrichs finite difference method.
Initial computation region
Extended computation region
38
6. STABILITY OF THE LAX-FRIEDRICHS METHOD
This section describes the conditions that must be satisfied to ensure that the solution
of the finite difference equation is reasonably accurate. A von Neummann stability
analysis can be used to predict the stability of the Lax-Friedrichs method presented
previously. In order to perform the analysis, the system is rewritten. Recall that
r ⎡h ⎤
U = ⎢ ⎥ and apply the chain rule to the flux term in Equation (24) to obtain
⎣hu ⎦
r
r
r
∂F ∂F ∂U
= r⋅
.
∂x ∂U ∂x
With this Equation (24) can be rewritten as
r
r
r
∂U ∂F ∂U r
+ r⋅
= S.
∂t ∂U ∂x
r
r ⎡h⎤
∂F
To obtain an expression for r , let v = ⎢ ⎥ and apply the chain rule again
∂U
⎣u ⎦
r
r
r
∂F ∂F ∂v
r = r⋅ r
∂U ∂v ∂U
( )
r
r
∂F ∂U
= r⋅ r
∂v ∂v
−1
⎛ r
⎞
⎜ ∂v
1 ⎟
Since ⎜ r = r ⎟
⎜ ∂U ∂U r ⎟
∂v ⎠
⎝
h ⎞ ⎛1 0⎞
⎛ u
⎟⎟ ⋅ ⎜⎜
⎟⎟
= ⎜⎜ 2
⎝ u + λ h 2hu ⎠ ⎝ u h ⎠
−1
h ⎞ ⎛ 1 ⎡ h 0⎤ ⎞
⎛ u
⎟⎟ ⋅ ⎜⎜ ⎢
= ⎜⎜ 2
⎥ ⎟⎟
⎝ u + λ h 2hu ⎠ ⎝ h ⎣− u 1⎦ ⎠
39
=
0
1⎛
⎜⎜
2
h ⎝ h(u + λ h) − 2hu 2
⎛ 0
= ⎜⎜
2
⎝λ h − u
h ⎞
⎟
2hu ⎟⎠
1⎞
⎟ ≡ A.
2u ⎟⎠
(28)
With this definition of the matrix A , Equation (27) can be written as
(
r j +1 1 r j
U n = 2 U n +1 + U
j
n −1
)− σ2A [U
r
j
n +1
r
−U
j
n −1
]+ k Sr
j
n
.
(29)
The von Neummann method can be used to examine the stability of the finiter
difference method above without the non-homogeneous term, S nj . Afterward some
conclusions can be drawn about the complete finite-difference method that includes the
forcing term. DuChateau (1989) gives a complete analysis of the von Neummann
stability analysis for the Lax-Freidrichs method. The following encapsulates his analysis.
r
To perform the von Neummann stability analysis, it is assumed that U nj and therefore the
error can be expanded in a Fourier series with terms
r iβ x
ξ je n ,
r
where 0 < β ≤ 2π and ξ j is the vector of error amplitudes. Substituting this expression
into the Lax-Friedrichs scheme yields
(
) (
)
r j +1 iβ xn
σA r j iβ xn +1 r j iβxn −1
1 r j iβ xn +1 r j iβ xn −1
e
=
ξ e
+ξ e
−
ξ e
−ξ e
.
2
2
ξ
(30)
Now using the fact that q is the grid size, and q = xn+1 − x n
e
iβ x n +1
=e
iβ x n iβ xq
e
,
eiβx n−1 = eiβx n e − iβxq
(31)
40
Equation (30) can be rewritten as
) (
(
r j +1 iβx
1 r j iβ x iβxq r j iβ xn −iβxq σA r j iβ xn iβxq r j iβ xn −iβ xq
e n = ξ e ne
e
ξ e
e
+ξ e
−
−ξ e
e
2
2
ξ
r j +1
ξ
(
)
(
1
⎡1
= ⎢ e iβ q + e − iβ q I − σ A e iβ q − e iβ q
2
⎣2
)⎤⎥ξr
⎦
j
,
)
(32)
r
where the vector of error amplitudes ξ j have been factored out and the nonzero
term e
iβ x n
has been canceled. Now recalling from Euler’s formula that
cos βq =
e iβq + e −iβq
2
and
sin βq =
e iβ q − e − iβ q
,
2i
Equation (32) can be rewritten as
r j +1
ξ
r
= [cos(β q )I − iσA sin (βq )]ξ j
r j +1
ξ
(33)
r
= Gξ j ,
where G = cos(βq )I − iσA sin(βq ) is the amplification matrix for the finite difference
method. If we let λ1 and λ 2 denote the eigenvalues of the amplification matrix G, then the
von Neumann necessary condition for stability of the finite difference method is
λ p ≤ 1,
p = 1,2.
The eigenvalues of the matrix G can be related to those of A as follows. The
matrix G has an eigenvalue λ if and only if
det[G − λI ] = 0
det[(cos(βq )I − iσA sin (βq )) − λI ] = 0
det [(cos (β q ) − λ )I − iσA sin (β q )] = 0
(34)
41
⎡ cos (β q ) − λ
⎤
det ⎢
I − A⎥ = 0 .
⎣ iσ sin (β q )
⎦
Equating Equation (34) to Equation (35) it is seen that ς =
(35)
cos(βq ) − λ
must be an
iσ sin (β q )
eigenvalue of the matrix A. This gives a way to relate the eigenvalues of the
amplification matrix G with the eigenvalues of the matrix A. Solving for λ , it is found
that
λ = cos(βq ) − iσς sin (βq )
(36)
from which it follows that
λ = cos 2 (β q ) + σ 2ς 2 sin 2 (β q )
2
(
)
λ = 1 − sin 2 (β q ) + σ 2ς 2 sin 2 (β q )
2
(
)
λ = 1 + σ 2ς 2 − 1 sin 2 (β q ) .
2
From this last expression for λ 2, we see that all the eigenvalues of the
amplification matrix G satisfy λ ≤ 1 if and only if σ 2ς 2 ≤ 1 . Thus, the von Neumann
stability analysis condition for the Lax-Friedrichs method with no forcing term is
σς ≤ 1
k
for all eigenvalues ς of the matrix A where σ = . With the matrix A defined in
q
Equation (28) as
⎛ 0
A ≡ ⎜⎜
2
⎝λh − u
1⎞
⎟
2u ⎟⎠
42
the eigenvalues will be of the form ς 1 = u + λ h and ς 2 = u − λ h . For the velocities
and heights that are expected to be observed, the stability condition for the Lax-Friedrichs
method requires that the time step size satisfy k ≤ .001 for a spatial step size of
q = 0.01 which gives σ = 0.1 .
In the case of the finite difference method with the non-homogeneous term, the
stability can be tested numerically. Setting the model simulation to run at the boundary
of stability but with the non-homogeneous term included enabled us to draw the
conclusion that the non-homogeneous equation appears to be stable.
7. MODEL SIMULATIONS
The governing depth-averaged equations for the conservation of the mass and
momentum of an avalanche are solved numerically using the Lax-Freidrichs explicit
finite difference scheme for several choices of sliding friction parameter and viscous
friction parameter values. The goal is not to find the best parameter combinations but to
evaluate the model’s sensitivity to parameter values.
7.1 Simulation Design
To test the sensitivity of the model on the proposed parameters, several
simulations were run on a general curved avalanche path. First, a general flow is
displayed in Figure 12 showing the numerical avalanche at various stages of the flow.
Afterward several simulations are conducted to show the effect of the viscous friction
parameter and sliding friction parameters on the model. The avalanche path is defined by
Path profile=
65
.
1 + e 0.1x
In the release zone, Figure 12a, a parabolic cap-shaped snow mass is released
from rest with initial profile given by y = −
x2
+ 3 . The sliding friction μ has been taken
2
to be 0.49 and the internal friction parameter set to a value of 0.5 in the active zone and
2.5 in the passive zone. The flow height has been exaggerated for illustration
43
44
purposes by a factor of 3.5. In each plot there is a constant predefined flow width of 10
meters.
Shortly after the mass is released from rest, the head of the avalanche quickly
speeds up while the tail remains almost stationary. As the mass continues to move it
begins to elongate because the head of the avalanche is traveling faster then the tail. The
mass continues to elongate until the velocity of the tail of the avalanche exceeds the
velocity of the head of the avalanche. Afterwards the tail speed is greater then the head
speed and the mass of snow re-accumulates. Finally, it is observed that the avalanche
comes to rest when both the velocity at the head and tail reach zero. Note that the
maximum flow heights are always located within the body of the avalanche and decrease
smoothly towards the head and tail.
(12b)
Height (m)
Height (m)
(12a)
Length of Avalanche path (m)
Length of Avalanche path (m)
(12d)
Height (m)
Height (m)
(12c)
Length of Avalanche path (m)
Length of Avalanche path (m)
Figure 12: Development of a parabolic cap-shaped snow mass released from rest down a general sloped avalanche path. Fig.
12a shows the initial shape. After one second the mass is in the transition zone; the height of the mass has decreased as the
avalanche begins to elongate (Fig. 12b.) The mass three-seconds after release is shown in Fig. 12c. Here the mass is starting
to decelerate. Fig. 12d shows the mass six seconds after release when it has come to rest at the end of the avalanche path and
has re-accumulated in a pile.
45
46
time (sec)
Figure 13: Velocity at the tail and head of a finite volume of snow as it moves down the
general sloped avalanche path shown in Figure 12.
The graph of the velocity of the avalanche shows that the velocity of the tail
mimics the velocity of the head but delayed. The maximum velocity of 14 meters per
second is attained in the first 2 to 4 seconds when the avalanche reaches the end of the
acceleration zone. Afterward the velocity of the tail of the avalanche overtakes the
velocity of the head of the avalanche and the mass re-accumulates as shown in Figure
12c. Both the head and tail of the avalanche begin to decelerate almost linearly until the
mass comes to rest in the deceleration zone (Figure 12d).
The length of the avalanche is not constant over the avalanche path and is
predominantly influenced by the internal viscous friction parameter λ . Figure 14 shows
the progression and deformation of the mass as it moves down the avalanche path for
values of the friction parameters λactive / λ passive set to (a) 0 / 0.5, (b) 2 / 3 and (c) 4 / 5. In
47
all simulations, the maximum height of the avalanche is at the mass’s center and tapered
down toward the front and rear edges.
As the viscous friction parameters are decreased, the length of the flow increases.
This is to be expected if the flow of a thick maple syrup is considered. Observe that the
runout distance appears not to be affected by viscous friction, but is predominantly
controlled by the sliding friction parameter μ .
48
(14a)
Avalanche path (m)
(14b)
Avalanche path (m)
(14c)
Avalanche path (m)
Figure 14: Front and rear edges of an avalanche over time with friction parameters set to
μ = 0.49 and λ active / λ passive (a) 0 / 0.5, (b) 2 / 3 and (c) 4 / 5.
49
Increased sliding friction causes a decreased velocity and shorter runout distance
while less friction allows increased velocity and longer runout distance. Figure 15 shows
the relationship between the sliding friction parameter and the runout distance of the
numerical avalanche. The runout distance is measured using the head of the avalanche as
a reference. The head of the avalanche is defined as the location where the flow height is
a set minimum value. In all simulations this minimum value was set to 10 centimeters,
which was 3% of the maximum snow thickness.
Runout Distance (m)
Coeffecient
Friction
vs. Runout
Distance
Coefficient Sliding
Friction
vs. Runout
Distance
140
120
100
80
60
40
20
0.18
0.38
0.58
0.78
0.98
Coeffecient
Coefficient
SlidingFriction
Friction
Figure 15: Predicted runout distance of the avalanche vs. the sliding friction parameter.
The head of avalanche is taken from numerical simulations. Flow simulations are shown
for the diamond values in Figure 16.
Figure 16 shows the motion and deformation of the snow from initiation to rest
with specific sliding friction coefficients of (a) μ =0.29, (b) μ =0.49 and (c) μ =0.69.
50
Observe that the distance the avalanche flows into the runout zone depends greatly on the
size of the friction parameter. The rate at which the length of the mass initially spreads
appears to be independent of the choice of the sliding friction parameter and is primarily
controlled by the internal viscous term. This is not true in the runout zone though
because the length of the mass deposition is related to the location where the avalanche
stops.
51
(16a)
Downhill distance (m)
(16b)
Downhill distance (m)
(16c)
Downhill distance (m)
Figure 16: The motion and deformation of the snow from initiation to rest with specific
sliding friction coefficients of (a) μ =0.29, (b) μ =0.49 and (c) μ =0.69. Simulations
were conducted with internal viscous parameters λ active=0.5, λ passive = 2.5. Flow heights
have been exaggerated by a factor of 4.
52
Figure 17 shows the effects of the sliding friction parameter on the maximum
velocity of the head and tail of the avalanche. As expected, the avalanche flow reaches
higher velocities as the sliding friction parameter is decreased. It is observed that the
maximum tail velocity is always greater then the maximum head velocity of the
avalanche in all simulations. This is to be expected since the tail of the avalanche has
more time to reach a higher velocity. The maximum velocity obtained is dependent on
both the length of the acceleration zone and the size of the sliding friction parameter μ .
Coeffecient
Friction
vs. Maximum
Velocity
Coefficient
Sliding
Friction
vs. Maximum
Velocity
Velocity
m/sec
19
Max. Tail Velocity
17
Max. Head Velocity
15
13
11
9
7
5
0.18
0.38
0.58
0.78
0.98
Coeffecient Friction
Coefficient Sliding Friction
Figure 17: Effect of sliding friction on the velocity of the head and tail of the avalanche
for internal viscous friction parameters λ active = 0.5, λ passive = 2.5.
8. DEMONSTRATION
A demonstration was designed to test the general dynamics of the model when
applied to the flow down an inclined chute. Since snow cannot be used in my backyard,
the demonstration was conducted with sand. The demonstration involved the motion of
sand released from rest on a smooth inclined surface. The benefit to this demonstration is
that the geometry of the avalanche track is well defined and the sand will not melt. There
is a clear acceleration zone and runout zone to compare to numerical simulations. The
disadvantage is that the sand is not exactly like snow.
An energy conservation approach was used to approximate the value of the
sliding friction parameter used in the numerical simulations. The model does not use
energy conservation to determine the flow of the avalanche but this approach will provide
an approximate value for the sliding friction parameter. The internal viscous friction
parameter was determined by comparing several model simulations with observed sand
slide data to be 0.465.
The demonstration consisted of a finite mass of sand that was deposited in a
parabolic shaped pile behind a gate on top of an inclined chute. The chute consisted of
two straight tracks of 3.5 meters and 2 meters in length joined together. The first track
had an adjustable angle of inclination and the second track was flat. The track geometry
is shown in Figure 18. The 35 cm wide track was lined with shelf paper to reduce the
53
54
roughness of the sliding surface. In the experiment, the gate was quickly opened by
lifting it away from the sand, thus setting the sand in motion. Figure 19 shows a photo of
the track.
3.5m
2m
θ
Figure 18: Design of a demonstration involving the motion of a finite volume of sand
down an inclined plane. The total volume of sand released from rest is 4/1000 meters3.
The goal of this demonstration will be to show that the model correctly predicts
the evolution of the sand slide. There are two important observations to be made during
this demonstration. First, the length of the deposition of the mass and second the distance
traveled in the runout zone. The demonstration was conducted and the deposition of sand
was surveyed in the runout zone for angles of elevation of 33, 34, 35, 36, 37, 38 and 39
degrees. The length of the mass and the runout distance was measured and compared to
numerical predictions.
55
Figure 19: Avalanche chute constructed of two tracks of length 3.5 meters and 2 meters
adjoined with a smooth transition zone. The 35 cm wide track is lined with shelf paper to
reduce the roughness of the sliding surface.
Figure 20 shows the simulated avalanche at various locations on the track set at
33 degrees inclination. After release, the sand begins to move down the chute. It is seen
that the mass quickly spreads into a long thin layer with a shallow depth. The mass
continued to spread until it reached the flat section of track where it re-accumulated and
stopped 1.07 meters into the runout zone with a deposition length of 0.55 meters. The
sliding friction and viscous friction parameters were calibrated using this run. It was
determined that at a 33 degree angle the sliding friction parameter μ =0.465 and the
viscous friction parameters, λ active = 0.5, λ passive = 1.5, would accurately describe the
motion of the sand slide. The experiment was then conducted increasing the angle of
elevation of the track in one-degree increments. The observed deposition lengths and
distances were then compared to numerical results (Figures 21 and 22).
Height (m)
56
Height (m)
Length of Avalanche path (m)
Height (m)
Length of Avalanche path (m)
Runout
distance
Length
Length of Avalanche path (m)
Figure 20: Computer representation of the numerical avalanche. Flow heights have been
exaggerated by a factor of 3.
57
Runout Distance (m)
Total Runout
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
32
33
34
35
36
37
38
39
40
Angle of Inclination
Numerical Avalanche
Observed Avalanche
Figure 21: Comparison between observed runout distances (squares) and numerically
simulated runout distances (circles) at various angles of inclination.
Figure 21 compares the results of the numerical and observed sand slide runout
distances. The comparison gives some idea of the accuracy of the numerical avalanche.
There is good agreement between the observed and numerical runout distances of the
avalanche with a relative error less than 5%.
Figure 22 compares the results of the numerical and observed sand slide
deposition lengths. There is good agreement between the observed and numerical runout
lengths of the avalanche with a relative error of less then 3% for angles of inclination
between 33 and 36 degrees. The numerical model does predict slightly longer deposition
lengths in the runout zone than are observed for angles of inclination between 37 and 39
degrees with a relative error between 4% and 9%.
58
Deposition Length
Runout Length (m)
0.8
0.75
0.7
0.65
0.6
0.55
0.5
32
33
34
35
36
37
38
39
40
Angle of Inclination
Numerical Avalanche
Observed Avalanche
Figure 22: Comparison between observed runout lengths (squares) and numerically
simulated runout lengths (diamonds) at various angles of inclination.
9. DISCUSSION
A numerical dense-snow avalanche model has been developed and used to
describe the motion and deformation of an avalanche over a general two-dimensional
terrain from initiation to runout. It shows that the downhill motion of the avalanche can
be described by numerically solving the depth-averaged mass and momentum equations.
The model can predict runout distance, flow velocities and flow heights over the entire
avalanche path. An implementation of this model was developed using finite difference
methods. The numerical model was used to predict experimental results obtained from
an inclined chute demonstration involving sand. The results of the experiment showed
there is good agreement between the observed and numerical runout distances and the
lengths of the slide in the deposition zone.
The numerical model uses a simplification of the mathematics to describe the
behavior of the flow. It distinguishes between internal viscous flow resistance, which
controls avalanche length, and sliding friction, which controls the runout distance and
flow velocities. The model is sensitive to these flow parameters. Although the
mathematics in this model is a simplification of a flowing avalanche’s behavior, it is
sufficient to recover the basic properties of the snow avalanche.
59
60
9.1 Model Application
The current model can predict runout distance, flow velocities and flow heights
over the entire avalanche path. This knowledge could be beneficial to avalanche
practitioners, avalanche prone communities and avalanche mitigation specialists, who
must decide how best to protect human life from their exposure to avalanche events. For
example, Mear (2005) has investigated current avalanche mitigation techniques.
Avalanche mitigation techniques range from very simple to extremely complex and
costly. The expected size and destructive force of potential avalanches are important
factors in choosing the best measure. This avalanche model is useful since it can predict
important characteristics of the avalanche flow and thus can give insight to the
appropriate mitigation measure required using minimal parameters.
9.2 Future Model Improvements
This research presents a preliminary model to predict the flow of the dense
flowing core of an avalanche down an inclined plane. By collecting the essential features
presented both in experiments and other models, we have been able to capture the
important characteristics of the dense snow avalanche. Although the runout distances
and deposition lengths agree well in the experiment, there are many possible extensions
to the model that could be included. Future improvements to this model could include
mass transfer into and out of the dense flowing core, a second dimension, including
61
multiple components of the observed avalanche, or comparison to actual avalanche
runout data to provide a statistical validation of the model’s accuracy.
A further extension to the model could be to allow mass transfer into and out of
the dense flowing core, namely snow entrainment and loss. The size of such an
avalanche is not constant during its movement down the hill and thus additional terms
would need to be included in the model to reflect these processes. The addition of such
terms would account for how the total mass of the avalanche fluctuates as snow is
entrained or lost from the dense flowing core. The result of such features would imply
that the right hand side of Equation (7) would be nonzero. In order to include such terms,
more knowledge would be required on the exact mechanism of snow transfer into and out
of the flowing core.
An additional further improvement of the model could include adding a second
dimension. The model is currently a one-dimensional model, and describes the flow of
the dense flowing core down a general inclined plane. To make the transition to a twodimensional model, flow rates in the lateral direction would need to be considered to
describe the spread of the snow spatially as it slides down the hill. This would imply that
the flow of the avalanche would not be confined in a chute of constant width.
The current model describes only the dense-flowing core of the avalanche as it
moves down the avalanche path. Turnbull et al. (2002) describe the avalanche as having
multiple components: dense flowing core, powder cloud and a turbulent wake. The
inclusion of such components could be an additional extension of the model. The
62
addition of such components would create a system of more than just two equations and
would describe the movement of snow between each of the components of an avalanche.
Future work may also include a comparison of actual avalanche runout data.
Actual data would be valuable in assessing the accuracy of the model, its parameters and
the appearance of different phases of the flow. Actual data could provide a statistical test
of the full accuracy of this model’s ability to predict the flow of the dense flowing core.
Actual data would enable a test of the model parameters and the appearance of different
phases of the flow as it accelerated and decelerates along the avalanche path.
The model predicted well the observed sand avalanche down a confined chute.
There are many possibilities for extension to the model by adding additional aspects of an
avalanche. Improving the model would result in a more robust and useful tool for
predicting avalanche flow, preventing catastrophic incidents and determining mitigation
measures to protect human life.
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