A heuristic numerical model for three-dimensional timedependent glacier flow
BY
WILLIAM J. CAMPBELL and LOWELL A. RASMUSSEN
U.S. Geological Survey, Tacoma, Washington
ABSTRACT
A key problem in glacier dynamics is one that is of importance to many
areas of geophysics—the slow flow of a substance with a free surface.
Detailed analytic descriptions of glacier flow have only been possible with
one- or two-dimensional models which could not be used to study many
complex glacier features. As a first step in the development of a mathematically tractable model for three-dimensional, time-dependent glacier
flow, a Newtonian viscous flow law is assumed. The Navier-Stokes
equation, with acceleration and inertial terms omitted, is integrated
vertically to form a flow (volume transport) equation, which, alone with
the three-dimensional continuity equation for incompressible flow, forms
a system of three equations in four inknowns. By assuming a linear
relation between volume transport and bottom shear stress, one unknown
is eliminated.
Using an IBM-7094, the model is applied to two bed configurations—
valley and cirque—with the same transverse profile. Several steady-state
solutions are obtained by varying the mass-balance-versus-altitude input
function, the ice viscosity, and the bed friction coefficient.
Time-dependent motion is studied in two ways: (1) by imposing an
arbitrary wave on a steady-state solution for the valley glacier and observing its deformation and diffusion and (2) by imposing short period
(five year) climate changes on the same steady-state solution and observing
the return to the original node. With these simple causative mechanisms,
complex, three-dimensional, dynamic waves are generated.
Introduction
A glacier has a complicated, varying surface that results from the
interplay between the climate and the flow within and at the bed of the
glacier. If the multifaceted glacier climate is simply quantized, say in
terms of the mass balance of ice, then the remaining chief difficulties in
the mathematical analysis of glacier flow and response to varying climates
are encountered in (1) the complex relation between the shear stress on
the bed and the sliding velocity and (2) the nonlinear equations for freesurface flow. Much work has been done on the first of these difficult
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problems, whereas the second, which is of importance in many areas of
geophysics, has received attention mainly in hydrodynamics, where few
of the results can be applied to glaciers.
The authors first became interested in the slow-flow, free-surface
problem when studying photographs of the folded moraines of the
Malaspina Glacier and the waves on Nisqually Glacier. We felt that
these and many other glacier phenomena could be best understood if
the glacier were treated as a three-dimensional, dynamic whole. Our
experience in meteorology and oceanography told us that in the study of
motion involving frictional forces significant jumps in understanding were
made when the frictional forces were explicitly stated and treated in threedimensional motion; i.e., the Ekman spiral. Experience also told us that
in many slow-flow problems it is necessary to consider the nonlinear terms
in the equation of motion in order to model the flow properly, especially
where the scale of flow features is within an order of magnitude of the
smallest dimension of the flow system considered; i.e., the waves on
Nisqually Glacier which have amplitudes as large as one-third the thickness
of the glacier.
These considerations led us to work on a model which would treat
three-dimensional, time-dependent flow with explicitly stated frictional
forces. We soon became convinced that it would be very difficult to
create a mathematically tractable model if we used a nonlinear flow law
such as Glen's (1952). However, considering the uncertainties in the
boundary conditions at the bed of glaciers, we were prompted to make
what may be called a pragmatic backstep and use a Newtonian viscous
flow law for glacier flow, as did Somigliana (1921). We felt it worthwhile
to make a system in which the three-dimensionality of glacier features
could be modelled by sacrificing strict quantitative accuracy. Also, we
reasoned that if we could make a three-dimensional, time-dependent model
work for the Newtonian viscous case, we could then proceed to a model
with a more complex flow law. It was this key assumption that led the
authors to call the model "heuristic", considering that it is a demonstration
which is persuasive rather than logically compelling.
Development of model
The Navier-Stokes equation, omitting inertial and acceleration terms,
may be written in vector notation as follows:
- ^ + 1 + vV2V = 0
(1)
P
where V = ui + vj + wk is the three-dimensional velocity vector, v is
—>•
*
*
the kinematic viscosity, g = gxi + gyj + gzk is the gravity vector, p is
the pressure, and p is the ice density. If we assume that the velocities
normal to the bed are much smaller than the velocities parallel to the bed,
the above can be written in component form as
DYNAMICS
p3x
+
179
g
(2)
g
pöy
! - *
<3>
where TX and r y are the shearing stress components in the x and y directions,
respectively, and V22 is the two-dimensional Laplacian operator. Eq. (3)
is the hydrostatic equation, which may be integrated vertically to yield
the hypsometric equation
p = pg(h - z) + p a
(4)
where h is the thickness of the glacier measured normal to the bed, and
Pa is the atmospheric pressure.
Now let us define a volume transport vector
Q = Qxi + Qyj
h
where
Qx =
.
. _
u dz and
Qy =
o
f
.
v dz
>
(5)
o
This allows us to proceed with the vertical integration of Eq. (2) to
form the transport equation. Let us integrate the x-component equation
of Eq. (2) term by term. By using the hypsometric equation Eq. (4) we
can integrate the pressure term as follows:
ÇH dp ,
1 d fh
.,
..
1 3 [>gzh2~|
, ôh
- w- dz = - — pgz(n—z)dz = - —- r—— \ = gzh —
J pdx
pdxj
p dxl 2 J
dx
o
o
r
The gravity term is integrated simply
gx dz = gxh
o
If we assume the kinematic viscosity is independent of z, the horizontal
viscous term is integrated as
f vV 2 2 u dz = v V 2 2 I u dz = vV 2 2 Qx
o
o
Since at the glacier surface (z = h) the shearing stress vanishes, the
vertical viscous term integrates simply
p dz.
p
where TX' is the shearing stress in the x direction at the bed of the glacier.
Combining the above four terms we can now write the transport
equation in component form for both the x and y components by considering the symmetry in x and y.
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TX'
3h
p
Sx
— = ~gZh ï
p
+ gyh + VV22 Qy
°y
Or in vector form
- = - g z h V h + g*x,yh + vA22 Q
(6)
P
Turning now to the continuity equation for incompressible flow with
a forcing function a we have
V V = a(x,y,z,t)
By integrating this vertically, substituting Eq. (5), and assuming that
the forcing function is applied at the free surface z = h(x,y,t) we have
— = — V"Q + a(x,y,t)
(7)
In the present case the forcing function a(x,y,t) is the mass balance of
the glacier surface.
Eqs. (6) and (7) constitute a set of three equations in four unknowns.
The obvious way to proceed is to eliminate T' in terms of Q. The best
choice would be to use the Glen flow law in the way used by Nye (1960)
and Weertman (1958), such as
VP = ( —- j ; m s» 2-5, A = const.
where VH is the flow velocity parallel to the bed averaged over the
thickness h. However, such a choice would have formed a transport
equation which was nonlinear in Q as well as in h and which, at the-time
of the development of their theory, the authors felt incapable of solving.
Also, having assumed a Newtonian viscous flow law for the ice-to-ice
friction, we decided to treat the ice-to-rock friction in a similar manner.
However, it was felt that the bed shearing stress might be a function of
glacier thickness as well as velocity, as Bodvarsson (1955) maintained,
and decided to assume his flow law in which the volume transport is a
linear function of bottom shear stress.
Q = hVu = I
(8)
Thus, the ice-to-rock friction in this model is not Newtonian in the strict
sense. This quasi-Newtonian bed friction is the second key assumption
of this model.
DYNAMICS
181
Substituting this relation into Eq. (6) gives the flow equation
Q = T l - g z h V h + 6c,yh + vV22 Q !
(9)
We wish to apply the system of Eqs. (7) and (9) to an arbitrarily shaped
glacier bed. However, we wish to measure h along the true (earth)
vertical axis, rather than normal to the bed, because the mass-balance
forcing function is measured in the vertical. We therefore transform
according to
"vertical = g _ \
••normal
gz
c
The final form of the flow and continuity equation is then
-c 2 g z hVh + cgt.yh + vV22 Q~|
(11)
j t = - * V-Q + a(x,y,t)
where h is now the thickness of the glacier along the true vertical axis.
The dynamic flow equation (11) which is not based on perturbation
analysis is similar to Nye's (1963a) kinematic equation. It contains a
surface slope term that is nonlinear in h and a term in which frictional
forces are explicitly stated. The coefficients in this equation are composed
of physical constants (gravity, density) or parameters (viscosity) which
are held constant in time and space for each integration.
No varying empirical quantities, such as surface velocity, are used in
the flow equation. Therefore, it is necessary to specify only an arbitrary
three-dimensional bed configuration and a mass balance input so that
for each set of the parameters v and A a three-dimensional glacier solution
can be found.
It is important to bear in mind the character of the two key parameters
v and A. What in effect has been done, by vertically integrating the velocity vector to form a transport vector, is that the horizontal and vertical
viscous forces have been partitioned. Mathematically, the glacier is
viewed as an array of vertical columns of unit cross-sectional area and
varying height. The friction on the sides of these columns is controlled
by the viscosity parameter v, whereas the friction at the ice-bed interface
is controlled by the bed friction coefficient A. Physically, this means
that the lateral ice-to-ice friction is treated separately from the bottom
ice-to-rock friction, the magnitudes of the frictions depending upon the
choice of v and A.
Numerical solution
Equation (11) was solved by superimposing a planar, square finite
difference grid of 21 by about 50 points with a grid spacing S of 200 m.
The error thus committed is believed to be less than the truncation error
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in the finite difference approximation of the derivatives in equation (11).
Then, the two difference equations, corresponding to the two scalar
equations implicit in equation (11), can be written as
+
S2
(12)
4S
IV'^'J J
f o J O
V ' + 'J J I ' cijgyu
+O ( n ) +O (n ~ 1) -4O (n J
where the superscripts in n designate time, and the subscripts in i and j
denote grid points. These equations can then be solved for Q ^ and
Q W by explicit relaxation. When v ^ o the relaxation was usually
accomplished in four passes over the solution region. When v = o it
was accomplished (exactly) in one pass.
The boundary conditions for equations (12) were conveniently managed
by choosing the solution region to be of such a size and shape that the
glacier lay wholly within it for a given mass balance. Then, since h — o
all around the glacier, and by requiring Q = o wherever h = o, it was
possible to have a dynamic, fixed h, no-slip boundary condition precisely
at the glacier margin without having to specify in advance the location
of the margin.
This boundary treatment leads to trouble, though, if the solution
region is chosen too short; ice piles up extremely rapidly after the glacier
terminus touches the bottom of the solution region. However, an
intelligent choice of a solution region can be made by applying the rule
JJ a(x,y) dx dy = o
where the integration is performed over the entire extent of the steadystate glacier.
Equation (11) can be written in finite difference form as
'3hYn> = a(n)J _ J _ f o (n)
_
Vx
1
(n)
^t/y
2Sc ij L "+
from which the new h-field is formed by
, Q (n) _
'^Y>)At
Q (n)
1
(14)
t/
The stability criterion was developed by J. Holton (personal communication, 1968) in a stability analysis based on a perturbation analysis of
DYNAMICS
equation (11) in linearized form.
upper bound on the time step At
183
It is most usefully expressed as an
^
(15)
The usual time step used to achieve steady-state solutions was 30 days.
The steady-state condition for a particular bed topography, v, A, and
mass balance was obtained by continuing the integration until the height
change was less than 0-01 m/yr at all points except those adjacent to the
glacier margin. The instability at the edges was greatest at the glacier
terminus where erratic height changes of about 0-1 m/yr occurred.
All calculations were performed on an IBM-7094 by a modular program
written in FORTRAN IV. Two years of glacier flow were modelled in
one minute when a 30-day time step was used. Using three or four
restarts with manually extrapolated h-fields to conserve computer time,
steady-state solutions consumed about two hours of computer time each.
The program will operate on an arbitrary bi-rectangular solution region,
say to accommodate a narrow feeder glacier flowing into a broad piedmont
glacier; however, the examples presented in this paper all had simple
rectangular solution regions.
Glaciers modelled
The model was applied to two bed configurations—valley and cirque—
having the same roughly parabolic transverse profile. Several steady-state
solutions were obtained for each bed for various climates (mass-balanceversus-altitude inputs) and various choices of the key parameters v and A.
The solutions are given in terms of a three-dimensional thickness field
and a two-dimensional volume transport field. Four of these solutions
having the same mass-balance-versus-altitude input are shown in Figs.
1 to 4. In the lower left corner of each of these illustrations is shown a
square array of the numerical grid nodes so that the reader may relate
the node density to the scale of motion in the solutions. These solutions
show features observed on real glaciers, such as concave accumulation
areas with converging streamlines of flow and convex ablation areas with
diverging streamlines of flow. The character of the thickness and flow
fields shows marked sensitivity to the choice of v and A. In Fig. 1, a
high v value is used which gives a valley glacier solution having the observed
concave to convex transverse profiles from head to terminus of the
glacier. Fig. 2 shows the solution from the same bed, mass balance, and
A value, but with the v value set to zero. In this case, the transverse
profiles down glacier become flat, and the streamlines become markedly
more convergent and divergent with greater lateral shear of the transport
vector than in Fig. 1 where ice-to-ice friction is present. The valley
glacier solution with a high ice-to-ice viscosity (Fig. 1) bears a far greater
resemblance to real glaciers than does the one with zero viscosity (Fig. 2).
In Fig. 3 is shown the steady-state solution for the same bed, mass
balance input, and viscosity v as for Fig. 1, but with the bed friction A
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increased three times. This results in a much thicker glacier with a similar
volume transport streamline field and with concave to convex transverse
profiles going down glacier as shown in Fig. 1. However, one interesting
difference between these solutions is that in the high A case standing
transverse waves appear close to the glacier border in the lower accumulation area. These are real features in the numerical solution, and appear
to result from the dynamic interplay between the downslope transverse
and longitudinal flow. No solutions for a different transverse bed profile
were obtained, but it is believed that a lesser slope than that used at the
border of the solution region would result in transverse profiles similar to
those shown in Fig. 1.
A cirque glacier (one having a bed partly in the form of a closed
depression) solution is shown in Fig. 4 in which the mass balance input,
v, and A are the same as those used in Fig. 1. The maximum volume
transport occurs far down glacier from the area of greatest thickness.
Standing transverse waves similar to those in Fig. 3 occur along the
glacier border in the lower accumulation area.
Steady state valley glacier solution—A= 10', v=IO l s
2
Volume transport
Thickness (meters)
Streamlines
~
8
~
N
6
«
- 4 - 1
balance
3km
0
0
I
2
(meters)
Magnitude (cmîsec"1)
FIG. 1. Steady-state solution for a valley glacier with A = 10* and v =
3
41cm
185
DYNAMICS
Steady-state solutions were found for three different climates, referred
to as the one-, two-, and four-meter climates. The two-meter climate is
the net mass balance versus altitude curve shown in Figs. 1 to 4. The onemeter climate is obtained by shifting the curve to the left so that it is
parallel to its original position with the value at the head of the glacier
equal to one instead of two meters. The four-meter climate is obtained
by parallel shifting of the curve to the right with the value at the head of
the glacier equal to four rather than two meters. In Fig. 5 are shown the
steady-state longitudinal profiles for three climate solutions with the same
bed, v, and A values used in Fig. 1.
Having solved a number of steady-state cases, it was decided to study
the time-dependent motion in two ways—by imposing an arbitrary wave
on the steady-state solution for the valley glacier shown in Fig. 1, and then
on the same glacier imposing short period climate changes and studying
the return to the original steady-state shape. In Fig. 6 are shown height
change fields from steady-state conditions for four time steps of approximately 16 years. Only the left side of the glacier is shown because the h
Steady state valley glacier solution — A —10*. v — O
3km
Volume transport
Streamlines
Thickness (meters)
-8
0
I
6 - 4 - 2
0 12
Net balance (meters)
1
Magnitude (cm^sec" )
FIG. 2. Steady-state solution for a valley glacier with A = 10° and v = 0.
186
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changes were bilaterally symmetrical. A transverse wave with an
amplitude of 10 m having a straight wave front is placed on the steadystate solution. After 16 years the plane wave has formed a complex
pattern of height rises and falls on the glacier surface. A high, kidneyshaped wave with an amplitude 4 m greater than the original wave has
migrated downstream along the glacier centreline. A crescent-shaped
depression has formed upstream and laterally from the high. In the
subsequent time frames the high moves downstream decreasing in amplitude and spreading as it moves, with a complex series of minor height falls
and rises occurring upstream from it. This illustration shows how very
complex are the three-dimensional, time-dependent height changes
occurring on the surface of a glacier, even when the causative mechanism
is a simple plane wave. In order to better illustrate the character of this
dynamic wave, its centerline profile as a function of time is shown in Fig. 7.
The raison d'etre of this model is to elucidate the interplay between
glaciers and climate and, although the following two cases are merely a
brief first application of the model, they illustrate the complex character of
Steady state valley glacier solution — A = 3 x 10*. *•—1015
<^\\\\l
Volume transport
Thickness (meters)
— • Streamlines
1
— Magnitude (cm sec"1)
-6 - 4 - 2
0 12
Net balance (meters)'
FIG. 3. Steady-state solution for a valley glacier with A = 3 x 108 and v = 1015.
187
DYNAMICS
glacier response to climate change. Starting with the steady-state solution
shown in Fig. 1 for the two-meter climate, the climate was changed to a
one-meter climate for five years, and then the original two-meter climate
was reimposed and the glacier recovery to steady-state was observed.
This is shown in Fig. 8 where the centerline profiles are plotted. The
curve for zero years represents the glacier centerline profile at the end of
the five-year, one-meter climate change. The maximum thickness fall
occurs 16 years after the original climate has been restored. Then the
wave trough fills in as the wave moves down glacier and diffuses. A
series of smaller amplitude waves are generated at the head of the glacier
and rapidly diffuse down glacier. The original steady-state profile is
approached as the glacier fills in and the diffuse wave moves down glacier
from the accumulation zone.
A climate change with an increase in supplied mass is shown in Fig. 9.
A two-meter climate increase is imposed on the steady-state solution
shown in Fig. 1. The zero curve represents the glacier centerline at the
end of the five-year increase. The maximum thickness increase occurs
Steady state cirque solution—A--10*. v- IOIS
I
Volume transport
Thickness (meters)
— - Streamlines
Magnitude (cm^ec ')
-8
-6
-4
3km
2
2
0
0
I
2
3
12
Net balance (meters)
*
'
FIG. 4. Steady-state solution for a cirque glacier with A = 106 and v = 1015.
4km
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„TOO-
-500.
<n
j-400
y
10
II
12
13
Steody-Stale Glacier Profiles
for Three Climates, A-io'V" 1 0
FIG.
10 16 .
5.
Steady-state centerline glacier profiles for three climates, A = 10' and v =
Arbitrary Wove - A-10*, V'\0* .
Contours show height differences in meters from steady state
0
Yeors
i
FIG.
6. Deformation and diffusion of an arbitrary dynamic wave, A = 10", v =
1016.
]6 years after the original climate has been restored. The wave crest
then falls as the wave diffuses moving down glacier. A series of smaller
waves are generated at the glacier head, and similarly to the above case
rapidly diffuse down glacier.
DYNAMICS
189
Years after release
of arbitrary wave
(centreline)
FIG. 7. Centerline glacier profiles for an arbitrary dynamic wave, A = 10*, v = 1011
Steady-state
Years after
restoring original
climate
Climate change
I meter decrease for 5 years
A=IO', v = I O I !
FIG. 8. Centerline glacier profiles for a climate change of one meter decrease for five
years.
Climate change
2 meter increase for 5 years
A=IO'. V = I O I !
FIG . 9. Centerline glacier profiles for a climate change of two meters increase for
five years.
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Conclusions and future plans
This paper must be considered as simply a statement of a new model
along with a few applications indicating that it may be of use in the study ;
of the appallingly complex interplay between glaciers and climate. We
believe we have shown that complex dynamic waves are generated on
glacier surfaces by simple causative mechanisms. We believe the inherent
advantages in this kind of a model are that ice-to-ice and ice-to-rock
friction are treated separately and stated explicitly, and that solutions are
generated for an arbitrary bed and climate without empirical measurements. We believe we can possibly improve this model by working into
it a power flow law rather than the linear Newtonian one; however, in
light of the solutions thus far obtained, we believe that more applications
of the present model are called for. One we plan to do soon is to model
a periodically surging valley glacier, and, if successful, funnel it into a
piedmont glacier and see if we find any Malaspina-type folded structures.
Then, perhaps we should model an icecap and possibly surge it. After
that, maybe on to a new model where serendipity shall reign.
REFERENCES
BODVARSSON, GUNNAR.
1955.
On the flow of ice-sheets and glaciers.
Jökull,
No. 5, p. 1-8.
GLEN, J. W. 1952. Experiments on the deformation of ice. J. Glaciol., Vol. 2,
No. 12, p. 111-14.
NYE, J. F. 1960. The response of glaciers and ice-sheets to seasonal and climatic
changes: Proc. Royal Soc, A, Vol. 256, p. 559-84.
NYE, J. F. 1963a. The response of a glacier to changes in the rate of nourishment
and wastage. Proc. Royal Soc, A, Vol. 275, p. 87-112.
NYE, J. F. 1963b. On the theory of the advance and retreat of glaciers. Geophys.
Jour. Royal Astronomical Society, Vol. 7, No. 4, p. 413-56.
NYE, J. F. 1963C. Theory of glacier variations. In : W. D. Kingery, Ice and Snow,
ed., MIT Press, p. 151-61.
SONHGLIANA, C , 1921, Sulla profondita' dei ghiaccini: Atti della Reale Academia,
Nationale dei Lincei, Rendiconti, Classe di Scienze fisiche, matematiche e
naturali, v. 30, ser. 5.
WEERTMAN, J. 1957. On the sliding of glaciers. J. Glaciol., Vol. 3, No. 21, p.
33-38.
WEERTMAN, J. 1958. Traveling waves on glaciers. Intern. Assoc. Sei. Hyd.,
Chamonix Symposium, Pub. 47, p. 162-70.