THE LONGITUDINAL VELOCITY PROFILE
OF LARGE ICE MASSES
W. BUDD
Antarctic Division, Department of External Affairs, aUached to
Meteorology Department, University of Melbourne
ABSTRACT
The recent detailed measurements of ice movement, strain rates, ice thickness and
elevation of the Wilkes local ice cap have provided means of testing various theories
of ice movement, and thereby have provided information on the flow law of the ice.
An analysis has been made of 2 dimensional ice flow to determine the longitudinal
strain rate and velocity as a function of the ice mass dimensions and the flow law
parameters of ice. It has been necessary to use a flow law consisting of a linear term
plus a high power term to cover both the high shear at the base and the low but fluctuating longitudinal strain rates.
By the application of this theory to the analysis of the Wilkes dome data it is found
that the basal friction varies as the smoothed surface slope, and is related to the ice
thickness and velocity, depending on a high power in the flow law. Fluctuations in
surface slope over smaller distances give rise to small changes in longitudinal stress
and similar changes in the strain rate, but the magnitude of these changes at low stresses
involves only a small power in the flow law.
An extension of the analysis by Weertman (1957) for 3 dimensional ice shelf
spread has been carried out to derive the effect of an arbitrary transverse strain on the
longitudinal velocity gradient. This allows the above results to be applied to 3 dimensional diverging or converging flow.
This analysis has been found successful in explaining the lower longitudinal strain
rates over the "less active" side of the Wilkes dome where transverse extension is the
dominant strain rate.
1.
INTRODUCTION
The longitudinal velocity along a flow line from the centre to the edge is of major
importance in the study of the dynamics and history of an ice cap. This longitudinal
velocity should be determined solely by the dimensions of the ice mass, the flow law of
the ice and the velocity at the edge. In order to obtain insight into the relation between
these factors and the longitudinal velocity a programme of measurements was started
by the Australian National Antarctic Research Expeditions to obtain the velocity
distribution over the local medium scale ice cap (200 km diameter) near Wilkes (cf.
fig. 1). The first phase of this programme has now been completed. Surface velocities
and longitudinal strain rates have been determined around a large and a small triangular sector by repeated tellurometer traversing. The results of the initial surveys are
illustrated in figure 1. Associated measurements include a series of strain rosettes along
the triangle sides yielding lateral strain. Seismic and gravity readings over the region,
supplemented recently by radio echo sounding, have been carried out to determine the
bedrock elevations. Surface elevations have been measured by optical spirit levelling
to the dome summit and by theodolite and barometric levelling over the remainder of
the dome. Detailed accumulation results are available over the northern section from
stake readings over three years and over the southern section for two years. Repeated
gravity readings and optical levelling are at present being carried out to measure
directly the change in surface elevation of the ice cap.
A brief description of the field work of the first stage of the programme has been
given by McLaren (1966). A detailed report of the first stage by McLaren is to be
published (as an ANARE report) and another on the results of the first complete
remeasurements and evaluation is in preparation. A summary of the preliminary mass
58
budget estimates and state of change of the ice cap is to be published elsewhere. The
object of the present paper is to examine the theoretical principles governing longitudinal
velocity and strain rates with application especially to the profile on the " active" side
of the Wilkes ice cap from the summit to Cape Poinsett.
PE POINSETT
Lsnert
indicate poùrioniof s*àn grids
WILKES
LOCAL
ICE
All heights h
CAP
TOTTEN
ANTARCTICA
Fig. 1 — The Wilkes local ice cap showing movement vectors calculated from the
differences between tellurometer traverses carried out in 1965 and 1966.
To begin with we consider a two dimensional case, but as can be seen from figure 1
the flow lines curve and diverge and lateral strain becomes important; hence we then
extend our study to a simplified form of three dimensional flow. Finally we consider
the direct effect of the lateral strain rate on the longitudinal strain rate and velocity
profiles.
The major problems can be stated as follows:
(i) Given the dimensions of an ice cap and the flow law of ice, to calculate the velocity
profile;
(ii) Alternatively, from the measured dimensions and velocity distribution of an ice
cap to deduce information about the flow law of the ice.
Since the details of the flow law of ice in natural ice masses are still not well known
we restrict ourselves here to problem (ii).
59
GLACIER
2. THE FLOW LAW OF ICE
An examination of experimental results on the flow law of ice by many workers
including Glen (1955), Steineman (1958), Butkovitch and Landauer (1960), Meier
(1960), Mellor and Smith (1967) has revealed a generally clear picture of the relation
in the flow of ice between stress, strain and temperature. These results are depicted in
figure 2. The main features are an approximately linear flow law at low stress (below
0.1 bars) and a power law flow at high stresses (above 1 bar). The region we are most
concerned with in natural ice masses is for shear stresses between .1 and 1 bars, i.e.
largely in the transition region. We notice there is not a great deal of experimental data
available for this region of the flow law.
-I
:
O-«
Slwor Strtu Bor«
Fig. 2 — The empirical flow law of ice from the laboratory results of various workers
converted to the same units. The results of Mellor and Smith have been extrapolated to the density of pure ice. The numbers next to the curves indicate the
temperature in °C.
60
From these experimental results we may establish the following empirical relation
for the flow law of randomly orientated polycrystaline ice.
where
ti is the strain rate tensor
a'ij is the stress deviator tensor
T is the octahedral shear stress defined by
/ / 2 | ' 2 i / 2 \ |
/"O\
where a;, a'^, <r'3 are the principle stress deviators.
= Al<0 e~ke
AXfi
~ 5 x 1014 dynes cm" 2 sec" '
A2.e = ^2,0 e" M
A2fi
~ 3 x 108 dynes cm" 2 s e c l / 3 5
Aie
n
Ä
(3)
3.5
where 6 is the ice temperature in °C,
and
1- ~ —- °C~
l
10
We note that over a limited range of stress we can represent the flow law by
where the n and B values are chosen to match the curve of figure 2 over the appropriate
range.
3. TWO-DIMENSIONAL FLOW
We consider a vertical section through an ice mass along a flow line where there are
no transverse strain rates—typical of, say, the central flow line of a glacier with parallel
sides or of a flow line in an ice cap where horizontal divergence is negligible. We wish
to determine the longitudinal profile of velocity and strain rate from the dimensions of
the ice mass and theflowlaw of ice. The following approach is similar to that of Shoumskiy (1961, 1963) but with several major modifications and with particular emphasis on
the application to the study of measurements made on the ice masses.
We take orthogonal axes, x horizontal in the line of motion and z vertically downwards.
The following symbols are introduced
a — surface slope,
ß — basal slope,
Z— ice thickness,
p — ice density,
g — gravitational acceleration,
T& — the basal shear stress,
61
(Ox, tzz, Gz) — the stress components, and
(E*, £XJ. ez) — the strain rate components.
We assume that the bedrock slope is sufficiently small that we may take
sin ß = tan/? = ß
and
cos/? = 1.
The equations of motion can then be written as
ox
vz
doT o r , ,
-f + -f=~P9
ÖZ
(6)
OX
We integrate these equations with respect to z to obtain
^ = +pgZß-rb
ox
dx
(7)
(8)
where Fx is the total longitudinal force over the section.
We require as the basic quantity governing the longitudinal strain rate the longitudinal stress deviator
K = i (*,-*.)
(9)
where the bar denotes the average over the vertical section.
Now
äx = ^ = -{
pgZ(ß-f)dx
(10)
where we write/for the basal friction coefficient defined by
and
dz = a. dz
Z Jo
(12)
(13)
Z
62
dx
We now obtain
TX2 dz dz
dZa'x
(14)
pgz(ß-
dx
We note that
dZ
—
ox
(15)
and hence equation (14) becomes
xx: dz dz
S(Zà'x) ^
:«-/)-
dx
dx2
(16)
If we consider a flow law of the type
.«-
i
(17)
where
(18)
we note that with a linear flow law (w = 1) the longitudinal strain rate is directly
related to the longitudinal stress deviator independent of the vertical shear stress rxz.
When a higher power operates however the vertical shear becomes important.
We write the flow law as
(19)
where
2T,
(20)
y =
and we sec that the vertical shear is negligible when it is significantly smaller than
Nye (1959) showed in a calculation for the stress components through an ice cap
that at the surface rxy = 0 and T = (ax — ay)l2. Then as ax — ou decreases near the
base Txy increased and as a result the value of r did not vary greatly from (nz—Oy)/2,
(the value at the surface). Hence we make the assumption that vertically averaged the
flow law can be written
(21)
ßi
where èx is the longitudinal strain rate at the surface and Si is the simple linear average
of JS over the section.
Equation (16) now becomes
xxy dz dz
il/n
dx
pgZ(a-f)-
dx2
(22)
63
obtained from slope measurements over each mile (1.6 km) smoothed by a running
mean of the same distance as the average tellurometer line so as to be directly comparable with the averages over the same distance for the strain rate gradients. Figure 3
shows that equation (31) describes the situation if the basal friction/is taken to vary
as ä- the smoothed surface slope (over distances greater than the wavelengths of the
fluctuations). This means over large distances, with smoothed values, the formula for
basal stress
Tb = pgZoc
(32)
is a reasonable approximation.
From equation (31 ) and the results depicted in figure (3) we can determine the values
of «.and B by comparing the stress variations resulting from the slope changes with the
resultant strain rate variations i.e. from the equation
èl" = pg
2BZ
(33)
Z(oc— a) dx
where the integral is taken over a half wave. The results of this for each of the waves
illustrated infigure3 have been plotted infigure4. This figure shows an approximately
i
r
i
1 £
/
J
1
/
i
V
7 .
•
/
/
/
/ *
1
[2 x 10
Effcc iv« S war Str« >*
4
3
4
5
7
*. n« I / cm.
9
I09
2
3
Fig. 4 — From the fluctuations of figure 3, the variations in strain rate have been
plotted against the corresponding variation in stress over a i wave. . individual
values for peaks and troughs x mean values for adjacent peaks and troughs.
The figure suggests the strain rates arc proportional to the stress deviations.
66
linear relation between the stress and strain deviations. The range of the average stress
deviation over a half wave was from 0.06 to 0.25 bars. The average values of the flow
law parameters over the whole line were determined as
«i = 1.2
B\ = 1.2 x io 9 dyne cm"2 sec"1/3
Adjustment to the value in B due to the temperature variation of the ice over the
distance has been carried out according to equations (I) and (3), using the measured
values of temperature at the surface. Calculations of temperature profiles from the
surface to bedrock over the region show strong negative gradients at the surface with
the consequence that the surface temperature does not differ greatly from the average
temperature through the ice. The value of B above corresponds to an average temperature of about — 15"C and is compatible with values for B for different temperatures
shown by Budd (1966, fig. II) when values at low stresses from measurements by
Butkovich and Landauer (1960) and Mellor and Smith (1957) are incorporated.
We next consider the longitudinal velocity profile which on the larger scale is
determined by the general ice cap curvature rather than the minor fluctuations in slope.
To do this we first see how the velocity at the surface is related to the basal shear stress.
Nye (1957, 1959) has shown that the presence of a small longitudinal strain rate does
not significantly affect the velocity-depth profile.
We can express/ = T^lpgZ in terms of the velocity at the surface. Taking xXz =
pg <xz (23) and a flow law
we deduce
y
2(pgç£
t
( + l)Bn
where Vs, Vb are the velocities at the surface and base respectively and B is the weighted
integrated average dependent on the temperature distribution and defined by
Z
Jo ö
/
B is in effect the average value of B in the basal layers. With isothermal conditions
94% of the value of the integral for B is determined by the lower half of the ice mass.
Moreover the typical positive temperature gradient (~ 2-4°C/100m) near the base
gives smaller values of S which enhances this so that B is effectively determined by the
temperature in the lowest 10-20% of the ice. From equations (36) and (23) we can write
for the basal stress
l/n
(38)
For a cold ice mass (basal temperature well below zero) we may expect Vt = 0
67
Hence equation (31) can be written using (38) as
This is now a second order differential equation in velocity and the flow law parameters and the boundary dimensions of the ice mass (viz. ice thickness and surface
slope). If we knew the flow parameters precisely we could use this equation to calculate
the velocity and strain rate along the ice mass provided boundary values of velocity and
strain rate are known at an end point. On the other hand, if the strain rates and velocities
can be measured, along with the ice mass thickness and elevation profiles, then the
effective flow parameters can be calculated.
We note that the «and B values on the LHSand RHS may be quite different since
they refer to quite different regions of the ice mass with different stresses and temperatures. The values on the left are associated with the longitudinal strain right through the
ice mass while those on the right refer to the high shear in the basal layers.
For the Wilkes ice cap the empirical result/ = ä leads us to the following relation
for longitudinal velocity
CO)
This relation is equivalent to the result deduced by Haefeli (1961 equation (12)) and
used by him to determine the velocity along a flow line in Greenland.
By plotting VIZ against aZ/B, for the Dome-Poinsett line, taking into consideration the temperature variation of B without specifying its actual value, we see from
figure 4 that this relation appears to hold quite well over the Wilkes ice dome with n =
3.4. We may now determine B from these results, and converting to units dyne cm"2
sec1'3 according to Budd (1966) equation (21) we obtain Bo = 0.85 x 109 dyne cm"2
sec1/3 at a temperature of — 10°C and stress range 0.5 to 1.1 bars. This value also is
found to agree with the plot of S values against temperature as described for Si.
The results of this two dimensional analysis are applicable to ice sheets where
transverse strain rates are negligible. An extension to certain simple three dimensional
models may be carried out to cover the cases of glaciers where transverse shear is
important by considering averages over the cross section and appropriate shape factors
(cf. Nye 1965). This will be treated elsewhere. For the Wilkes dome data the transverse
shear was generally unimportant but because of the converging and diverging flow
transverse extension and compression were important. This makes it necessary to
examine the effect of the transverse stress and strain rates on the two dimensional
analysis given above.
4. EFFECT OF TRANSVERSE STRAIN
ON THE LONGITUDINAL VELOCITY PROFILE
Weertman (1957 appendix), considered a special case of three dimensional strain
in an ice shelf. For the two dimensional case i.e. with zero lateral strain èu = 0
êx =
-êz
Weertman obtained
èx^{2Ay\ax-az\"-i{ax-az)
68
(1)
For the case of an ice shelf expanding equally in all (horizontal) directions (3D)
— P — — 4 P — À"
— t'y —
2fcz — ^
We note that
ax-az
n-l
V3
(2)
\
3 y
«-*.)= (4=
V3VV3
and since
4=^-4=) = 1
for« = 4
V3VV3
we obtain the interesting result that the longitudinal strain for a given stress deviator
gradient is decreased with the presence of an equal lateral expansion for low values of
n ( < 4) but for high values of n it would be increased.
We shall now consider the effect on the longitudinal strain rate (êx) of an arbitrary
transverse strain rate ?.y.
We adopt a flow law of the form
Bx = )M'X
(4)
where
1
where
and
Â=.i4""t"~1
(6)
where
2S = a'x2 + a;2 + o'z2
(7)
T is the " effective shear stress" (equal to \/3Ï2 times the octahedral shear stress)
A, and n are parameters of the power law for flow.
Now we require the longitudinal strain rate èx in terms of the longitudinal and vertical stresses ax, az and the transverse strain rate au. From (4) and (5) we obtain
èx = M °x- -(ffx + ^y + ffj
= ^[2<7x-«7,-<rz]
(8)
We can obtain ay from
i, = K
= ^[2ff,- ffx -«7j
(9)
(10)
69
2 A.
Substituting this in equation (8) above we obtain
3
3 6,1
-r--ni
Next to obtain X from (6) we first require r from (7) for which we require the stress
deviators.
From 4 (4) and (12) we see
Ox = K f f x - f f , ) - 77
(13)
2A.
similarly
<=^ox-ax)-e^
(14)
*; = -?
(is)
and
Hence from (7) we obtain
Now let us write the lateral strain rate as a certain fraction (yX) of the stress difference
(ax — ay) i.e.
*-* = y(cx-oz)
(17)
A
Then from equation (12)
and from equation (16)
2T 2 = }[(<T,-<7 2 ) 2 (T + 7 2 ) ] + K - ( 7 Z ) 2 y2
(19)
\
70
2J
(20)
We now substitute this for T in equation (6) to obtain / which we can substitute in (18)
to give
(21)
Since we wish to know the relation between the longitudinal strain rate (£,:) and the
stress difference (ax — az) for a given transverse strain rate (èy) which is a certain fraction
(say v) of the longitudinal strain rate, we write
È, = vèx
(22)
then from (12) we obtain
èx = -{px-aù—
(23)
1+ V
By comparing this with equation (18) we see
1
v
(2-y) =
ory=
1+v
2v
orv=^2+ v
1-y
(24)
2
Finally then from (24) and (21) we can write
sx = (2A)-n(<Tx-czy<t>-x
(25)
where
1
2
"-1)/2
(26)
Hence as analagous to equation (I) for two dimensions, where the lateral strain rate is
zero, we now have for a lateral strain rate v time the longitudinal strain rate
(<j>êx) = (2A)-" (ox-ozy
(28)
In other words if a transverse strain èy = vèx is present we can incorporate its effect
into the association of the longitudinal stress and strain simply by using <j>lln èx1/n
instead of Ei 1 '".
Values of iß and <j>lln for various values of v and n are given in Tables 1 and 2.
From the tables we can see that for a flow law with n = 3 or 4 the presence of a
lateral strain rate which is a small fraction of the longitudinal strain rate makes no
71
appreciable difference. Even values of v = + 1, + 2, + 3 only cause slight variations for
large n values. For n = 1 or 2 the deviations become more significant.
TABLE I
+3
4
3
2
1
i
-i
-1
-2
-3
3
2.5
2
1.49
1.25
1.12
1
.876
.751
.5
0
-0.5
2 + V.
1.99
1.74
1.49
1.30
1.18
1.11
1
.853
.655
.25
0
-0.095
1.29
1.20
1.14
1.12
1.11
1.10
1
.825
.565
.125
0
-0.018
.83
.87
.97
1.06
1.06
1
.808
.488
.063
0
-0.005
TABLE 11
1/»
>JI
4
3
2
1
*
i0
~i
-1
-2
-3
72
3
2.5
2
1.49
1.25
1.12
1
.876
.751
.5
0
-0.5
.41
.32
.22
.14
.08
.05
.924
.810
.5
0
-0.31
1.09
1.06
1.05
1.04
1.03
1.03
1
.934
.823
.5
0
-0.26
.95
.95
.96
.99
1.01
1.01
1
.930
.935
.5
0
-0.26
When the lateral strain is of opposite sign however the deviations become very
important. In particular for v = - 2 we find the longitudinal stress difference is zero.
This may be easily seen from
êx + èy + èz = 0
and if
èy =
-2èx
then
£, = er
i.e. a longitudinal strain rate can exist even with ax-az = 0.
This means that the longitudinal motion is being dominated by the lateral stress,
which causes equal vertical and longitudinal strains simply to conserve volume. For
greater negative values of v ( - 3 , - 4 ...) the longitudinal mean stress difference is of
opposite sign to the longitudinal strain rate.
Finally our general equation of motion for three dimensions may now be written
as
ôZB(4>èxy
(29)
dx
a>
-g
PO
Mtan Valu«)
01
#
-
8'
TX"
!
00
c/ ec
1- t
°l
co
?
0
i
•
a
m
* \ «
•
•
1 1 1
u»
[
1
I
1
v
Fig. 5 — For the increase in velocity V from the dome summit to Cape Poinsett
we plot ( VIZ)f(8) against ocZ where a is the surface slope, Z is the ice thickness
and J(0) is a factor to account for the variation in temperature along the line.
The results suggest a value of the flow law parameter of n = 3.4.
73
and we can relate the longitudinal slope a to the longitudinal strain rate èx and include
the effect of the transverse strain in <j>.
As an application of these results we consider the relatively stagnant line on the
Wilkes ice cap from the summit to Cape Folger. It can be seen from figure 1 that along
this line the velocity increases only slightly. The strain measurements in five grids
along this line indicated that the transverse extension was on the average four times the
magnitude of the longitudinal extension. Owing to the large distances between the
strain grids and the wide local variations along the line this figure requires further
confirmation. However in view of the relatively large transverse extension we may
expect from equation (29) that the longitudinal strain rate, and hence the velocity, will
be considerably smaller, for the same slopes and ice thickness, than along the DomePoinsett line, where the transverses train rates are generally considerably smaller than
the longitudinal strain rates.
From table I we see that for a transverse extension four times the longitudinal
extension (v = 4) the function <j> has the value 3 when n = 1. This suggests that the
value of Bi when computed from equation 3(40) should come out about 3 times as
large as that found for the Dome-Poinsett leg for equivalent temperatures. The actual
figures obtained confirmed this but a fuller discussion would require an analysis of the
temperature depth profiles over the ice cap, which will be treated elsewhere. We
conclude that the observed discrepencies are closely accounted for by the theory of
this section if we taken n ~ 1 in the longitudinal stress-strain relation for these low
strain rates, in agreement with the result obtained earlier (fig. 4) for the longitudinal
extension.
ACKNOWLEDGEMENTS
The Wilkes local ice cap project has been a combined effort by many workers of
the ANARE, in particular P. Morgan 1964, A. McLaren 1965, L. Pfitzner 1966, and
D. Carter 1967. Valuable comments have been received in particular from J.F. Nye,
M.S. Patterson, G. de Q. Robin, and U. Radok. The paper is published by permission
of the Director of the Antarctic Division, Department of External Affairs.
REFERENCES
BUDD, W., (1966): The dynamics of the Amery Ice Shelf. Journal of Glaciology,
Vol. 6, No. 45, pp. 335-58.
BuTKOvrcn,T.R.,and LANDAUER, J.K., (1960): Creep of ice at low stresses. U.S. Army
CRREL Research Report 72.
GLEN, J.W., (1955): The creep of polycrystalinc ice. Proceedings of Royal Society.
Ser. A, No. 1175, Vol. 228, pp. 519-38.
HAEFEU, R., (1961): Contribution to the movement and the form of ice sheets in the
Artie and Antarctic. Journal of Glaciology, Vol. 3, No. 30, pp. 1133-1150.
MCLAREN, A., (1966): Ice cap study, Wilkes, Antarctica. "Antarctic". New Zealand
Antarctic Society, Vol. 4, No. 8.
MCLAREN, A., (unpublished) A study of the local ice cap near Wilkes, Antarctica,
1965. To be published as an ANARE research report.
MEIER, M.F., (1960): Mode of flow of Saskatchewan Glacier, Alberta, Canada.
U.S. Geological Survey. Professional Paper 351.
MELLOR, M., and SMITH, J.H., (1966): Creep of snow and ice. U.S. Army CRREL
Research Report 220.
NYE, J.F., (1957): The distribution of stress and velocity in glaciers and ice sheets.
Proceedings of the Royal Society, Ser. A, Vol. 239, No. 1216, pp. 559-84.
NYE, J.F., (1959): The motion of ice sheets and glaciers. Journal of Glaciology, Vol. 3,
No. 26, pp. 493-507.
NYE, J. F., (1965): Theflowof a glacier in a channel of rectangular, elliptic or parabolic
cross-section. Journal of Glaciology, Vol. 5, No. 41, pp. 661-690.
74
SHUMSKIV, P.A., (1961): On the theory of glacier motion. IVGG, IASH, General
Assembly of Helsinki, Symposium on Antarctic Glaciology, pp. 142-9.
SHUMSKIY, P.A., (1963): On the theory of glacier variations. WGG, IASH, Bulletin,
VIII» Année, No. 1, pp. 45-66.
STEINEMAN, S., (1958): Résultats expérimentaux sur la dynamique de la glace et leurs
corrélations avec le mouvement et la pétrographie des glaciers. IUGG, IASH,
Symposium of Chamonix, pp. 184-98.
WEERTMAN, J., (1957): Deformation of floating ice shelves. Journal of Glaciology,
Vol. 3, No. 21, pp. 38-42.
DISCUSSION
W.S.B. PATERSON
I should like some more information on the derivation of equation (1) in the extended summary. This equation seems to have been obtained by using the flow law to eliminate the longitudinal stress, ax, from the stress equilibrium equation. But three points
are not clear to me:
1. The equilibrium equation contains the stress ax, whereas the flow law contains
the stress deviator ax' (which equals h(<yx — ay) in the present two-dimensional case).
How is the transition between the two?
2. The flow law involves the octahedral shear stress r. (In the present case 2 r 2 =
(ax — Oy)2 + 4 Txy'.) Why does T, or else the octahedral strain rate e, not appear in the
left h and side of equation ( 1 ) ?
3. What is the justification for taking the multiplying factor Z inside the derivative <r/<7T on the left hand side of equation (1)?
W.F.
BUDD
1. The mean stress deviator ax has been obtained from \ the difference of ax
and Oz, which are obtained from the integration of the equilibrium equations.
2. Since the longitudinal strain rate here has only been measured at he surface it is
necessary to make some assumption as to how it varies with depth. If the velocity at the
base is zero, so also is the longitudinal strain rate there, and hence also the longitudinal stress deviator. However if most of the shear in the ice takes place near the base
(as suggested by the calculated temperature and velocity depth profiles) then the average longitudinal strain rate does not differn greatly from the longitudinal strain rate at
the surface.
Also, except when xXy is large compared to ax — ay
T"- 1
a'x « (a'xY
Hence in absence of more data about the vertical variation of these parameters
the approximate assumption is made that the average longitudal stress deviator ax'
is proportional to the surface longitudinal strain rate ex to the power 1/«, (equation 21),
and the error in this is given by equations ( 19) and (20), and by the deviation of the average strain rate from the surface value.
3. The factor Z is required inside the derivative in equation (22) in order to eliminate the integral in equation (10) for the longitudinal stress ax which contains the factor 1/Z.
G. DE Q. ROBIN
In presenting the paper, Dr. Radok referred to apparently similar conclusions in
my paper {Nature, Vol. 215, No. 5105, pp 1029-1032, 1967) on "Surface Topography
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of Ice Sheets". Altough similar, there are some basic differences between the conclusions of the two analyses. I estimated mean strain rates throughout the ice thickness from
a very detailed knowledge of bottom topography (obtained by radio echo sounding)
and use of equilibrium theory and knowledge of rates of accumulation. Budd used direct
observations of surface strain rates between stakes generally spaced 3 to 5 km apart.
Although my estimates are inferior to direct observation in certain respects, they do
indicate probable strain variations along the line of flow in much greater detail than
do observations giving mean values over several kilometres.
I found that the detailed variations of surface slope, and especially major variations over distances of one or two kilometres, were well explained by use of ä flow law
<ix° == Be * based on Glen's laboratory experiments and on the variation of temperature
with depth in the ice sheet. A flow law of the type derived by Budd, in which stress has
an almost linear relationship to strain rate would not have explained the major slope
variations observed in Greenland. One explanation might be that Budd is dealing with
lower stresses than for my Greenland data, and some laboratory experiments suggest,
as Budd states, that the stress-strain rate relationship becomes closer to linear at low
steresses.
A further reason for the above difference is likely to be that Budd's analysis deals
only with values of surface slope variations which are smoothed over distances of
several times the ice thickness, which are similar to the distances over which mean strain
rates were measured. Thus rapid variations of slope over two or three kilometres arc
not dealt with in detail by Budd, and something may have been lost in this smoothing
of data.
Certainly the main contrast between the two studies is that Budd is studying minor
fluctuations of surface slope which do not generally exceed the mean surface slope,
whereas my conclusions are based mainly on slope variations an order of magnitude
greater than the mean surface slope which were present in Greenland.
W. F. BUDD
I agree in general whith Dr. Robin's comments and am pleased to add that a detailled radio echo sounding profile has recently been obtained by D. Carter (using a SPRI
designed echo sounder) over the line studied in this paper, which should allow the analysis of finer detail. However, two points are emphasised here. Firstly extrapolation of
Glen's flow law to strain rates 3 orders of magnitude lower appears to give strain rates
much lower than the measured values. Secondly to calculate velocities (or strain rates)
from the steady state assumption can lead to considerable error in regions such as this
which are changing form.
In order to determine the index of the power law which is appropriate to these low
strain rates it is necessary to have data spread over a wide range. The results here covered strain rates from 1.5 to 7 x 1011 sec"1 and stresses .06 to .25 bars, and appeared to be
linear.
The comparison of values over longer distances does not give information on the
shorter distance fluctuations, but information has not been 'lost' by this comparison,
since strain rates over a long distance incorporate the variations within it and the corresponding slopes were obtained by integrating closely spaced values.
L. LLIBOUTRY
Je m'étonne que le calcul du profil vertical des vitesses ait été fait sans tenir compte
de la variation de température avec la profondeur. Profil des températures et profil des
vitesses doivent être calculés simultanément car il y a interaction entre la température
et la vitesse de déformation.
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W.F. BUDD
The effect of the temperature variations both in the vertieal and in the horizontal
have been taken into account. The temperature calculations, referred to in the paragraphs after equation (33), also consider the frictional heating produced by varying
stress, strain rate and temperature within the ice mass and the associated positive feedback. This will be the subject of a separate paper. The formula for ß 2 (equation 37) as a
function of B (which is itself a function of z and 0) emphasises the importance of the
temperature but shows B2 is highly weighted to the basal layer.
Along the horizontal the variation of B is largely governed by the warming of the
surface mean temperature as the coast is approached.
J.W. GLENN
Your correction of the general equation by the insertion of a factor 0 suggests that
the power \jx is not changed by a lateral strain, however if the octahedral shear strain
rate is a function of the octahedral shear stress, then as the lateral stress becomes large
one would expect the power to tend to unity. Presumably lateral stress is realated to the
lateral strain and so I wonder whether it is accurate to use simply the factor 0 without
changing x.
W. F. BUDD
For a power flow law of the form r = (r/S)" where e and r are the octahedral shear
strain rate and shear stress, and n and B are constants for a given e and a, the value of
n is not dependent on the configuration of the stress, but only on the variation in the
magnitude of £ with varying a, (which are both invariants). Hence there is no reason to
suppose n—>l as the transverse strain rate becomes large. The lateral stress deviator is
dependent on the lateral strain rate and the octahedral stress according to
A similar relation holds for the stress - strain rate relation in the other two directions.
All these do vary with the stress configuration and this has been taken into consideration. What has not been treated here is the effect of vertical shear. However if most of
this shear takes place near the basal layers the present result is applicable to most of the
ice thickness.
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