216 ISAGE

PATERSON, W. S. B., and SAVAGE, J. C. 1963. Measurements on Athabasca Glacierrelating to the flow law of ice. J. o/Geoph. Res., Vol. 68, No. 15, p. 4537-43.

STEINEMANN, S. 1958. Resultats expérimentaux sur la dynamique de la glace etleurs correlations avec le mouvement et la pétrographie des glaciers. UnionGéodésique et Géophysique Internationale. Association Internationale d'Hydro-logie Scientifique, Publication No. 47, Symposium de Chamonix, p. 184-98.

WEERTMAN, J. 1968. Diffusion law for the dispersion of hard particles in an icematrix that undergoes simple shear deformation. J. Glaciol., Vol. 7, No. 50,p. 161-65.

Ice flow law from ice sheet dynamics

BY

L. A. LLIBOUTRY

Laboratoire de Glaciologie, Centre National de la Recherche Scientifique,2, rue Très-Cloîtres, Grenoble, France

ABSTRACT

The usual assumptions in ice flow theory are listed. Laboratory valuesof the constants involved in the ice flow law are reviewed : they are soconflicting that values deduced from the flow of actual ice sheets aredesirable. Present studies of the dynamics of grounded ice sheets andice shelves are then reviewed. All the data are reduced to referencevalues of — 13°C and 0-5 bar. It appears that the strain rate of very oldGreenland ice is larger than the strain rate of artificial isotropic ice or ofice shelves by a full order of magnitude, the values for small local ice capslying between both. It is concluded that previous anistropy (not inducedby local stresses) must be taken into account in ice sheet flow theory.Some hints are given about this problem.

Importance of the subject

A better knowledge of ice sheet dynamics is needed in order to tackleseveral important problems, for instance:

(1) Thickness changes which have followed past variations of climate,and modified the sea level during the Pliocene and Pleistocene. Thecurrent assumption, that the thickness of the Eastern Antarctic or theGreenland ice sheets is insensitive to climatic variations is wrong. Forinstance an increase in the precipitations cools the ice, owing to the Robineffect, and thus increases the thickness. (Lliboutry 1964-65, p. 780-85).

(2) Past precipitation as inferred from core analysis. This must becompared to the actual precipitation in the area from where the ice came.This area is often dubious, since ice divides and lines of flow are difficultto dertermine in Antarctica.

Unfortunately the inaccuracy in the ice flow law may give errors in thecomputed creep rate of a full order of magnitude.

DYNAMICS 217

Usual assumptionsIn theoretical treatments of ice sheets the following assumptions are

made:

(1) Infinitesimal strains of only first order apply. The strain ratetensor (eij) is sufficient to describe the flow at a given point. Somesupport to this assumption is given by the fact that the ice fabric at a givenpoint is related to the stresses at this point, although ice foliation, comingfrom upper regions, may be discordant (Vallon, 1967).

(2) Linear tensorial relationship between the strain rate and stress tensors.Theoretically, the ice flow law may involve any matrix inferred from thematrices (eij) and (crij) by the operations of matrix calculus, (êij) =S éikëkj for instance. The assumption is that only (éij), (cru), the scalarinvariants of these matrices and the unity matrix (Sij) intervene.

(3) Incompressibility, which means not only that changes in ice densityare neglected, but that the influence of ice elasticity on flow is negligible

(4) Isotropy. From this assumption and the previous one Saint-Venant's relations are deduced:

(7|j = — 2 7}ëi] (1)

(CT'IJ) = (aij) — p (Su) denotes the deviatoric stress; p the hydrostatic(or octahedral) pressure 1/3 (ax + oy + az) — IL/3 ; -7 denotes the viscosity,a scalar which may vary from one point to another.

Compressive stresses are taken as positive, as is done in soil mechanics,while in most handbooks of mechanics they are taken as negative. Theminus sign in (1) follows, since 77 is a positive scalar.

(5) Viscosity. 77 is a function of the second invariant of the deviatoricstress, I'2 = — T2, and not of the third one (Generalized Von Misescriterion). Putting:

h = -*2 = - r 7 4 (2)it follows from Saint-Venant's relations that

T = r1y (3)

So the viscosity may be expressed either as a function of the effectiveshear stress T or the effective shear strain rate y.

(6) Non-influence of the hydrostatic pressure, for a given temperaturedifference 8 from melting point (Rigsby's law).

(7) One unique flow law for all kinds of ice regardless of their age andhistory. This assumption is grounded upon the already quoted factthat the ice fabric at a given point seems to be a function of the stressesat the same point.

(8) Glen's law, which will be written in the two following forms:

y = Bo . T" = 2 (r/A0)" (4)

218 ISAGE

According to assumptions (6) and (7), Bo and Ao are functions of 6only.

(9) Temperature dependence following the law:

B9 = B o exp( -Q/RT) (5)

where R = 1-986 cal/mole and Q = activation energy.

Within the small range of temperatures found in ice sheets the law (5)may be approximated either by B6 s» B0/(l — aö) or by

Bo «* Bo exp (k0) with k = Q/RT2 (6)

Laboratory values of n, Bo, Q

Although Glen's law is more satisfactory than Stokian viscosity (stillused by several theorists) there is no general agreement for the valuesof its parameters n, Bo and Q. Steinemann (1958) found n increasingfrom 1-85 to 4-16 when T increases from 1 to 10 bars; the results of Higashi(1959) indicate that n increases when T increased from 0-6 to 2-5 bars.Mel lor and Smith (1966) with compacted snow or very bubbly ice foundthe best fit with the polynomial formula

y = Bo T + Bx r3-5.

Dillon and Anderland (1967) found a good fit with y = y0 sinh (T/T0)while Wakahama (1962) found

y = B (T - TC)3 with TC = 0-2 bar.

On the other hand the careful experiments of Glen (1955), who triedto eliminate the transient flow effects, gave a smaller variation of n(n fa 4-2). Butkovich and Landauer (1958), who used all kinds of ice,found a better fit with Glen's law (n = 2-96) than with the sinh law or apolynominal law y = a T + b T3.

There is no one theory related to the motion of dislocations, upon whicha unique value of n may be grounded. According to Weertman (1957 b,1962) dislocation climb as controlling factor brings n = 4-5; dislocationdamping by the Eshelby-Schoeck mechanism, n = 3; the motion ofdislocation lines in a Peierls stress field n = 2-5. According to Friedel(1964, p. 279 and 314-15) the motion of dislocations piled up against grainboundaries, by diffusion along these boundaries or by recrystallization,would give a law y = Cx T2 + C2 T4 -•- . . . . The Newtonian term(n = 1) which seems to prevail at stresses <^ 1 bar (Butkovich andLandauer, 1960) could be explained by bulk diffusion following theNabarro-Herring mechanism (Bromer and Kingery, 1968).

Q was measured by Nakaya (1959) on various ice samples by thedamping of mechanical vibrations. Values ranging from 12-7 to 18-7kcal/mole (the higher the older the ice) were found. Values of Q inferredfrom the creep of polycrystalline samples by Glen (1955) or by Lliboutry(1964-65, p. 86-89) range from 25-4 to 43-3 kcal/mole.

DYNAMICS 219

This discrepancy may be explained by distinct controlling factors in theflow process, namely migration of the dislocations within the crystals inthe first case, and along the grain boundaries in the second one. Dillonand Anderland found such a high value (25-5 kcal/mole) for high strainrates ( > 550 per cent year"1) and a lower value (11-4 kcal/mole) at lowstrain rates. Very careful experiments by Mellor and Testa (1969) haverecently shown that a strong temperature dependence is found withpolycrystalline ice only above — 10°C, when local pressure melting mayhappen owing to stress concentration. Below — 10°C, Q = 16-4 kcal/mole fits nicely the data.

Since in polar ice caps 255 < T < 273, the values of Q (in kcal/mole)must be divided by 138 ± 10 in order to have k in (°C)-1. The presentauthor has used k = 0-25, while Budd (1968), uses k = 0-10.

The uncertainty of Q affects the precision of Bo. Glen's law may bewritten in the metre-bar-year system:

y = 0-226 e0-227» T4-2

After discussing all literature available up to 1963, and distinguishingwhether secondary creep or tertiary creep had been measured, the presentauthor adopted the following values for effective shear stresses of aboutone bar (0-8 < T < 3 bars) (Lliboutry, 1964-65). The value of n wastaken equal to 3 :

—For secondary flow, which for some obscure reason seems to prevailin temperate valley glaciers, Bo = 0-164 year-1 bar~3 and Q = 43-3 ± 4kcal/mole

—For tertiary flow, which would prevail in cold ice sheets, Bo = 1-0year-1 bar~3 and Q = 38-4 ± 4 kcal/mole.

Shumsky (1967) referring to Steinemann's experiments suggests fortertiary creep (in the metre-bar-year system):

y = [0-265 T + 0-0146 T4]e°-2820 (8)So for T < 1 bar, ice would behave as a stokian viscous fluid.

In view of these discrepancies, it seems obvious that there is a need toestablish the ice flow law directly from field data. This supposes thatproper ice sheets are studied, and the field data processed with the purposeof finding the ice flow law, instead of taking it as the point of departure.

Grounded ice sheet theory, without solving the heat equation

Let us consider an ice sheet of almost uniform thickness Z, with asuperficial slope a which varies very slowly. The function on the bedrockis then

f=pgZa (9)where p g denotes the weight of an unit volume of ice.

(1) Assuming no bottom sliding, and that B and n have constant valuesthroughout the ice sheet, the surface velocity is:

us = Bf" Z/(n + 1) (10)

220 ISAGE

and its mean value:U = Bf»Z/(n + 2) (11)

This solution, used by Haefeli (1961, 1963 b) is unsound for coldglaciers, unless they are isothermal. This is the case for some Arcticice caps, owing to a warming of the climate between 1880 and 1940:Meighen. Ice Cap (Paterson, 1968) and Jackson Ice Cap (Razoumeiko,1963). Unfortunately thicknesses and surface velocities have not beenmeasured in places where equation (10) holds. For instance on the JacksonIce Cap mechanical drilling gave the thickness of the ice sheet: (a) near thesummit, where the flow lines sink swiftly; (b) in stagnant ice; (c) near thecatchment area of an outlet, where the surface velocity is more or lessperpendicular to the line of maximum slope (Vinogradov and Grosswald,1962). Extrapolated values are Z = 80 m, tg a = 0-10, f = 0-70 bar,us = 0-5 to 2-0 m/yr. It follows at 0 = -11-2°C, B . u . 2 = 0-05 to 0-07.

(2) A widely used relation U = Cfm (Nye, 1959; Weertman, 1961,1966 a; Haefeli, 1963 a) completely lacks any experimental or theoreticalbasis.

(3) When the temperature profile T (z) has been measured, Bo may becomputed from:

Us = Bo fe-QIRT (pgza)« dz (12)

This has been done by Weertman (1966 b) for the Camp Centuryborehole (North-eastern Greenland) where Z = 1387*4 m, a = 3-6 x 10~3,and us = 3-3 m/y. Weertman uses p = 0-917, Q = 13-8 kcal/moleand n = 3-2. He finds then, for the bottom where 0 — — 13°C andT = 0-45 bar, y = 0-030 yr-1, that is to say B_13. = 0-194 ba r -^y i " 1 -

(4) Budd (1968) studying the Wilkes ice cap, splits the ice sheet intotwo superimposed layers: a main body, which is subject to mild extension,and a bottom shear layer, where a strong pure shear prevails.

Since measured surface velocity comes almost entirely from the bottomshear layer, equations (9) and (10) may be used, taking for a a smoothedvalue a, and for B the value corresponding to some "bottom temperature".The field values give him (as seen in Fig. 5 of Budd's paper), in CGS units:

whence (taking p = 0-90g/cm3) Bo = 0-86 bar"3-32 year"1.

Since the mean bottom temperature is —10° we must retain only thevalue a t - 1 0 ° C :

B_10 = 0-078 bar-3"32 year"1 (0-5 < r < 1-1)

(Budd gives n = 3-4, but reduces B to a standard value of n = 3 andfinds B_10 = 0-102 bar"3 year"1).

In the upper layer, the flow law is computed from the fluctuations of

DYNAMICS 221

the surface slope. Taking the x axis parallel to the bedrock (slope ß),the following relation holds:

? ! I I ^ (14)

Putting Txz = 0 at the surface (with the chosen axes this boundary valueis not exactly true), this relation becomes

{ago-,«.-«-a 5^5The transverse strain is negligible on the more active side of the local icecap, according to Budd's computation. The comparison of the stressdeviator CTX' with the measured strain rate ëa (corrected for temperatureeffects according to the law Bo = Bo e

0'100) gives, in the range 0-06 < T<0-25, n = 1-2 and B_i5 = 0-025 bar"1"2 year-1. (This value comesdirectly from Budd's figure 4. In order to compare different B valuesBudd uses n = 3 and so he finds B_i5 = 0-0366 bar-3 yr-1.)

Solution of the heat equation

In the body of the ice sheet of uniform thickness Z, the steady-statetemperatures at a depth z are (Robin, 1955):

where b = net accumulation in equivalent height of ice andh = heat diffusity of ice = 38 m2/yr (or 48-5 according to

Cameron, 1964).When, following the ice in its movement, the altitude of the surface

decreases, and accordingly the surface temperature 93 rises, this profileis altered (Robin, 1955; Jenssen and Radok, 1963). As a consequencethe mean temperature of the upper main body of the ice sheet is oftencloser to the surface temperature (as pointed out by Budd). The tempera-ture profile in the bottom shear layer has been given by the present author(Lliboutry, 1961, 1963, 1964-65, 1966, 1968). A steady state, and noRobin warming are assumed. Let G and Gx be the thermal gradientsat the lower and upper limits of this bottom shear layer. Then

G{ = G2 -|- 2fy0 exp(k0b)/kK (17)where K = thermal conductivity of ice = 700 bar m2/yr deg (or 865according to Cameron, 1964).

The thickness of the bottom shear layer being negligible in comparisonwith Z, y0 = Bfn. Since bZ/2h is often small in actual ice sheets,erf(bZ/2h)1/2 will be taken as & 1. With this approximation the bottomtemperature 0b, at the lower limit of the cold ice sheet, has been proved tobe approximately

DT-l-ö^ö

222 ISAGE

Three cases must be distinguished :(a) Cold ice sheet with a cold ice bedrock interface. Here G = geo-

thermal gradient Go «a l°C/44 m. dt, is a negative unknown quantity,which may be eliminated from the set of equations (17-18) in order toobtain R = GJ/GQ. We obtain :

where c = - k 0 s + ln(kKG20/fy0) and m = kG0(TThZI2byi2.

Two values for R are found. The higher one is unstable; that is to say,the temperature régime becomes unsteady, the term in dd/dt in the heatequation can no more be neglected, and the temperature rises progressivelyuntil the melting point is reached.

On the other hand considering the energy balance we have the meanvelocity across the entire ice sheet:

U = K(GX - G0)/f = KG0(R - l)/f (20)Since R may be computed from surface measurements and an assumed

flow law, the velocity U ensues. Conversely, if mass balance is assumed,this velocity U may be determined independently, and the flow lawadjusted in order to find the actual surface profile. This has been donefor the central part of Greenland near the ice divide (Lliboutry, 1968).Assuming n = 3 and k = 0-25, a value Bo = 2-18 bar"3 yr - 1 is obtained,f rises quickly to about 0-5 bar and 6b decreases almost linearly from-10-4°Cto -6-5°C.

(b) Cold ice sheet with a temperate ice bedrock interface.

Here Ob = 0, G > 0; unknown. Gj is given by

-Os = Gx (77hZ/2b)V2 - (2/k) inCkUGj/yo) (21)and U by

KGX_ /KKÎS 2Ky0u ~ f V -p kf yzi)

This value is inaccurate, since the basal shear layer gets several hundredmetres thick and the Robin warming effect is no longer negligible.

(c) Cold ice sheet with a temperate bottom layer (Lagally's Ubergang-styp). Lagally (1932) made no allowance for vertical strain as dealt within Robin (1955) and assumed the viscosity to be independent of both Tand 6.

Therefore 0b = 0; G = 0; and Gx = (2f7o/Kk)1/2 (23)

Then U = (2Kyo/kf)1/2 -f the sliding velocity coming from the shearingof the temperate bottom layer and the pressure-melting process (Lliboutry,1967).

The second and third kind of ice sheets do not seem to provide valuesof the ice flow law parameters.

DYNAMICS 223

Ice shelf dynamics(1) The creep rate of an ice sheet with free edges has been given by

Weertman (1957 a). However, the application made by this author toMaudheim is inaccurate, since he assumed a linear increase in the tempera-ture between the surface and the bottom. Crary (1961) took the realtemperature profile into account, and also the variation of density withdepth.

We denote by h the thickness of the ice sheet, by h' the submergedportion of it, by pw the density of ocean water and counting z from thelower surface of the ice sheet upwards

h > w = fh/>(z).dz (24)J o

There is an uniform extension èxx from the surface to the bottom. Fromthe general equations and considering the compressive stresses as beingpositive:

J o

Whence

C7XX = -2A0(ëxx)Vn + J* p(£)gd£ ( 2 5)

ium boundary condition is:

axxdz = /j„gh'2/2 (26)

At the free edge, the equilibrium boundary condition is:

fh

Aodz = ^- [ £ p(7.)dz]2 - g j" j* P(C)dzd£ (27)

Thus the quantity

j ; (^)1/nJh-n^dz (28)

where «xx, h and 0(z) are known, may be computed from the field data.Assuming Glen's values of n = 4-2 and Bo = 0-226 bar~4-2 yr~* Crary

found Q = 18 kcal/'mole for the Maudheim ice shelf and Q = 21-5kcal/mole for the Ross Ice Shelf at Little America. The weighted meansof the temperature are respectively — 14-TC and —17-8°C (—18-4 and—23-4°C at the surface, —2°C at the bottom). The maximum strain rateis the same, ëxx = 0-0013 yr~\ the corresponding effective shear stressesare 0-58 and 0-66 bar.

(2) For a shelf restrained by both sides, a theory has been given byBudd (1966). The mean through a vertical of the longitudinal stress is:

^x = ^ g H - 2 A ' a 1 / n (29)

Assuming that erx is almost independent of the transverse co-ordinate y,and denoting 2 Y as the width, it follows from Sax/dx + ÔTXy/3x = 0 that

u =

224 ISAGE

By eliminating Sâx/ôx«aAôx/Ax this becomes

A =u n + l)u/2 Yn+ !] VnAx+2 A(aJ/n) ^31 ̂

Rather than following Budd's intricate and approximate treatment,we may compute A directly for the segments GjG2 and G2G3 (Table I).

TABLE 1COMPUTATION OF THE FLOW LAW PARAMETER A FROM THE AMERY ICE SHELF DATA

n:

A for G, G2A for G2 G 3

B23/B12 = (A,,/A2 ,)n

Budd's approximate A

2

21-21410-295

16-2

2-4

12-012-55

1-12

2-5

10-711-71-25

3

6-938-782-04

6-33

From the temperature difference, B23/Bi2 = 1-14 to 1-17, then n = 2-43and B_12 ^ 0-0048 bar-" yr-1. Since a varies from 6-0 X 10"3 at G t

to 0-5 X 10~3 at G3, the corresponding effective shear stress r = Aia!1/"varies from 1-46 to 0-53 bar.

Discussion

All these results gathered from cold ice sheets are summarized in Table2. In order to compare them we must refer to the same temperature andeffective stress. Since neither the temperature dependence nor the stressdependence are well known, the extrapolations must be the shortestpossible. Therefore reference values of — 13°C and 0-5 bar have beenchosen. According to Haefeli, Jaccard and De Guervain (1967) Rigsby'slaw is valid for polycrystalline ice even at —7°C, if we assume k =0-125 deg-1 (Q ^ 16-8 kcal/mole). Thus the field values of 0 have beencorrected for the hydrostatic pressure ( + 0-66 X 10~3 deg/m of over-lying ice), and the value k = 0-125 adopted. The reduced values of theeffective shear strain rate y are collected in Table 3.

The reduced strain rates are larger in the ice caps than in the ice shelvesby a full order of magnitude. In spite of the inaccuracy of these calcula-tions, they seem also to be significantly lower in the small Jackson andWilkes Ice Caps than in the Indlandsis of Greenland.

The discrepance is still higher if we compare these values with thelaboratory ones. At — 13CC and for 0-5 bar Glen's data give y = 0-00064;Lliboutry's suggested laws give y = 0-0023 for tertiary creep andy = O-OOO38 for secondary creep; Shumsky's suggested law for the tertiarycreep gives y = 0-0033.

The softness of ice seems to be the higher the older the ice. Now,owing to continuous recrystallization in a temperature gradient duringmillennia, very old ice has certainly a more perfect lattice, with less salt

DYNAMICS 225

TABLE 2VALUES OF THE ICE FLOW LAW PARAMETERS FROM FIELD DATA

Jackson Ice Cap

Indlandeis at CampCentury

Wilkes Ice Cap(upper part)

Wilkes Ice Cap(bottom shear layer)

Central Greenland

Maudheim Ice Shelf

Ross Ice Shelf atLittle America

Amery Ice Shelf

T

bars

0-70

0-45

006-0-25

0-05-11

«s0-5

0-58

0-66

1-46-0-53

ecc

-11-2

- 1 3 *

- 1 5

- 1 0

-10-4* to-6-5

- 1 4 "

-17-8**

- 1 7 * *

n

3(assumed)

3-2(assumed)

1-2

3-32

3(assumed)

1 4"2 11 (assu- 1f med) f

J J2-45

Be*t

005 to0-070194

0025

0078

0-267at -8 -4

0-226at0°C

(assumed)

00048

Qkcal

13-8(assumed)

14(assumed)

12-7 at-10°

(assumed)35

(assumed)18

21-5

t Note that B has units bar~° yr"1 and for more detailed comparisons should bestandardized for a common value of n.

* Bottom temperature.** Weighted mean temperature.

TABLE 3VALUES OF THE EFFECTIVE SHEAR STRAIN RATE y (IN YEAR-') AT — 13°C FROM MELTING

POINT, FOR AN EFFECTIVE SHEAR STRESS r = 0 -5 BAR

Jackson Ice CapIndlandeis at Camp CenturyWilkes Ice Cap (upper part)Wilkes Ice Cap (bottom

shear layer)Central GreenlandMaudheim Ice ShelfRoss Ice ShelfAmery Ice Shelf

9 from meltingpoint, °C

-11-2- 1 2 0- 1 5

- 9-4- 6-4- 1 4-17-8- 1 7

y (0, 0-5 bar)y r -

00063-000870021100109

00078003340002040000770-00088

y (-13°C, 0-5 bar)yr-1

00050-000700018600140

0-0050001470001850-00140000145

inclusions. This fact would diminish the number of dislocations andrather increase the hardness, while the reverse is observed. Then theonly explanation seems to be that the very old ice is softer owing to a verystrong and favourable ice fabric. This strong fabric has been observedat the Tuto and Red Rock tunnels by Butkovich (1959).

It seems then that usual assumptions (1) and (7), namely that only thelocal stresses and temperature determine the strain rate of ice, are wrong.Polycrystalline ice has a strong anisotropy which is only partly causedby the local factors.

226 1SAGE

We may introduce this anisotropy of ice into the flow equations in thefollowing way. In the general two-dimensional case, we may generalizeHill (1950) who studied the perfectly plastic anisotropic body, by changingSaint-Venant's relations and Von Mises criterion to the following ones.Here p denotes the hydrostatic (octahedral) pressure, and the compressivestresses are taken as positive. C is the "anisotropy factor"

ax — p = — 2rj -^ (32)

<jz — p = —2.7] — (33)

* " * (34)

1 A T » - (35)

' ( ! £ ) • (36)y = (1 + C) Th (37)

We may assume that the law IJ(T) holds for anisotropic ice, and thusGlen's law turns into:

y = (1 + C) BT« (38)

C increases with the previous anisotropy of the ice. It must be relatedwith the fact that a material plane (that means a plane which follows theice in its movements) has for a long time remained a plane of maximumshear. This fact, even in the simplest case, leads to exceedingly difficultmathematics. Before tackling the theoretical problem, it seems wise toobtain directly, from measurements-on deep cores, direct values of theanisotropy factor C.

REFERENCES

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BUDD, W. 1966. The dynamics of the Amery Ice Shelf. / . Glaciol., Vol. 6, p.335-58.

BUDD, W. 1968. The longitudinal velocity profile of large ice masses. AssembléeGénérale de Berne, Commission des Neiges et des Glaces (IASH Publ. No. 79),p. 58-77.

BUTKOVICH, T. R. and LANDAUER, J. K. 1958. The flow law for ice. In: Sympo-sium of Chamonix. IASH Publ. No. 47, p. 318-27.

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CAMERON, R. L. 1964. Glaciological studies at Wilkes Station, Budd Coast,Antarctica. In: Antarctic snow and ice studies (M. Mellor, ed.), AntarcticResearch Series, Vol. 2, p. 1-36.

DYNAMICS 227

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