The Antarctic ice sheet
BY
P. A. SHUMSKY
ABSTRACT
The most important results of the analysis of glaciological data published
up to 1966 are presented. Dimensions, form, and regime of the Antarctic
ice sheet are characterized. Distributions of velocity, strain rate, shear
stress, temperature, rate of melting at the bottom, and of the age of ice
are computed for a stationary axi-symmetrical model of the ice sheet.
Introduction
The Antarctic ice sheet has scientific interest from three points of view:
(a) as the largest accumulation of ice which determines the basic nature
of Antarctica and exerts an influence on natural processes and conditions
of the entire Southern Hemisphere and even the whole earth;
(b) as the only existing ice sheet of continental size, providing opportunities for the study of processes which occurred during the glaciation of other
continents;
and (c) as a unique container of cosmic dust and world wide terrestrial
precipitation, almost without local sources of terrestrial materials, and
containing precipitation a few hundred thousand years old.
The glacial cover of Antarctica completely changes the relief and the
radiational properties of the surface, creating a powerful centre of cooling
and, through atmospheric and oceanic circulations, exerts a considerable
influence on the atmospheric and hydrospheric conditions far beyond the
Antarctic region. This ice sheet contains an ice mass which is sufficient
to change the level of the world oceans by a few tens of metres, and changes
of its ice load must also create considerable isostatic and maybe tectonic
movements of the earth's crust and sub-crustal mass.
State oj knowledge
Although a series of studies has been devoted to the Antarctic glaciation
up to the International Geophysical Year, including the classical monographs of Drygalski (1921) and Wright and Priestley (1922), there were
very few factual data and much of the existing information was highly
Dr. Shumsky's paper is based on an extensive review entitled Oledenenie Antarktidy
(The Antarctic ice sheet) published in General results of research in Antarctica during
10 years by the Academy of Sciences of the U.S.S.R. (Moscow, 1967). The full text
has been translated by U. Radok and V. J. Vinocuroff, the extract here printed covers
the material presented by Dr. Shumsky to IS-AGE. Ed.
327
328
ISAGE
speculative. Since then the position has changed, mainly as a result of
the extensive research of expeditions from more than ten countries who
carried out an agreed programme with the help of the most modern
technical means.
Now the main surface relief patterns of the ice sheet are well known;
rather less is known of the thickness of the ice and the bottom relief, the
rate of nourishment, the temperature and physical properties of a thin
surface layer (and at a few points—layers down to tens or the first few
hundred metres), the velocity of ice movement at some points along the
coast, the structure of seaward edges and of the bottom layers (at the edge
of oases and in overturned icebergs) and lastly morain deposits and other
traces of geological activity of the ice in ice-free regions. There is also
rather scant information on changes in the sizes of the glaciers with time,
the rate of their nourishment and the surface temperatures for the last few
decades and longer. Thus, there is far less information than that available
to meteorologists and oceanographers who can make continuous measurements of the atmosphere and oceans. Absence of actual measurements
creates a need to obtain the required data by way of calculation.
Results of investigations have been published extensively and in the
form of observations, stored in the International Data Centres. A series
of many generalized works has been published on specific problems and
on Antarctic glaciation as a whole (Kotliakov, 1961 ; Evteev, 1964; Robin
and Adie, 1964; Gow, 1965). Part of the glaciological results has been
presented in map form, especially in the Soviet Antarctic Atlas, published
in 1966.
Climatic snow limit and ice free space
In the Southern Hemisphere this boundary descends from 1800 to
1000 m above sea level in New Zealand and Terra del Fuego, to 900 to
200 m on the sub-Antarctic islands (Werth, 1908), 200 to 50 m at the
north-west coast of the Antarctic Peninsula (Robin and Adie, 1964), and
down to sea level further south. All of Antarctica with the exception of
the west coast of the Antarctic Peninsula north of 69° is above the climatic
snow boundary. However, about 200,000 sq km, or 1-4 per cent of
Antarctica is free of ice (G. M. Braslavsky's measurements), including
about one half on the Great Antarctic Horst and on the Antarctic Peninsula, at heights of from 2000 to 3000 m ; the remainder is evenly distributed
between sea level and 5000 m. Coincidence of ice free areas with the
sea coast is accidental.
The reason for the existence of ice free areas above the climatic snow
limit is that, in the dry continental climate of Antarctica, the thermal
regime of rock surfaces differs sharply from the regime of ice-snow
surfaces, due to local radiational heating in summer. Therefore in areas
where ice flow is deflected by the rock surface relief, non-continuous windblown deposits of snow melt and evaporate, leaving only snow-flakes and
snow-drift glaciers at the foot of the slopes.
MASS BUDGETS : ICE AGES
329
Dimension and types of glaciation
The ice of Antarctica and adjacent islands includes continental ice
sheets, a series of isolated island ice caps, mountain and snow-drift
glaciers, subterranean ice, lake ice and seasonal snow. In turn, the
continental ice sheet consists of land and floating parts and includes also,
a number of island ice caps connected by floating glaciers. The areas
and volumes of the major components of the Antarctic ice sheet (compiled
from the cartographic data of 1. A. Suetova and G. M. Braslavsky) are
shown in Table 1.
Land glacial cover—sizes, forms and distribution
The area of the land based ice sheet in Antarctica is larger than that of
any of the quaternary ice sheets of the Northern Hemisphere (Table 1).
The average diameter of the Antarctic Ice Sheet is 3934 km. It has a
minimum diameter of 2900 km and a maximum diameter of 5600 km.
The Antarctic continental ice sheet consists of five large (Fig. 1) and
many small peripheral ice caps and ice sheets (Table 2). The circumference
of the ground-based ice sheet is 2-46 times longer than the circumference
of a circle of the same area (2TTR). The large ice caps of Ellsworth Land
and Marie Byrd Land are contiguous with other ice caps over a considerable part of their boundary, while the complex Graham Land ice sheet is
connected with the others only by a narrow saddle.
Form oj surface
Larger areas of the central part of large ice caps have an almost ideal,
even gently sloping, surface which in places contains hardly noticeable
corrugations. The ice sheets of the Antarctic Peninsula especially in its
northern regions, are different because of their very dissected relief. Some
of them are piedmont glaciers, bordering mountain massifs; occasionally
between valley glaciers flowing out from the ice sheet are ice free projections bordered along the coast by lines of snow-drift glaciers. From
mountain ranges above ice caps, valley glaciers and other types of glaciers
descend and sometimes form dendritic systems of valley glaciers. Where
mountain ranges project above the ice sheets they are frequently intersected by valley glaciers, which rejoin below in the lower region of the ice
sheet or in ice shelves. A characteristic feature of the peripheral zone
are the so-called outflow glaciers which are ice rivers embedded in the
ice sheet. They occupy about 15 per cent (4500 km) of the length of the
outer boundary. Most of them differ very little from the surrounding
ice sheet as regards surface and bottom relief (which contain shallow
depressions only), but they move at much greater velocities and are
bordered by systems of crevasses. Short glaciers of this type form
crevassed amphitheatres at the edge of the ice sheet, but the longest
extend over hundreds of kilometres (the length of the Lambert Glacier
system and IGY Valley equals 700 km). The depressions whose sides
TABLE 1
COMPARATIVE DATA OF ANTARCTIC ICE SHEET AND GLACIERS OF THE WHOLE EARTH
AREA
VOLUME OF ICE
%of
Subject
1000
km 2
Ice sheets
Ice, land based
Ice, floating
Ice, islands
Oases and nunataks
Total Antarctic
Glaciers of the earth
Total ice of globe
13779
12150
1460
169
196
13975
16080
Antarctic
ice sheet
100
Antarctic
surface
Earth
glaciers
km3
10'
Antarctic
ice sheet
Earth
glaciers
Whole
earth ice
98-6
85-7
75-6
23-7-31-3
23-0-30-4
100
91-5-91-8
88-8-89-7
2-3- 1-8
0-4- 0-3
88-8-91-2
86-2-89-1
2-2- 1-8
0-4- 0-3
100
97-0-99-4
88-2
10-6
870
1-2
1-2
1-4
100
—
—
—
%of
10-4
91
10
_
100
0-6
008
25-9-33-9
26-7-34-1
97-1-97-8
2-5- 1-9
0-4- 0-3
100
a
TABLE 2
COMPARATIVE DATA OF ICE CAPS AND ICE SHEETS OF ANTARCTICA*
BOUNDARY LINE LENGTH
AREA
Outer
Inner
Ice cap or ice sheet**
1000
km2
East Antarctic
Ellsworth
Marie Byrd
Palmer
Graham
Land bound ice sheet
Total ice sheet
10225
973
582
270
100
12150
13779
%
841
80
4-8
2-3
0-8
1000
1000
km
17-4
3-3
30
30
3-6
30-3
300
•>/
/o
88-8
57-9
71-4
88-2
98-4
91-8
1000
km
/o
1-2
2-4
1-2
0-4
01
2-7
11-2
42-1
28-6
11-8
1-6
8-2
2irR
1000
km
1
12-4
13-2
• The table is compiled from G. M. Braslavsky's data; the areas of small adjacent ice caps and ice sheets are included in the larger ones.
** Names are given by author.
o
8
332
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40
20
\
0
60
<
c
80 —-—
Sx
00
_ — - — •
or
20
40
60
/
\ 1 ?»
80
Cr—-,
I0O
•A
120
\
120
m
;
U0
/
-//
160
180
160
140
Fio. 1. I. East Antarctic ice cap. II. Ellsworth ice cap. III. Marie Byrd ice cap.
IV. Palmer ice cap. V. Graham ice sheet. 1. ice shelves. 2. boundaries between
ice caps.
approach one another in the upper reaches of outflow glaciers separate
numerous small ice domes by sloping saddles at the edge of the ice sheet.
The hypsographic curve of the land based ice sheet, excluding nunataks
and oases and constructed from the data of I. A. Suyetova's and F. M.
Braslavsky's measurements on a map with 1:3,000,000 scale, represents
a curve of the third degree (Fig. 2), not a parabola as was assumed by
W. Meinardus (1926). With a circular base this corresponds to the
surface form marked F in Fig. 3. Within the height limits 1000 and 3000
m, the surface is close to ellipsoidal, corresponding to the equation
1903 2
3-63 2
= 1
(The origin of the co-ordinate system is at sea level in the centre of the ice
sheet with the z axis directed vertically upward). In the stated limits,
deviations do not exceed ±18 m, but at the centre and at the edge the
MASS BUDGETS : ICE AGES
333
Z(km)
5
3 )
2
1
V
\v
- ^-—
0
-
1
^
~-
" " • —
"
2.5
1
-2
—-^
3
5
1
7.5
~r.~ "Vio
^»
'
"-;~-IÎ
S 106kn\
\
\\\
1
1
FIG. 2. Hypsographic curves of surface and bottom of the land-based ice sheet of
Antarctica. 1. Hypsographic curve of surface. 2. Hypsographic curve of Antarctica
without ice (based on the map of basal contours in the Antarctic Atlas, with correction
for isostatic lifting). 3. Hypsographic curve of Antarctica without ice based on
relief data for the continents of Gondwana Land (reduced to the size of Antarctica).
true surface is 400 to 500 m higher. This part of the cross section is
accurately represented by part of an ellipsoid
x2 , y2
_1
10962
2-2922
with the centre at point r == 780 km, z = 0-949 km and with axes inclined
by 0 = rx = zy = — 0°02' (see Fig. 3). The actual profile can be
sufficiently approximated by Hermite polynomials only; using a sixth
degree polynomial
F(r) = 4-038 - 3-633 x 10"6r2 + 5-554 x 10"9r3 - 4-251 X 10"12r4 +
1-659 x 1 0 - 1 5 r 5 - 3-008 X 10- 19 r 8
(1)
the deviation at the points of known height on the hypsographic curve
does not exceed 36 m.
Bottom relie/
The form of the lower surface of the land-based ice sheet is determined
by the complex dissected relief of the subglacial rock surface. A map of
the Antarctic Atlas (1966) shows the basic subglacial contours, specifically,
forms with cross section not less than 150 km in west Antarctica and not
less than 500 km in east Antarctica (Kapitsa, 1965).
In east Antarctica there exists a central rise (the Gamburtsev and
Vernadski Mountains), a series of valley depressions and an interrupted
ring of peripheral heights. The summit of the ice sheet is located above
334
ISAGE
Z (km)
r.km
<
\
»3
FIG. 3. Radial profiles of surface and bottom of land-based ice sheet with circular
base. F. actual profiles. 1. Ellipse with centre at sea level and centre of ice sheet.
2. Inclined ellipse with centre at point r = 780 km, z = 0-949 km. 3. Bottom
profile according to hypsographic curve of subglacial relief according to the Antarctic
Atlas. 4. Bottom profile according to hypsographic curve of Antarctic, constructed
from data for the contours of the continent of Gondwana Land with corrections for
isostatic sinking under the ice load, a, b. axis of ellipse 2.
the highest massif of the Gamburtsev Mountains. Coincidence of the
summits and rises of the bottom is typical for all the five large ice caps of
Antarctica. Thus the phenomenon of ice cap summits displacement to
the adjacent depressions, the so-called "migration of glaciation centres",
is not observed in Antarctica, in contrast to Greenland and the Northern
Hemisphere continental ice sheets of the quaternary period. The
explanation apparently is that in Antarctica the depressions are located
closer to the regions of ice removal than are the central highs. (Such
regions of removal are the coasts, including parts which are covered by
ice shelves.)
The highest measured point of the Gamburtsev Mountains rises to
3390 m. The highest point of the continent is the peak of the Vinson
Massif in the Sentinel Mountains, 5140 m above sea level and 3 km above
the Ellsworth ice cap surface. The lowest measured bottom point of the
ice sheet is 2555 m below sea level, located in the centre of west Antarctica,
below the southern slope of the Ellsworth ice cap where the surface level
is 1750 m.
Bottom irregularities exceeding one tenth of the ice thickness are
reflected in very smoothed form at the surface of the ice sheet. However,
the nature of the relationship between surface and bottom relief has not
yet been established, since it is determined by a system of equations which
has no analytic solution. Investigations in the region between the South
Pole and the Horlick Mountains have proved that this relationship differs
MASS BUDGETS : ICE AGES
335
from the solution for the condition of simple shear (Robinson, 1966).
Ice free areas of the surface have the characteristic relief of "roches
moutonnées" with height differences of a few tens of metres, exactly
similar to the relief of the Baltic and Laurentian Shields.
The map of the sub-glacial relief of the Antarctic (Antarctic Atlas,
1966) gives a height profile quite different from those of all other continents
including Gondwana Land and especially plateau-shaped Africa (Fig. 2).
The average height of the bottom is 410m, which if in isostatic equilibrium
without the ice, corresponds to the average height;
Go = G + £ (F - G) = 927 m,
Pc
where pc is the density of the subcrustal rock and F the average height of
the surface.
This does not agree with the observed relationship between the area of
continents and their average elevation, and would be possible only if the
earth's crust were thickened in the Antarctic (Voronov, 1964) in contradiction to geophysical data. The form of the ice sheet surface also
does not correspond to a bottom relief with similar height profile, average
height, and the approximately known heights at the centre and edges.
The simplest explanation for these contradictions is that the average
height of the bed of the Antarctic ice sheet has been overestimated by
interpolation of inadequate data (on the average, one seismic sounding
point for every 16,000 km 2 and one gravimetric point for every 2000 km2
(Kapitsa, 1965)).
Another possible way of determining the average height of the continent
is based on the use of an empirical relation (with physical explanation)
between that height and the area of the continent (Voronov, 1964). That
relationship when applied to the continents of Gondwana Land gives an
average height Go = 485 m for Antarctica without its ice. Consequently
the present height in isostatic equilibrium condition must be equal to
G =
PÄ
Pe-P
=
_198m>
The true average level of the bottom, possibly, lies between 410 and
— 198 m; in our opinion it should be closer to the last figure.
Thickness and volume of ice
The volume of land based ice in Antarctica, assuming an average
bottom level of 410 m, is 23-0 mil km3. This would increase to 30-4
mil km 3 for an average bottom level of —198 m (see Table 1). The
difference between these two numbers (approx. 25 per cent) characterizes
the accuracy of present knowledge. The thickness of ice, measured far
away from the edges of the ice sheet and nunataks, ranges from 640 m
above the Gamburtsev Mountains to 4335 m on the slope of the Ellsworth
ice cap above a depression that stands —2538 m below sea level.
336
ISAGE
Melting of the continental ice sheet, without isostatic compensation and
assuming that the temperature of the melt water temperature remains at
0°, would raise the mean ocean level of the world by
A_
M1—(pw—p)M2
where Mi is the mass of ice now located above sea level; M 2 is the ice
mass below sea level; p w = 1 g cm" 3 the density of melt water; So =
361,059 mil km2 the area of the world ocean; AS0 the areas of Antarctic
rock below sea level; and 0 = 1-50 x 1013 cm the slope of the hypsographic curve.
With the volume of ice 23 million km3, ( p w - p ) M 2 = 2-031 x 1020 g,
AS0 = 4-204 million km2 and AZ = 50 m, and with the volume 30-4
million km 3 accordingly, (P„-p)M2 = 2-406 X 1020 g, AS0 = 10-724
million km2 and AZ = 65 m. The total increase of ocean area
AS = v / (S o +AS°) 2 +20[M 1 -( / >w-p)M 2 ]/ P w - So
in the first case would be 11-8 million km2 and in the second 20-7 million
km2.
Surface layer, accumulation and ablation regime
The distribution of the specific rate of surface accumulation by atmospheric precipitation is shown in the relevant map of the Antarctic Atlas
(1966). The scale of the map and the lack of data do not permit showing
the irregularity of the accumulation in the hilly peripheral portions with
strong winds. The same applies to mountainous regions, where there
are widespread patches of ablation due to drift and evaporation, and to
areas around oases and nunataks where melting and drainage of melt
water occur. Especially large fluctuations of net accumulation exist on
the ice sheets of the Antarctic Peninsula; these are not shown on the map
referred to above. In the far north of the east coast the net accumulation
is equal to 35-40 g cm" 2 year"1. On some islands on the north-west
coast, at elevations of a few hundred metres above sea level the accumulation can register as much as 100-200 g cm" 2 year"1, while on the lower part
of the glacier surface melting may be predominant (Robin and Adie,
1964). In spite of some differences between meridional sectors (caused
by the latitude differences of the ice sheet edge and also by peculiarities
of relief and of atmospheric circulation) the map shows a basic pattern
of distribution of accumulation ; this consists of a decrease in accumulation
from the edge to the centre of the ice sheet that can be correlated with
increasing distance from the sources of atmospheric moisture and
decreasing temperature.
From measurements of areas between lines of equal net accumulation
rate, carried out by G. M. Braslavsky, a curve was obtained for the average
accumulation rate along a radial profile (here the entire area of the
MASS BUDGETS : ICE AGES
337
2
Antarctic Peninsula has been counted in the range of 30-70 g cm" year" 1 ).
This curve corresponds to be a hyperbola
x2
v2
x
y
__ i
8752
82-162
with the centre located at point r = 2838 km, a = —17-6 g cm" 2 year" 1
and with the main axes in the plane ra (Fig. 4). Accordingly, the dependence of accumulation rate on distance from the centre of the continental
ice sheet is
a(r) = 259-44 - 0-09761 r - 0-09479 Vr 2 -5677r+7-297x 106
(2)
The curve a(r) intersects the abscissa of the ice sheet edge (R = 1967
km) at the points a — 64-9 and a = 70 g cmr 2 year" 1 ; the trend of the
curve at the edge zone towards the point with co-ordinates r = 1967 km
and a = 64-9 g cm~2 year" 1 corresponds within the limits of accuracy of
existing data of the mean accumulation rate after deducting ablation.
The absence of a maximum on the curve at the level of condensation of
atmospheric moisture may be connected with wind displacement of snow
or with lack of precise data.
The area of surface ablation can be roughly estimated as 150,000 km2
and its total loss as Js B a dS = —15±10 km3 of water in a year; this
corresponds to a = —1 g cm~2 year"1.
However, the major part of this area is represented by patches within
the region of net accumulation created by snow drift and evaporation.
The surface area with more or less intensive melting around oases and
nunataks, represents merely a small proportion of the area of ablation.
Because of ice flow around ice free areas a large part of the melt region
does not coincide with the ends of the ice flow lines; these continue to
coastal ice cliffs which often lie again in the region of net accumulation.
The part of the accumulation area in which flow lines end on land and
fulfil the equation
Js A adS + JsBadS = 0
(A = accumulation, B = ablation)
without losses by icebergs, is very small, as is the total area S = SA + SB;
to evaluate it is not yet possible. In any case the ablation on land
practically plays no part in limiting the ice sheet in regions with climatic
snow limits at sea level.
The total accumulation rate, calculated from the formula:
= 27T J \ ( r ) d r = 2TT ((*A - ^B)r2 - C[^(D3/2 - q»/t) ^p[(2r+p)D 1 /2 _ pqi/2) _ • p (4q-p 2 )ln 2D* - 2r + p ]
8
16
(3)
in which A, B, C are numerical coefficients of three terms of the right side
of the equation (2), and D = r2 + pr + q, is shown in Fig. 4.
338
ISAGE
gern yr
FIG. 4. Rate of nourishment of1 the ice sheet, a is average rate of accumulation
along
a radial profile, g cm"' yr" . b is total accumulation rate JS^ads (equation 9),
km 3 of water in a year. 1 is real axis of hyperbola a(r). 2 is centre of hyperbola.
For the whole continental ice sheet the total accumulation rate is
2-1680 X 1018 g year"1, i.e. 2168 km 3 of water per year. This corresponds
to an average accumulation rate of a = 17-84 g cm" 2 year"1. This result
is very close to that obtained by Giovinetto (1964), for the ice sheet that
included ice shelves but excluded the Antarctic Peninsula (2100 ± 400
km 3 of water a year). Compared with Giovinetto's estimate our estimate
of possible error of calculation must be larger because of the absence of
data for the Antarctic Peninsula.
According to curve b (Fig. 4) one half of all the accumulation is received
by a coastal zone 260 km wide, and only one quarter by the central region
with radius of 1400 km which represents more than one half of the entire
continental ice sheet area.
Temperature and zones of ice formation
The distribution of the temperature in the snow-névé cover at the depth
of vanishing yearly fluctuations (15-20 m below the surface) is shown in
the relevant map of the Antarctic Atlas (1966). Except for regions of
melting with drainage, this temperature is very close to the annual average
temperature of the surface and the adjacent air. The temperature variation along the coast depends largely on latitude; the temperature decreases
with distance from the sea and increases with the surface elevation. The
minimum temperature at the level of vanishing annual fluctuations at
the centre of the east Antarctic plateau equals —60-5°, the maximum (at
the west coast of the Antarctic Peninsula north of 68° south latitude)
equals 0°. The curve of the average temperature at this level along a
radial profile, according to G. M. Braslavsky's measurements of areas
between isotherms is shown in Fig. 5. The average surface temperature
of the entire continental ice sheet equals —35-5°.
MASS BUDGETS : ICE AGES
500
1000
1500
10
20
30
40
-50
-60
339
2000
r,km
FIG. 5. Surface temperature of the Antarctic continental ice sheet.
Density and thickness of snow-névé cover
The average density of the upper annual snow layer, the distribution
of which is shown in the relevant map of the Antarctic Atlas (1966),
depends mainly on wind velocity and therefore increases from the centre
of the ice sheet to the edge. The maximum density can be observed in
the zone of katabatic winds on the ice sheet slope. The relationship
between the average density of the surface layer (g cm"3) and distance
(km) from the centre of the ice sheet can be expressed by a straight line
/>8(r) = 0-292 + 7-8 x 10"6 r
The average thickness of the upper annual layer a/p increases from 11-6
cm at the centre to 157 cm at the edge. The density increases with depth,
reaching 0-815-0-830 gm cm~3 at the boundary with the ice. The thickness of the snow-névé cover sharply increases from zero on the outer
limit of the névé (infiltration) zone to 50-80 m in the re-crystallizationregelation zone, and then increases slowly to 160-170 m at the centre of
the east Antarctic ice sheet. This increase in thickness is caused mainly
by lowering of temperature and the deformation rate of ice crystals.
Internal regime
The measured velocity of the surface ice in the peripheral zone of the
continental ice sheet ranges from a value close to zero to 1250 m yr - 1
(Antarctic Atlas, 1966). A small velocity is observed above bottom
projections, specially near rock outcrops. Movement is much faster on
outflow glaciers. Along sections of lower velocity the latter increases
with increasing distance from the edge while for the outflow glaciers it
decreases. The outflow glaciers and the drainage basins feeding them
are regions of flow line convergence ; they are divided by gently sloping
ice hills with divergent flow lines. In many cases such regions form
peripheral ice domes with independent centrifugal movement, separated
from the slope of the main dome by saddles in the upper drainage basins
of the outflow glaciers. Thus in the peripheral zone a differentiation of
ice flow occurs in which outflow glaciers are regions of longitudinal
340
ISAGE
tension and lateral compression, while the intervening regions are
marked by longitudinal compression and lateral tension. In the absence
of visible differentiation far away from nunataks the velocity at the edge
fluctuates between 120-400 m year~l (Dolgushin, 1966). A lack of data
prevents any reliable calculation of the total ice loss through the edge of
the continental ice sheet.
Below the layer with annual temperature fluctuation the temperature
decreases more slowly and to a greater depth in central regions than at
the edge of the ice sheet. At Byrd Station, located on the gentle slope
of the Ellsworth dome, close to the boundary with the Marie Byrd dome,
the maximum temperature gradient (Sö/Sz) with depth at 68 m is equal
to 2-7 X 10"5 degree cm" 1 , and at a depth of 170-300 m a constant
gradient of 2-6 X 10"6 degree cm" 1 is observed (Gow, 1963). Within
5 km of the edge of the ice sheet close to Mirny station the gradient at
20 m depth reaches 1-875 x 10-" degree cm- 1 ; at a depth of 170 m it
decreases to zero; then the temperature begins to increase and at a depth
of 352 m the gradient is —4-52 X 10~4 degree cm"1, so that at the bottom
(540 m) the temperature must reach 0° (Bogoslovsky, 1960). The
character of the temperature distribution along a vertical radial cross
section of the ice sheet is shown in profile 144A of the Antarctic Atlas
(1966). The increasing negative temperature gradient (temperature
decreasing with depth) as the edge of the ice sheet is approached is
explained by the large effect of advection of deep cold ice due to the
increasing velocity and temperature gradient along the surface, and the
decreasing thickness of the layer with temperature inversion—corresponds
to the decreasing total ice thickness in the presence of powerful heat
sources at the bottom.
The ice density increases very slowly with depth and (theoretically) must
reach 0-92 g cm" 3 at a depth of 4 km. The average density of an entire
vertical cross section including the snow-névé layer together is everywhere
close to 0-9 g cm" 3 (Shumsky, 1963).
Regime oj a stationary axi-symmetric ice cap
Because of the lack of data on the regime of internal processes we shall
try below to find it by calculation. Without a knowledge of the velocity
or stress distribution at the outer boundary the problem can be solved
only for an axi-symmetric model of the ice sheet with a given distribution
of the height changes rate h/t. Analogous determinations for different
profiles require the knowledge of the flow line direction in a plane which
is not yet known. The strict solution of this problem is possible only by
numerical methods. As a first approximation analytical solutions are
summarized below for conditions which are simplified but not far from
reality. In particular the rate of change of all dimensions of the section
is taken as zero, and the ice density as 0-9 g cm"3. Cylindrical coordinates r, 0 , z, are used with the origin at sea level and at the centre
of the shield, and with axis OZ directed vertically upward.
MASS BUDGETS: ICE AGES
341
Average bottom profile
From a number of assumptions the equation of a radial profile has been
determined analytically, which is adequately close everywhere to the
profile derived from the hypsographical curve of the Gondwana Land
continents (Fig. 3). It has the form
G(r) = 3-5 - 9-009 x 10-3r + 6-430 x 10-6r2 - 1-453 X 10"9r3 (4)
The equation of ice thickness along the average radial profile of the ice
sheet is obtained by subtraction of (4) from (1), which gives
H(r) = F(r) - G(r)
(5)
The thickness increases from 438 m at the centre to 3537 m at 870 km
from it and then decreases to 340 m at the edge.
Velocity of movement
Integrating the equation of mass continuity in the form
£' + !' + «* o
or
r
oz
with r and z obeying the boundary conditions implied by the surface mass
balance equation
a ,
h
.
- + Vn
= 0
p
t
where vn is the ice velocity component along the external normal to the
surface, a is the accumulation, p is the ice density, and dh/dt the velocity
of displacement of the surface, yields the following expression for the
vertically averaged radial velocity components:
1 f
Vr(r) = —ri
(a s + ai,)r dr
(6)
where a s and ab are the values of the accumulation on the upper and
lower glacier surfaces. In the first approximation, ignoring ab and
substituting the value of H and jro a(r)r dr from equations (3) and (5), the
curve for vr(r) shown in Fig. 6 is obtained. Fig. 6 shows also the result
of a calculation of vertical component of the surface velocity by means of
the formula
Vz(F) = v r tga — —
Ps
where tga = F'(r) is the surface slope.
Taking into consideration the large difference in surface and bottom
temperatures, and the exponential dependence of ice viscosity on temperature, it can be assumed that a predominant part of the shear deformation
is concentrated in the bottom layer, and that the rest of the ice moves with
about the same horizontal velocity. Therefore, for the given approximation, the changes of the horizontal velocity component along the vertical
342
U r » m yr
600
ISAGE
573
'
/2000
r, km
FIG. 6. Curves of ice movement velocities. 1. Radial component. 2. Vertical
component. 3. Average velocity at edge taking into account its true length.
4. Velocity in regions of raised bedrock at the edge of the ice sheet between outflow
glaciers.
can be ignored. We have therefore, a movement in the form of a sliding
of an extending ice mass along the bottom.
Rate of deformation and basal shear stress
The result of shear stress calculations is presented on Fig. 7, where the
stress produced by the mass forces (pgHtga) is also shown.
The basal shear stress grows from zero at the centre to 0-72 kg cm" 2
at a distance of 1750 km and then decreases to 0-586 kg cm" 2 at the edge.
Close to the edge the major part of the shear stress is produced by plane
forces which do not play an important part in other parts of the profile.
343
MASS BUDGETS : ICE AGES
T,kg
1:00 r
0.75
aso
0.25
500
1000
FIG. 7. Shear stress on the bottom of an ice sheet.
produced by mass forces (pgHtga).
1500
2000
r, km
1. shear stress T. 2. stress
Sources oj heat, temperature and melting on the bottom
The results of calculation of heat flux from the bottom into the glacier,
A(d0/dz)z =o. loss of heat due to bottom melting Lab, and of temperature
and melting rate at the bottom are shown in Fig. 8. With increasing
radial distance r from the centre the magnitude of the basal heat flow into
Q.cal cm yrH
1000 r
-500-
FIG. 8. Power of sources and sinks of heat Q, Lati, temperature Ob and rate of
melting at> at the bottom of a continental ice sheet.
344
ISAGE
the glacier gradually increased from 38 to 209 cal crrr 2 year- 1 . The
quantity of heat expended for bottom melting increases from zero at
approximately 1400 km from the centre to 617 cal cm" 2 year" 1 at the edge.
The temperature increases from —33-1° at the centre to —8-77° at a
distance of 500 km from it and to the melting point at a distance of 1400
km from the centre. The specific rate of bottom melting increases from
zero at a distance of 1400 km from the centre to 7-7 g cm" 2 year" 1 at the
edge, and the total quantity of bottom melt water is 76-3 km 3 a year,
i.e. 3-5 per cent of the net accumulation of the continental ice sheet. In
reality, the temperature at the bottom must be somewhat lower and the
melt rate at the edge smaller, due to the fact that part of the heat arises
not from bottom friction but from deformations inside the ice and at
the edge from advection, (vr dö/dr), of cold ice from the interior of the
ice sheet, factors not considered in our solution.
Age of the ice
The average turnover period of the ice in a continental ice sheet and,
therefore, the average age of ice in steady state conditions
= (
Js
is equal to 9-55 — 12-60 X 103 years, for an ice volume of 23-0 or 30-4
million km 3 respectively. The age of the ice at a given point is determined
from the equation
d r
- = ^ = dt
Vr
VZ
Since in the assumed model all variables are independent of t, and vr is
independent of z,
where integration is carried out from the beginning of the flow line
(r0, F(z0)), passing through the given point.
The flow line slope in the radial cross sections is determined from the
equation
tg0(r, z) =
^
Vr
Because the integrals in equation (7) cannot be expressed as elementary
functions, the integration is carried out approximately in finite differences.
The result of the calculation is shown in Fig. 9.
The age of the ice grows with the depth according to a logarithmic law.
The oldest ice occurs on the bottom at the boundary of bottom melting.
If this boundary is located at 1400 km from the centre then the ice there
must reach an age of about 466,000 years. In reality this figure may be
higher, due to the absence of bottom sliding when temperatures are below
345
MASS BUDGETS : ICE AGES
\
FIG. 9. Flow lines and age of ice of the continental ice sheet. 1. Flow lines.
2. Isochrones. Numbers on the graph give the age of the ice in 1000 years, numbers
below the graph give the abscissa (r) scale.
the melting point, but the thickness of the very old ice layer at the bottom
is vanishingly small. Movement of the ice from the melt limit to the edge
of the ice sheet lasts 10,700 years. During this period about 56 m of
ice is melted, with the result that at the edge of the sheet the ice apparently
is not more than 60-78,000 years old. It must be mentioned, however,
that the calculation is approximate and that space differences and change
of regime and dimensions with time can alter the above picture.
Problems for further in vestigations
The knowledge of Antarctic glaciology obtained up-to-date, as shown
by this review and extensive material not included in its abridgement, is
very limited and incomplete. Especially tangible blanks exist in the
knowledge of ice thickness, movement velocity and internal regime of
the ice sheet, and also of its changes in time and the causes for these.
Up-to-date the major part of the field investigations in the Antarctic
and other regions are being used for hypothetical qualitative considerations. Quantitative methods of analysing the data received may be used
for statistics and correlation of different characteristics or for analysing
physical processes. Both these directions are useful and mutually
supplementary; the second one is more difficult, but considerably more
effective for solving basic problems of glaciology—finding the connections
between size and form, regime and conditions of glacier existence.
The full solution of this problem requires first of all the solution of the
system of equations governing the mass and energy exchange for the
general three-dimensional non-stationary case. Not long ago to attack
this problem was not realistic. Now, the achievements of computing
mathematics with the use of fast computers make it possible to tackle such
problems. As the first stage, in the Institute of Geography of the U.S.S.R.
346
ISAGE
Academy of Science preparations were made for solving the entire system
of equations for the simplest exi-symmetric three-dimensional problem,
modelled on the Antarctic ice sheet. Preliminary results of this preparatory work have constituted a part of the present study. If the problem
is to be solved satisfactorily then the question of application and check
of the theory will arise. For this it is necessary to have a knowledge of
boundary conditions, and, possibly, of the precise parameters, characterising ice of different structures in a wide range of thermodynamic conditions, typical of the Antarctic ice sheet.
If the basal shear stress of the glaciers were included in the number of
known boundary conditions as some function of temperature, basal sliding
velocity and other variables determined by the equations, then for a
solution it would be sufficient basically to know the relief of the bottom
and the free surface, and also the accumulation rate and the surface
temperature. That is the position with floating glaciers, where the basal
friction equals zero. They are the simplest objects, convenient for a more
precise definition of ice properties, and deserve study in the first stage of
exploiting the theory. The results of this stage should be the possibility
to calculate the regime and rate of the present changes of floating glaciers
in accordance with the given large-scale topography and information on
accumulation and surface temperature.
Lack of knowledge of the basal shear stress of land glaciers makes
necessary, for an application of the quantitative theory, the preliminary
determination of the distribution of the ice surface movement velocity, and
this greatly increases the volume of field work needed. The need for these
data must be considered in planning any complex field research, and
measurement of the strain rate tensors or ice velocity vectors must be
included in such a programme.
REFERENCES
Atlas of Antarctica (in Russian) 1966.
BOGOSLOVSKIY, V. N. 1960. The temperature regime and movement of the
Antarctic ice sheet (in Russian). Info. Bull. Sov. Ant. Exped. No. 10.
DOLG usHiN, L. D. 1966. New data on the speeds of Antarctic glaciers (in Russian).
Info. Bull. Sov. Ant. Exped. No. 56.
DRYGALSKI, E. 1921. Das Eis der Antarktis und der subantarktischen Meere.
Deutsche Südpolarexpedition 1901-1903 Bd. T.H.4.
EVTEEV, S. A. 1964. Geological activity of the East Antarctic ice sheet (in Russian).
Glaziologicheskie Issledovaniya No. 12.
GIOVINETTO, M. B. 1964. The drainage systems of Antarctica. American
Geophysical Union, Antarctic Research Series, Vol. 2.
Gow, A. J. 1963. Results of measurements in the 309 metre bore hole at Byrd
Station, Antarctica. J. Glac. Vol. 4, No. 36, p. 771-84.
Gow, A. J. 1965. The ice sheet. Antarctica, N.Y.
KAPITSA, A. P. 1965. The reliefof Antarctica below the ice (in Russian). Doctoral
thesis, Moscow University.
KOTLIAKOV, V. M. 1961. The snow cover of Antarctica and its role in the present
glaciation of the continent (in Russian). Glaciologicheskie Issledovaniya, No. 7.
MEINARDUS, W. 1926. Die hypsographischen Kurven Grönlands und der Antarktis
und die Normalform der Inlandeisoberfläche. Petermanns geogr. Mitt., Vol.
72, H5/6.
MASS BUDGETS : ICE AGES
347
ROBIN, G. DE Q. and ADIE, R. J. 1964. The ice cover. Antarctic Research.
ROBINSON, E. S. 1966. On the relationship of ice surface topography to bed
topography on the South Polar Plateau. J. Glaciology, Vol. 6, No. 43, 43-54.
SHUMSKY, P. A. 1963. The fields of pressure and density in glaciers. Clac.
Issled., No. 9.
VORONOV, P. S. 1964. On the dimensions of the Antarctic continent and the
character of its denudation (in Russian). Problems of Arctic and Antarctic 17.
WERTH, E. 1908. Aufbau und Gestaltung von Kerguelen. Deutsche Südpolarexpedition 1901-1903 Bd. II, H.2.
WRIGHT, C. S. and PRIESTLEY, R. E. 1922. Claciology. British Antarctic
Expedition, 1910-1913.
The Antarctic ice sheet and its probable bi-modal response to
climate
BY
MARIO B. GIOVINETTO
Department of Geography, University of California-Berkeley, Ca., U.S.A.
Contribution No. 225, Geophysical and Polar Research Center, Department of Geology, University of Wisconsin, Madison, Wise, U.S.A.
ABSTRACT
The net mass budget estimates reported elsewhere for the Amery Ice
Shelf drainage system and the eastern and western parts of the Ross Ice
Shelf system are combined with (i) an alternate estimate for the Amery
Ice Shelf system, and (ii) alternate estimates for the eastern part of the
Filchner Ice Shelf system. These systems make up the interior province
of Antarctica and their combined net budget is estimated to be positive
and in the order of (3 ± 1) 1017 g yr"1. The Ross Ice Shelf system as a
whole is the only system of the interior province for which the estimate of
a positive net budget is significant ((18 ± 5 ) 1016 g yr- 1 ); direct and
indirect evidence confirms that the western part of the system is a region
within the interior province where the net budget is positive. The interior
province accounts for approximately one half of both the area and the
mass of the ice sheet, and one third of the total mass input; engulfed ice
shelves are the agents of drainage, and the net mass gain is equivalent to
approximately one half the annual input in the province. The remaining
drainage systems are split into three groups and make up the peripheral
province. This province accounts for the remaining one half of both
the area and the mass of the ice sheet, and two thirds of the total mass
input; the agents of drainage are marginal ice shelves, glacier tongues
and grounded ice termini. A comparison of the net budget for the
interior province with data on sea level change during the last 100 years,
indicates that the net budget in the peripheral province should be negative.
Empirical and heuristic two-province models of the ice sheet suggest that