Hooke R.L. Principles of glacier mechanics

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Basal temperatures in Antarctica 141

-5

Areas at the pressure

melting temperature

Areas below -10

0 1000 km

-10

-5

-20

-30

-20

-10

Areas at the pressure

melting temperature

Areas below -10

0 1000 km

Ice

shelf

Ice

shelf

(a)

(b)

-10

Figure 6.15. (a) Basal

temperatures,

◦

C, from the

Column model; (b) basal

temperatures,

◦

C, calculated

by Ph. Huybrechts, especially

for this book, using the model

described in Huybrechts

(2002). ((a) Modiﬁed from

Budd et al., 1971. Reproduced

with permission of the

Australian Antarctic Division.)

142 Temperature distribution in polar ice sheets

Budd et al.were using balance velocities that were an order of magni-

tude higher than Huybrechts in this sector, thus overemphasizing strain

heating.

The differences between the temperature distributions in Figure 6.15

emphasize the need for caution in using the Column model. However,

the Column model does illustrate the basic physical factors affecting the

temperature distribution in ice sheets, and can be used to obtain reason-

able estimates of the temperature distribution with only a calculator or

spreadsheet. For geomorphologists wishing to test ideas on the origin of

certain landforms, for example, the errors introduced by the simplifying

assumptions made in the Column model are probably no greater than

the uncertainties in the ice age values of parameters like b

, H, β

and u

that are used in it.

Geomorphic implications

Temperature distributions such as that in Figure 6.12 have implications

for glacial erosion and deposition and the origin of some glacial land-

forms. Erosion rates are likely to be highest where basal melt rates are

low, and particularly where meltwater is refreezing to the glacier sole.

Thus, we might expect to ﬁnd that erosion was most intense some dis-

tance from the divide. Conversely, the formation of lodgment till by

subglacial melting should be most prevalent beneath the ablation zone.

Both are consistent with observation.

The calculations shown in Figure 6.12 also suggest that zones of

frozen bed, a couple of kilometers wide, should develop along ice sheet

margins in regions where mean annual temperatures are sufﬁciently low.

This, indeed, seems to have been the case in North Dakota and adjacent

areas of Alberta and Saskatchewan. Here, blocks of bedrock, tens to

hundreds of meters on a side, became frozen to the base of the glacier

and were moved outward a kilometer or so (Figure 6.16). Detachment

may have been facilitated by high pore-water pressures in the unfrozen

rock beneath the frozen zone. Upon deposition, these blocks formed

hills. When the ice eventually receded, the basins from which the blocks

were plucked became lakes (Moran et al., 1980).

In some places, subglacial frozen bed conditions persisted through-

out the last glacial period. The best studied such areas are in western

Sweden, in the divide region of the Weichselian ice sheet. These cold

zones were probably a consequence of a combination of high accu-

mulation rates, cold temperatures, and thin ice in the topographically

high divide region. Relict periglacial landforms like patterned ground

(Kleman and Borgstr¨om, 1994) and weathering features such as tors

(Kleman and H¨attestrand, 1999) are found in these areas. These features

Geomorphic implications 143

Detachment

Unfrozen

Frozen

2–3 km

Lake

Hill

Former ice sheet

Figure 6.16. Hill–lake pair formed by thrusting at a frozen margin.

developed under ice-free conditions during the last interstadial. Around

the edges of these zones of relict landscape, frozen bed conditions per-

sisted on higher ground while lower areas were at the pressure melting

point. Kleman and Borgstr¨om (1994)have described a distinctive set of

landforms in such situations (Figure 6.17). Up- and downglacier from the

higher areas the ice was sliding, whereas it remained frozen to the sub-

strate over the hill itself. This led to longitudinal compression on the stoss

side of the hill, and longitudinal extension on the lee side. Till dragged

by the ice thus became stacked in transverse moraines on the stoss side.

On the lee side it was pulled away, forming an abrupt scarp (Figure 6.17).

Along the lateral boundaries, there is a narrow transition separating the

relict surface from an area of glacially modiﬁed topography.

Ribbed moraine is another distinctive landform that is commonly

present around the edges of areas where frozen bed conditions either per-

sisted throughout a glaciation or perhaps developed during deglaciation

(H¨attestrand and Kleman, 1999). Ribbed moraines are anastomosing,

somewhat sinuous ridges oriented transverse to ice ﬂow (Figure 6.18a).

The ridges typically consist of glacial drift similar to that in adjacent

areas without ridges. The drift is usually till but may be glacioﬂuvial

sediment or a combination of the two. In troughs between ridges, seis-

mic investigations and limited exposures suggest that the drift sheet is

generally thin or missing (Figure 6.18c). If one could decouple the ridges

from the substrate and slide them together, they would ﬁt remarkably well

(Figure 6.18b). These characteristics suggest that the ridges were formed

by pull-apart of a once-continuous drift sheet at the boundary between

zones of thawed and frozen bed (Figure 6.18d). H¨attestrand and Kleman

have argued convincingly that this is the case. They have shown that

ribbed moraine is conﬁned almost exclusively to the core areas of late

Pleistocene ice sheets. They ﬁnd that the ridges are transverse to ice ﬂow

directions during deglaciation and not to ﬂow directions during the Late

Glacial Maximum, suggesting that they formed during deglaciation.

As our understanding of the origin and distribution of features such

as hill–hole pairs, relict surfaces, and ribbed moraine improves, they

will become increasingly valuable in constraining numerical models

144 Temperature distribution in polar ice sheets

(b)

Stoss side

moraines

Patterned ground

Compressive

flow

Frozen

Unfrozen

Extending flow

Transverse

till scarp

Ice flow

Transverse

till scarp

Lateral sliding boundary

Stoss side

moraines

Ice

flow

(a)

Patterned

ground

Figure 6.17. Schematic map (a) and cross section (b) through a low hill that

remained frozen throughout a glacial advance and on which periglacial

landforms are thus preserved. Stoss side moraines were formed in the zone of

compressive ice ﬂow on the stoss side of the hill, and transverse till scarps were

formed on the lee side.

of Pleistocene ice sheets. Some modeling studies have already used

the distribution of these features for this purpose (see, for example,

Fastook and Holmlund, 1994; Moran et al., 1980)but much remains to be

done.

Summary

We began this chapter by deriving the energy balance equation. Given

boundary conditions appropriate for a polar ice sheet, solutions to this

equation yield the temperature distribution in the ice sheet. The boundary

conditions most commonly used are: (1) the temperature at the surface,

which is approximated by the mean annual temperature, perhaps with a

correction for heating by percolating melt water; and (2) the temperature

gradient at the bed. The latter is based on estimates of the geothermal

Summary 145

Decollément

Ice flow

Unfrozen

Frozen

(a)

(d)

Ice flow

(c)

(d)

60% extension

Figure 6.18. Formation of ribbed moraine. (a) Map of ribbed moraine ridges

near Lake Rogen in west-central Sweden. The line pattern on the ridges shows

the direction of faint ﬂuting. (b) Inferred original relative positions of ridges.

Areas of overlap are attributed to streamlining after the ridges were pulled apart.

material above decoll

ement. (d) Schematic diagram showing how a layer of

frozen soil overlying thawed material could be pulled apart by extensional ﬂow

in the ice. ((a) and (b) from H

attestrand and Kleman, 1999. Reproduced with

permission of the authors and Elsevier Science.)

ﬂux. If calculations suggest that the bed is at the pressure melting point,

the temperature gradient is adjusted to ensure that calculated basal tem-

peratures do not exceed the melting point.

By using appropriate simpliﬁcations, we studied solutions to the

energy balance equation for the situation at an ice divide, for the situa-

tion near the glacier surface but some distance from the divide, and for

a column of ice extending through an ice sheet some distance from the

divide. Two key assumptions in the latter, the so-called Column model,

are that w decreases linearly with depth and that longitudinal advection

can be approximated by assuming a warming rate at depth that equals that

at the surface. Both of these lead to basal temperatures that are too cold,

and the second may result in physically impossible temperature distri-

butions in areas where surface temperatures are changing rapidly in the

longitudinal direction. With suitable caution, however, we found that the

146 Temperature distribution in polar ice sheets

Column model was useful for illustrating how various physical processes

that affect the temperature distribution were reﬂected in calculated tem-

peratures along a ﬂowline of the Laurentide Ice Sheet. Comparison of

basal temperatures beneath the Antarctic ice sheet calculated by using

the Column model and with a state-of-the-art numerical model served

to emphasize both the value and the limitations of the former.

Finally, we discussed some geomorphic features formed at bound-

aries between zones of thawed and frozen bed, and noted that these could

be used to constrain numerical models of ice sheets.

Chapter 7

The coupling between a glacier and

its bed

In Chapter 4 we found that the rate of deformation of ice, ˙ε

, could be

related to the applied stress, σ

,by:˙ε

(

)

(Equation (4.5)). The

rigorous basis for this ﬂow law will not be developed until Chapter 9,

but some indications of the complexities involved in applying it have

already been mentioned. Despite these complexities, calculations using

it are reasonably accurate. Computed deformation proﬁles are an exam-

ple. This is, in large part, because ice is a crystalline solid with relatively

uniform properties. The principal causes of inaccuracy in such calcu-

lations are a consequence of impurities in the ice, including water, of

anisotropy associated with the development of preferred orientations of

crystals, and of incomplete knowledge of the temperature and boundary

conditions.

As mentioned brieﬂy in Chapter 5 (Figure 5.5), glaciers also move

over their beds readily when the basal temperature is at the pressure

melting point. However, the rate at which this movement occurs is far

more difﬁcult to analyze. This is again, in part, because the boundary

conditions, principally the water pressure and the morphology of the

bed, are not known. However, a more fundamental problem is the fact

that granular rock debris is usually present, either in the ice or between

the ice and the bed, or both. There is considerable uncertainty surround-

ing the processes involved in the deformation of such material and the

appropriate constitutive relations describing its deformation and that of

ice containing it. Furthermore, unlike the situation with pure ice, the

properties of the rock debris vary, not only from glacier to glacier, but

also from point to point beneath a single glacier.

147

148 The coupling between a glacier and its bed

Figure 7.1. Bed geometry

used in Weertman’s (1964)

analysis of basal sliding.

Although it may be impossible to know the boundary conditions

well enough to predict accurately the rate of movement of a glacier over

its bed, it is nevertheless important to understand the processes in order

to place limits on the rate. Thus, a signiﬁcant effort has been made to

analyze these processes, using judicious assumptions where necessary

as a substitute for detailed data. This analysis has led to relations between

the rate of movement and measurable quantities such as water pressure

and driving stress that can be tested with ﬁeld data.

We start this discussion by looking at the movement of clean ice

over an irregular hard rigid bed – the traditional sliding problem. Some

of the principal shortcomings of the analysis are then discussed. Finally

we take up the problem of deformation of the granular materials over

which many glaciers move.

Sliding

The basic processes by which ice moves past an obstacle on a rigid bed,

regelation and plastic ﬂow, were ﬁrst discussed by Deeley and Parr (1914)

and later quantiﬁed by Weertman (1957a, 1964). Regelation involves

melting of ice in the region of high pressure on the upglacier or stoss side

of the obstacle and refreezing of that water in the region of lower pres-

sure on its lee face. Plastic ﬂow is simply deformation of ice in a three-

dimensional ﬂow ﬁeld around the obstacle. In his analysis, Weertman

used a simpliﬁed model of the bed geometry, sometimes called the tomb-

stone model, consisting of uniformly spaced rectangular blocks on a ﬂat

surface (Figure 7.1). This model has been roundly criticized as being

Sliding 149

Ice flow

Higher pressure,

Colder

Lower pressure,

Warmer

Figure 7.2. Pressure and

temperature on stoss and lee

sides of a rectangular bump

on a glacier bed.

unrealistic, and inappropriately defended by arguing that fudge fac-

tors can be inserted to make it applicable to real situations. The real

value of the model is that the physical principles involved in the sliding

process are illustrated without resorting to sophisticated mathematical

techniques.

Consider the bed shown in Figure 7.1.For simplicity we will use

cubical obstacles with sides of length, , instead of Weertman’s rectan-

gular ones. The mean spacing between obstacles is

L, and we therefore

deﬁne r = /

L as the roughness of the bed. The mean drag on the bed is

τ .Asthe ice is separated from the bed by a thin ﬁlm of water, we assume

that faces parallel to the ﬂow cannot support a shear stress. Thus the

entire drag over an area

must be supported by one obstacle. The total

force on the obstacle is then F =τ

,sothe stress difference across the

obstacle, σ

− σ

, is:

− σ



(7.1)

where the subscripts, s and l, refer to stoss and lee, respectively.

Regelation

Let us deal with regelation ﬁrst. The average temperature at the bed

is constrained to be at the pressure melting point determined by the

average pressure in the water layer. The average pressure is a function

of the local glacier thickness. The pressure in the water ﬁlm on the

stoss face of the obstacle is higher than the average, and the pressure

on the lee face is lower (Figure 7.2). We denote the pressure difference

across the obstacle, σ

− σ

,byP. This pressure difference results in a

temperature difference, T = CP,where C is the change in melting

temperature with pressure (Equation (2.2)).

The temperature difference across the obstacle, from Equation (7.1),

is thus:

T = C





(7.2)

150 The coupling between a glacier and its bed

and the temperature gradient through it is T/. The heat ﬂow through

the obstacle is thus:

Q =

T



= TK

 (7.3)

where 

is the cross-sectional area of the obstacle and K

is the ther-

mal conductivity of rock. K

has the dimensions J m

−2

−1

(K/m)

−1

,or

−1

.Atypical value for rock or ice is 2.2. Q has the dimensions

−1

This heat ﬂow can melt ice at a rate Q/Hρ,where H is the heat of

fusion, 3.3 × 10

Jkg

−1

, and ρ is the density of ice, ∼900 kg m

−3

Thus, the melt rate is expressed in m

−1

.Dividing this rate by the

cross-sectional area of the obstacle, 

,gives the speed with which ice

can move past the obstacle by regelation, S

. Thus, using Equations (7.2)

and (7.3):



Hρ

Cτ K

Hρr

(7.4)

In reality, some heat also ﬂows from the low-pressure region to the high

pressure region through the ice above the obstacle and through the rock

beneath it, so this relation slightly underestimates S

The water formed by melting in the high-pressure area on the stoss

side of the obstacle ﬂows either upglacier or downglacier to areas of

lower pressure. The area of low pressure in the lee of the obstacle that

we have been analyzing is one such sink. Because heat is conducted

away from this area by the temperature gradient through the obstacle,

this water refreezes. Thus, to complete the regelation cycle, the water

ﬂux to the lee of the obstacle must equal the melt on the stoss side. We

will examine the consequences of a failure of this condition later, and

consider plastic ﬂow next.

Plastic ﬂow

Plastic ﬂow (or creep) of ice around an obstacle clearly must contribute to

ﬂow of a glacier past the obstacle. Weertman suggested that this plastic

ﬂow is enhanced owing to the high stresses on the obstacle. He thus

assumes that the speed with which ice moves past the obstacle by this

process, S

,isproportional to  and to the creep rate obtained by using

the stress difference from Equation (7.1)intheﬂow law:

= b˙ε = b





 (7.5)

where b is a dimensionless constant of proportionality. Note that dimen-

sionally ˙ε is a speed.