Hooke R.L. Principles of glacier mechanics

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Sliding 161

experiments at −9.1

◦

C, Hooke et al.(1972) found that B was ∼85%

higher in ice with 20 volume percent dispersed ﬁne sand than it was in

clear ice. They also found that B increased approximately linearly with

sand content. They concluded that sand particles inhibited movement of

dislocations, and attributed the strengthening of the ice to development

of dislocation tangles in the vicinity of the particles.

The role of normal pressure

Another effect that is overlooked in the sliding theories discussed above

is that of normal stresses. Budd et al.(1979) carried out some laboratory

experiments in which ice blocks upon which a normal load, N, had been

placed, were dragged across rough rock surfaces. Temperature control

was achieved by immersing the ice and rock surfaces in an ice-water

bath. They found that S ∝ τ

/N. The cubic dependence on τ might

suggest that plastic ﬂow was the dominant sliding process, and this may

very well have been the case as it was possible for melt water formed at

the interface to escape to the surrounding bath. Thus, heat released by

refreezing of this melt water in the lee of the obstacle might not have

been available. In that case, the only heat available for melting would

be frictional heat and heat conducted from the bath to the interface.

(The interface would be colder than the bath owing to depression of the

melting point.)

More puzzling is the inverse dependence on N; this is what one

expects in a purely frictional system, such as would be provided by

a rock being dragged across a bedrock surface. To the extent that the

ice–rock interface in the experiments was perfectly lubricated by a melt

ﬁlm, however, we would presume that no tractions could have been sup-

ported parallel to the surface. In this case, the sliding speed should have

been independent of the normal pressure. However, some erosion of the

rock surface occurred during the experiments, and the erosion rate was

proportional to N

2/3

. Rock particles entrained in the basal ice and in

contact with the bed would have increased the drag. It seems doubtful,

however, that the small amount of rock debris involved could account

for a signiﬁcant reduction in sliding speed.

Budd et al. suggested that in studies of real glaciers, N should

be replaced by the effective normal pressure, N

,orthe normal pres-

sure minus the water pressure, a factor ﬁrst vigorously emphasized by

Lliboutry (1968 and earlier). The importance of water pressure on sliding

speed is now widely recognized (see, for example, Figure 7.8), but, as

we will discuss next, some of the mechanisms involved are not frictional

as ﬁrst suggested.

162 The coupling between a glacier and its bed

Figure 7.8. Diurnal variations in surface speed on Storglaci

aren, Sweden,

measured with the use of a computer-controlled laser distance meter. The

distance from a point off the glacier to a stake on the glacier was determined

every 10 min. The dashed line shows corresponding water pressures measured

in nearby boreholes. Only the major peaks in speed are clearly related to water

pressure peaks. (Modiﬁed from Hanson et al., 1998. Reproduced with

permission of the author and the International Glaciological Society.)

Cavities and the effect of water pressure

Elevated water pressures increase the sliding speed in two ways: (1) by

increasing the degree of separation of ice from the bed, thereby increasing

the shear traction on parts of the bed still in contact with the ice; and

(2) by exerting a net downglacier force on ice that bridges cavities.

In addition, they weaken any deforming subglacial till over which the

glacier is moving, thus increasing u

(Figure 5.5). Here, we consider the

ﬁrst two of these. The third will be addressed in connection with our

discussion of subglacial till deformation.

The degree of separation of ice from the bed in the lee of obstacles

is increased when water pressures remain elevated for periods of a few

days or weeks. Let us brieﬂy examine the conditions required for such

separation in an idealized situation. The pressure at the bed is:

P(x, y) = σ

+ P

(x , y)

Sliding 163

where σ

is the ice overburden pressure and P

(x, y)isaﬂuctuating

contribution that is positive on the stoss sides of bumps and negative on

lee sides (Iken and Bindschadler, 1986). The basal drag due to this effect

can be expressed in terms of P

, thus:



(x , y)

∂z

∂x

dx dy (7.11)

Here, the x-axis is directed downglacier and z is normal to the mean bed

and positive upward, so a positive ∂z/∂x is a slope opposing ﬂow, while

negative values indicate downglacier-sloping surfaces. A

is an area of

the bed large enough to be representative of average conditions.

If we consider a sinusoidal bed of amplitude a and wavelength λ,

will vary sinusoidally; Iken and Bindschadler (1986) ﬁnd that its

maximum amplitude is:

max

√





1/2

λτ

a

(7.12)

where P

 is the root mean square of P

. The minimum pressure will

occur at inﬂection points on the downglacier faces of the undulations,

and is:

min

= σ

−

λτ

a

(7.13)

Note that P

min

decreases as τ

increases, and hence as the sliding speed

increases. If the water pressure exceeds this minimum value, separation

occurs and cavities will grow to a size determined by the degree to which

the water pressure exceeds P

min

. Roughness elements on the bed that are

bridged by such cavities no longer exert any drag on the ice. The task

of balancing the driving force, ρghα,isthus shifted to places where

the ice is still in contact with the bed. S increases, and this results in

the necessary increase in shear traction on these surfaces. (Note that S

is the independent variable in this situation, while τ

is dependent. This

is a subtle but important distinction.)

The second mechanism by which elevated water pressures lead to

acceleration of a glacier is a type of hydraulic jacking. If the subglacial

drainage system is reasonably well connected to cavities on the lee sides

of bumps, increasing water pressures in the drainage system result in

increased water pressures in the cavities. (Pressures in the water ﬁlm

on the stoss sides of the bumps are always in excess of the overbur-

den pressure, and thus are not affected appreciably by changes in the

(lower) pressure in the cavities.) The water in the cavity thus pushes

upglacier against the bedrock and downglacier against ice. The result is

adownglacier force that is added to the downglacier component of the

body force. Drag forces on the bed must then increase to balance this

164 The coupling between a glacier and its bed

24.3

23.6

3.2

3.7

20 mm d

−1

3.7

3.0

2.9

3.0

3.2 3.7

(b)

(a)

2.5

3.0

3.1

−1.25

−0.21

+0.17

+0.32

3.3

3.2

+0.19

−0.47

+0.07

3.0

2.5

0.21

1.26

−

Figure 7.9. Flow around a

subglacial cavity in the lee of a

sinusoidal bump based on a

numerical model using the

ﬁnite element method.

(a) The water pressure in the

cavity, 2.36 MPa, is too low,

so the cavity is shrinking. (b)

The water pressure in the

cavity is too large, so the

cavity is expanding.

Light-faced numbers show

pressure at bed. Bold numbers

show change in pressure

following a 0.07 MPa change

in water pressure in the cavity.

The basal drag was 0.103

MPa in both experiments, and

the mean pressure at the bed

was 2.72 MPa. The cavity

would be stable at a pressure

of 2.41 MPa. (After

othlisberger and Iken, 1981,

Figure 3. Reproduced with

permission of the authors and

the International Glaciological

Society.)

additional downglacier force. Again, an increase in sliding speed results

in this increase in drag.

Over time spans of the order of hours, cavity sizes cannot change

appreciably because such changes require ice ﬂow. Thus, on bedrock

beds, diurnal changes in speed resulting from input of meltwater or from

storms (Figure 7.8) must be a result, principally, of hydraulic jacking.

Of course, cavity size increases during hydraulic jacking as a result of

the increased ﬂow rate, so if high speeds are sustained the degree of sep-

aration will increase sufﬁciently to result in a signiﬁcant further increase

in speed.

The effect of changing water pressures in a lee-side cavity is nicely

illustrated by a numerical modeling study conducted by R¨othlisberger

and Iken (1981)(Figure 7.9). When the pressure in the model cavity

was 0.05 MPa lower than the 2.41 MPa necessary to support it, velocity

vectors were toward the cavity (Figure 7.9a), tending to close it. An

increase in pressure of only 0.07 MPa was sufﬁcient to start enlarging the

Sliding 165

Mean bed

slope

Cavity

Glacier surface

rgd

rgdl sin (b

−

rgd cos a

rgd

rgd sin a

l sin b

−

Figure 7.10. Diagram illustrating calculation of P

crit

cavity at a rate of about 10 mm d

−1

(Figure 7.9b). Note, in particular, the

substantial reductions in normal pressure, indicated by the bold numbers

in Figure 7.9b; the decrease of over 1.2 MPa at the crest of the bump could

easily have resulted in freezing there in a real situation, as suggested by

Robin (Figure 7.6b).

Iken (1981) notes that for a given adverse bed slope, β, measured with

respect to the average slope of the bed, there is a critical pressure, P

crit

above which the glacier may accelerate without bound. Such acceleration

would occur if (Figure 7.10):

λ sin β>ρgdλ sin(β − α) (7.14)

Here, P

is the pressure in the cavity and λ sin β is the projected area

of the cavity face, normal to the back slope of the bump, against which

acts. Thus, this is the force trying to push the glacier up the back

slope of the bump. The right-hand side of the equation is the compo-

nent of the body force acting parallel to the back slope of the bump

and in the upglacier direction. Using the expressions for σ

and τ

shown in Figure 7.10 and the trigonometric identity sin (β − α) =

sin β cos α − cos β sin α,weobtain:

crit

= σ

−

tan β

(7.15)

On an actual glacier bed consisting of a variety of sizes and shapes of

obstacles, P

crit

would not be exceeded everywhere simultaneously. Thus,

for most situations, P

crit

≈ σ

is probably more realistic.

Working on Findelengletscher (Findelen glacier), Iken and Bind-

schadler (1986)have collected an outstanding set of ﬁeld data on the

166 The coupling between a glacier and its bed

Horizontal speed at surface, mm h

−1

140180 100 60 20

Water level in borehole, m below glacier surface

Speed in fall, 1982

Figure 7.11. Speed of a

stake on the surface of

Findelengletscher as a

function of water pressure,

here represented by the water

level in a borehole. Were the

water level to rise to 18 m

below the surface, the glacier

would ﬂoat. (After Iken and

Bindschadler, 1986, Figure 6.

Reproduced with permission

of the authors and the

International Glaciological

Society.)

relation between water pressure and surface speed (Figure 7.11). Here,

the expected exponential increase in speed with increased water pressure,

with water pressure asymptotically approaching the limit σ

,isclearly

demonstrated.

Iken and Bindschadler suppose that the character of the bed in front of

Findelengletscher is similar to that beneath the glacier, and thus are able

to calculate sliding speeds using Kamb’s (1970) theory. For wavelengths

and roughnesses that they believe to be appropriate, the theory gives

sliding speeds that are too large, compared with the surface speed, to be

realistic. They attribute the discrepancy largely to failure of the theory

to take rock-to-rock friction into consideration.

Jansson (1995) has studied the relation between effective pressure,

, and surface speed, u

,onFindelengletscher and Storglaci¨aren, using

Iken and Bindschadler’s (1986) data (Figure 7.11) for Findelengletscher,

and ﬁnds that a relation of the form

= CN

−0.4

(7.16)

ﬁts the data well (Figure 7.12). Note that τ

does not vary signiﬁcantly

within either of the two data sets, so its effect is incorporated into the

Sliding 167

Figure 7.12. Relation

between surface speed, u

sliding speed, S, and effective

pressure, N

,on

Findelengletscher and

Storglaci

aren. Dashed lines

show sliding speed estimated

by subtracting internal

deformation from u

. (After

Jansson, 1995. Reproduced

with permission of the author

and the International

Glaciological Society.)

constant factor, C,which is more than an order of magnitude higher

on Findelengletscher (Figure 7.12). Even after subtracting the contribu-

tion of internal deformation, estimated with the use of Equation (5.16),

Jansson found that Findelengletscher still seemed to be sliding more than

ten times as fast as Storglaci¨aren under comparable effective pressures.

Such a difference in sliding speed would have resulted in a basal drag

beneath Findelengletscher that was two to three times that beneath Stor-

glaci¨aren, but driving stresses (albeit uncorrected for longitudinal stress

gradients; see Figure 12.7)atthe sites of the measurements were nearly

equal on the two glaciers.

More recent data from Findelengletscher (Iken and Truffer, 1997)

serve only to further emphasize our lack of understanding of the effect of

water pressure on sliding. By 1985, three years after the measurements

shown in Figures 7.11 and 7.12, the surface speed had decreased 25%

for comparable water pressures. By 1994 there had been an additional

35% decrease. There have not been any changes in the geometry of the

glacier, and hence in driving stress, that could explain this deceleration.

Iken and Truffer suggest that the basal water system was better connected

in 1982, so that high water pressures reached more subglacial cavities.

Thus, in effect, there may have been more subglacial hydraulic jacks

urging the glacier forward in earlier years.

168 The coupling between a glacier and its bed

Deformation of subglacial till

We have known for decades that ice moving over granular subglacial

materials can deform these materials. (Herein, the term “granular mate-

rial” should be understood to include materials with signiﬁcant amounts

of clay, although a distinction between granular materials and clays is

usually made in the soil mechanics literature.) Commonly, the granular

material is till, either formed by erosion during the present glacial cycle,

or left from a previous one. Recently it has become clear that a large frac-

tion of the surface velocity of a glacier may be a result of deformation

of such till (Figure 5.5).

Intense interest in the rheology of till dates from work on Whillans

Ice Stream in Antarctica where studies of seismic velocities suggested

that a layer with high porosity, saturated with water under high pressure,

and 2–13 m thick was present beneath the ice (Blankenship et al., 1986).

The high porosity suggested active deformation, facilitated by the high

water pressure. Thus, the high speed of the ice stream, about 450 m a

−1

was attributed to deformation of the till. Subsequent drilling revealed

that the ice stream was, indeed, underlain by till, and also conﬁrmed

that the water pressure was close to the overburden pressure (Engelhardt

et al., 1990). A key question, then, is whether the till is deforming, or

alternatively whether the high water pressures have simply decoupled

the ice stream from the till. Experiments addressing this question will

be described later in this chapter.

More recently, some scientists studying the Quaternary period have

suggested that the large volumes of material found in till sheets in the

midwestern United States and the large volumes of glacigenic material

found in some submarine fans surrounding the Barents Sea could only

have been transported to their present locations in deforming subglacial

till layers (e.g. Alley, 1991; Hooke and Elverhøi, 1996). It is estimated

that the amount of material that could be transported in basal ice or by

subglacial melt streams is too low to account for the volumes of these

deposits in the time inferred to be available for their formation. In the

Barents Sea case, calculated basal melt rates are so high that little material

is likely to have been entrained by basal ice, and yet they are too low to

provide the water volumes required for signiﬁcant ﬂuvial transport.

Because glacial till is a granular material, its rheology is quite dif-

ferent from that of ice. Granular materials normally have a yield strength

below which they deform only elastically. This yield strength, s,isrelated

to two physical properties of the material, the cohesion, c, and the angle

of internal friction, ϕ,bythe classical Mohr–Coulomb relation:

s = c + N

tan ϕ (7.17)

Deformation of subglacial till 169

200

150

100

Shear stress, s, kPa

3002001000

Effective normal pressure, N

, kPa

8 kPa

Apparent c

True c

Figure 7.13. Relation

between s and N

obtained

from a laboratory test on a

sample of till from beneath

Storglaci

aren (Iverson,

unpublished data). Inset

shows, schematically, how s

may actually vary with N

low effective pressures.

where N

is the effective normal pressure. To determine c and ϕ, labora-

tory tests are conducted in which the stress needed to initiate deformation

of a material is measured at various effective normal pressures. When s

varies linearly with N

(Figure 7.13), the slope of the line is tan ϕ, and

the intercept is the apparent cohesion.

The term apparent cohesion is used because detailed measurements

often show that the variation of s with N

is not linear at low effective

normal pressures, but rather is as shown by the dashed line in the inset

in Figure 7.13. The true cohesion is the value of s at the intercept of

this dashed line with the ordinate. Because the apparent cohesion varies

directly with the true cohesion, however, we normally will not draw a

distinction between the two quantities.

Let us now examine the physics involved in cohesion, and the phys-

ical signiﬁcance of ϕ.

Cohesion

True cohesion in soils is a consequence of cementation, of electromag-

netic forces between clay particles, and of electrostatic forces resulting

from charge imbalances among ions absorbed on clay minerals (Mitchell,

1993, pp. 125, 373–374). Cementation is the major source of cohesion

in subaerial soils, but would not be signiﬁcant in continuously deform-

ing subglacial tills. Thus, the magnitude of c in such tills is determined

primarily by the amount and species of clay minerals present.

In situ deforming subglacial tills formed by erosion in the current

cycle of glaciation do not seem to have much clay-sized material unless

the glacier has moved over a bed containing such material. Furthermore,

170 The coupling between a glacier and its bed

most of the clay-sized particles that are present in such clay-poor tills are

not clay minerals. Thus, c may be small in such tills. For example, records

from tiltmeters emplaced in till beneath Storglaci¨aren demonstrate that

this till is deforming. However, only ∼5.4% (by weight) of the particles

in samples collected through boreholes are less than 2 µm, and these

particles are largely quartz and hornblende (Iverson, unpublished data).

Laboratory tests yield c = 8kPa for this till (Figure 7.13). Similar tests

on till samples collected from beneath Whillans ice stream, containing

∼35% clay, gave c = 3 ± 1.3 kPa (Tulaczyk et al., 2000a). (In this case,

the clay-sized material, which does consist of clay minerals, is inferred

to have been derived from Tertiary glaciomarine sediments (Tulaczyk

et al., 1998; Kamb, 2001).) For comparison, values for silty and clayey

sands are typically between 20 and 75 kPa (Hausmann, 1990).

The absence of clay-sized material in deforming tills is likely to be

largely a consequence of ﬂushing by subglacial streams. In addition,

however, it is noteworthy that the deviatoric stress required to fracture a

grain increases as the particle size decreases, and that in the limit very

ﬁne grains deform plastically rather than fracture into still smaller par-

ticles (Kendall, 1978). That the clay-sized particles present in such tills

tend not to be clay minerals is the result of the absence of subaerial

weathering processes. Higher concentrations of clay-sized particles and

of clay minerals in Pleistocene tills may be a consequence either of sub-

aerial weathering after retreat of the ice, or of incorporation of previously

weathered material over which the ice moved.

Cohesion is not increased by saturation by water unless clay minerals

are present. The well-known fact that the walls of wet sand castles stand

up better than dry ones is, rather, a result of surface tension. Surface ten-

sion effects are present when the sand is wet but pore spaces still contain

air. This is because surface tension is a result of stresses associated with

the air–water interface.

Consolidation

When a granular material accumulates gradually, it compacts under its

ownweight. Such a material is called normally consolidated.Ifanaddi-

tional load, such as a shear stress, is then placed on the material, it

becomes overconsolidated. The term overconsolidated is also used to

describe a granular material which, after being normally consolidated,

experiences a reduction in overburden pressure due to erosion or, per-

haps, to melting of an overlying glacier. The highest past effective stress

to which a sample has been subjected is called the preconsolidation

stress.