Hooke R.L. Principles of glacier mechanics

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Tunnel valleys 241

Figure 8.29. Sketch of small

daughter esker diverging from

and later rejoining parent

esker. This may occur near the

margin where the ice is thin

and conduit closure rates are

low. (Bruce Hooke assisted

with artwork.)

massive wedge-shaped deposit that increases in height downstream.

Steep distal slopes mark the downstream ends of these wedges, which I

call ramp structures.

Esker nets were apparently ﬁrst described by Stone (1899), who

called them reticulate eskers. A particularly good example is found in a

valley called Atnedalen in Norway. Following the esker down-ﬂow, one

ﬁrst encounters several situations in which a small esker drops down

off the side of the parent, parallels it for a short distance, and then

climbs back up onto it again (Figure 8.29). This is probably because

the ice thickness required to keep the conduit on top of the esker was

becoming marginal. Closer to the ice margin, where the ice was still

thinner, these daughter eskers grow to nearly the same size as the parent,

forming an anastomosing pattern of ridges. Still further downstream, the

ridge character becomes indistinct and the system merges into a kettled

outwash plain.

Esker nets and ramps are characteristic of the Katahdin esker. A

typical segment of the esker begins as a relatively small, sharp-crested

ridge. Down ﬂow, the ridge increases in size, at ﬁrst rather gradually,

but near the end more rapidly. Some segments end in ramps. Others

end in esker nets or in a combination of the two forms (Figure 8.30).

Downstream from ramps and esker nets there is commonly either a fan

or a delta built into the sea in which the glacier terminated. These fans

and deltas are composed of sand and ﬁne gravel that was ﬂushed through

the enlarged conduit as coarser material was deposited. The transition

from ramp to fan has been described well by Ashley et al. (1991,p.112).

The next segment down ﬂow may develop from a continuation of one of

the ridges in an esker net, may emerge from a fan or delta, or may pick

up after a short gap.

Tunnel valleys

The margins of many ice lobes of the southern Laurentide Ice Sheet

are marked by broad, relatively straight valleys, oriented normal to

242 Water ﬂow in and under glaciers

Figure 8.30. Map of a section of the Katahdin esker showing a ramp and esker

net. There is a fan down-ﬂow from the ramp. (The site is 4.5 km NNE of Lincoln

Center, Maine.)

Tunnel valleys 243

the former ice margin. These features, apparently ﬁrst described by

Ussing (1903) from Jylland in Denmark, are called tunnel valleys. They

are typically 10–20 km long, although they may be shorter or longer;

afew hundred meters wide; and ∼10 m deep, though depths as great

as 70 m have been reported (Wright, 1973) and the depth of any ﬁll

is generally not known. Small eskers are occasionally associated with

the valleys (Johnson, 1999; Mooers 1989), and some valleys have tribu-

taries or distributaries. The valleys end at former ice margins, where they

debouch onto proglacial gravel fans. Thus, they must have been formed

subglacially. The lack of evidence for modiﬁcation by overriding ice

implies that any subsequent ice movement was probably not very vigor-

ous. Some tunnel valleys in Minnesota trend across the regional gradient

and are cross cut by present stream courses (Wright, 1973). Others ascend

adverse gradients near their termini (Cutler et al., 2002; Mooers, 1989).

Both characteristics suggest that they were excavated by pressurized sub-

glacial water. Mooers describes one situation in which a tunnel valley

terminating at one recessional moraine is cross cut by a younger tunnel

valley terminating at a younger recessional moraine. The process of for-

mation thus must have been intermittent, with the earlier valley becoming

completely closed, presumably by inﬁlling with ice, before the later one

formed. Tunnel valleys are rare or absent along the most southerly margin

of the Laurentide Ice Sheet, suggesting that they may be characteristic

of colder environments where frozen bed conditions were likely to have

been present along the ice margin (Cutler et al., 2000). This, however, is

controversial.

The origin of the water and the nature of the ﬂows that cut the tunnel

valleys are actively debated. Was the water from basal melt alone, or

was there a contribution from the surface? Was the water ﬂow contin-

uous over a period of decades or centuries, or was the water released

catastrophically? Wright argued for a catastrophic release of subglacial

water because he thought that a number of tunnel valleys were carved

simultaneously and that the ice was cold enough to preclude drainage

from the surface. Cutler et al. found that some fans at the ends of tun-

nel valleys consisted of tens of meters of sand and gravel overlain by

a bed, 3–5 m thick, containing boulders up to 2 m in diameter. They

thought the gravel might have been deposited by superglacial melt-

water reaching the bed through moulins near the margin. The boulder

bed, however, seemed to require a water ﬂux greater than could be sup-

plied by superglacial water or by steady-state basal melt. Accordingly,

they suggested that it was deposited by subglacial water that was released

when a seal was breached. They believed that the seal was a marginal

zone in which the ice was frozen to the bed. Relict permafrost features,

244 Water ﬂow in and under glaciers

Figure 8.31. Bed geometry and parameters used in Hallet’s (1996) analysis of

glacier quarrying.

pollen evidence, and numerical modeling all suggest that a wedge of

frozen ground might well have extended several kilometers in under the

ice sheet at this time (as in the left diagram in Figure 6.16). Patterson

(2002) thinks that sapping, driven by the high gradient in hydraulic head

across a marginal seal, may be an important process in initiating an

outburst.

Water pressure and glacier quarrying

Quarrying is an important process of glacier erosion. In quarrying, blocks

of bedrock must ﬁrst be loosened, either along preglacial joints or along

fractures formed by subglacial processes. Then they must be entrained

by the glacier. Because rock fragments that have been loosened but not

removed are uncommon on deglaciated bedrock surfaces, Hallet (1996)

argues that loosened blocks are readily entrained. He thus concludes that

fracture must be the rate-limiting process.

To analyze the stresses causing fracture, Hallet considers an idealized

bed consisting of steps as shown in Figure 8.31. The distance between

steps is L. The treads slope upglacier at an angle β. The crests of the steps

are at the same elevation, so the average bed slope is 0. Ice sliding over

the crest of one step separates from the bed, forming a water-ﬁlled cavity

under pressure P

. The ice regains contact with the bed a distance S from

the crest. The slope of the cavity roof is α. The glacier is supported in

part by water in the cavity and in part by a vertical stress, σ

,onthe crest

of the step, so summing forces:

L = P

S + σ

(L − S) (8.28)

The slopes α and β are related by:

S tan α = (L − S) tan β (8.29)

From Equation (8.28), noting that the effective pressure (or pressure

causing closure of the cavity), P

,isP

− P

,weobtain:

= P

+ P

L − S

(8.30)

Water pressure and glacier quarrying 245

Now σ

acts on the crest of a step, promoting fracture, but P

supports

the rock face, resisting fracture. Thus, the total stress causing fracture

(the non-hydrostatic or deviatoric stress – see Chapter 2) is:



= σ

− P

= P



L − S



(8.31)

To determine σ



we need to know S.

Hallet assumes that the rate of closure of the cavity roof, u

, can be

approximated by a relation of the form of Equation (8.3), thus:

= kSP

(8.32)

where k is a constant involving, among other things, the ice viscosity

parameter, B.Ifu

is considered to be vertical and uniform over the

cavity roof and u

is the horizontal sliding speed, then:

tan α =

(8.33)

Combining Equations (8.29), (8.32), and (8.33) then yields

+ S tan β − L tan β = 0 (8.34)

Thus, for a given L, P

, u

, and β, Equation (8.34) can be solved for S

and Equation (8.31) then gives σ



Hallet then uses principles of fracture mechanics (Chapter 4), into

which we will not delve further here, to estimate the rate of crack growth.

In his calculations, deviatoric stresses in the ice are constrained to be less

than the tensile strength of ice. By assuming that the crack is initiated

at a distance (L − S) from the crest of the step (Figure 8.31), he then

calculates the quarrying rate (Figure 8.32) for two rock types, a gran-

ite and a marble. The softer marble erodes 1000 times faster than the

granite!

As expected, quarrying rates increase with sliding speed. In addi-

tion, as u

increases maximum quarrying rates occur at higher effective

pressures. To understand the latter, note in Equation (8.31) that as S →L,

the term in brackets increases without bound. An increase in S can result

either from an increase in u

or from a decrease in P

. This is because

both of these changes cause ice in the roof of the cavity to regain contact

with the bed further from the crest of the preceding step. If an increase

in S is a consequence of a decrease in P

, σ



does not increase so much

because the two effects offset one another in Equation (8.31) (note that

S = L when P

= 0 because the glacier is then aﬂoat). Thus, quarrying

This differs from Hallet’s analysis because he takes u

to be normal to the cavity roof.

Considering the approximations in Equation (8.32) and the simpliﬁcations achieved by

using Equation (8.33), taking u

to be vertical seems justiﬁed.

246 Water ﬂow in and under glaciers

Figure 8.32. Theoretical

rates of quarrying of marble

(solid lines) and granite

(dashed lines) on a bed like

that shown in Figure 8.31.

Parameters used in the

calculation were L = 10 m,

β =11.5

, and k =5a

−1

MPa

−3

(Based on Hallet, 1996,

Figure 2. Reproduced with

permission of the author and

the International Glaciological

Society.)

rates have maxima at some fairly low P

(Figure 8.32). On the other

hand, if the increase in S is a consequence of an increase in u

, then

the maxima in σ



and the quarrying rate can occur at a higher P

. Also

noteworthy in Figure 8.32 is the sensitivity of the quarrying rate to P

;

under 450 m of ice, for example, P

may vary from 0 to ∼4 MPa, but

signiﬁcant erosion occurs over <10% of that range.

Because stresses that can be generated in the rock in this way are

limited by the tensile strength of ice, the maximum steady σ



that can

be applied to a rock is about 10 MPa, while tensile strengths of strong

crystalline rocks without macroscopic ﬂaws generally range from 10 to

20 MPa (Hallet, 1996). Water-pressure ﬂuctuations in cavities in the lee

of a bump on a glacier bed provide a mechanism for fracturing this strong

rock. They also accelerate propagation of fractures in weaker rocks or

those with macroscopic ﬂaws and play a role in the entrainment process

(R¨othlisberger and Iken, 1981;Iverson, 1989). Let us consider fracture

ﬁrst.

Water-pressure ﬂuctuations are particularly rapid where moulins

provide connections from the surface of a glacier to the bed. Water

inputs from rain or melt can then cause subglacial cavities to ﬁll and

drain faster than they can adjust by ﬂow of the ice. The resulting pres-

sure ﬂuctuations transfer the weight of the glacier ﬁrst to and then from

the tops of bumps. Under 250 m of ice, for example, the pressure could

Water pressure and glacier quarrying 247

vary from a relatively uniform 2.2 MPa on all faces of a bump to over

12 MPa, say, on the top, and nearly zero on the lee face.

All rocks contain microcracks, fractions of a millimeter to a few

millimeters in length, and the stress variations resulting from these water

pressure ﬂuctuations can lead to propagation of tensile fractures at the

tips of favorably oriented cracks (Grifﬁth, 1924). This can occur even

at stresses well below the experimentally determined tensile strength of

the rock (Atkinson and Rawlings, 1981; Atkinson, 1984;Segall, 1984).

The likelihood of crack growth increases when the water pressure within

cracks remains elevated while that in an adjacent cavity drops, or when

stress corrosion resulting from repeated pressure changes reduces the

strength of the rock (Iverson, 1991). Even higher and more concentrated

stress differences can result when a cobble or boulder is dragged over a

bump by the ice. Thus, it now seems safe to conclude that even sound

crystalline rocks can be fractured subglacially through the action of ice

and water, despite the fact that the ice is weaker than the underlying

rock.

Boulders with smooth stoss faces and plucked lee faces that are

embedded in till, called bullet boulders, provide convincing ﬁeld evi-

dence for subglacial fracture (see, for example, Sharp, 1982). These boul-

ders must have been transported by the ice and become lodged in the till as

the basal ice melted. They would not have had their characteristic shape

prior to lodgement, nor could they have been transported to their present

location, intact, if they had pre-existing fractures. Thus their shape

must have been produced by the overriding glacier after they became

lodged.

Once a block of rock is isolated by fractures, bed-parallel forces

tending to slide it out of position must exceed frictional forces tending

to hold it in place in order for entrainment to occur (Iverson, 1989). Both

the bed-parallel and the frictional forces are affected by ﬂuctuations in

water pressure. As noted earlier (Figure 7.6), pressure-release freezing

may occur on tops of bumps when increases in water pressure in cavities

transfer part of the weight of a glacier away from the bumps (Robin,

1976), and similar cold patches can also develop owing to simple ﬂow

of the ice from the stoss side of a bump to its crest. Both processes

increase the drag exerted by the ice on the block. The latter process

is more effective at higher sliding velocities, so increases in subglacial

water pressure that cause increases in sliding speed should increase its

effectiveness.

Frictional forces resisting dislodgement of loosened blocks are

reduced as water pressures rise (Iverson, 1989). This is because the

normal pressure that ice exerts on a bedrock surface upglacier from a

248 Water ﬂow in and under glaciers

1700

1500

1300

1100

Elevation, masl

3.53.02.01.00

Distance from head, km

Riegel

Bergschrund

Crevasse

zones

Ver tical exaggeration 2x

w.e.l.

Main over-

deepening

Figure 8.33. Longitudinal section of Storglaci

aren, Sweden, approximately

along a ﬂowline showing cirque, overdeepened basins, water-input points

(crevasse zones and bergschrund), and inferred locations of quarrying (indicated

by ). Here w.e.l. = water equivalent line; circles (o) show heights of water

in boreholes. (Modiﬁed from Hooke, 1991, Figure 2.)

cavity is reduced, thus decreasing the friction along fractures that bound

loosened blocks. In addition, once fractures are well-developed and in

hydraulic communication with cavities, increases in water pressure in

the fractures themselves reduce the effective pressure across fracture

surfaces.

In summary, it appears that steady ﬂow and low effective pressures

can fracture weaker rocks or rocks with macroscopic ﬂaws. However,

ﬂuctuations in subglacial water pressure and associated transient changes

in glacier sliding speed facilitate quarrying, particularly of more resis-

tant lithologies. Abrupt reductions in water pressure promote subglacial

fracture while increases, whether rapid or more gradual, promote the

dislodgement of loosened blocks.

Origin of cirques and overdeepenings

Cirques and overdeepened basins in glacier beds, such as those in Figure

8.33, are similar in form. Both have steep headwalls and both tend to

have beds with adverse slopes. We will discuss the headwalls ﬁrst.

Headwalls have ragged surfaces, apparently resulting from frac-

ture and removal of blocks of rock. This morphology suggests that

they are eroded by glacial quarrying. As we have just discussed, quar-

rying appears to be a result of water-pressure ﬂuctuations on time

scales of hours to days. These ﬂuctuations seem to be most pronounced

close to areas of water input (Hooke, 1991). In the case of cirques,

the water input is localized by the bergschrund, and in the case of

overdeepenings, by crevasses that form over the convexities at their heads

Origin of cirques and overdeepenings 249

(Figure 8.33). Thus, these water inputs and resulting pressure ﬂuctua-

tions occur at precisely the points where erosion is necessary to maintain

the headwalls.

In the case of the headwall of an overdeepening, a positive-feedback

process appears to be operating. Crevassing over a minor convexity in

the bed, an initial perturbation, localizes water input and hence erosion.

The erosion is concentrated on the downglacier side of the convexity.

Thus, as the erosion progresses, the convexity is ampliﬁed, resulting in

further crevassing.

The other deﬁning characteristic of cirques and overdeepenings is

the gentle adverse slope of their beds. The steepness of this slope may

be limited by the ability of water to ﬂow along the bed. For example, if

k = 0.309 and ρ = 916 kg m

−3

in Equation (8.12), it is easy to show (by

using Equation (8.5)) that

m = 0when dz/ds =−1.7 dH/ds. (This is

left as an exercise for the reader. Note that the constant of proportionality

is lower if the water is partially or completely saturated with air and a

higher value of k is thus used.) In other words, when the adverse bed

slope, dz/ds,is1.7 times the surface slope, all of the energy dissipated

in the ﬂowing water is needed to warm the water to keep it at the pressure

melting point as the ice thins in the downglacier direction, and none is

available to melt more ice (see Equation (8.10) and discussion of Figure

8.11c). Where the adverse slope is steeper, Equation (8.12) predicts,

mathematically, that

m should become negative, so water should freeze

in the conduit, and Equation (8.24) then predicts that P

> P

. Indeed,

measured water pressures in the main overdeepening of Storglaci¨aren

are quite close to the overburden pressure (Figure 8.33).

Actually, ﬁeld data suggest that the water, rather than remaining in the

conduit, spreads out in a maze of linked water pockets with ﬂow velocities

that are too low to move signiﬁcant amounts of sediment (Figure 8.22).

If the adverse slope is steep enough, frazil ice (platelets of ice that form

in the ﬂowing water) may form and further inhibit ﬂow. Because the

subglacial drainage system is thus disrupted in overdeepenings, water is

forced to follow englacial conduits through them (Hooke and Pohjola,

1994).

Where subglacial streams are thus not available to ﬂush out the

products of erosion, a layer of till must accumulate. Substantial amounts

of this sediment can be entrained if frazil ice forms and is eventually

incorporated into the glacier sole (Lawson et al., 1998). Continuity con-

siderations suggest that the till layer will increase in thickness until the

downglacier mass transfer by deformation within it and entrainment by

frazil ice at its surface equals sediment production by erosion. Such a

sediment layer would protect the bed throughout the downglacier reaches

250 Water ﬂow in and under glaciers

of an overdeepening, thus concentrating erosion at its head. This is prob-

ably why overdeepenings exist, and why their longitudinal proﬁles are

characteristically asymmetrical with the deepest point at their upglacier

ends (Hooke, 1991).

Summary

In this chapter we have investigated the glacier hydraulic system, start-

ing with the vein network along lines of intersection among three ice

crystals and progressing through the englacial to the subglacial system.

We found that theoretically water ﬂow in the englacial system should be

normal to equipotential surfaces that dip upglacier at an angle roughly

11 times the slope of the glacier surface. Along the bed, water ﬂow

is normal to the intersections of these equipotential surfaces with the

bed.

Water moving through conduit systems in or beneath glaciers

releases viscous energy. This energy melts ice on conduit walls. However,

the pressure in the water is generally less than that in the surrounding

ice, so the conduits close by creep. In the steady state, the water pressure

is everywhere adjusted so that u =

m. Along the bed u is inhibited by

drag. In addition, melting is concentrated low on the walls when tunnels

are not full of water. Thus, the steady-state conduit shape is likely to be

broad and low.

The water pressure in conduits increases upglacier, approximately

in proportion to the increase in ice thickness over the conduit. In conduit

systems consisting of tunnels, P

decreases as Q increases, but in sys-

tems consisting of cavities in the lee of bedrock steps linked by oriﬁces,

increases as Q increases. Thus, tunnel systems are believed to be

arborescent, while linked-cavity systems are distributed. In all probabil-

ity, these are end members of a continuum of drainage system types.

Tunnel systems are likely to collapse when discharges are low, tun-

nels are shrinking, and water pressures are increasing. If the tunnel sys-

tem cannot regenerate when discharges increase again, water pressures

may rise high enough to initiate a surge.

In conduits between a glacier and a deformable bed, the steady-state

condition is one in which, in addition to u =

m, erosion of sediment by the

ﬂowing water must balance the ﬂux of sediment into the conduit. Under

these conditions, conduit systems are likely to be distributed. Analysis

suggests that this condition is most likely when glacier surface slopes

are low and water pressures high, as is commonly the case beneath ice

sheets. Beneath valley glaciers, water pressures are typically lower, so

bed deformation is not as likely.