Hooke R.L. Principles of glacier mechanics

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Examples 311

climate model tuned with ice core records. With an enhancement factor

(Equation (4.10)) of 6 and without basal sliding, the model simulates

the modern Greenland Ice Sheet well, and is in close agreement with

other EISMINT models of Greenland. However, when it is then used

to model the Laurentide Ice Sheet, it produces an LGM ice mass with

avolume of 36.3 × 10

,whereas studies of moraines, LGM sea

level, and post-glacial isostatic recovery suggest that the actual volume

was only ∼22.5 × 10

.Totry to bring the calculated volume into

better agreement with that observed, Marshall et al. ﬁrst added a calv-

ing routine. With reasonable calving rates, this reduced the volume only

modestly. Higher calving rates led to inconsistencies with the known

ice sheet extent in Hudson Bay and Hudson Strait. They then added a

sliding routine but did not retain the calving algorithm. This reduced

the volume to 29.1 × 10

and resulted in an ice sheet extent

that was in reasonable agreement with observations of glacial geolo-

gists (Figure 11.9). A further reduction in calculated ice volume could

have been achieved by increasing the enhancement factor or by reducing

precipitation in the climate forcing, but these options were not explored

quantitatively.

The enhancement factor that Marshall et al. used for Greenland

probably reﬂects: (1) development of strong single-maximum crystal

fabrics, and (2) sliding, which occurs in Greenland but is not allowed

in their model. Paterson (1991) has argued that the former is facili-

tated by impurities because impurities inhibit grain-boundary migration,

resulting in smaller crystals, and smaller crystals recrystallize readily,

leading to strong preferred orientations. Pleistocene ice tends to have

more impurities, smaller grain sizes, and higher strain rates. The softness

of Pleistocene ice was apparently ﬁrst recognized in borehole deforma-

tion studies on Barnes Ice Cap (Hooke, 1973b) and later in similar studies

on Devon (Paterson, 1977) and Agassiz (Fisher and Koerner, 1986) Ice

Caps and then in Greenland (Dahl-Jensen and Gundestrup, 1987). During

the Holocene the basal layers of Pleistocene ice in these four ice masses

have thinned, so they now have less inﬂuence on the proﬁle than was

likely to have been the case during the Wisconsinan in the Laurentide Ice

Sheet. Thus, it is reasonable to expect that the appropriate enhancement

factor for the Laurentide Ice Sheet would be higher than that for Green-

land today. Other modelers have also found that relatively high enhance-

ment factors were required to model the accepted volume of the Lau-

rentide Ice Sheet (Huybrechts and T’Siobbel, 1995;Tarasov and Peltier,

1999).

In conclusion, the Laurentide Ice Sheet can be modeled successfully

if algorithms for sliding and calving are included, if Pleistocene ice is

312 Numerical modeling

2000

1600

2000

1600

2000

1200

2000

1800

2600

2000

Contour intervals on ice sheet:

400 m below 1600 m

200 m above 1600 m

Figure 11.9. Surface topography of the Laurentide Ice Sheet at 20 ka

calculated with a thermomechanical model that included a basal sliding routine.

(Modiﬁed from Marshall et al., 2000, Figure 9b. Used with permission of the

authors and the Canadian Journal of Earth Sciences.)

assumed to have been softer than Holocene ice, and if the Pleistocene

climate is assumed to have been somewhat drier than the Holocene cli-

mate. Marshall et al.(2000)donot pursue combinations of these effects,

arguing sensibly that the physics of the calving and sliding processes are

not known well enough. Sliding, for example, depends on pore water

pressures in the substrate, and pore water pressures, in turn, depend

Summary 313

upon basal temperatures and the efﬁciency of the basal drainage system.

Models of the basal drainage system are even more tenuous than those

of calving and, as noted above, basal temperature distributions are not

modeled consistently.

In short, we can choose between viewing the glass as half empty or

half full. We have a long way to go to write calving and sliding algorithms

that are well-supported by physics. On the other hand, the models seem

to have already revealed important characteristics of the Laurentide Ice

Sheet and of ice in it.

Summary

In this chapter we have reviewed the two main numerical modeling tech-

niques commonly used in glaciology: the ﬁnite-difference and ﬁnite-

element techniques. For simple problems, ﬁnite-difference approaches

can be implemented with a spread sheet or simple computer program.

Finite-element approaches and ﬁnite-difference solutions of more com-

plex problems both require more sophisticated computer programs.

While numerical details of these advanced techniques are beyond the

scope of this book, we have introduced some of the vocabulary com-

monly associated with them.

Owing to their complexity, numerical models can easily have ﬂaws

that result in realistic but incorrect predictions. Parts of models can often

be tested against analytical solutions before the model as a whole is

applied to problems without such solutions. Output of models of entire

ice sheets can be compared with the EISMINT benchmarks. The lat-

ter tests have shown that models generally do well at predicting the

size and shape of an ice sheet, but relatively subtle differences among

the models result in signiﬁcant differences in solutions for basal temper-

ature distribution, and in particular in the fraction of the bed that is at

the pressure melting point.

In the last part of the chapter, we explored three applications of

models to problems of glaciological and geomorphological signiﬁcance.

In the ﬁrst, a ﬁnite-element model was used to study the stress distribution

in a calving face in order to see if a physical explanation could be found

for the commonly cited proportionality between water depth and calving

rate. In the second, another ﬁnite-element model predicted that when

an ice sheet advances over permafrost, a frozen margin of appreciable

width is likely to develop and to persist for centuries or millennia. Certain

landforms have been interpreted as being a consequence of such a frozen

margin. Finally, we discussed a ﬁnite-difference model of the Laurentide

Ice Sheet and found that with certain reasonable assumptions relatively

314 Numerical modeling

good agreement probably could be obtained between the size calculated

by the model and that determined from moraines, sea level records, and

isostatic rebound. The results point to the need for a better quantitative

understanding of sliding, calving, and the development of basal drainage

systems, and of climate history, while at the same time they constrain

the ranges of these parameters that will yield agreement between model

and probable reality.

Chapter 12

Applications of stress and deformation

principles to classical problems

In this chapter, we will study some glaciologically signiﬁcant problems

for which an appreciation of the material presented in Chapters 9 and 10

is required. Our objective is not to provide a comprehensive overview of

theoretical developments in glaciology, but rather to solidify the gains

made in these preceding two chapters by applying the principles devel-

oped therein. In the course of this discussion, the student will be intro-

duced to some deﬁnitive studies, frequently referenced in the glaciolog-

ical literature.

Let us ﬁrst consider the problem of closure of a cylindrical borehole,

in part because this is relevant to our earlier discussion of glacier hydrol-

ogy. Then we will investigate efforts to calculate basal shear stresses

using a force balance model, followed by study of the creep of ice shelves.

Finally, the problem of using borehole deformation experiments to obtain

estimates of the values of the parameters in the ﬂow law will round out

the chapter.

Collapse of a cylindrical hole

The ﬁrst problem we address is that of the closure of a cylindrical hole

in ice. This problem was studied by Nye (1953)inthe context of using

closure rates of tunnels in ice to estimate the constants in Glen’s ﬂow law,

and our development is based on Nye’s paper. More recently, the theory

has been used to analyze two problems in water ﬂow at the base of a

glacier: (1) the closure of a water conduit, and (2) leakage of water into

or away from a subglacial conduit. We used the ﬁrst of these analyses in

Chapter 8 (Equation (8.3)).

315

316 Applications of stress and deformation principles

Figure 12.1. Stresses on the

wall of a cylindrical hole in a

weightless medium.

Our approach is very similar to that used in Chapter 10 to obtain

stresses and velocities in a “glacier” consisting of a slab of ice resting

on a bed with a uniform slope. We ﬁrst reduce the problem to one of

plane strain by setting it up so that there is no deformation parallel to

the axis of the hole. This reduces the number of unknowns from nine –

the three components of the velocity vector and six components of the

stress tensor – to ﬁve. Expressions for the stresses and strain rates are

then obtained and inserted into the momentum and continuity equations.

Finally, we use a constitutive relation between stress and stain rate, Glen’s

ﬂow law, to obtain the desired solution.

Consider, ﬁrst, the closure of a hole of radius a in an inﬁnite weight-

less medium (Figure 12.1). There is no strain parallel to its axis, the

z-direction. Once the solution to this problem is obtained, we will mod-

ify it to apply to a real glacier. On the surface of the hole, at r = a,an

internal tension, σ

,is(somehow) applied. Eventually, this stress will be

equated with that resulting from the difference between the pressure in

the ice and that in the hole.

As there is no deformation in the z-direction, ˙ε

= λσ



= 0. Thus,



= 0 = σ

−

(

+ σ

θθ

+ σ

)

(

+ σ

θθ

)

(12.1)

The mean stress thus becomes

P =



(

+ σ

θθ

)





(

− σ

θθ

)

(12.2)

and



θθ

=−

(

− σ

θθ

)

(12.3)

This gives us expressions for the three deviatoric stresses and the mean

stress.

Collapse of a cylindrical hole 317

As stated, there is no deformation in the z-direction and, in addition,

the hole wall cannot support shear tractions. Thus, in view of the radial

symmetry, there can be no shear stresses in the θ - and z-directions in the

medium away from the hole. Therefore, σ

, σ

θθ

, and σ

are principal

stresses. The effective stress is then



(

− σ

θθ

)



σ =

− σ

θθ

(12.4)

Let us now obtain another relation between σ

and σ by considering

rdq

dq/2

Figure 12.2. Stresses on a

segment of the wall of a

cylindrical hole.

the condition for stress equilibrium. From Figure 12.2,wehave

(rdθ) −



dσ



(

r + dr

)

dθ + 2σ

θθ

dθ

= 0

Canceling like terms of opposite sign, dividing by dr dθ , and ignoring

the term still containing a differential yields

dσ

+ σ

− σ

θθ

= 0

or, by using Equation (12.4)

dσ

2σ

= 0 (12.5)

At any radius r ≥ a the stress is σ

.Atinﬁnity σ

= θ . Thus, Equation

(12.5)may be integrated:



dσ

=−

∞



2σ

∞



2σ

dr (12.6)

In order to integrate Equation (12.6), we must express r in terms

of σ .Wewill do this by determining the velocity ﬁeld, and hence ˙ε,

and inserting a ﬂow law. Let u be the velocity in the radial direction.

Other velocities vanish owing to the radial symmetry and the absence

of deformation in the z-direction. Conservation of mass requires that the

mass ﬂux through two concentric cylindrical surfaces with radii r and

r + dr (Figure 12.3) must be the same in an incompressible medium.

Thus, we have

2ur = 2



u +



(r + dr)

318 Applications of stress and deformation principles

∂u

∂r

Figure 12.3. Radial velocity

ﬁeld around a cylindrical hole.

which may be simpliﬁed, thus:

=−

(12.7)

and integrated to yield

ln u =−ln r + ln c

Because we obtained this without referring to the direction of u,aminus

sign is now inserted to indicate that u is in the negative r-direction.

Thus,

u =−

So u is directed toward the hole and is a maximum on the hole wall, at

r = a.Itdecreases to 0 at r =∞.

From its deﬁnition in terms of velocity derivatives (Equations

(9.21)), we see that ˙ε

= ∂u/∂r, so from Equation (12.7):

˙ε

=−

and by continuity, since ˙ε

= 0,

˙ε

θθ

=−˙ε

The effective strain rate is thus:

˙ε





As u =−c/r, this becomes:

˙ε =

(12.8)

We can retain the generality of the solution a little longer before

inserting a ﬂow law. Because ˙ε = f (σ ), we have

f (σ ) =

(12.9)

Collapse of a cylindrical hole 319

To obtain r in terms of σ , note that (using Equation (12.9)):

df(σ )

dσ

df(σ )

dσ

=−

dσ

Again using Equation (12.9)toeliminate c:

df(σ )

dσ

=−

f (σ )

dσ

so:

dr =−

df(σ )

f (σ )

Substituting this into Equation (12.6) yields

=−



2σ

df(σ )

f (σ )

=−



f (σ )

df(σ ) (12.10)

It is now necessary to make use of a ﬂow law, speciﬁcally Glen’s

ﬂow law, to obtain an analytical expression for the functional relation in

Equation (12.9), thus:

˙ε = f (σ ) =





(12.11)

whence, differentiating:

df(σ ) =

nσ

n−1

dσ

With the use of these last two relations, Equation (12.10) becomes



(

σ/B

)

nσ

n−1

dσ =



ndσ

so,

= nσ (12.12)

In other words, the radial stress at any distance r ≥a from the hole wall is

simply n times the effective stress. One cannot help but be impressed by

the simplicity and elegance of this result, considering the effort required

to obtain it. Unfortunately, real life is rarely so conveniently uncom-

plicated. Furthermore, we still have some way to go before obtaining

relations that can be applied to real glaciers.

In preparation for relaxing the assumption that the medium is weight-

less, let us now scale this solution to the normal stress, σ

,onthe hole

wall. Using the last equality in Equation (12.11)toobtain an expression

for σ , σ

and σ

become:

= nB





1/n

and σ

= nB





1/n

whence:





2/n

(12.13)

320 Applications of stress and deformation principles

Solutions for the remaining stresses can now be written similarly, as

follows. From Equations (12.4) and (12.12):

σ =

(

− σ

θθ

)

(

nσ − σ

θθ

)

whence, transposing and again using the last equality in Equation

(12.11):

θθ

(

n − 2

)

σ =

(

n − 2

)





1/n

so:

θθ

n − 2





2/n

(12.14a)

Then, from Equations (12.1), (12.13), and (12.14a):



θθ









2/n

n − 2





2/n









2/n



1 +

n − 2



n − 1





2/n

(12.15)

Finally, from Equations (12.4 ), (12.13), and (12.14a):



−

θθ









2/n

−

n − 2





2/n



Thus,





2/n

(12.16)

We can now relax the assumption that the medium is weightless.

Suppose we have a horizontal hole at atmospheric pressure at a depth

in a real glacier. The hydrostatic pressure in the glacier is P = ρgh,

and around the hole it is P = ρgh

. Note that P is not equal to the

mean stress, P(=

). If h

 a, P will be nearly uniform around the

hole. We have not previously speciﬁed the magnitude of σ

,solet us

now solve Equations (12.13) through (12.16) for σ

(i = r, θ, z) with

= P. Furthermore, because P is hydrostatic, let us add a compres-

sive stress, −P,tothe solutions. This is valid because a hydrostatic

pressure inﬂuences all of the stresses equally, and therefore does not

affect the local differences among the stresses given by Equations (12.13)