Hooke R.L. Principles of glacier mechanics

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Analysis of borehole-deformation data 341

this. Expanding Equation (12.44) for i = x and dividing by dλ yields



∂u

∂x

+ 

∂u

∂y

+ 

∂u

∂z

dλ



dλ + u

∂

∂x

+ ν

∂

∂y

+ w

∂

∂z

∂

∂t

(12.46)

Because the inclination of the casing is a function of z alone, ∂

/∂x = 0

and ∂

/∂y =0. Expanding the right-hand side of Equation (12.45), using

the result to replace the term involving D(dλ)/Dt in Equation (12.46),

and rearranging terms, we obtain:



∂u

∂x

+ 

∂u

∂y

+ 

∂u

∂z

− 



∂u

∂x

− 



∂u

∂y

− 



∂u

∂z

− 



∂v

∂x

− 



∂v

∂y

− 



∂v

∂z

− 



∂w

∂x

− 



∂w

∂y

− 



∂w

∂z

∂

∂z

∂

∂t

(12.47)

Using the summation convention, this can be written as



∂u

∂x

− 



∂u

∂x

= w

∂

∂z

∂

∂t

or even more compactly as

(δ

− 



)

∂u

∂x

= w

∂

∂z

∂

∂t

(12.48)

Because i is not repeated in any of the terms in Equation (12.48), this

equation represents three separate equations (for i = x,y,z). However,

only two of these equations are independent because only two of the

direction cosines are independent.

If the inclination of a borehole is known at two separate times, and

if seven of the nine velocity derivatives in Equation (12.47) can be mea-

sured or estimated, Equations (12.48) can be solved for the two remain-

ing velocity derivatives. Equations (12.48) are exact, but approximations

have to be made in calculating the 

, ∂

/∂t, and ∂

/∂z from observa-

tional data that are obtained at discrete points in time and space.

Two alternative approaches taken to this problem in two separate ﬁeld

experiments on Barnes Ice Cap (Hooke, 1973b; Hooke and Hanson,

1986) are outlined in Table 12.2. Strain nets were placed around the

tops of the boreholes, so that some of the velocity derivatives could be

measured directly at the surface. Assumptions were then made about

how they varied with depth. In the ﬁrst experiment, the boreholes were

closely spaced so ∂u/∂x could be determined, as a function of depth,

from the successive borehole proﬁles. As can be seen from Table 12.2,

the two velocity derivatives that were calculated were ∂u/∂z and ∂v/∂z.

One might expect that measurements of the rate of tilting of the borehole

would give these velocity derivatives directly, but this is not the case. Yet,

342 Applications of stress and deformation principles

Table 12.2. Calculation of velocity derivatives in borehole deformation studies of Hooke

(1973b) and Hooke and Hanson (1986)

Derivative 1973 1986

∂u/∂x Two boreholes We assumed that this

decreased with depth in

proportion to the decrease in

u with depth

∂u/∂y 0 (assumed) This was obtained from the

measured u at the surface

and the radius of curvature

of the ﬂow line

∂v/∂x 0 (assumed) ˙ε

was measured at the

surface and assumed to

decrease with depth in

proportion to the decrease in

u with depth. Then, from

Equation (9.21):

∂v/∂x = 2˙ε

– ∂u/∂y

∂v/∂y This was measured at the surface, and we assumed that it decreased with

depth in proportion to the decrease in ∂u/∂x

∂w/∂z =−∂u/∂x – ∂v/∂y by continuity Same as 1973

∂w/∂z was then integrated over depth to obtain w as a function of depth,

using either a no slip boundary condition at the bed where temperatures are

well below the melting point, or the measured w at the surface

∂w/∂x Two boreholes This was measured at the

surface; we assumed that it

decreased with depth in

proportion to the decrease in

w with depth

∂w/∂y We set ˙ε



∂ν

∂z

∂w

∂y



the surface, and let ∂w/∂y

decrease linearly with depth.

∂v/∂z was calculated (see

below), so an iterative procedure

was required

Same as ∂w/∂x

∂u/∂z,

∂v/∂z

These derivatives were then

calculated from Equations

(12.48)

Same as 1973

Analysis of borehole-deformation data 343

∆ t

Figure 12.16. Effect of

vertical advection on borehole

inclination.

as one would expect and as implied by our opening discussion, ∂u/∂z is

one of the most important velocity derivatives.

Sensitivity studies suggest that the solutions obtained in these two

Barnes Ice Cap experiments do not depend strongly on the assumptions.

The most important term is ∂

/∂t.Ininstances where the casing bends

abruptly, as at joints, w ∂

/∂z also becomes important. In experiments on

other glaciers, the results might be more sensitive to some of the other

velocity derivatives, and hence to any assumptions made in obtaining

them.

Further insight into Equation (12.48) can be achieved by consider-

ing the situation in plane strain. Assuming incompressible ﬂow and a

uniform longitudinal strain rate, r,wethen have ∂u/∂x =−∂w/∂z = r,

∂w/∂x = 0, 

= sin θ, 

= cos θ, and 

= 0, where θ is the inclination

of the borehole from the vertical. Equation (12.48) then reduces to

∂u

∂z

∂

∂t

tan θ − 2r tan θ + w

∂

∂z

tan θ (12.49)

The ﬁrst term on the right is the obvious one, involving a change in

inclination of the borehole with time. The second is the one illustrated

in Figure 12.12 and discussed earlier. The third is an advection effect.

In an area of non-zero vertical velocity, a section of a borehole at depth

, measured with respect to some constant datum, and with inclination



) will, at the end of a time interval t,beat, say, depth z

(Figure

12.16). If the initial inclination of the borehole at depth z

was different

from 

), our measurements would show that the inclination at depth

had changed, and this would be true even if ∂u/∂z were 0. This is why

w ∂

/∂z becomes important near some joints, as just mentioned.

The results of the borehole deformation experiment reported by

Hooke and Hanson (1986) will be used to illustrate an application of

this analysis. Four boreholes, located approximately along a ﬂowline on

Barnes Ice Cap (Figure 12.17), were drilled and cased and inclinom-

etry data were obtained from them over a period of up to four years.

Figure 12.18a shows the deformation proﬁles, and Figure 12.18b shows

values of ∂u/∂z calculated from Equations (12.48).

The deformation proﬁles in most of the holes end at the top of a zone

of white ice (Figure 12.18a). Oxygen isotope data demonstrate that this

ice is of Pleistocene age (Hooke and Clausen, 1982). The ice is white

344 Applications of stress and deformation principles

Pleistocene bubbly white glacier ice

Holocene deformed superimposed ice

Holocene glacier ice

Distance from divide, km

Weakly-oriented

Broad single-maximum

Multiple maximum

Strong single maximum

Vertical exaggeration, 5x

Small

circle

T020

T061

T081

T0975

Equilibrium line

510

Two maximum

Figure 12.17. Longitudinal section along a ﬂowline on Barnes Ice Cap showing

types of ice encountered in boreholes. The deformed superimposed ice near the

margin was overridden during an advance of the glacier (see Figure 5.17). (After

Hooke and Hanson, 1986, Figure 2. Reproduced with the kind permission of

Elsevier Science.)

because it contains a lot of air bubbles. As a result of these bubbles,

the density of this ice is only 870 kg m

−3

, compared with a density of

920 kg m

−3

in the overlying blue ice. We presume that the high concen-

tration of air bubbles is a result of two processes.

1. When the climate warmed at the end of the Pleistocene, meltwater

percolation increased, and ice lenses formed. These lenses trapped air

in the underlying porous ﬁrn.

2. As basal meltwater escaped into the underlying permeable bedrock,

air may have been left behind in a sort of physical fractionation pro-

cess.

As noted in Chapter 11 (p. 311), it is commonly found that such

Pleistocene ice is softer than Holocene ice, apparently because impurities

lead to smaller grain sizes that then develop strong single-maximum

fabrics (Paterson, 1991). The high strain rates implied by the dotted

extrapolations of the deformation proﬁles for holes T061 and T081 in

Figure 12.18a are indicative of this weakening. The value of B obtained

for this ice from the deformation proﬁle in hole T0975 is 0.1 MPa a

1/3

(at −10.1

◦

C), which is much lower than those ranging from 0.23 to

0.30 MPa a

1/3

in the overlying blue Holocene ice in holes T061 and

T020 (Table 12.3) and also much lower than other experimental values

at this temperature (Figure 12.5).

Also of interest are the values of the parameter , deﬁned by (see

Equation (9.29))

2 =



˙ε

n−1

(12.50)

Analysis of borehole-deformation data 345

Height above bed, m

400

300

200

100

02468

Velocity, m a

−1

T020

Bot

T0975

T081

(a)

T061

Height above bed, m

(b)

400

300

200

100

−0.06 −0.02 −0.12 −0.006 0.0

∂u/∂z

T0975

T081

T020

T061

Bot = hole bottom

Figure 12.18. (a) Velocity

proﬁles in boreholes; and

(b) ∂u/∂z as a function of

depth. (After Hooke and

Hanson, 1986, Figure 3.

Reproduced with the kind

permission of Elsevier

Science.)

˙ε

is obtained from the velocity derivatives (Table 12.2) using Equation

(9.21), while σ

is estimated with the use of

=−ρghα −

∂

∂x



˙ε

n−1

(2˙ε

+ ˙ε

)dz − T (12.51)

which is derived from Equation (12.26)inmuch the same way that

we derived Equation (12.29)except that we now retain the T term and

also assume that changes in the transverse direction are negligible in an

346 Applications of stress and deformation principles

Table 12.3. Values of B in MPa a

−1/3

for different ice types

Borehole T0975 T081 T061 T020

Ice type

Weakly oriented — — 0.23 (−10.2) —

Broad single maximum

— 0.46 (−9.0)

0.26 (−8.4) 0.30 (−7.4)

Two maxima — 0.44 (−8.6) 0.24 (−7.5) 0.26 (−6.5)

Three or four maxima — 0.50 (−8.3) 0.30 (−6.8) —

White ice (clean) 0.10 (−10.1) 0.18

(−7.8) 0.10

(−6.4) —

0.18

(−7.8)

White ice (dirty) 0.13 (−9.8)

Values given are for zones in which fabric is well developed, and thus exclude transition regions.

Equivalent fabric in T020 is small circle.

Numbers in brackets are mean temperatures in

◦

Velocity proﬁle calculated by assuming no slip on the bed.

Measured over two weeks, starting three weeks after completion of hole in 1977. No smoothing

used in calculation.

ice cap. Equation (12.27)was used to evaluate the T term. If B and n

are constant, as might be expected,  should vary inversely with ˙ε. The

awkward fact is that near the surface where ˙ε

is low, this does not appear

to be the case. Figure 12.19a shows that  is effectively independent of

˙ε.Even the direction of change of  with depth is not consistent from

one hole to the next, as indicated by the arrows on the curves in Figure

12.19a. This problem is not unique to Barnes Ice Cap; Raymond (1973)

also found that  was independent of ˙ε near the surface of Athabasca

Glacier.

Somewhat deeper in the glacier the situation improves, and 

decreases steadily with increasing ˙ε (Figure 12.19b). Here, the slope

and intercept of the log  –log ˙ε line can be used to determine B and

n (Equation (12.50)). In the present case, however, Hooke and Hanson

(1986) chose, instead, to assume that n = 3; they then calculated B as

a function of depth, and related changes in B to changes in crystallo-

graphic fabric. The results are shown in Table 12.3. Although there was

quite a lot of noise in the record, it appears that B is slightly lower in

fabrics containing only two maxima, and increases in fabrics with three

or four maxima. This stiffening can be seen at the bottoms of the defor-

mation proﬁles in Figure 12.18a.Itisconsistent with expectation, as it is

the third and fourth maxima in these multiple-maximum fabrics that are

inclined to the direction of shear (Figure 4.14f). In other words, the basal

planes of crystals with these orientations dip either up- or downglacier,

Analysis of borehole-deformation data 347

-3

-4

T081

Λ, MPa

Octahedral strain rate,

-1

(a)

(b)

Λ, MPa

-3

-4

Octahedral strain rate,

-1

T061

T020

T081

T061

T020

T061

T0975

(Scale to right)

Figure 12.19. Octahedral shear strain rate, ˙ε

, plotted against . (a) The

upper 50 m of holes T061 and T081, and the upper 150 m of hole T020. Arrows

show direction of increasing depth. (b) The lower parts of the holes. Depth

increases from upper left, following lines of points. Reversals in trends reﬂect

hardening of ice in zones where fabric is changing. (After Hooke and Hanson,

1986, Figure 4. Reproduced with the kind permission of Elsevier Science.)

348 Applications of stress and deformation principles

whereas the basal planes in the ﬁrst two maxima that form dip in the

transverse direction (Figure 4.14e).

Values of B in hole T081 are nearly double those in the other holes.

Hooke and Hanson assumed that this was because stresses at this location

on the glacier were overestimated. However, they were unable to isolate

the apparently erroneous assumption that led to this error, even though

they undertook calculations with a ﬁnite-element model.

Summary

In this chapter we have studied four classical problems in glacier mechan-

ics: closure of cylindrical holes, calculation of force balances, creep of

ice shelves, and deformation of boreholes. As examples of applications

of the theory presented, we discussed problems such as the ﬂow of water

to, from, and in subglacial conduits, the mechanics of glacier accelera-

tions, the stability of ice shelves, and the extraction of ﬂow-law param-

eters from borehole deformation data. From these examples, we gained

insights into the dynamic and kinematic behavior of glaciers. These,

however, were secondary objectives.

The primary objective of the chapter was to help students develop

facility with the mathematics of stress and deformation as applied to

problems in glacier mechanics. Such analyses are complicated because

multiple stresses, strains, and strain rates are involved, and even more

so because the strains in which we are interested are a consequence of

deviatoric, not total stresses. In many cases, once the physics of a prob-

lem have been formulated, prescribed mathematical procedures must

be followed before a result with clear physical signiﬁcance reappears.

Students who have mastered the material in this chapter will be able to

understand many papers in the glaciological literature that would other-

wise be impenetrable.

Chapter 13

Finite strain and the origin of foliation

Suppose one were to drop a deformable sphere into the accumulation

area of a glacier. As the sphere became buried deeper and deeper in the

glacier, it would be deformed by the ﬂow. Because the accumulation area

is normally an area of longitudinal extension and vertical compression,

it would become an ellipsoid, elongated in the direction of ﬂow and com-

pressed vertically. If there were also signiﬁcant transverse compression

it would become a prolate ellipsoid, whereas if there were transverse

extension, it would become oblate. Figure 13.1 illustrates schematically,

in two dimensions, how the shape of a cross section through the sphere

would change as the sphere passed through the glacier.

Our objective in this chapter is to discuss the cumulative deforma-

tion experienced by ice as it is advected through a glacier, and to show

the relation between this deformation and the banded structure, called

foliation, that is characteristic of glaciers.

The strain ellipse

The ellipsoid we have just discussed is called the strain ellipsoid or, in

two dimensions, the strain ellipse. Following normal convention, we will

denote the greatest, intermediate, and shortest principal semi-axes of the

ellipsoid by subscripts 1, 2, and 3, and the directions of these axes by X,

Y, and Z, respectively. These axes will rotate with respect to our ﬁxed x,

y, z coordinate system as the strain ellipsoid rotates (Figure 13.2). The

lengths of the axes of the ellipsoid are a measure of the strain it has

experienced. As in Equation (9.16)wedeﬁne the strain by:

e =

 − 



− 1 (13.1)

349

350 Finite strain and the origin of foliation

Equilibrium

line

Figure 13.1. Schematic illustration of the deformation that would be

experienced by an object that is initially spherical (or circular in two dimensions)

as it moves through a glacier from the accumulation zone to the ablation zone.

(From Hooke and Hudleston, 1978, Figure 3B. Reproduced with permission of

the International Glaciological Society.)

Figure 13.2. Sketch showing

parameters used to describe,

in two dimensions, the

cumulative strain experienced

by an initially circular object.

(Modiﬁed from Hooke and

Hudleston, 1980, Figure 8C.)

If 

, the radius of the initial sphere, is taken as 1, then 

= 1 + e

is the length of the ith semi-axis of the ellipsoid (Figure 13.2). From

Equation (9.23) the logarithmic strain is then:

= ln



= ln(1 + e

) (13.2)

Let us now take 

/

(i, j = 1, 2, 3 and i = j)asameasure of the

strain in the i, j plane. Then

− ε

= ln



(13.3)