Hooke R.L. Principles of glacier mechanics

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Appendix

Problems

Chapter 3

3.1. Determine the changes in b

, R, and T

that would, if they occurred alone,

result in a 100 m increase in equilibrium line altitude. Assume a 120 d melt

season and a lapse rate of −0.007

◦

−1

Chapter 4

4.1. Determine the activation energy for creep for the following two sets of data.

Data set 1 Data set 2

˙ε,a

−1

◦

C˙ε,a

−1

◦

18.65 −5.5 1.33 −30.4

9.06 −9.9 0.0047 −61.0

All experiments were run at the same stress. Express the activation energy in

kJ mol

−1

4.2. (a) The temperature dependence of ice creep can be represented by an

Arrhenius-type relation:

˙ε





−

Rθ

By differentiating this with respect to θ

and expressing the result in terms

of differentials, determine the fractional change in ˙ε, d ˙ε/˙ε, due to a

change, dθ

,inθ

(b) In a laboratory experiment run at a temperature of −15

◦

C, what would be

the approximate percentage variation in ˙ε if the temperature were allowed

to vary by 0.5

◦

C? Use Q = 79 kJ mol

−1

4.3. Demonstrate analytically that the temperature dependence of ice creep can be

reasonably approximated by:

˙ε = ˙ε

kθ

where θ is the temperature in degrees Celsius and ˙ε

is the strain rate at 0

◦

What are reasonable values of k for different temperatures?

4.4. Calculate and plot the variation of crevasse depth with tensile stress up to a

tensile stress of 0.2 MPa.

391

392 Problems

Chapter 5

5.1. Calculate the difference between the surface velocity and the bed velocity

in a glacier 300 m thick with a surface slope of 0.046. Use n = 3

and B = 0.141 MPa a

1/n

. Use:

the inﬁnitely wide approximation,

an approximation based on Raymond’s estimate of the appropriate shape

factor for Athabasca Glacier (0.58), and

the semicircular approximation.

Which result comes closest to the values measured by Raymond on Athabasca

Glacier, and why?

5.2. At the equilibrium line on the Barnes Ice Cap Trilateration Net, the surface

velocity is 6.7 m a

−1

, the ice is 185 m thick, and the surface slope is 0.07.

Using B = 0.317 MPa a

1/n

(appropriate for ice at about −5

◦

C) and n = 3,

calculate and plot a velocity proﬁle through the ice cap. What is the basal

velocity. Is your result consistent with the above ice temperature?

5.3. An inﬁnitely wide glacier has a velocity of 1 m a

−1

at the surface and

0.7ma

−1

at a depth of 16 m. Determine the thickness of the glacier. Assume

= 0 and n = 3.

5.4. (a) An ice sheet has a surface proﬁle given by h =

√

cx,where h is the height

in meters and x is the distance from the margin, also in meters.

Differentiate this to obtain an expression for the surface slope, S.By

inserting this in the expression for the basal drag, τ = ρghS, show that τ is

independent of x. Obtain a numerical value for τ if c = 16 m.

(b) Let the ablation rate be b

−1

.Byequating the discharge through any

cross section to the volume of ice lost by melting downglacier from that

cross section, show that the average horizontal velocity in the ablation zone

is: u = 

√

x where  = b

√

c. (Note that  is negative because b

negative. Thus, u is negative, consistent with the fact that it is in the

−x-direction.)

quarries a cobble from the lee slope of the bump. The ice closes under the

cobble, so at the start of its journey to the margin it is 0.5 m above the bed.

Determine the x and z coordinates (z vertical) of the point where the cobble

will melt out, and its time en route. Plot its path. Assume plug ﬂow and

incompressibility. Use b

= 0.6ma

−1

and c = 16 m, and assume the

ablation zone is 2 km wide. Hint: use the incompressibility condition,

du/dx=−dw/dz, and the result from Problem 5.4b to get w(x).

Then use the deﬁnition of velocity, u = dx/dt, and the initial condition,

x(t = 0) = x

,tointegrate the expression for u to get

x(t) = (x

1/2

t)

. Then use w = dz/dt, and the initial condition,

z(0) = z

to obtain z(t) = z

/x)

1/2

(d) The accumulation zone in the above problem is 10 km wide (Figure P1). At

the end of the Pleistocene a mammoth dies 500 m from the divide.

Determine the x and z coordinates of the point where he melts out, and his

Chapter 6 393

Accumulation zone

Ablation

zone

10 km 2 km

Divide Equilibrium line Margin

Figure P1.

time en route. Assume the glacier has had a balanced budget for the last

10 000 a, and that the ablation rate is 0.6 m a

−1

over the 2 km wide ablation

zone, as before. Plot the path. Hint:doasinProblems 5.4b and 5.4c,

remembering that the horizontal velocity is now u =−b

(L − x)/

√

cx,

where L is the distance from the margin to the divide. You will encounter

an integral



√

x/(L − x) dx which may be transformed using r = L − x

and then evaluated using the tabulated integral:



√

L −r

dr = 2



(L − r ) + L



√



(L − r ) −

√



(L − r ) +

√



(e) If the mammoth was 3 m long and, when he died, he was lying down with

his tail 3 m closer to the divide than his head, determine the time required

for his body to pass completely beneath the equilibrium line, and his

approximate length when he is at this point in his journey.

Chapter 6

6.1. Calculate and plot a temperature proﬁle for an ice sheet that is 1368 m thick,

assuming that θ

=−24

◦

C, b

= 0.35 m a

−1

, κ = 37.2 m

−1

, u = 0, and

=−0.0228

◦

−1

. Obtain temperatures at least at 0, 200, 500, and

900 m above the bed.

6.2. (a) Determine the inﬂuence of strain heating on a temperature proﬁle by

integrating the energy balance equation, simpliﬁed with the use of the

following assumptions:

• horizontal temperature gradients are negligible,

• κ = constant,

• steady state,

• zero accumulation (or w

= 0), and

•˙ε = (σ/B)

Note: this is easier if the z-axis points downward. If you retain a z-axis

pointing upward, the sign of β

below must be changed.

(b) Plot the proﬁle for a glacier that is 1000 m thick with:

=−35

◦

C ρ = 900 kg m

−3

α = 0.01

= 0.0228

◦

−1

K = 7.1 × 10

−1

−1 ◦

−1

n = 3

B = 0.397 MPa a

1/3

394 Problems

To see the effect, you will need to make calculations at about 25 m

intervals between 900 and 1000 m depth, and you will need to use a long

temperature axis.

6.3. Solve Problem 6.2 but with the additional assumption that strain heating is

negligible, and calculate the temperature at the base of this same 1000 m thick

glacier. How much does strain heating increase the basal temperature?

6.4. (a) To get a sense of the inﬂuence of longitudinal advection, calculate

and plot a temperature proﬁle for the glacier in Problem 6.1, assuming

u = 15ma

−1

. Use the Column model with the values of θ

, b

, κ, H, and

given in Problem 6.1, the value of K in Problem 6.2, α =−0.01,

λ =−0.01

◦

−1

, and w

= 0ma

−1

(b) Compare the result with that from Problem 6.1 in detail.

6.5. (a) Obtain an expression for the temperature gradient and the temperature

distribution in a stagnant sheet of ice of inﬁnite horizontal extent, and

thickness H. Assume that the climate has been warming at a rate of

θ, and

that the interior of the glacier is warming at the same rate.

(b) By examining the original differential equation after simpliﬁcation, explain

how the uniform warming rate is accomplished.

Chapter 8

8.1. Water ﬂowing along a glacier bed must warm up as the ice thins and the

pressure melting point increases. Water ﬂowing up an adverse bed slope must

warm up more rapidly, as the ice is thinning more rapidly. The energy needed

to warm the water comes from viscous dissipation. Determine how steep the

bed slope can get, relative to the surface slope, without exceeding the amount

of viscous energy available. Obtain a numerical value for the constant of

proportionality between the two slopes.

8.2. The discharge in a horizontal subglacial conduit with a circular cross section is

0.025 m

−1

. The water pressure in the conduit is 1.5 MPa and the hydrostatic

pressure in the adjacent ice is 2.0 MPa. The Manning roughness of the conduit

is 0.1 m

−1/3

s and the viscosity parameter, B,is0.16 MPa a

1/3

. Determine the

pressure gradient in the conduit, the radius of the conduit, the water velocity in

the conduit, and the melt rate on the conduit walls (or closure rate).

8.3. An esker splits as shown in Figure P2. Stratigraphic relations suggest that the

branch around the end of the ridge is younger. Explain why the esker changed

course, and estimate the basal shear stress at the time of the change in course.

Assume that the glacier had a parabolic proﬁle, h =

√

cx. Assume further that

water ﬂow down the potential gradient could be maintained even though some

water might be forced to refreeze to keep the temperature at the pressure

melting point.

8.4. Consider a glacier with a parabolic proﬁle, h =

√

16x ,where x is the

horizontal coordinate in meters and h is the surface elevation. Assume that the

glacier is 2 km long and is on a horizontal bed. It is drained by a circular

conduit at the bed. Calculate and plot the height of the hydraulic grade line as a

function of distance from the terminus for discharges of 0.015 m

−1

,awinter

Chapter 9 395

05 km

Contour interval 10 m

Esker

Ice

flow

Moraine

Outwash

fans

Figure P2.

discharge, and 1.0 m

−1

,asummer discharge. Use a channel Manning

roughness of 0.1 m

−1/3

s and ice viscosity parameter, B = 0.06 MPa a

1/3

Assume that the conduit is at atmospheric pressure within 50 m of the margin.

(As the integration has to be carried out numerically, you might want to write a

short program to do the calculations.)

Chapter 9

9.1. Use Equation (9.2.) for σ

in terms of σ

, σ

, and θ to do the following.

(a) Determine the angle θ of the planes on which σ

is a maximum.

(b) Determine the orientation of these planes relative to those on which σ

is a

maximum.

,onthe plane on which σ

is a maximum.

(d) Determine the magnitude of σ

Smax

Express all answers in terms of σ

, σ

, and σ

9.2. Show that J

and J

in three dimensions (Equations 9.8a) reduce to

Equations (9.5)intwo dimensions.

9.3. We have shown (Chapter 6) that

˙ε

is the total work done per

unit time in a unit volume of ice subjected to simple shear. It is also true that

˙ε



is the work done by a normal stress. Thus the total work done is

W =

˙ε



. Show that because ˙ε

= λσ



, W = ˙ε



9.4. A laboratory ice deformation experiment is run using biaxial compression with

applied stresses σ

and σ

on the faces of a cube. Stresses in the third direction

are atmospheric. Strain rates are ˙ε

and ˙ε

in the σ

and σ

directions.

Determine the effective stress and the effective strain rate.

396 Problems

9.5. An experimental system is designed to run tests in combined uniaxial

compression and simple shear (Figure P3). Determine the effective stress and

effective strain rate for this stress conﬁguration.

Figure P3.

9.6. The third invariant of the stress tensor, J

, can be interpreted in terms of the

stress conﬁguration. To do this, we deﬁne a stress conﬁguration parameter, ξ,

by ξ = J

∗

. Here, J

∗

is a normalized value of J

obtained by dividing all of the

stresses by a constant factor, c,which you will derive below. The motivation

for doing this is that it simpliﬁes the expression for ξ.Weproceed as follows.

The octahedral shear stress is deﬁned by: σ



. Let us normalize σ

by dividing all stresses by a constant factor, c, thus: σ

∗

= (1/3c

)(σ



(Here, the

∗

is used to indicate the normalized value.) Let us further select c

such that σ

∗

= 2

1/6

. Obtain an expression for c in terms of the second

invariant of the stress tensor, J

. Because the normalized stresses must retain

the sign of the original stresses, use |c| where necessary.

Show the following.

(a) The deviatoric stresses in uniaxial compression under a compressive stress,

, are: (−σ

/3, −σ

/3, 2σ

/3), and (remembering that σ

is negative) that

ξ =−1 for this case.

(b) The deviatoric stresses in pure shear under stresses −σ

, σ

, are: (−σ

,0,

), and that ξ = 0 for this case.

ξ in terms of σ

and σ

, and evaluate this for:

=−0.1MPa,σ

= 0.1MPa, and

=−0.1MPa,σ

= 0MPa.

Chapter 10

10.1. Calculate and plot velocity proﬁles for an inﬁnitely wide glacier that is 300 m

thick with a surface slope of 0.046. Use n = 3 and B = 0.141 MPa a

1/n

Calculate one proﬁle for ˙ε

= 0.0 and one for ˙ε

= 0.1 a

−1

. Assume

a sliding velocity, u

,of20ma

−1

. Explain the difference between the

proﬁles.

Chapter 11

11.1. Calculate a temperature proﬁle in the ablation area of a glacier in a place

where the ice is 500 m thick, the vertical velocity at the surface, w,is

0.25 m a

−1

(upward), β

is −0.017

◦

−1

, and the temperature at the surface

is −20

◦

C. Assume that w decreases linearly with depth.

11.2. Using Equation (11.9b) and an initial condition in which θ =−20

◦

Catthe

surface, varies linearly to −10

◦

Catadepth of 4 m, and is then independent

of depth to a depth of 15 m, calculate a temperature proﬁle for the end of June

(t =

year). Use κ = 16 m

−1

and a sinusoidal variation in temperature

Chapter 13 397

with time at the surface, with θ

= 18

◦

C. The initial condition applies on

January 1 (t = 0). Compare your solution to the analytical solution given by

Equation (6.31).

Chapter 12

12.1. For comparison of borehole deformation rates, with shear stress σ

and shear

strain rate ˙ε

, with deformation rates in uniaxial compression, the stress in

uniaxial compression must be multiplied by 1/

√

3 and the strain rate by

√

3/2 (Nye, 1953). Show that this is true by calculating σ

and ˙ε

for the two

stress conﬁgurations.

12.2. Starting with Equation (12.48), verify Equation (12.49). To do this it is

necessary to use only one of Equations (12.48) (i.e. for i = 1).

12.3. Starting with ˙ε = (σ/B)

and ˙ε

= λσ



, show that the constants n and B in

Glen’s ﬂow law can be evaluated if one can determine only one component of

the stress tensor, the corresponding component of the strain rate tensor, and ˙ε

at two or more places. Here n and B are determined either algebraically when

there are only two points, or by plotting 1/λ against ˙ε and drawing a straight

line through the points. Explain.

12.4. Determine n and B in MPa a

1/n

for the following two sets of velocity

derivatives.

Depth Velocity xyz

9.4 m u −0.004 45 0.000 00 −0.014 17

v 0.000 00 0.000 13 0.003 61

w 0.000 99 −0.000 40 0.004 32

12.5 m u −0.004 14 0.000 00 −0.022 96

v 0.000 00 0.000 12 0.003 93

w 0.000 99 −0.004 68 0.004 02

The surface slope is 0.152. Depths are in meters and velocities in m a

−1

. The

y-axis is transverse and the z-axis is normal to the surface and directed

downward. Data are from the Barnes Ice Cap boreholes.

12.5. Verify that the vertically downward forces on the top of the semi-cylinder of

ice in Figure 12.6 balance the vertically upward forces on the bed when the

pressure in the ice is P and the pressure in the conduit is P.

Chapter 13

13.1. Verify Equation (13.5).

13.2. Calculate

for the mammoth in Problem 5.4e.

13.3. Verify that Equation (13.13) follows from Equation (13.12) and that

Equation (13.14)isasolution to Equation (13.13).

398 Problems

Chapter 14

14.1. We found (Equation (5.7)) that one could obtain a ﬁrst-order estimate of the

surface velocity (assuming no sliding) from:

n + 1



ρg sin α



n+1

where H is the ice thickness and α is the surface slope. By expressing this in

differential form and using the approximation α

∼

sin α (

∼

tan α), show that:

= n

dα

+ (n + 1)

(P14.1)

14.2. Use Figure 14.13 to do the following.

(a) Compare the observed change in velocity of Nisqually Glacier from 1949

to 1951 with that calculated from Equation (P14.1) above. Use h ≈ 80 m.

(b) Estimate the change in ﬂux, q, and use Equation (P14.1)toestimate the

fraction of the change that is due to the change in thickness and that due

to the change in surface slope, respectively. Then calculate c

and D

from their deﬁnitions, expressed in ﬁnite difference form. (Note that q, c

and D

are deﬁned in terms of a unit width.)

14.3 Consider a perturbation, b

= 0.1ma

−1

on Storglaci¨aren, which is 2.8 km

long and has a maximum velocity, u

0max

≈ 35ma

−1

. Calculate  and the

response time based only on the time needed to accumulate the additional

mass. Use the J´ohannesson et al.’s model generalized to three dimensions. For

this you will need to know the geometry of the glacier (Figures 8.13 and

12.9), and you will need:

glacier area = 3.03 × 10

width at terminus, W (

) = 400 m

ablation rate at terminus = 1.3ma

−1

(W(

)islarger than shown in Figure 12.9 because the topography is such that

the width would expand signiﬁcantly as the glacier thickened here.) Assume

that the average ice thickness over a cross section is ∼

the thickness shown

on the proﬁle. (Remember that V =

h W  because W > W (

).)

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