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Principles of Glacier Mechanics
Second Edition

This book provides students and practicing glaciologists with the tools they need
to understand modern glaciology. Relatively simple concepts are introduced
first, followed by mathematically more sophisticated chapters. A knowledge of
basic calculus is assumed, but important equations describing physical processes
are developed from elementary principles. Emphasis is placed on connections
between modern research in glaciology and the origin of features of glacial land-
scapes. Student exercises are included. This new edition builds on the successful
first edition: it has been completely updated, and important new sections and
whole chapters have been added. Principles of Glacier Mechanics is designed to
be used as a primary textbook in upper division and graduate courses in glacio-
logy, and can be used as either a primary or supplementary text in courses in
glacial geology. Practicing glacial geologists and glaciologists will also find it
useful as a reference book.

R     L  B. H     is Research Professor in the Department of Earth Sci-

ences and the Climate Change Institute, University of Maine. He has been
involved in glaciological research for over 30 years, focusing on processes rele-
vant to the origin of glacial landforms. In addition to the first edition of Principles
of Glacier Mechanics, he has published over 80 refereed research papers in jour-
nals such as the Geological Society of America Bulletin, Geology, the Journal
of Glaciology, Quaternary Research, and the Journal of Geology.
Principles of Glacier Mechanics
Second Edition
Roger LeB. Hooke
Research Professor
Department of Earth Sciences
and Climate Change Institute
University of Maine, Orono
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge  , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521836098

© in the Second edition R. LeB. Hooke 2005

This book is in copyright. Subject to statutory exception and to the provision of

relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.

First published in print format 2005

- ---- eBook (NetLibrary)

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First edition © 1998 by Prentice Hall Simon and Schuster / A Viacom Company
Upper Saddle River, New Jersey 07458
 Contents ix

Effect of a frozen bed 193

Summary 194

8 Water flow in and under glaciers:

geomorphic implications 197
The upper part of the englacial hydraulic system 197
Equipotential surfaces in a glacier 201
Melt rates in conduits 205
Water pressures in subglacial conduits on hard
beds 208
Types of subglacial drainage system 215
Surges 230
Subglacial drainage paths and the formation of
eskers 232
Tunnel valleys 241
Water pressure and glacier quarrying 244
Origin of cirques and overdeepenings 248
Summary 250

9 Stress and deformation 252

Stress 252
Momentum balance 261
Deformation 262
Condition that principal axes of stress and strain
rate coincide 267
Summary 269

10 Stress and velocity distribution in an

idealized glacier 271
Solutions for stresses and velocities in plane
strain 271
Comparison with real glaciers 286
Summary 287

11 Numerical modeling 288

Goals of modeling 289
Numerical integration 289
Finite-difference models 291
Finite-element models 298
Initial conditions and forcing 299
Validation 301
It is with a deep sense of gratitude that I dedicate this book to those who,
at various times through the formative stages of my life, guided me into
the most exciting and rewarding career I can imagine: the study of our
Earth.
To my parents, who opened many doors for me;
to my older brother, Richard, who led me through a door leading to
the wilderness;
to John Muir who opened my eyes to the spirituality in wilderness;
to my wife, Ann, who introduced me to Geology;
to John P. Miller who focused my attention on processes at the Earth’s
surface; and
to Robert P. Sharp who taught me that basic physical principles could
be used to understand these processes.
Contents

Preface to the first edition page xi

Preface to the second edition xiii
Physical constants relevant to ice xiv
Derived SI units and conversion factors xvii

1 Why study glaciers? 1

2 Some basic concepts 5

A note on units and coordinate axes 5
Glacier size, shape, and temperature 6
The condition of incompressibility 9
Stresses, strains, and strain rates 10

3 Mass balance 17
The transformation of snow to ice 18
Snow stratigraphy 20
Mass balance principles 23
Climatic causes of mass balance fluctuations 26
The budget gradient 29
Other modes of ice loss from valley glaciers 31
Mass balance of polar ice sheets 34
Effect of atmospheric circulation patterns on
mass balance 37
Global mass balance 40
Summary 41

4 Flow and fracture of a crystalline material 43

Crystal structure of ice 43
Dislocations 44
Rate-limiting processes 48
Internal stresses 53
Recrystallization 54
Deformation mechanism maps 63

vii
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viii Contents

A flow law for glacier ice 66

Fracture 70
Summary 74

5 The velocity field in a glacier 76

Measurement of velocity 77
Balance velocity 78
Shear stress distribution 79
Horizontal velocity at depth in an ice sheet 81
Horizontal velocity in a valley glacier 83
Mean horizontal velocity and ice flux 87
Vertical velocity 88
Submergence and emergence velocities 91
Flow field 92
Transverse profiles of surface elevation on a
valley glacier 94
Radar stratigraphy 96
Effect of drifting snow on the velocity field 98
Ice streams 105
Summary 110

6 Temperature distribution in polar ice sheets 112

Energy balance in an ice sheet 112
Dependence of K on temperature 117
The steady-state temperature profile at the
center of an ice sheet 117
Temperature profiles in the ablation zone 127
Temperature profiles near the surface of an ice
sheet 127
Temperature distributions far from a divide 131
Englacial and basal temperatures along a flow
line calculated using the Column model 135
Basal temperatures in Antarctica – comparison
of solutions using the Column model and a
numerical model 138
Geomorphic implications 142
Summary 144

7 The coupling between a glacier and its bed 147

Sliding 148
Deformation of subglacial till 168
Stability of ice streams 190
x Contents

Intercomparison of models 301

Sensitivity testing and tuning 302
Coupling thermal and mechanical models 303
Examples 304
Summary 313

12 Applications of stress and deformation

principles to classical problems 315
Collapse of a cylindrical hole 315
Calculating basal shear stresses using a force
balance 326
Creep of floating ice shelves 333
Analysis of borehole-deformation data 338
Summary 348

13 Finite strain and the origin of foliation 349

The strain ellipse 349
Simple and pure shear 351
Parameters describing cumulative deformation 352
Calculating cumulative strain 353
Components of foliation 356
Summary 364

14 Response of glaciers to changes in mass

balance 365
Positive feedback processes 366
Response of a temperate glacier 367
Elementary kinematic wave theory 368
Analysis of the effect of a small change in mass
balance using a perturbation approach 371
Effect of diffusion 375
The problem at the terminus 376
Further study of the response time 376
Numerical modeling of glacier responses 381
Comparison with observation 383
Summary 390

Appendix Problems 391

References 399
Index 421
Preface to the first edition

One might well ask why one should write a book about so specialized a
subject as glacier mechanics when there are already other good books on
this subject written by eminent glaciologists. This book is an outgrowth
of a course that I teach to students who, in many cases, do not have
any background in continuum mechanics. Consequently, it was neces-
sary to start at a level considerably less advanced than that at which
other similar books begin, and to develop the theoretical principles one
step at a time. Thus, unlike other books on the subject and the general
scientific literature, in which space is at a premium, the steps leading
from one equation to another are, in most cases, easily seen. In addition,
qualitative interpretations of the equations are often provided to clarify
the physics behind the mathematics. Capable students with a solid back-
ground in basic physics and in differential and integral calculus, and with
some modest exposure to differential equations, will have little difficulty
understanding the concepts and derivations presented.
My goal in writing this book was not to produce a comprehensive
treatise on glacier mechanics, but rather to develop the basic foundation
upon which the modern literature on this subject rests. Thus, many topics
are not covered, or are treated in less detail than some readers might wish.
However, students who have a full appreciation for the concepts in this
book will have the background they need to understand most of the
current literature.
Beginning students in glaciology will find that this book will save
them many long hours of searching through the background literature
to clarify basic concepts. Glacial geologists and geomorphologists will
also find much of value, including applications of glacier physics to the
origin of some glacial landforms. Structural geologists and others with
interest in stress and deformation will likewise discover that glaciers
are, in fact, monomineralic rock masses that are deforming at the Earth’s
surface where they can be observed in detail. The book is, thus, appro-
priate for upper division and graduate level courses in glaciology, and
as a supplementary text for courses in glacial geology and in structural
geology.

xi
xii Preface to the first edition

In the preliminary pages, readers will find a compilation of physical

constants relevant to ice, and a list of SI units and conversion factors. A
series of problems keyed to individual chapters is also included.
The encouragement I have received in this undertaking from many
present and former students, as well as from other glaciologists, has been
a major stimulus in bringing it to completion. I trust the final product
is worthy of their confidence. The book has benefited from the critical
comments of R. W. Baker at the University of Wisconsin, River Falls;
C. R. Bentley at the University of Wisconsin, Madison; G. K. C. Clarke
at the University of British Columbia; E. M. Grace, and B. Hanson at the
University of Delaware; N. R. Iverson at the University of Minnesota; T.
Jóhannesson at the Icelandic Meteorological Office; M. Kuhn at the Uni-
versity of Innsbruck, Austria; M. F. Meier at the University of Colorado;
J. F. Nye at the University of Bristol, England; C. F. Raymond at the
University of Washington; R. L. Shreve at the University of California,
Los Angeles; J. Weertman at Northwestern University, and especially I.
Whillans at Ohio State University.

June 25, 1996

Preface to the second edition

When I wrote the preface to the first edition of this book seven years ago,
nothing was further from my mind than a second edition. The first edition
was well received, however, and on numerous occasions colleagues have
lamented the fact that it was no longer available. When Cambridge Uni-
versity Press agreed that a new edition was desirable, little did I realize
what I had gotten into.
When I told Matt Lloyd (my editor at Cambridge) that my goal was
to have the text ready by a certain time, he graciously gave me a target
date that was nearly double that time. I told him that his time schedule
was fine, but that I did not want to be held too strictly to it. As it happens,
I had an unrealistic view of the volume of new material that needed to
be sifted through, absorbed, and translated into language appropriate for
the upper-division undergraduate and graduate-level students for whom
this book is written. As with the first edition, my goal is not to provide an
encyclopedia of research in glaciology, as other books do that well, but
rather to give students the basic background they will need to understand
the modern literature. At the same time, the book has proven to be a
useful reference for professionals who don’t keep all of the equations
and conversion factors stored for instant recall. I myself use it for that
purpose frequently.
I am indebted to many who have encouraged me in this undertak-
ing, and especially to those who have generously given their time to
review new sections or entire chapters, who have resurrected archived
computer files to provide images or data files from which new fig-
ures were produced, or who have made new calculations especially
for this volume. The following have assisted me in this effort: Richard
Alley, Bob Bindschadler, Ginny Catania, Chris Clark, Lee Clayton, Paul
Cutler, Gordon Hamilton, Brian Hanson, Bruce Hooke, Peter Hudleston,
Kolumbian Hutter, Philippe Huybrechts, Neal Iverson, Peter Jansson,
Susan Kaspari, Katie Leonard, Paul Mayewski, Shawn Marshall, Howard
Mooers, Nadine Nereson, Felix Ng, Charlie Raymond, Vandy Spikes,
Slawek Tulaczyk, and Joe Walder.

xiii
Physical constants relevant to ice

Symbol Parameter Value

g Acceleration of gravity 9.81 m s−2
ρ Density of bubble-free ice 916 kg m−3
ρ Density of water at 0 ◦ C 999.84 kg m−3
θm Melting point at atmospheric pressure 0.0 ◦ C
273.15 K

C Heat capacity 2093 J kg−1 K−1

(For temperatures above ∼−0.5 ◦ C the
effective heat capacity increases owing
to the presence of a liquid phase, the
amount of which depends upon the
concentration of chemical impurities.)
Cw Heat capacity of air-free water at constant 256.9 J kg−1 K−1
pressure
C Depression of the melting point due to
pressure
Pure ice and air-free water 0.074 K MPa−1
Pure ice and air-saturated water 0.098 K MPa−1
(Harrison, 1972)
ζ Depression of the melting point due to 1.86 ◦ C kg mol−1
solutes
L Heat of fusion 3.34 × 105 J kg−1
K Thermal conductivity at −1 ◦ C 7.1 × 107 J m−1 a−1 K−1
K varies with temperature, thus:
K = 7.10 × 107 − 1.96 × 105 θ =
3.63 × 103 θ 2
where θ is in degrees Celsius (a
negative number) (Ratcliffe, 1962)

xiv
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