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Geophysical Continua presents a systematic treatment of deformation in the Earth from
seismic to geologic time scales, and demonstrates the linkages between different aspects of
the Earth™s interior that are often treated separately.
A unified treatment of solids and fluids is developed to include thermodynamics and
electrodynamics, in order to cover the full range of tools needed to understand the interior of
the globe. A close link is made between microscopic and macroscopic properties manifested
through elastic, viscoelastic and fluid rheologies, and their influence on deformation.
Following a treatment of geological deformation, a global perspective is taken on
lithospheric and mantle properties, seismology, mantle convection, the core and Earth™s
dynamo. The emphasis throughout the book is on relating geophysical observations to
interpretations of earth processes. Physical principles and mathematical descriptions are
developed that can be applied to a broad spectrum of geodynamic problems.
Incorporating illustrative examples and an introduction to modern computational
techniques, this textbook is designed for graduate-level courses in geophysics and
geodynamics. It is also a useful reference for practising Earth Scientists. Supporting
resources for this book, including exercises and full-colour versions of figures, are available
at www.cambridge.org/9780521865531.

B R I A N K E N N E T T is Director and Distinguished Professor of Seismology at the Research
School of Earth Sciences in The Australian National University. Professor Kennett™s
research interests are directed towards understanding the structure of the Earth through
seismological observations. He is the recipient of the 2006 Murchison Medal of the
Geological Society of London, and the 2007 Gutenberg Medal of the European Geosciences
Union, and he is a Fellow of the Royal Society of London. Professor Kennett is the author of
three other books for Cambridge University Press: Seismic Wave Propagation in Stratified
Media (1983), The Seismic Wavefield: Introduction and Theoretical Development (2001),
and The Seismic Wavefield: Interpretation of Seismograms on Regional and Global Scales
(2002).
H A N S -P E T E R B U N G E is Professor and Chair of Geophysics at the Department of Earth and
Environmental Sciences, University of Munich, and is Head of the Munich Geo-Center.
Prior to his Munich appointment, he spent 5 years on the faculty at Princeton University.
Professor Bunge™s research interests lie in the application of high performance computing to
problems of Earth and planetary evolution, including core, mantle and lithospheric
dynamics. A member of the Bavarian Academy of Sciences, Bunge is also President of the
Geodynamics Division of the European Geosciences Union (EGU).
Geophysical Continua
Deformation in the Earth™s Interior




B.L.N. KENNETT
Research School of Earth Sciences, The Australian National University

H.-P. BUNGE
Department of Geosciences, Ludwig Maximilians University, Munich
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521865531

© B. L. N. Kennett and H.-P. Bunge 2008


This publication 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 2008


ISBN-13 978-0-511-40890-8 eBook (EBL)

ISBN-13 978-0-521-86553-1 hardback




Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents




1 Introduction page 1
1.1 Continuum properties 2
1.1.1 Deformation and strain 3
1.1.2 The stress-¬eld 3
1.1.3 Constitutive relations 3
1.2 Earth processes 6
1.3 Elements of Earth structure 8
1.3.1 Mantle 12
1.3.2 Core 13
1.4 State of the Earth 14

PART I: CONTINUUM MECHANICS IN GEOPHYSICS 19
2 Description of Deformation 21
2.1 Geometry of deformation 21
2.1.1 Deformation of a vector element 23
2.1.2 Successive deformations 24
2.1.3 Deformation of an element of volume 24
2.1.4 Deformation of an element of area 25
2.1.5 Homogeneous deformation 25
2.2 Strain 27
2.2.1 Stretch 27
2.2.2 Principal ¬bres and principal stretches 28
2.2.3 The decomposition theorem 29
2.2.4 Pure rotation 30
2.2.5 Tensor measures of strain 32
2.3 Plane deformation 34
2.4 Motion 36
2.5 The continuity equation 38
2.A Appendix: Properties of the deformation gradient determinant 39

v
vi Contents

3 The Stress-Field Concept 41
3.1 Traction and stress 41
3.2 Local equations of linear motion 44
3.2.1 Symmetry of the stress tensor 44
3.2.2 Stress jumps (continuity conditions) 46
3.3 Principal basis for stress 48
3.4 Virtual work rate principle 51
3.5 Stress from a Lagrangian viewpoint 53
4 Constitutive Relations 54
4.1 Constitutive relation requirements 54
4.1.1 Simple materials 55
4.1.2 Material symmetry 56
4.1.3 Functional dependence 57
4.2 Energy balance 57
4.3 Elastic materials 60
4.4 Isotropic elastic material 61
4.4.1 Effect of rotation 61
4.4.2 Coaxiality of the Cauchy stress tensor and the Eulerian triad 62
4.4.3 Principal stresses 62
4.4.4 Some isotropic work functions 63
4.5 Fluids 64
4.6 Viscoelasticity 67
4.7 Plasticity and ¬‚ow 69
5 Linearised Elasticity and Viscoelasticity 71
5.1 Linearisation of deformation 71
5.2 The elastic constitutive relation 72
5.2.1 Isotropic response 73
5.2.2 Nature of moduli 73
5.2.3 Interrelations between moduli 74
5.2.4 An example of linearisation 74
5.2.5 Elastic constants 75
5.2.6 The uniqueness Theorem 76
5.3 Integral representations 79
5.3.1 The reciprocal Theorem 80
5.3.2 The representation Theorem 81
5.4 Elastic Waves 83
5.4.1 Isotropic media 83
5.4.2 Green™s tensor for isotropic media 85
5.4.3 Interfaces 86
5.5 Linear viscoelasticity 88
5.6 Viscoelastic behaviour 91
5.7 Damping of harmonic oscillations 92
Contents vii

6 Continua under Pressure 95
6.1 Effect of radial strati¬cation 95
6.1.1 Hydrostatic pressure 96
6.1.2 Thermodynamic relations 97
6.2 Finite strain deformation 100
6.3 Expansion of Helmholtz free energy and equations of state 102
6.4 Incremental stress and strain 105
6.4.1 Perturbations in stress 106
6.4.2 Perturbations in boundary conditions 107
6.5 Elasticity under pressure 107
7 Fluid Flow 110
7.1 The Navier“Stokes equation 110
7.1.1 Heat ¬‚ow 111
7.1.2 The Prandtl number 112
7.2 Non-dimensional quantities 113
7.2.1 The Reynolds number 115
7.2.2 Stokes Flow 115
7.2.3 Compressibility 115
7.2.4 The P´ clet number
e 117
7.3 Rectilinear shear ¬‚ow 117
7.4 Plane two-dimensional ¬‚ow 118
7.5 Thermal convection 121
7.5.1 The Rayleigh and Nusselt numbers 121
7.5.2 The Boussinesq approximation 121
7.5.3 Onset of convection 122
7.5.4 Styles of convection 125
7.6 The effects of rotation 126
7.6.1 Rapid rotation 127
7.6.2 The Rossby and Ekman numbers 128
7.6.3 Geostrophic ¬‚ow 128
7.6.4 The Taylor“Proudman theorem 129
7.6.5 Ekman layers 129
8 Continuum Equations and Boundary Conditions 131
8.1 Conservation equations 131
8.1.1 Conservation of mass 132
8.1.2 Conservation of momentum 132
8.1.3 Conservation of energy 133
8.2 Interface conditions 134
8.3 Continuum electrodynamics 135
8.3.1 Maxwell™s equations 135
8.3.2 Electromagnetic constitutive equations 136
8.3.3 Electromagnetic continuity conditions 137
viii Contents

8.3.4 Energy equation for the electromagnetic ¬eld 138
8.3.5 Electromagnetic disturbances 139
8.3.6 Magnetic ¬‚uid dynamics 141
8.4 Diffusion and heat ¬‚ow 145
8.4.1 Equilibrium heat ¬‚ow 146
8.4.2 Time-varying problems 148

PART II: EARTH DEFORMATION 151
9 From the Atomic Scale to the Continuum 153
9.1 Transport properties and material defects 153
9.1.1 Grains and crystal defects 153
9.1.2 General transport properties 156
9.1.3 Atomic diffusion 157
9.2 Lattice vibrations 158
9.3 Creep and rheology 162
9.3.1 Crystal elasticity 162
9.3.2 Deformation behaviour 163
9.4 Material properties at high temperatures and pressures 166
9.4.1 Shock-wave techniques 166
9.4.2 Pressure concentration by reduction of area 168
9.5 Computational methods 171
9.5.1 Electronic structure calculations 171
9.5.2 Atomistic simulations 174
9.5.3 Simulation of crystal structures 175
9.5.4 Finite temperature 176
9.5.5 In¬‚uence of defects 178
10 Geological Deformation 180
10.1 Microfabrics 181
10.1.1 Crystal defects 181
10.1.2 Development of microstructure 182
10.1.3 Formation of crystallographically preferred orientations 184
10.2 Macroscopic structures 186
10.2.1 Multiple phases of deformation 187
10.2.2 Folding and boudinage 189
10.2.3 Fractures and faulting 193
10.2.4 Development of thrust complexes 204
11 Seismology and Earth Structure 207
11.1 Seismic Waves 207
11.1.1 Re¬‚ection and refraction 208
11.1.2 Attenuation effects 209
11.2 Seismic sources 212
11.3 Building the response of the Earth to a source 216
Contents ix

11.3.1 Displacements as a normal mode sum 218
11.3.2 Free oscillations of the Earth 220
11.4 Probing the Earth 231
11.4.1 Seismic phases 231
11.4.2 Normal mode frequencies 239
11.4.3 Comparison with observations 243
11.4.4 Imaging three-dimensional structure 247
11.5 Earthquakes and faulting 253
12 Lithospheric Deformation 257
12.1 De¬nitions of the lithosphere 257
12.2 Thermal and mechanical structure 258
12.2.1 Thermal conduction in the oceanic lithosphere 258
12.2.2 Mechanical deformation 261
12.2.3 Estimates of the elastic thickness of the lithosphere 265
12.2.4 Strength envelopes and failure criteria 266
12.3 Plate boundaries and force systems 271
12.3.1 Nature of plate boundaries 271
12.3.2 Plate boundary forces 272
12.4 Measures of stress and strain 274
12.4.1 Stress measurements 274
12.4.2 Strain measurements 276
12.5 Glacial rebound 280
12.6 Extension and convergence 283
12.6.1 Extension 283
12.6.2 Convergence 288
13 The In¬‚uence of Rheology: Asthenosphere to the Deep Mantle 294
13.1 Lithosphere and asthenosphere 294
13.1.1 Seismic imaging 295
13.1.2 Seismic attenuation 297
13.1.3 Seismic anisotropy 299
13.1.4 Asthenospheric ¬‚ow 302
13.1.5 The in¬‚uence of a low-viscosity zone 302
13.2 Subduction zones and their surroundings 308
13.2.1 Con¬guration of subduction zones 309
13.2.2 Flow around the slab 311
13.2.3 Temperatures in and around the subducting slab 314
13.2.4 Subduction and orogeny 316
13.3 The in¬‚uence of phase transitions 319
13.4 The deeper mantle 323
13.4.1 Viscosity variations in the mantle and the geoid 323
13.4.2 The lower boundary layer 328
x Contents

14 Mantle Convection 330
14.1 Convective forces 330
14.1.1 Boundary layer theory 330
14.1.2 Basic equations 332
14.1.3 Boundary conditions 333
14.1.4 Non-dimensional treatment 334
14.1.5 Computational convection 335
14.2 Convective planform 339
14.3 Thermal structure and heat budget 342
14.3.1 Thermal boundary layers and the geotherm 342
14.3.2 Plates 346
14.3.3 Hot spots and plumes 348
14.4 Circulation of the mantle 353
14.4.1 Present-day and past plate motion models 354
14.4.2 Implications of plate motion models for mantle circulation 357
14.4.3 Mantle circulation models 361
14.5 Mantle rheology 371
14.5.1 Temperature dependence 372
14.5.2 Strain dependence 373
14.6 Coupled lithosphere“mantle convection models 375
14.7 Thermochemical convection 377
15 The Core and the Earth™s Dynamo 379
15.1 The magnetic ¬eld at the surface and at the top of the core 380
15.2 Convection and dynamo action 384
15.2.1 Basic equations 384
15.2.2 Boundary conditions 387
15.2.3 Interaction of the ¬‚ow with the magnetic ¬eld 388
15.2.4 Deviations from the reference state 389
15.2.5 Non-dimensional treatment 390
15.3 Numerical dynamos 392
15.4 Evolution of the Earth™s core 397
15.4.1 Energy balance 397
15.4.2 Thermal and compositional effects 399
15.4.3 Inner core growth in a well-mixed core 400
Appendix: Table of Notation 407
Bibliography 413
Index 423
Preface




Geophysical Continua is designed to present a systematic treatment of deformation
in the Earth from seismic to geologic time scales. In this way we demonstrate the
linkages between different aspects of the Earth™s interior that are commonly treated
separately. We provide a coherent presentation of non-linear continuum mechanics
with a uniform notation, and then specialise to the needs of particular topics such
as elastic, viscoelastic and ¬‚uid behaviour. We include the concepts of continuum
thermodynamics and link to the properties of material under pressure in the deep
interior of the Earth, and also provide the continuum electrodynamics needed for
conducting ¬‚uids such as the Earth™s core.
Following an introduction to continuum methods and the structure of the Earth,
Part I of the book takes the development of continuum techniques to the level
where they can be applied to the diverse aspects of Earth structure and dynamics
in Part II. At many levels there is a close relation between microscopic properties
and macroscopic consequences such as effective rheology, and so Part II opens
with a discussion of the relation of phenomena at the atomic scale to continuum
properties. We follow this with a treatment of geological deformation at the
grain and outcrop scale. In the subsequent chapters we emphasise the physical
principles that allow understanding of Earth processes, taking a global perspective
towards lithospheric and mantle properties, seismology, mantle convection, the
core and Earth™s dynamo. We make links to experimental results and seismological
observations to provide insight into geodynamic interpretations.
The material in the book has evolved over a considerable time period and has
bene¬ted from interactions with many students in Cambridge, Canberra, Princeton
and Munich. Particular thanks go to the participants in the Geodynamics Seminar in
Munich in 2005, which helped to re¬ne Part I and the discussion of the lithosphere
in Part II.
In a work of this complexity covering many topics with their own speci¬c
notation it is dif¬cult to avoid reusing symbols. Nevertheless we have have tried to
sustain a uni¬ed notation throughout the whole book and to minimise multiple use.
We have had stimulating discussions with Jason Morgan, John Suppe and Geoff
Davies over a wide range of topics. Gerd Steinle-Neumann provided very helpful

xi
xii Preface

input on mineral properties and ab initio calculations, and Stephen Cox provided
valuable insight into the relation of continuum mechanics and structural geology.
Special thanks go to the Alexander von Humboldt Foundation for the Research
Award to Brian Kennett that led to the collaboration on this volume.


Acknowledgements
We are grateful to the many people who have gone to trouble to provide us
with ¬gures, in particular: A. Barnhoorn, G. Batt, J. Besse, C. Bina, S. Cox, J.
Dawson, E. Debayle, U. Faul, A. Fichtner, S. Fishwick, J. Fitz Gerald, E. Garnero,
A. Gorbatov, B. Goleby, O. Heibach, M. Heintz, G. Houseman, R. Holme, G.
Iaffeldano, M. Ishii, A. Jackson, I. Jackson, J. Jackson, M. Jessell, J. Kung, P.
Lorinczi, S. Micklethwaite, M. Miller, D. Mueller, A. Piazzoni, K. Priestley, M.
Sandiford, W. Spakman, B. Steinberger, J. Suppe, F. Takahashi, and K. Yoshizawa.
1
Introduction




The development of quantitative methods for the study of the Earth rests ¬rmly
on the application of physical techniques to the properties of materials without
recourse to the details of atomic level structure. This has formed the basis of
seismological methods for investigating the internal structure of the Earth, and for
modelling of mantle convection through ¬‚uid ¬‚ow. The deformation behaviour
of materials is inextricably tied to microscopic properties such as the elasticity
of individual crystals and processes such as the movement of dislocations. In
the continuum representation such microscopic behaviour is encapsulated in the
description of the rheology of the material through the connection between stress
and strain (or strain rate).
Different classes of behaviour are needed to describe the diverse aspects of the
Earth both in depth and as a function of time. For example, in the context of
the rapid passage of a seismic wave the lithosphere may behave elastically, but
under the sustained load of a major ice sheet will deform and interact with the
deeper parts of the Earth. When the ice sheet melts at the end of an ice age, the
lithosphere recovers and the pattern of post-glacial uplift can be followed through
raised beaches, as in Scandinavia.
The Earth™s core is a ¬‚uid and its motions create the internal magnetic ¬eld
of the Earth through a complex dynamo interaction between ¬‚uid ¬‚ow and
electromagnetic interactions. The changes in the magnetic ¬eld at the surface on
time scales of a few tens of years are an indirect manifestation of the activity in
the core. By contrast, the time scales for large-scale ¬‚ow in the silicate mantle are
literally geological, and have helped to frame the con¬guration of the planet as we
know it.
We can link together the many different facets of Earth behaviour through the
development of a common base of continuum mechanics before branching into the
features needed to provide a detailed description of speci¬c classes of behaviour.
We start therefore by setting the scene for the continuum representation. We then
review the structure of the Earth and the different types of mechanical behaviour
that occur in different regions, and examine some of the ways in which information

1
2 Introduction

at the microscopic level is exploited to infer the properties of the Earth through both
experimental and computational studies.


1.1 Continuum properties
A familiar example of the concept of a continuum comes from the behaviour of
¬‚uids, but we can use the same approach to describe solids, glasses and other more
general substances that have short-term elastic and long-term ¬‚uid responses. The
behaviour of such continua can then be established by using the conservation laws
for linear and angular momentum and energy, coupled to explicit descriptions of
the relationship between the stress, describing the force system within the material,
and the strain, which summarises the deformation.
We adopt the viewpoint of continuum mechanics and thus ignore all the ¬ne
detail of atomic level structure and assume that, for suf¬ciently large samples,:

• the highly discontinuous structure of real materials can be replaced by a
smoothed hypothetical continuum; and
• every portion of this continuum, however small, exhibits the macroscopic
physical properties of the bulk material.

In any branch of continuum mechanics, the ¬eld variables (such as density,
displacement, velocity) are conceptual constructs. They are taken to be de¬ned at
all points of the imagined continuum and their values are calculated via axiomatic
rules of procedure.
The continuum model breaks down over distances comparable to interatomic
spacing (in solids about 10’10 m). Nonetheless the average of a ¬eld variable
over a small but ¬nite region is meaningful. Such an average can, in principle,
be compared directly to its nominal counterpart found by experiment “ which will
itself represent an average of a kind taken over a region containing many atoms,
because of the physical size of any measuring probe.
For solids the continuum model is valid in this sense down to a scale of order
10’8 m, which is the side of a cube containing a million or so atoms.
Further, when ¬eld variables change slowly with position at a microscopic level
∼10’6 m, their averages over such volumes (10’20 m3 say) differ insigni¬cantly
from their centroidal values. In this case pointwise values can be compared directly
to observations.
Within the continuum we take the behaviour to be determined by
a) conservation of mass;
b) linear momentum balance: the rate of change of total linear momentum is equal
to the sum of the external forces; and
c) angular momentum balance.
The continuum hypothesis enables us to apply these laws on a local as well as a
global scale.
1.1 Continuum properties 3

1.1.1 Deformation and strain
If we take a solid cube and subject it to some deformation, the most obvious change
in external characteristics will be a modi¬cation of its shape. The speci¬cation of
this deformation is thus a geometrical problem, that may be carried out from two
different viewpoints:
a) with respect to the undeformed state (Lagrangian), or
b) with respect to the deformed state (Eulerian).
Locally, the mapping from the deformed to the undeformed state can be assumed
to be linear and described by a differential relation, which is a combination of pure
stretch (a rescaling of each coordinate) and a pure rotation.
The mechanical effects of the deformation are con¬ned to the stretch and it is
convenient to characterise this by a strain measure. For example, for a wire under
load the strain would be the relative extension, i.e.,

change in length
= , (1.1.1)
initial length

The generalisation of this idea requires us to introduce a strain tensor at each point
of the continuum to allow for the three-dimensional nature of deformation.



1.1.2 The stress ¬eld
Within a deformed continuum a force system acts. If we were able to cut the
continuum in the neighbourhood of a point, we would ¬nd a force acting on a cut
surface which would depend on the inclination of the surface and is not necessarily
perpendicular to the surface (Figure 1.1).


δS
n
Figure 1.1. The force vector „
acting on an internal surface
P „ speci¬ed by the vector normal n
will normally not align with n.




This force system can be described by introducing a stress tensor σ at each point,
whose components describe the loading characteristics, and from which the force
vector „ can be found for a surface with arbitary normal n.
For a loaded wire, the stress σ would just be the force per unit area.
4 Introduction

1.1.3 Constitutive relations
The speci¬cation of the stress and strain states of a body is insuf¬cient to describe
its full behaviour, we need in addition to link these two ¬elds. This is achieved by
introducing a constitutive relation, which prescribes the response of the continuum
to arbitrary loading and thus de¬nes the connection between the stress and strain
tensors for the particular material.
At best, a mathematical expression provides an approximation to the actual
behaviour of the material. But, as we shall see, we can simulate the behaviour
of a wide class of media by using different mathematical forms.
We shall assume that the forces acting at a point depend on the local geometry
of deformation and its history, and possibly also on the history of the local
temperature. This concept is termed the principle of local action, and is designed
to exclude ˜action at a distance™ for stress and strain.

Solids
Solids are a familiar part of the Earth through the behaviour of the outer layers,
which exhibit a range of behaviours depending on time scale and loading.
We can illustrate the range of behaviour with the simple case of extension of
a wire under loading. The tensile stress σ and tensile strain are then typically
related as shown in Figure 1.2.

σ
hardening



- yield
point




Figure 1.2. Behaviour of a
wire under load

µ


Elasticity
If the wire returns to its original con¬guration when the load is removed, the
behaviour is said to be elastic:
(i) linear elasticity σ = E “ usually valid for small strains;
(ii) non-linear elasticity σ = f( ) “ important for rubber-like materials, but not
signi¬cant for the Earth.
Plasticity
Once the yield point is exceeded, permanent deformation occurs and there is no
1.1 Continuum properties 5

unique stress“strain curve, but a unique dσ “ d relation. As a result of microscopic
processes the yield stress rises with increasing strain, a phenomenon known as work
hardening. Plastic ¬‚ow is important for the movement of ice, e.g., in glacier ¬‚ow.
Viscoelasticity (rate-dependent behaviour)
Materials may creep and show slow long-term deformation, e.g., plastics and
metals at elevated temperatures. Such behaviour also seems to be appropriate to
the Earth, e.g., the slow uplift of Fennoscandia in response to the removal of the
loading of the glacial ice sheets.
Elementary models of viscoelastic behaviour can be built up from two basic
building blocks: the elastic spring for which
σ=m , (1.1.2)
and the viscous dashpot for which
σ = · ™. (1.1.3)

EK
·M
EM




·K
(b)
(a)

Figure 1.3. Mechanical models for linear viscoelastic behaviour combining a spring and
viscous dashpot: (a) Maxwell model, (b) Kelvin“Voigt model.

The stress-strain relations depend on how these elements are combined.

(i) Maxwell model
The spring and dashpot are placed in series (Figure 1.3a) so that
σ = EM( ™ + /„M);
™ (1.1.4)
this allows for instantaneous elasticity and represents a crude description of a
¬‚uid. The constitutive relation can be integrated using, e.g., Laplace transform
methods and we ¬nd
t
σ(t) = EM (t) + dt (t ) exp[’(t ’ t )/„M] , (1.1.5)

so the stress state depends on the history of strain.

(ii) Kelvin“Voigt model
The spring and dashpot are placed in parallel (Figure 1.3b) and so
σ = EK( ™ + /„K), (1.1.6)
6 Introduction

which displays long-term elasticity. For the initial condition = 0 at t = 0 and
constant stress σ0, the evolution of strain in the Kelvin“Voigt model is
σ0 t
= 1 ’ exp ’ , (1.1.7)
2EK „K
so that the viscous damping is not relevant on long time scales.
More complex models can be generated, but all have the same characteristic that
the stress depends on the time history of deformation.

Fluids
The simplest constitutive equation encountered in continuum mechanics is that for
an ideal ¬‚uid, where the pressure ¬eld p is isotropic and depends on density and
temperature
σ = ’p(ρ, T), (1.1.8)
where ρ is the density, and T is the absolute temperature. If the ¬‚uid is
incompressible ρ is a constant.
The next level of complication is to allow the pressure to depend on the ¬‚ow of
the ¬‚uid. The simplest such form includes a linear dependence on strain rate ™ “ a
Newtonian viscous ¬‚uid:
σ = ’p(ρ, T) + · ™ . (1.1.9)
Further complexity can be introduced by allowing a non-linear dependence of stress
on strain rate, as may be required for the ¬‚ow of glacier ice.


1.2 Earth processes
The Earth displays a broad spectrum of continuum properties varying with both
depth and time. A dominant in¬‚uence is the effect of pressure with increasing
depth, so that properties of materials change as phase transitions in minerals
accommodate closer packed structures. Along with the pressure the temperature
increases, so we need to deal with the properties of materials at conditions that are
not simple to reproduce under laboratory conditions.
The nature of the deformation processes within the Earth depends strongly on
the frequency of excitation. At high frequencies appropriate to the passage of
seismic waves the dominant contribution is elastic, with some seismic attenuation
that can be represented with a small linear viscoelastic component. However, as
the frequency decreases and the period lengthens viscous ¬‚ow effects become
more prominent, so that elastic contributions can be ignored in the study of
mantle convection. This transition in behaviour is illustrated in Figure 1.4, and
is indicative of a very complex rheology for the interior of the Earth as different
facets of material behaviour become important in different frequency bands. The
observed behaviour re¬‚ects competing in¬‚uences at the microscopic level, and
1.2 Earth processes 7


Time scale [yr]
10 -6 10 -3 10 0 10 3 10 6 10 9

Elastic


Viscoelastic LM
Seismic Waves
UM
and
Normal Modes
Post-seismic Glacial
Deformation Rebound
Plate
Earth Tectonics
Tides Tectonic Mantle
Strain Convection
Viscous flow


10 0 10 3 10 6 10 9 10 12 10 15
Time scale [s]

Figure 1.4. Spectrum of Earth deformation processes indicating the transition from
viscoelastic to fully viscous behaviour as the frequency decreases. The upper curve refers
to the lower mantle (LM), and the lower curve to the upper mantle (UM), indicating the
differences in viscosity and deformation history.




varies signi¬cantly with depth as indicated by the two indicative curves (UM, LM)
in Figure 1.4 for the transition from near elastic behaviour to fully viscous ¬‚ow
behaviour, representing the states for the upper and lower mantle.
Further, the various classes of deformation occur over a very wide range of
spatial scales (Figure 1.5). As a result, a variety of different techniques is needed
to examine the behaviour from seismological to geodetic through to geological
observations. There is increasing overlap in seismic and space geodetic methods
for studying the processes associated with earthquake sources that has led to new
insights for fault behaviour. Some phenomena, such as the continuing recovery
of the Earth from glacial loading, can be studied using multiple techniques that
provide direct constraints on rheological properties.
Our aim is to integrate understanding of continuum properties and processes
with the nature of the Earth itself, and to show how the broad range of terrestial
phenomena can be understood within a common framework. We therefore now
turn our attention to the structure of the Earth and the classes of geodynamic and
deformation processes that shape the planet we live on.
In Part I that follows, we embark on a more detailed examination of the
development of continuum methods, in a uniform treatment encompassing solid,
¬‚uid and intermediate behaviour. Then in Part II we address speci¬c Earth issues
building on the continuum framework
8 Introduction


Length scale [km]
10 -6 10 -3 10 0 10 3
Mantle 10 9
Convection
10 15
Plate Tectonics
10 6

10 12
Glacial Rebound
10 3
Tectonic
10 9 Strain
Time scale [s]




Time scale [yr]
Core Motions 10 0

Earth Tides
10 6
Post-seismic
10 -3
Deformation

10 3
Earthquake
Displacements
10 -6

10 0
Laboratory
Experiments
10 -9
In-situ
Studies
10 -3
10 -3 10 0 10 3 10 6
Length scale [m]


Figure 1.5. Temporal and spatial spectrum of Earth deformation processes.


1.3 Elements of Earth structure
Much of our knowledge of the interior of the Earth comes from the analysis of
seismological data, notably the times of passage of seismic body waves at high
frequencies (≈ 1 Hz) and the behaviour of the free oscillations of the Earth at lower
frequencies (0.03 “ 3 mHz). Such studies provide information both on the dominant
radial variations in physical properties, and on the three-dimensional variations in
the solid parts of the Earth. Important additional constraints are provided by the
mass and moments of inertia of the Earth, which can be deduced from satellite
observations. The moments of inertia are too low for the Earth to have uniform
density, there has to be a concentration of mass towards the centre that can be
identi¬ed with the seismologically de¬ned core.
The resulting picture of the dominant structure of the Earth is presented in Figure
1.6. The ¬gure of the Earth is close to an oblate spheroid with a ¬‚attening of
0.003356. The radius to the pole is 6357 km and the equatorial radius is 6378 km,
1.3 Elements of Earth structure 9

ρ [Mg/m3 ]
β, ± [km/s]
AK135
2 4 6 8 10 12 14
0


1000


2000




Depth [km]




Depth [km]
3000
tle
an
M




4000
er
Co




5000
β ± ρ
Co ner




6000
re
In




Figure 1.6. The major divisions of the radial structure of the Earth linked to the radial
reference Earth model AK135, seismic wave speeds ± (P), β (S): Kennett et al. (1995);
density ρ: Montagner & Kennett (1996). The gradations in tone in the Earth™s mantle
indicate the presence of discontinuities at 410 and 660 km depth, and the presence of the
D near the core“mantle boundary.


but for most purposes a spherical model of the Earth with a mean radius of 6371
km is adequate. Thus reference models for internal structure in which the physical
properties depend on radius can be used. Three-dimensional variations can then be
described by deviations from a suitable reference model.
Beneath the thin crustal shell lies the silicate mantle which extends to a depth
of 2890 km. The mantle is separated from the metallic core by a major change of
material properties that has a profound effect on global seismic wave propagation.
The outer core behaves as a ¬‚uid at seismic frequencies and does not allow the
passage of shear waves, while the inner core appears to be solid.
The existence of a discontinuity at the base of the crust was found by
Mohoroviˇ i´ in the analysis of the Kupatal earthquake of 1909 from only a limited
cc
number of records from permanent seismic stations. Knowledge of crustal structure
from seismic methods has developed substantially in past decades through the use
of controlled sources, e.g., explosions. Indeed most of the information on the
oceanic crust comes from such work. The continental crust varies in thickness
from around 20 km in rift zones to 70 km under the Tibetan Plateau. Typical values
are close to 35 km. The oceanic crust is thinner, with a basalt pile about 7 km thick
whose structure changes somewhat with the age of the oceanic crust.
Earthquakes and man-made sources generate two types of seismic waves that
propagate through the Earth. The earliest arriving (P) wave has longitudinal
motion; the second (S) wave has particle motion perpendicular to the path. In
the Earth the direct P and S waves are accompanied by multiple re¬‚ections and
10 Introduction

conversions, particulary from the free surface. These additional seismic phases
follow the main arrivals, so that seismograms have a quite complex character with
many distinct arrivals. Behind the S wave a large-amplitude train of waves builds up
from surface waves trapped between the Earth™s surface and the increase in seismic
wavespeed with depth. These surface waves have dominantly S character and are
most prominent for shallow earthquakes. The variation in the properties of surface
waves with frequency provides valuable constraints on the structure of the outer
parts of the Earth.
The times of arrival of seismic phases on their different paths through the globe
constrain the variations in P and S wavespeed, and can be used to produce models
of the variation with radius. A very large volume of arrival time data from stations
around the world has been accumulated by the International Seismological Centre
and is available in digital form. This data set has been used to develop high-quality
travel-time tables, that can in turn be used to improve the locations of events. With
reprocessing of the arrival times to improve locations and the identi¬cation of the
picks for later seismic phases, a set of observations of the relation between travel
time and epicentral distance have been produced for a wide range of phases. The
reference model AK135 of Kennett et al. (1995) for both P and S wave speeds,
illustrated in ¬gure 1.6, gives a good ¬t to the travel times of mantle and core
phases. The reprocessed data set and the AK135 reference model have formed the
basis of much recent work on high-resolution travel-time tomography to determine
three-dimensional variations in seismic wavespeed.
The need for a core at depth with greatly reduced seismic wave speeds was
recognised at the end of the nineteenth century by Oldham in his analysis of the
great Assam earthquake of 1890, because of a zone without distinct P arrivals (a
˜shadow zone™ in PKP). By 1914 Gutenburg had obtained an estimate for the radius
of the core which is quite close to the current value. The presence of the inner core
was inferred by Inge Lehmann in 1932 from careful analysis of arrivals within the
shadow zone (PKiKP), which had to be re¬‚ected from some substructure within the
core.
The mantle shows considerable variation in seismic properties with depth, with
strong gradients in seismic wavespeed in the top 800 km. The presence of
distinct structure in the upper mantle was recognised by Jeffreys in the 1930™s
from the change in the slope of the travel time as a function of distance from
events near 20—¦ . Detailed analysis at seismic arrays in the late 1960s provided
evidence for signi¬cant discontinuities in the upper mantle. Subsequent studies
have demonstrated the global presence of discontinuities near 410 and 660 km
depth, but also signi¬cant variations in seismic structure within the upper mantle
(for a review see Nolet et al., 1994).
The use of the times of arrival of seismic phases enables the construction of
models for P and S wavespeed, but more information is needed to provide a
full model for Earth structure. The density distribution in the Earth has to be
1.3 Elements of Earth structure 11

inferred from indirect observations and the main constraints come from the mass
and moment of inertia. The mean density of the Earth can be reconciled with the
moment of inertia if there is a concentration of mass towards the centre of the Earth;
which can be associated with a major density jump going from the mantle into the
outer core and a smaller density contrast at the boundary between the inner and
outer cores (Bullen, 1975).

With successful observations of the free oscillations of the Earth following the
great Chilean earthquake of 1960, additional information on both the seismic wave
speeds and the density could be extracted from the frequencies of oscillation.
Fortunately the inversion of the frequencies of the free oscillations for a spherically
symmetric reference model provides independent constraints on the P wavespeed
structure in the outer core. Even with the additional information from the normal
modes the controls on the density distribution are not strong (Kennett, 1998), and
additional assumptions such as an adiabatic state in the core and lower mantle have
often been employed to produce a full model.

The reference model PREM of Dziewonski & Anderson (1981) combined the
free-oscillation and travel-time information available at the time. A parametric
representation of structure was employed in terms of simple mathematical functions
to aid the inversion; thus a single cubic was used for seismic wavespeed in the
outer core and again for most of the lower mantle. The PREM model forms the
basis of much current global seismology using quantitative exploitation of seismic
waveforms at longer periods (e.g., Dahlen & Tromp, 1998).

In order to reconcile the information derived from the free oscillations of the
Earth and the travel time of seismic phases, it is necessary to take account of the
in¬‚uence of anelastic attenuation within the Earth. A consequence of the energy
loss of seismic energy due to attenuation is a small variation in the seismic wave
speeds with frequency, so that waves with frequencies of 0.01 Hz (at the upper limit
of free-oscillation observations) travel slightly slower than the 1 Hz waves typical
of the short-period observations used in travel-time studies. The differences in
the apparent wavespeeds between travel-time analysis and free-oscillation results
thus provides constraints on the attenuation distribution with depth. The density
and attenuation model shown in ¬gure 1.6 was derived by Montagner & Kennett
(1996) to satisfy a broad set of global information with a common structure based
on the wavespeed pro¬les of the AK135 model of Kennett et al. (1995).

The process of subduction brings the cold oceanic lithosphere into the upper
mantle and locally there are large contrasts in seismic wave speeds, well imaged
by detailed seismic tomography, that extend down to at least 660 km and in some
zones even deeper. Remnant subducted material can have a signi¬cant presence in
some regions, e.g., above the 660 km discontinuity in the north-west Paci¬c and in
the zone from 660 down to 1100 km beneath Indonesia.
12 Introduction

1.3.1 Mantle
The nature of the structure of the silicate mantle varies with depth and it is
convenient to divide the mantle up into four major zones (e.g., Jackson & Ridgen,
1998)

Upper Mantle (depth z < 350 km), with a high degree of variability in seismic
wavespeed (exceeding ± 4%) and relatively strong attenuation in many
locations.
Transition Zone (350 < z < 800 km), including signi¬cant discontinuities in P
and S wavespeeds and generally high velocity gradients with depth.
Lower Mantle (800 < z < 2600 km) with a smooth variation of seismic
wavespeeds with depth that is consistent with adiabatic compression of
a chemically homogeneous material.
D layer (2600 < z < 2900 km) with a signi¬cant change in velocity gradient
and evidence for strong lateral variability and attenuation.

As the pressure increases with depth, there are phase transformations in the
silicate minerals of the minerals as the oxygen coordination varies to accommodate
denser packing. The two major discontinuities in seismic wavespeeds near depths
of 410 and 660 km are controlled by such phase transitions The changes in seismic
wave speed across these two discontinuities occur over just a few kilometres, and
they are seen in both short-period and long-period observations. Other minor
discontinuities have been proposed, but only one near 520 km appears to have some
global presence in long-period stacks, although it is not seen in short-period data.
This 520 km transition may occur over an extended zone, e.g., 30“50 km, so that it
still appears sharp for long-period waves with wavelengths of 100 km or more. A
broad ranging review of the interpretation of seismological models for the transition
zone and their reconciliation with information from mineral physics is provided by
Jackson & Ridgen (1998).
Frequently a de¬nition for the lower mantle is adopted that begins below the
660 km discontinuity. However, strong gradients in seismic wavespeeds persist to
depths of the order of 800 km and it seems appropriate to retain this region within
the transition zone. There is increasing evidence for localised sharp transitions
in seismic properties at depth around 900 km that appear to be related to the
penetration of subducted material into the lower mantle.
Between 800 km and 2600 km, the lower mantle has, on average, relatively
simple properties which would be consistent with the adiabatic compression of
a mineral assemblage of constant chemical composition and phase. Although
tomographic studies image some level of three-dimensional structure in this region
the variability is much less than in the upper part of the mantle or near the base of
the mantle.
The D layer from 2600 km to the core“mantle boundary has a distinctive
character. The nature of seismic wavespeed distribution changes signi¬cantly with
1.3 Elements of Earth structure 13

a sharp drop in the average velocity gradient. There is a strong increase in the
level of wavespeed heterogeneity near the core“mantle boundary compared with
the rest of the lower mantle. The base of the Earth™s mantle is a complex zone with
widespread indications of heterogeneity on many scales, discontinuities of variable
character, and shear-wave anisotropy (e.g., Gurnis et al., 1998; Kennett, 2002).
The results of seismic tomography give a consistent picture of the long-wavelength
structure of the D region: there are zones of markedly lower S wavespeed in the
central Paci¬c and southern Africa, whereas the Paci¬c is ringed by relatively fast
wavespeeds that may represent a ˜slab graveyard™ arising from past subduction.
A discordance between P and S wave results suggests the presence of chemical
heterogeneity rather than just the effect of temperature (e.g., Masters et al., 2000).


1.3.2 Core
The core“mantle boundary at about 2890 km depth marks a substantial change
in physical properties associated with a transition from the silicate mantle to the
metallic core (see Figure 1.6). There is a signi¬cant jump in density, and a dramatic
drop in P wavespeed from 13.7 to 8.0 km/s. The major change in wavespeed arises
from the absence of shear strength in the ¬‚uid outer core, so that the P wave speed
depends just on the bulk modulus and density. No shear waves can be transmitted
through the outer core.
The process of core formation requires the segregation of heavy iron-rich
components in the early stages of the accretion of the Earth (e.g., O™Neill & Palme,
1998). The core is believed to be largely composed of an iron“nickel alloy, but
its density requires the presence of some lighter elemental components. A wide
variety of candidates has been proposed for the light components, but it is dif¬cult
to satisfy the geochemical constraints on the nature of the bulk composition of the
Earth.
The inner core appears to be solid and formed by crystallisation of material from
the outer core, but it is possible that it could include some entrained ¬‚uid in the top
100 km or so. The shear wave speed for the inner core inferred from free-oscillation
studies is very low and the ratio of P to S wavespeeds is comparable to that of a
slurry-like material at normal pressures. The structure of the inner core is both
anisotropic and shows three-dimensional variation (e.g., Creager, 1999). There
is also some evidence to suggest that the central part of the inner core may have
distinct properties from the rest (Ishii & Dziewonski, 2003), but this region is very
dif¬cult to sample adequately.
The ¬‚uid outer core is conducting and motions within the core create a
self-sustaining dynamo which generates the main component of the magnetic ¬eld
at the surface of the Earth. The dominant component of the geomagnetic ¬eld is
dipolar but with signi¬cant secondary components. Careful analysis of the historic
record of the variation of the magnetic ¬eld has led to a picture of the evolution
of the ¬‚ow in the outer part of the core (e.g., Bloxham & Gubbins, 1989). The
14 Introduction

presence of the inner core may well be important for the action of the dynamo,
and electromagnetic coupling between the inner and outer cores could give rise
to differential rotation between the two parts of the core (Glatzmaier & Roberts,
1996). Efforts have been made to detect this differential rotation using the time
history of different classes of seismic observations but the results are currently
inconclusive.


1.4 The state of the Earth
The complexity of the processes within the Earth giving rise to the presence of
three-dimensional structure is indicated in Figure 1.7. We discuss many of these
processes in Part II.
Heterogeneity in the mantle appears to occur on a wide range of scale lengths,
from the kilometre level (or smaller) indicated by the scattering of seismic waves
to thousands of kilometres in large-scale mantle convection. The mantle in Figure
1.7 is shown with large-scale convective motions (large arrows), primarily driven
by subduction of dense, cold oceanic lithosphere (darker outer layer, and dark
slabs). The different con¬gurations re¬‚ect conditions in various subduction zones;
including the possibility of stagnant slabs on top of the 660 km discontinuity,
penetration into the lower mantle and ultimately cumulation at the core“mantle
boundary. Such downwelling needs to be matched by a return ¬‚ow of hotter
material, this is most likely to be localised plume-like features which tend to entrain
mantle material in their ascent towards the surface. Plumes which traverse the
whole mantle are expected to form near or above the hottest deep regions, possibly
guided by topographical features in the structure near the core-mantle boundary.
The dominant upper mantle phase boundaries near 410 and 660 km depth are
expected to be de¬‚ected by thermal effects or chemical heterogeneity (e.g., slabs
and plumes). Other boundaries have also been detected but might not be global
(e.g., the 220 and 520 km discontinuities, dashed).
The dominant lower mantle mineral structure, magnesium-silicate perovskite, is
predicted to transform to a denser phase, post-perovskite (ppv), in the lowermost
few hundred kilometres of the mantle (D ). If slab material is also dominated
by perovskite chemistry, then subducted material may independently transform to
ppv (white dashed lines near D in slabs). The pressure“temperature behaviour of
the phase transition has yet to be fully established and is likely to be noticeably
in¬‚uenced by minor components. Complex structure exists near the core“mantle
boundary. Large scale features with lowered seismic wavespeed are indicated by
seismic tomography that are inferred to have higher density and are likely to be
chemically distinct from the rest of the mantle. These dense thermo-chemical piles
(DTCP in Figure 1.7) may be reservoirs of incompatible elements and act as foci for
large-scale return ¬‚ow in the overlying mantle. Seismological studies characterize
signi¬cant reductions in shear velocity in such regions, which may well be the
hottest zones in the lowermost mantle, and thus related to partially molten material
1.4 The state of the Earth 15




Figure 1.7. Schematic cross-section of the Earth™s interior indicating a range of processes
that have been indicated by recent studies [courtesy of E. Garnero].


that comprises ultra-low velocity zones (ULVZ) right at the core“mantle boundary
(CMB).
Abundant evidence now exists for seismic wavespeed anisotropy (stippled or
grainy areas in Figure 1.7) near the major boundary layers in the mantle: in the
top few hundred kilometres below the surface, and in the lowermost few hundred
kilometres of the mantle (the D region). The inner core is also anisotropic in
its seismic properties, and has been characterized as having a fast propagation
direction aligned similar to, but slightly offset from the Earth™s rotation axis. The
100“200 km immediately below the inner core boundary (ICB) appears to have
much reduced anisotropy compared with the rest. The innermost inner core may
have its own unique subdivision (slightly darker shading).
The convective motions in the conducting outer core that give rise to the
16 Introduction

geodynamo are expected to have a signi¬cant component of columnnar behaviour.
This Taylor roll convection is depicted in the outer core (spiral arrows). Lower
mantle heterogeneity may affect the heat ¬‚ow from the core and hence in¬‚uence
the pattern of convective ¬‚ow within the core.


Deformation regimes and Earth dynamics
The different segments of Earth structure are subject to varying stress regimes, and
respond rheologically in different ways. The most direct information is for the near
surface, but a combination of careful experimentation and modelling has provided
insight into the nature of behaviour at depth.


Lithosphere:
The lithosphere is characterised by instantaneous elasticity, but is also capable
of long-term deformation, such as the deformation around oceanic islands and
post-glacial rebound following ice-load.
The oceanic lithosphere thickens away from mid-ocean regions where new
oceanic crust is generated. This is dominated by thermal cooling processes with
thickness approximately proportional to t1/2 (at least out to an age of 85 Ma). The
base of the lithosphere may be quite sharp in the oceanic environment, with distinct
changes in seismic wavespeed and electrical conductivity.
The mantle component of the oceanic lithosphere appears to be relatively strong
since it survives the transition into subduction relatively intact to form the distinct
subduction zones well-imaged by seismic tomography. The lithosphere is bent as
it descends into the subduction zone and this produces shallow earthquakes near
the trench. Earthquakes are generally concentrated near the top of the subducting
plate close to the division between the former oceanic crust and mantle component.
However, in some subduction zones such as northern Japan there is a second deeper
zone of earthquakes near the centre of the subducting material.
The relative uniformity of the oceanic lithosphere is in striking contrast to the
complexity of the continental environment, where the crust re¬‚ects a complex
amalgamation of units dating back 3 Ga or more. Lithospheric properties
are somewhat variable, but the lithosphere is signi¬cantly thinner (< 120 km)
under Phanerozoic belts than for the Precambrian. The resilience of the ancient
components beneath the shield is achieved because they are underlain by slightly
lowered densities in the lithospheric mantle; this material is highly refractory (and
hence dif¬cult to melt), but is intrinsically weak if stretched. The base of the
lithosphere is only locally sharp.
The crustal component of the lithosphere is the most accessible and exhibits a
range of character. In the near surface the materials are relatively brittle, but plastic
deformation becomes more signi¬cant with depth. As a result earthquakes occur
predominantly in the top 15 km above the brittle“ductile transition.
1.4 The state of the Earth 17

Asthenosphere:
Beneath the lithosphere in the upper mantle lies the asthenosphere that is more
susceptible to shorter-term deformation and thus can sustain ¬‚ow.
The asthenosphere generally has lowered shear wavespeed, enhanced attenuation
of seismic wavespeeds and lowered apparent viscosity. These properties were
originally ascribed to the presence of partial melt, but recent studies suggest that
enhanced water content could produce the requisite change in physical properties.
The rate of change of elastic moduli and attenuation increase signi¬cantly with
temperature, and for temperatures above 1200 K the effects are noticeable even
though there is no actual melt.
Seismological studies of the properties of shear waves and surface waves indicate
the presence of anisotropy in mantle materials, manifested either by differences in
the arrival times of shear waves of different polarisation or by angular variations
in the apparent propagation speed of surface waves. The shear-wave-splitting
measurements do not allow localisation of the source of anisotropy and there
has been considerable debate as to whether the observations are best explained
by ˜frozen™ anisotropy in the lithosphere re¬‚ecting past deformation or current
asthenospheric ¬‚ow.
Transition zone:
The properties of the transition zone are dominated by the in¬‚uences of the
various phase transformations in the silicate minerals of the mantle. The dominant
in¬‚uence comes from the transformations of olivine, but the minor minerals can
play a signi¬cant role in modifying behaviour. Further, many nominally anhydrous
minerals appear to be capable of incorporating signi¬cant amounts of water in their
crystalline lattices, and the presence of water at depth may have a strong local
in¬‚uence on the behaviour of materials.
Lower mantle
The dominant mineral in the lower mantle is ferro-magnesian perovskite
[(Fe,Mg)SiO3] with an admixture of magnesiowustite [(Fe,Mg)O] and much
smaller amounts of calcium- and aluminium-bearing minerals, which nevertheless
may have an important in¬‚uence on the seismic properties. A small fraction
of the lower mantle is occupied by material that has arrived through the action
of past subduction. There are relatively coherent sheet-like features as beneath
the Americas, associated with the extinct Farallon plate. Elsewhere, such as
in the Indonesian region, there is ponding of material down to 1000“1100
km depth. Distinct, but enigmatic, wavespeed anomalies occur to substantial
depth (1800“2000 km beneath present-day Australia) but have no connection to
subduction in the last 120 Ma.
This seismic evidence provides a major argument for the presence of some form
of whole-mantle convection, even though some classes of geochemical information
favour some degree of segregation between the upper and lower mantle.
18 Introduction

No major phase transition occurs within the lower mantle, but there is a
possibility of a change of iron partitioning with depth associated with a spin-state
transition in magnesiowustite. The consequences of such subtle changes in density
on convective processes have yet to be explored.
Core“mantle Boundary zone “ D :
This region just above the core“mantle boundary is highly heterogeneous on both
large and small scales. The recent discovery of a post-perovskite phase transition
(see, e.g., Murakami et al., 2004) provides a possible mechanism for explaining
the presence of seismic discontinuities. However, the constraints on the pressure
and temperature characteristics of this transition are still not tight enough for us to
be con¬dent that such a transition actually occurs within the silicate mantle. With
large scale chemical heterogeneity suggested by seismic tomography, the regimes
in the regions with lowered wavespeeds beneath southern Africa and the Paci¬c
that appear to be related to major upwellings may well differ from the rest of the
D layer.
Outer core:
The outer core is a conducting ¬‚uid with a complex pattern of ¬‚ow, and is the
seat of the internal magnetic ¬eld of the Earth. Direct evidence for variation in
the core comes from the variations in the magnetic ¬eld at the Earth™s surface,
¬rst recognised through an apparent westerly drift of the magnetic pole. Careful
work on reconstructing the magnetic ¬eld patterns over the last few centuries (e.g.,
Bloxham & Gubbins, 1989) has been exploited to map ¬‚ow patterns at the top of
the core. There is not quite suf¬cient information to make a direct mapping, but
different classes of approximation give similar results.
We have little information on the way in which the deeper parts of the outer core
behave, although the analysis of the free oscillations of the Earth suggests that the
overall behaviour is very close to an adiabatic state. The convective motions in the
internal dynamo induce small, and time-varying, ¬‚uctuations about this state.
Inner core:
The crystallisation of the solid inner core provides substantial energy that is
available to drive the ¬‚ows in the outer core. The assymmetry and anisotropy of
the seismic properties of the inner core suggests that the formation of crystalline
material is not uniform over the surface and may re¬‚ect a rather complex pattern of
growth.
Part I
CONTINUUM MECHANICS IN
GEOPHYSICS
2
Description of Deformation




In this Part we will introduce the concepts of continuum mechanics, starting
with the description of the geometry of deformation and the notion of strain.
We introduce the force ¬eld within the continuum through the stress tensor and
then link it to the rheological properties of the medium through the appropriate
constitutive equations. The treatment is based on the concepts of ¬nite deformation
and the results are derived in a general fashion so that the links between the
descriptions of solids, ¬‚uids, and intermediate properties have a common basis.
Having established the general results, we specialise to the important special cases
of small deformation in the treatment of linearised elasticity and viscoelasticity.
The materials deep within the Earth exist under states of both high pressures
and high temperatures so we examine the way in which we can provide a
suitable description that can tie to both laboratory experiments and seismological
observations. We then treat the evolution of ¬‚ow in a viscous ¬‚uid and the
introduction of non-dimensional variables; we present some simple examples
including the description of the onset of convection. We bring this Part to a close
by bringing together the differential representations of the conservation of mass,
momentum and energy with the necessary boundary conditions. The active core of
the Earth produces the internal magnetic ¬eld that we perceive at the surface, so we
need to be able to consider the interaction of continua with the electromagnetic ¬eld
to describe both the highly conducting core ¬‚uid and the much lower conductivity
of the silicate mantle. A section is therefore devoted to the development of
continuum electrodynamics and comparisons with the simpler cases discussed in
the earlier chapters.


2.1 Geometry of deformation
The pattern of deformation within a medium can be described by the geometry
imposed by the change to the medium which can be recognised through the
behaviour of points, lines and volumes. Such a description of deformation
can be based on the transformation from the reference state to the current
deformed state, or alternatively by relating the deformed state back to the

21
22 Description of Deformation

reference con¬guration from which it was derived. This distinction between
a viewpoint based on the initial (reference) con¬guration often called a
material description, and the alternative spatial description based on the current
state plays a important role in the way that different aspects of the properties of the
continuum are studied.
After an arbitrary deformation of a material continuum, the amounts of
compression (or expansion) and distortion of material vary with position
throughout the continuum.




P(x)
P(ξ )



Before Deformation After Deformation

Figure 2.1. Representation of the deformation of a continuum by the relation of a grid in
the current state derived from a simple con¬guration in the reference state.

We need to look at the deformation on a local basis and so examine the geometrical
aspects in the neighbourhood of a point P. We consider any two con¬gurations of
a material continuum. One of these is then taken as a reference state relative to
which the deformation in the other is assessed.


.
3
x
P




.
ξ Figure 2.2. The relation of a point P in
P
the reference state ξ and the
2 current, deformed state x.




1


We take a set of rectangular background axes and use these to specify the
coordinates of a material point P (Figure 2.2):
i) in the reference state ξ ≡ (ξ1, ξ2, ξ3),
ii) in the deformed state x ≡ (x1, x2, x3).
2.1 Geometry of deformation 23

The nature of the deformation from the reference state to the current, deformed
state is speci¬ed by knowing
x = x(ξ, t),
ξ (2.1.1)
as a function of x, or alternatively ξ. When the functions x(ξ, t) are linear, the
ξ
deformation is said to be homogeneous; in this case planes remain planes and lines
remain lines.


2.1.1 Deformation of a vector element
We can describe the local properties of the deformation, even when it varies with
position, by looking at the way in which a vector element transforms between the
reference and current states (Figure 2.3).


x
x + δx

Figure 2.3. Transformation of a
vector element between the
reference and deformed states.

ξ
ξ + δξ

In general, near the point P, if ξ + δξ ’ x + δx and x(ξ, t) is differentiable,
ξ ξ
‚xi
dxi = dξj, (2.1.2)
‚ξj
or, symbolically,
dx = F(ξ, t)dξ.
ξ ξ (2.1.3)
The matrix F ≡ (‚xi/‚ξj) is called the local deformation gradient, and for a (1,1)
mapping the Jacobian J = det F = 0.
Note: under change of background coordinates F transforms like a second-rank
tensor. However, since F relates vectors in two different spaces it is strictly a two-
point tensor.
The inverse transformation from the vector element in the current state to the
corresponding element in the reference state is given by
dξ = F’1(ξ, t)dx, with F’1 = ‚ξk/‚xl.
ξ ξ (2.1.4)
kl

The local deformation gradient F plays an important role in summarising the
nature of deformation. The combination FT F is the metric for the deformed state
relative to the reference state. (FFT )’1 is the corresponding metric for the inverse
24 Description of Deformation

transformation. The deviations of these metric tensors from the unit diagonal tensor
provide measures of strain.


2.1.2 Successive deformations
The result of successive deformations is to compound the effects of the two
transformations, so that the total deformation gradient between the reference and
¬nal state is the product of the deformation gradients for the successive stages of
deformation.
If x = x(y) with deformation gradient F1 = ‚x/‚y, and y = y(ξ) with ξ
deformation gradient F2 = ‚y/‚ξ, then x = x(y(ξ)) = x(ξ) with deformation
ξ ξ ξ
gradient F = ‚x/‚ξ, where
ξ
F = F1F2 (matrix multiplication). (2.1.5)


2.1.3 Deformation of an element of volume
The way in which an element of volume deforms can be determined by looking at
the transformation of a local triad of vector elements (Figure 2.4).

δξ

δx




dx
P




. 1
( 16)



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