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

dξ

dx

P