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Overview
A valuable synthesis of the physics of magmatism for students and scholars
Magma genesis and segregation have shaped Earth since its formation more than 4.5 billion years ago. Now, for the first time, the mathematical theory describing the physics of magmatism is presented in a single volume. The Dynamics of Partially Molten Rock offers a detailed overview that emphasizes the fundamental physical insights gained through an analysis of simplified problems. This textbook brings together such topics as fluid dynamics, rock mechanics, thermodynamics and petrology, geochemical transport, plate tectonics, and numerical modeling. End-of-chapter exercises and solutions as well as online Python notebooks provide material for courses at the advanced undergraduate or graduate level.
This book focuses on the partial melting of Earth’s asthenosphere, but the theory presented is also more broadly relevant to natural systems where partial melting occurs, including ice sheets and the deep crust, mantle, and core of Earth and other planetary bodies, as well as to rock-deformation experiments conducted in the laboratory. For students and researchers aiming to understand and advance the cutting edge, the work serves as an entrée into the field and a convenient means to access the research literature. Notes in each chapter reference both classic papers that shaped the field and newer ones that point the way forward.
The Dynamics of Partially Molten Rock requires a working knowledge of fluid mechanics and calculus, and for some chapters, readers will benefit from prior exposure to thermodynamics and igneous petrology.
- The first book to bring together in a unified way the theory for partially molten rocks
- End-of-chapter exercises with solutions and an online supplement of Jupyter notebooks
- Coverage of the mechanics, thermodynamics, and chemistry of magmatism, and their coupling in the context of plate tectonics and mantle convection
- Notes at the end of each chapter highlight key papers for further reading
Product Details
| ISBN-13: | 9780691232645 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 01/18/2022 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 368 |
| File size: | 14 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Table of Contents
Preface xiii
List of Symbols xvii
Chapter 1 Introduction 1
1.1 Motivation 1
1.2 Basic Physical Considerations 2
1.3 Research Questions and Applications 6
1.4 About This Book 12
1.4.1 Overview of the Organization and Content 12
1.4.2 References to the Literature 14
1.4.3 Mathematical Notation 14
1.5 The Way Forward? 16
Chapter 2 A Condensed History of Magma/Mantle Dynamics 18
2.1 Foundation 18
2.2 Axial Age 21
2.3 Exploration 24
2.4 Generalization, Extension, and Future History 26
Chapter 3 A Review of One-Phase Mantle Dynamics 29
3.1 Governing Equations 29
3.2 Mantle Convection 31
3.3 Kinematic Solutions for Corner Flow 33
3.4 Literature Notes 35
3.5 Exercises 36
Chapter 4 Conservation of Mass and Momentum 39
4.1 The Representative Volume Element and Phase-Averaged Quantities 39
4.2 Conservation of Mass 42
4.3 Conservation of Momentum 44
4.3.1 Stress and Pressure 45
4.3.2 The Interphase Force 46
4.3.3 Constitutive Equations in the Magma-Mantle Limit 48
4.4 A Note about Disaggregation 51
4.5 The Full Mechanical System, Assembled 52
4.6 Special, Limiting Cases 53
4.6.1 No Porosity, No Melting 53
4.6.2 Partially Molten, Rigid Medium 53
l4.6.3 Constant, Uniform Solid Viscosity 53
4.7 Literature Notes 54
4.8 Exercises 56
Chapter 5 Material Properties 58
5.1 Microstructure 58
5.1.1 Grain-Size Change 59
5.1.2 Textural Equilibration 62
5.2 Permeability 66
5.3 Viscosity 68
5.3.1 Shear Viscosity of the Aggregate 68
5.3.2 Compaction Viscosity of the Aggregate 69
5.3.3 Shear Viscosity of the Liquid 73
5.4 Thermodynamic Properties 73
5.4.1 Density 73
5.4.2 Solid-Liquid Phase Change 74
5.5 Literature Notes 75
5.6 Exercises 78
Chapter 6 Compaction and Its Inherent Length Scale 80
6.1 The Compaction-Press Problem 81
6.2 The Permeability-Step Problem 83
6.3 Propagation of Small Porosity Disturbances 86
6.4 Magmatic Solitary Waves 88
6.5 Solitary-Wave Trains 93
6.6 The Compaction Length in the Asthenosphere 94
6.7 Literature Notes 96
6.8 Exercises 97
Chapter 7 Porosity-Band Emergence under Deformation 100
7.1 Governing Equations 101
7.2 Linearized Governing Equations 102
7.3 Viscosity 105
7.4 The (In)Stability of Perturbations 105
7.4.1 Pure Shear 105
7.4.2 Simple Shear 108
7.5 Wavelength Selection by Surface Tension 114
7.6 Literature Notes 116
7.7 Exercises 119
Chapter 8 Conservation of Energy 121
8.1 The Internal-Energy Equation 122
8.2 The Enthalpy Equation 124
8.3 The Temperature Equation 126
8.4 The Entropy Equation 126
8.5 Boussinesq and Lithostatic Approximations 127
8.6 Dissipation-Driven Melting and Compaction 130
8.7 Decompression Melting 134
8.8 Literature Notes 137
8.9 Exercises 139
Chapter 9 Conservation of Chemical-Species Mass 141
9.1 Thermodynamic Components 143
9.1.1 Congruent Melting 145
9.1.2 Incongruent Melting 145
9.2 Trace Elements 146
9.2.1 Equilibrium Transport Model 147
9.2.2 Disequilibrium Transport Model 148
9.3 Radiogenic Trace Elements and Their Decay Chains 151
9.4 Closed-System Evolution of a Decay Chain 153
9.4.1 Evolution with Melting Only 154
9.4.2 Evolution due to Ingrowth Only 154
9.4.3 Evolution by Both Melting and Ingrowth 156
9.5 Literature Notes 158
9.6 Exercises 159
Chapter 10 Petrological Thermodynamics of Liquid and Solid Phases 161
10.1 The Equilibrium State 162
10.1.1 Partition Coefficients from Ideal Solution Theory 163
10.1.2 Computing the Equilibrium State 167
10.1.3 Application to a Two-Pseudo-Component System 167
10.1.4 Application to a Three-Pseudo-Component System 169
10.1.5 Approaching the Eutectic Phase Diagram 170
10.1.6 Linearizing the Two-Component Phase Diagram 173
10.1.7 Degree of Melting 173
10.2 Thermochemical Disequilibrium and the Rate of Interphase Mass Transfer 174
10.2.1 Affinity as the Thermodynamic Force for Linear Kinetics 175
10.2.2 Linearized Melting Rates 177
10.3 Computing the Melting Rate at Equilibrium 177
10.4 Remarks about Mantle Thermochemistry 179
10.5 Literature Notes 180
10.6 Exercises 181
Chapter 11 Melting Column Models 182
11.1 Fluid Mechanics 182
11.2 Melting-Rate Closures 186
11.2.1 Prescribed Melting Rate 186
11.2.2 Thermodynamically Consistent Melting Rate 187
11.3 The Visco-Gravitational Boundary Layer 196
11.4 The Decompaction Boundary Layer 199
11.5 Isotopic Decay-Chain Disequilibria in a Melting Column 203
11.5.1 Constant Transport Rates 205
11.5.2 Variable Transport Rates 208
11.6 Literature Notes 210
11.7 Exercises 211
Chapter 12 Reactive Flow and the Emergence of Melt Channels 214
12.1 Governing Equations 215
12.2 The Melting-Rate Closure 215
12.3 Problem Specification 217
12.4 Scaling and Simplification 218
12.5 Linearized Stability Analysis 221
12.5.1 The Base State 222
12.5.2 The Growth Rate of Perturbations 223
12.5.3 The Large-Damkohler Number Limit 226
12.5.4 A Modified Problem and Its Analytical Solution 227
12.6 Physical Mechanisms 231
12.7 Application to the Mantle 233
12.8 Literature Notes 234
12.9 Exercises 236
Chapter 13 Tectonic-Scale Models and Modeling Tools 238
13.1 Governing Equations in the Small-Porosity Approximation 239
13.2 Corner-Flow with Magmatic Segregation 241
13.3 Melt Focusing through a Sublithospheric Channel 243
13.3.1 Lateral Transport in Semi-Infinite Half-Space 243
13.3.2 Lateral Transport to a Mid-Ocean Ridge 246
13.4 Coupled Dynamics and Thermochemistry with the Enthalpy Method 252
13.5 Literature Notes 255
13.6 Exercises 257
Chapter 14 Numerical Modeling of Two-Phase Flow 258
14.1 The One-Dimensional Solitary Wave (Instantaneous) 259
14.1.1 Finite Difference Discretization 260
14.1.2 Finite Element Discretization 263
14.2 The One-Dimensional Solitary Wave (Time-Dependent) 267
14.3 A Two-Dimensional Manufactured Solution (Instantaneous) 269
14.4 Magmatic Solitary Waves as a Benchmark for Numerical Solutions 275
14.5 Porosity Bands as a Benchmark for Numerical Solutions 276
14.6 Literature Notes 277
14.7 Exercises 278
Chapter 15 Solutions to Exercises 280
15.1 Exercises from Chapter 3: One-Phase Mantle Dynamics 280
15.2 Exercises from Chapter 4: Conservation of Mass and Momentum 284
15.3 Exercises from Chapter 5: Material Properties 287
15.4 Exercises from Chapter 6: Compaction and Its Inherent Length Scale 293
15.5 Exercises from Chapter 7: Porosity-Band Emergence under Deformation 300
15.6 Exercises from Chapter 8: Conservation of Energy 303
15.7 Exercises from Chapter 9: Conservation of Chemical-Species Mass 308
15.8 Exercises from Chapter 10: Petrological Thermodynamics of Liquid and Solid Phases 310
15.9 Exercises from Chapter 11 ; Melting Column Models 310
15.10 Exercises from Chapter 12: Reactive Flow and the Emergence of Melt Channels 315
15.11 Exercises from Chapter 13: Tectonic-Scale Models 318
15.12 Exercises from Chapter 14: Numerical Modeling of Two-Phase Flow 319
Bibliography 321
Index 337
What People are Saying About This
“It is no exaggeration to say that magmatism is the reason why the Earth is as it is today. Proceeding systematically from simple to complex models, Richard Katz comprehensively surveys the physics and chemistry of partial melting with elegance, pedagogical clarity, and deep physical insight. This book will be the definitive treatment of its subject for many years to come.”—Neil Ribe, University of Paris-Saclay“The twin processes of partial melting and melt migration are essential features of the evolution of all terrestrial planets. Yet, the theory of partially molten rock, developed through a series of dense papers in the primary literature, has hitherto only been available to experts. In this appealing and accessible book, Katz brings all the necessary information together to introduce the field to newcomers and guides them to the most cutting-edge developments.”—Paul D. Asimow, California Institute of Technology“Igneous petrology, the study of melts produced from the Earth's interior, has a long history. Yet until about forty years ago, petrologists had almost no interest in how such melts formed. Since then a new branch of fluid dynamics has developed that explores how melts separate from solid residue. Deriving the governing equations and their solutions, this excellent book provides the basic fluid dynamical understanding that will underpin all future work in this important field.”—Dan McKenzie, emeritus professor of earth sciences, University of Cambridge"Filling an important gap in textbooks that describe Earth dynamics, this book provides an overview of the mathematical description of magma dynamics that is broad, encompassing, and detailed. Katz describes with clarity fairly complex mathematical physics and the text is well written. A pleasure to read."—Peter van Keken, Carnegie Institution for Science"Covering the two-phase formalism that has been widely used in geophysics to discuss magma extraction, this book makes a significant contribution to the field. No other books exist on this subject."—Yanick Ricard, CNRS/ENS-Lyon