The Dynamics of Partially Molten Rock

Richard F. Katz

A valuable synthesis of the physics of magmatism for students and scholars

Hardcover

Price:: $90.00/£65.00
ISBN:
Published (US):: Jan 18, 2022
Published (UK):: Mar 15, 2022
Copyright:: 2022
Pages:: 368
Size:: 6.12 x 9.25 in.
Illus:: 69 b/w illus. 7 tables.
Main_subject:: Earth Science

ebook

Price:: $90.00/£65.00
ISBN:
Published (US):: Jan 18, 2022
Published (UK):: Mar 15, 2022
Copyright:: 2022
Pages:: 368
Size:: 6.12 x 9.25 in.
Illus:: 69 b/w illus. 7 tables.
Main_subject:: Earth Science

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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

Preface
List of Symbols
CHAPTER 1 Introduction
- 1.1 Motivation
- 1.2 Basic Physical Considerations
- 1.3 Research Questions and Applications
- 1.4 About This Book
  - 1.4.1 Overview of the Organization and Content
  - 1.4.2 References to the Literature
  - 1.4.3 Mathematical Notation
- 1.5 The Way Forward?
CHAPTER 2 A Condensed History of Magma/Mantle Dynamics
- 2.1 Foundation
- 2.2 Axial Age
- 2.3 Exploration
- 2.4 Generalization, Extension, and Future History
CHAPTER 3 A Review of One-Phase Mantle Dynamics
- 3.1 Governing Equations
- 3.2 Mantle Convection
- 3.3 Kinematic Solutions for Corner Flow
- 3.4 Literature Notes
- 3.5 Exercises
CHAPTER 4 Conservation of Mass and Momentum
- 4.1 The Representative Volume Element and Phase-Averaged Quantities
- 4.2 Conservation of Mass
- 4.3 Conservation of Momentum
  - 4.3.1 Stress and Pressure
  - 4.3.2 The Interphase Force
  - 4.3.3 Constitutive Equations in the Magma–Mantle Limit
- 4.4 A Note about Disaggregation
- 4.5 The Full Mechanical System, Assembled
- 4.6 Special, Limiting Cases
  - 4.6.1 No Porosity, No Melting
  - 4.6.2 Partially Molten, Rigid Medium
  - 4.6.3 Constant, Uniform Solid Viscosity
- 4.7 Literature Notes
- 4.8 Exercises
CHAPTER 5 Material Properties
- 5.1 Microstructure
  - 5.1.1 Grain-Size Change
  - 5.1.2 Textural Equilibration
- 5.2 Permeability
- 5.3 Viscosity
  - 5.3.1 Shear Viscosity of the Aggregate
  - 5.3.2 Compaction Viscosity of the Aggregate
  - 5.3.3 Shear Viscosity of the Liquid
- 5.4 Thermodynamic Properties
  - 5.4.1 Density
  - 5.4.2 Solid–Liquid Phase Change
- 5.5 Literature Notes
- 5.6 Exercises
CHAPTER 6 Compaction and Its Inherent Length Scale
- 6.1 The Compaction-Press Problem
- 6.2 The Permeability-Step Problem
- 6.3 Propagation of Small Porosity Disturbances
- 6.4 Magmatic Solitary Waves
- 6.5 Solitary-Wave Trains
- 6.6 The Compaction Length in the Asthenosphere
- 6.7 Literature Notes
- 6.8 Exercises
CHAPTER 7 Porosity-Band Emergence under Deformation
- 7.1 Governing Equations
- 7.2 Linearized Governing Equations
- 7.3 Viscosity
- 7.4 The (In)Stability of Perturbations
  - 7.4.1 Pure Shear
  - 7.4.2 Simple Shear
- 7.5 Wavelength Selection by Surface Tension
- 7.6 Literature Notes
- 7.7 Exercises
CHAPTER 8 Conservation of Energy
- 8.1 The Internal-Energy Equation
- 8.2 The Enthalpy Equation
- 8.3 The Temperature Equation
- 8.4 The Entropy Equation
- 8.5 Boussinesq and Lithostatic Approximations
- 8.6 Dissipation-Driven Melting and Compaction
- 8.7 Decompression Melting
- 8.8 Literature Notes
- 8.9 Exercises
CHAPTER 9 Conservation of Chemical-Species Mass
- 9.1 Thermodynamic Components
  - 9.1.1 Congruent Melting
  - 9.1.2 Incongruent Melting
- 9.2 Trace Elements
  - 9.2.1 Equilibrium Transport Model
  - 9.2.2 Disequilibrium Transport Model
- 9.3 Radiogenic Trace Elements and Their Decay Chains
- 9.4 Closed-System Evolution of a Decay Chain
  - 9.4.1 Evolution with Melting Only
  - 9.4.2 Evolution due to Ingrowth Only
  - 9.4.3 Evolution by Both Melting and Ingrowth
- 9.5 Literature Notes
- 9.6 Exercises
CHAPTER 10 Petrological Thermodynamics of Liquid and Solid Phases
- 10.1 The Equilibrium State
  - 10.1.1 Partition Coefficients from Ideal Solution Theory
  - 10.1.2 Computing the Equilibrium State
  - 10.1.3 Application to a Two-Pseudo-Component System
  - 10.1.4 Application to a Three-Pseudo-Component System
  - 10.1.5 Approaching the Eutectic Phase Diagram
  - 10.1.6 Linearizing the Two-Component Phase Diagram
  - 10.1.7 Degree of Melting
- 10.2 Thermochemical Disequilibrium and the Rate of Interphase Mass Transfer
  - 10.2.1 Affinity as the Thermodynamic Force for Linear Kinetics
  - 10.2.2 Linearized Melting Rates
- 10.3 Computing the Melting Rate at Equilibrium
- 10.4 Remarks about Mantle Thermochemistry
- 10.5 Literature Notes
- 10.6 Exercises
CHAPTER 11 Melting Column Models
- 11.1 Fluid Mechanics
- 11.2 Melting-Rate Closures
  - 11.2.1 Prescribed Melting Rate
  - 11.2.2 Thermodynamically Consistent Melting Rate
- 11.3 The Visco-Gravitational Boundary Layer
- 11.4 The Decompaction Boundary Layer
- 11.5 Isotopic Decay-Chain Disequilibria in a Melting Column
  - 11.5.1 Constant Transport Rates
  - 11.5.2 Variable Transport Rates
- 11.6 Literature Notes
- 11.7 Exercises
CHAPTER 12 Reactive Flow and the Emergence of Melt Channels
- 12.1 Governing Equations
- 12.2 The Melting-Rate Closure
- 12.3 Problem Specification
- 12.4 Scaling and Simplification
- 12.5 Linearized Stability Analysis
  - 12.5.1 The Base State
  - 12.5.2 The Growth Rate of Perturbations
  - 12.5.3 The Large-Damköhler Number Limit
  - 12.5.4 A Modified Problem and Its Analytical Solution
- 12.6 Physical Mechanisms
- 12.7 Application to the Mantle
- 12.8 Literature Notes
- 12.9 Exercises
CHAPTER 13 Tectonic-Scale Models and Modeling Tools
- 13.1 Governing Equations in the Small-Porosity Approximation
- 13.2 Corner-Flow with Magmatic Segregation
- 13.3 Melt Focusing through a Sublithospheric Channel
  - 13.3.1 Lateral Transport in Semi-Infinite Half-Space
  - 13.3.2 Lateral Transport to a Mid-Ocean Ridge
- 13.4 Coupled Dynamics and Thermochemistry with the Enthalpy Method
- 13.5 Literature Notes
- 13.6 Exercises
CHAPTER 14 Numerical Modeling of Two-Phase Flow
- 14.1 The One-Dimensional Solitary Wave (Instantaneous)
  - 14.1.1 Finite Difference Discretization
  - 14.1.2 Finite Element Discretization
- 14.2 The One-Dimensional Solitary Wave (Time-Dependent)
- 14.3 A Two-Dimensional Manufactured Solution (Instantaneous)
- 14.4 Magmatic Solitary Waves as a Benchmark for Numerical Solutions
- 14.5 Porosity Bands as a Benchmark for Numerical Solutions
- 14.6 Literature Notes
- 14.7 Exercises
CHAPTER 15 Solutions to Exercises
- 15.1 Exercises from Chapter 3: One-Phase Mantle Dynamics
- 15.2 Exercises from Chapter 4: Conservation of Mass and Momentum
- 15.3 Exercises from Chapter 5: Material Properties
- 15.4 Exercises from Chapter 6: Compaction and Its Inherent Length Scale
- 15.5 Exercises from Chapter 7: Porosity-Band Emergence under Deformation
- 15.6 Exercises from Chapter 8: Conservation of Energy
- 15.7 Exercises from Chapter 9: Conservation of Chemical-Species Mass
- 15.8 Exercises from Chapter 10: Petrological Thermodynamics of Liquid and Solid Phases
- 15.9 Exercises from Chapter 11: Melting Column Models
- 15.10 Exercises from Chapter 12: Reactive Flow and the Emergence of Melt Channels
- 15.11 Exercises from Chapter 13: Tectonic-Scale Models
- 15.12 Exercises from Chapter 14: Numerical Modeling of Two-Phase Flow
Bibliography
Index

“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

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