Dynamics of Plate Tectonics and Mantle Convection 9780323857338

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DYNAMICS OF PLATE TECTONICS AND MANTLE CONVECTION

DYNAMICS OF PLATE TECTONICS AND MANTLE CONVECTION Edited by

JOA˜O C. DUARTE

Assistant Professor, University of Lisbon, Lisbon, Portugal; Researcher, Faculty of Sciences, Instituto Dom Luiz, University of Lisbon, Lisbon, Portugal

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2023 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-85733-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals

Publisher: Candice G. Janco Acquisitions Editor: Amy M. Shapiro Editorial Project Manager: Helena Beauchamp Production Project Manager: Rashmi Manoharan Cover Designer: Christian J. Bilbow Typeset by STRAIVE, India

Contributors

Rakib Hassan EarthByte Group, School of Geosciences, University of Sydney, Camperdown, NSW; Geoscience Australia, Canberra, ACT, Australia

David Bercovici Department of Earth and Planetary Sciences, Yale University, New Haven, CT, United States Magali I. Billen Department of Earth and Planetary Sciences, UC Davis, Davis, CA, United States

M.J. Hoggard Research School of Earth Sciences, The Australian National University, Acton, ACT, Australia

Doris Breuer DLR, Institute of Planetary Research, Berlin, Germany

Shun-ichiro Karato Department of Earth and Planetary Sciences, Yale University, New Haven, CT, United States

Sascha Brune Helmholtz Centre Potsdam—GFZ German Research Centre for Geosciences, Potsdam; Institute of Geosciences, University of Potsdam, Potsdam-Golm, Germany

Derek Keir Universita degli Studi di Firenze, Dipartimento di Scienze della Terra, Florence, Italy; University of Southampton, School of Ocean and Earth Science, Southampton, United Kingdom

Susanne J.H. Buiter Tectonics and Geodynamics, RWTH Aachen University, Aachen; Helmholtz Centre Potsdam—GFZ German Research Centre for Geosciences, Potsdam, Germany
Nicolas Coltice Laboratoire de G
eologie, Ecole Normale Sup
erieure, CNRS UMR 8538, PSL Research University, Paris, France

Matthew G. Knepley Computer Science and Engineering, University at Buffalo, Buffalo, NY, United States Adrian Lenardic Department of Earth Science, Rice University, Houston, TX, United States Diogo L. Lourenc¸ o Department of Earth and Planetary Science, University of California, Berkeley, CA, United States; Department of Geographic Engineering, Geophysics and Energy, Faculty of Sciences, University of Lisbon, Lisbon, Portugal; Department of Earth Sciences, Institute of Geophysics, ETH Zurich, Zurich, Switzerland

D.R. Davies Research School of Earth Sciences, The Australian National University, Acton, ACT, Australia Joa˜o C. Duarte Faculty of Sciences, Instituto Dom Luiz, University of Lisbon, Lisbon, Portugal Taras V. Gerya Department of Earth Sciences, ETH-Zurich, Zurich, Switzerland

Sarah J. MacLeod EarthByte Group, School of Geosciences, University of Sydney, Camperdown, NSW, Australia

S. Ghelichkhan Research School of Earth Sciences, The Australian National University, Acton, ACT, Australia

Dave A. May Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA, United States

Richard G. Gordon Department of Earth, Environmental and Planetary Sciences, Rice University, Houston, TX, United States Michael Gurnis Seismological Laboratory, California Institute of Technology, Pasadena, CA, United States

Louis Moresi Research School of Earth Sciences, The Australian National University, Canberra, ACT, Australia

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xii Jason P. Morgan

Contributors

SUSTech, Shenzhen, China

R. Dietmar M€ uller EarthByte Group, School of Geosciences, University of Sydney, Camperdown, NSW, Australia

Johnny Seales Department of Earth Science, Rice University, Houston, TX, United States Alisha Steinberger Canada

BRON Studios, Vancouver, BC,

Elvira Mulyukova Department of Earth and Planetary Sciences, Northwestern University, Evanston, IL, United States

Bernhard Steinberger GFZ German Research Centre for Geosciences, Potsdam, Germany; CEED, University of Oslo, Oslo, Norway

Takashi Nakagawa Department of Earth and Planetary System Science, Hiroshima University, Higashihiroshima, Japan; Previous affiliation: School of Earth and Environment, University of Leeds, Leeds, United Kingdom; Department of Planetology, Kobe University, Kobe, Japan

Robert J. Stern Geosciences Department, University of Texas at Dallas, Richardson, TX, United States

Jean-Arthur Olive Laboratoire de G
eologie, CNRS—Ecole Normale Sup
erieure—PSL University, Paris, France Gwenn Peron-Pinvidic Geological Survey of Norway (NGU); Norwegian University of Science and Technology (NTNU), Trondheim, Norway C
esar R. Ranero

ICM-CSIC, Barcelona, Spain

F.D. Richards Department of Earth Science and Engineering, Imperial College London, South Kensington, London, United Kingdom Antoine B. Rozel Department of Earth Sciences, Institute of Geophysics, ETH Zurich, Zurich, Switzerland W.P. Schellart Department of Earth Sciences, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands

Pietro Sternai Department of Earth and Environmental Sciences (DISAT), University of MilanoBicocca, Milano, Italy Paul J. Tackley Department of Earth Sciences, ETH Zurich, Z€ urich, Switzerland A.P. Valentine Department of Earth Sciences, Durham University, Lower Mountjoy, Durham, United Kingdom Paola Vannucchi University of Florence, Florence; National Institute of Oceanography and Applied Geophysics, Trieste, Italy Simon E. Williams EarthByte Group, School of Geosciences, University of Sydney, Camperdown, NSW, Australia; Department of Geology, Northwest University, Xi’an, Shaanxi Province, China Masaki Yoshida Volcanoes and Earth’s Interior Research Center, Research Institute for Marine Geodynamics; Agency for Marine-Earth Science and Technology (JAMSTEC), Yokosuka, Japan

Preface

Plate tectonics is the unifying theory of the solid Earth that describes the surface of our planet as being fragmented into rigid plates that move in relation to each other. Many fundamental geological processes, such as earthquakes, volcanism, and the formation of mountains, occur at the plates’ boundaries. In its initial formulation, plate tectonics was a kinematic theory that described and quantified these movements and associated deformation. Plate tectonics evolved into a dynamic theory, describing the forces driving the surface deformation and the internal dynamics of our planet. We also came to realize that plate tectonics is only one of the possible tectono-magmatic styles in which a planet can operate and that these can change over time. In the past years, the geosciences community has started to develop a theory that integrates plate tectonics and mantle convection as part of a single system. This was possible because of the ever-increasing high-quality imaging of Earth’s interior and our computational capabilities. But while the progress has been enormous, there are still many uncertainties and challenges ahead. I have asked some of the leading scientists in the field to write perspective review papers from the vantage point of their work. The idea was to capture the state of the art of our understanding of the dynamics of plate tectonics and mantle convection. I hope this book can glimpse what

we have learned so far, where we are now, and what are the future challenges. Editing this book was one of the best professional experiences I ever had. It was fun and exciting, and it was a privilege. It was a privilege to work with all the authors to produce this final product. This is your book. I must thank you all for having accepted my invitation to be part of this project. I have learned so much from all of you and I am sure the reader will too. I thank my family, Noemie, Maria de Lourdes, Joaquim, Helena, Pedro, Diogo, and Manuela, for the constant support. I also thank my colleagues Filipe Rosas, Nicolas Riel, Wouter Schellart, Mattias Green, Hannah Davies, Anto´nio Ribeiro, Marc-Andre Gutscher, Pedro Terrinha, Ricardo Trigo, Pedro Miranda, Rui Dias, Vasco Valadares, Henrique Duarte, Michael Way, Christian Hensen, Santanu Bose, Diogo Lourenc¸o, Susana Custo´dio, Luı´s Matias, Taras Gerya, Sierd Cloetingh, Catherine Meriaux, David Boutelier, Alexander Cruden, Jacques Malavieille, Paola Vannucchi, Jason P. Morgan, Susanne Buiter, Vincent Strak, Chiara Civiero, So´nia Silva, Luı´s Pinheiro, Luis Matias, Vitor Magalha˜es, Marta Neres, Nathan Mayne, Joa˜o Mata, Sophie Wilmes, Steven Balbus, Chris MacLeod, Cristina Roque, Jaime Almeida, Zhihao Chen, and many other colleagues with whom I have learned and discussed so much over the years. A special thanks to the Faculty of Sciences of the

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Preface

University of Lisbon, the Geology Department, the Instituto Dom Luiz (IDL), and Celia Lee for providing me with the grounds to build this project. I also would like to thank Nicolas Coltice for producing the incredible cover image that perfectly captures the efforts in trying to

understand the dynamics of plate tectonics and mantle convection. Joa˜o C. Duarte University of Lisbon 30 September 2022

C H A P T E R

1 Introduction to Dynamics of Plate Tectonics and Mantle Convection Joa˜o C. Duarte Faculty of Sciences, Instituto Dom Luiz, University of Lisbon, Lisbon, Portugal

One cannot tell exactly when it all started, but it is quite amazing that the cartographer Abraham Ortelius noted in 1596 that the margins of the Atlantic fitted like a puzzle. One can’t imagine what crossed Ortelius’ mind, but in 1858, Antonio Snider-Pellegrini drew a “supercontinent” fitting Europe and Africa with the Americas. While Snider-Pellegrini did not have a compelling scientific theory to explain it, this was a bold statement, suggesting that large portions of the planet’s surface have moved thousands of kilometers. Yet, it was only a few decades later that Alfred Wegener put together evidence to sustain this hypothesis, which became known as the continental drift theory (Wegener, 1929). Probably Wegener was not aware of it, but this was the first revolution in the solid earth sciences. It completely revolutionized how we looked at and interpreted geological processes. Notwithstanding, at the time it probably felt more like a controversy than a real revolution. Most of his peers strongly contested the idea. After all, how could continents move over solid oceanic crust? During the following years, a few earth scientists took the idea seriously and started working

A good way to start a book project is to think of a book the community would like to read. But that is not enough. One must also understand the community needs and make sure the book is timely. As I will try to explain, we are undergoing a key moment in our comprehension of plate tectonics and mantle convection. To help us understand where we came from and where we are going, I asked a number of experts in the field to write perspective papers from their vantage point and, whenever possible, to tell a story, either their story or the story of an idea. The book that the reader has in front is a combination of perspective and review papers dealing with some of the most exciting and timely topics in the field of plate tectonics and mantle convection. Plate tectonics is a relatively young theory compared to Darwin’s theory of evolution, Einstein’s theory of general relativity, or the theory of quantum mechanics by Bohr. But it did not come out of nowhere. Very few things do in science. The Greeks and other ancient civilizations had theories about the Earth, trying to explain the origins of earthquakes and volcanoes.

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1. Introduction to Dynamics of Plate Tectonics and Mantle Convection

on mechanisms that could explain continental drift. Arthur Holmes, in 1931, proposed that continental break-up could be caused by mantle convection, related to the radiogenic heating of the mantle (Holmes, 1931). Although in Holmes’ model convection was driving surface deformation, the ascending cells did not rise to the surface. There were no plate boundaries and no plates yet. In 1962, Herry Hess proposed a daring idea. The seafloor was spreading. New crust was forming at spreading ridges and being consumed at subduction zones. This was confirmed in the following year by Lawrence Morley, Fred Vine, and Drummond Matthews (Vine and Matthews, 1963; Morley and Larochelle, 1964), based on the linear pattern of seafloor magnetic anomalies. One could interpret this moment as a second revolution in the solid earth sciences, but I like to think of it as the start of what was about to come next. Toward the end of the same decade, Tuzo Wilson published a series of seminal papers (e.g., Wilson, 1965, 1966). In one of these papers, he came up with the idea of transform faulting, realizing the importance of transform faults in surface motion and deformation. This was the last piece of a puzzle that allowed connecting spreading centers to subduction zones and, by doing so, define the limits of tectonic plates. This epiphany immediately led to the realization that the surface of the planet was divided into several tectonic plates that moved in relation to each other. This was, without a doubt, the key moment of the second revolution in the solid earth sciences, which later became known as the theory of plate tectonics. In the following years, a series of seminal papers revolutionized our understanding of how planet Earth works (e.g., Morgan, 1968; McKenzie and Parker, 1967; Le Pichon, 1968; Isacks, Oliver and Sykes, 1968). The theory of plate tectonics was, in its original form, a quantitative kinematic theory of the Earth that described how tectonic plates moved and interacted at the Earth’s spherical surface. But

soon after, geoscientists started to enquire what were the forces driving the motions of the tectonic plates. Although the idea that mantle convection drags tectonic plates still persists to this day in some educational environments, it was soon realized that, in a certain way, plates drive themselves as a result of their negative buoyancy. It is mostly the sinking of dense slabs at subduction zones that pulls plates apart (Jacoby, 1970; Elsasser, 1971; Turcotte and Oxburgh, 1972; Forsyth and Uyeda, 1975). This was a powerful realization, but it was still a first approach. Over the years, tectonics became one of the most prolific fields in science. It completely changed how geology was studied and opened the door such that the fields of solid earth geophysics and geodynamics could flourish. But tectonics also became a specialized discipline in itself. This had a side effect. Tectonics and geodynamics became somewhat distinct disciplines. This is obvious in several congresses and geological societies, where tectonics and geodynamics are often separated. In some institutions, they are even based in different departments. In the last couple of decades, our imaging of mantle structure and our capacity of modeling large-scale geodynamic processes have dramatically improved, in terms of both quantity and quality. And while there were hurdles along the way, we now see plate tectonics and mantle convection as part of an integral system. Plate tectonics is the surface expression of a convective planet. The oceanic plates are part of and drive mantle convection. This integrated dynamic theory of plate tectonics and mantle convection can, in my view, be regarded as a third revolution in the solid earth sciences. Or at least, a second major step in the theory of plate tectonics. Some colleagues even named it Plate Tectonics 2.0. Whatever it is, we are just passing through it. It may seem pretentious to affirm that we are going through a revolution. To be certain, we will need some historical detachment. Future generations will tell. I would, however, argue

References

that some time has passed since this revolution was initiated. A few decades maybe. Many scientists that kick-started it are already established researchers. Notwithstanding, there are new exciting developments every day, and there is a whole new generation of earth scientists riding the wave. The hidden question in this book is: can we capture an ongoing revolution from the inside, while it is ongoing? I will leave the answer to the reader and to the next generations to tell. I was (and I am) aware of the risks involved in such projects, but I always thought that only something good could come out of them. It became a certainty when the first chapters started to arrive. This was the book I always wanted to read, and I am sure many colleagues will feel the same. But this book is not my book. This book was written by a group of colleagues that I deeply admire and to whom I am indebted. I must thank them from the depths of my heart for having accepted this challenge. The guidelines were simple. I asked them to write perspective papers. Start with some historical introduction, provide context, and, when possible, provide a glimpse of their personal path as scientists and/or of their fields. Then explore some of the timely ideas concerning the topics on which they have been working and finish with some future perspectives. I soon realized that all chapters had to be different, as authors opted for different perspectives and writing styles. This only made the book even richer. There are many distinct perspectives within this book. They are all looking at the same overarching topic, but from different vantage points. This tells me that developing a unified dynamic theory of plate tectonics and mantle convection does not need to be a goal in itself. It may even

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lessen the prosperity of the field. What matters most is that we are making exciting progress in our understanding of how the planet Earth works. And there is still so much to do. One last note. The advantage of coordinating a project like this is that one can see behind the scenes. I asked the authors to write freely and try to open a bit the window. By doing so, they told me they wrote the manuscript they always wanted to write. I hope the reader finds in these chapters what they are looking for.

References Elsasser, W.M., 1971. Sea-floor spreading as thermal convection. J. Geophys. Res. 76, 1101–1112. Forsyth, D., Uyeda, S., 1975. On the relative importance of the driving forces of plate motion. Geophys. J. R. Astron. Soc. 43, 163–200. https://doi.org/10.1111/j.1365-246X.1975. tb00631.x. Holmes, A., 1931. XVIII. Radioactivity and Earth Movements. Trans. Geol. Soc. Glasgow. Geol. Soc. Lond. 18 (3), 559–606. https://doi.org/10.1144/transglas.18.3.559. Isacks, B., Oliver, J., Sykes, L.R., 1968. Seismology and the new global tectonics. J. Geophys. Res. 73, 5855–5899. https://doi.org/10.1029/JB073i018p05855. Jacoby, W.R., 1970. Instability in the upper mantle and global plate movements. J. Geophys. Res. 75, 5671–5680. Le Pichon, X., 1968. Sea-floor spreading and continental drift. J. Geophys. Res. 73, 3661–3697. https://doi.org/10.1029/ JB073i012p03661. McKenzie, D., Parker, R.L., 1967. The North Pacific: an example of tectonics on a sphere. Nature 216, 1276–1280. Morgan, W.J., 1968. Rises, trenches, great faults, and crustal blocks. J. Geophys. Res. 73, 1959–1982. https://doi.org/ 10.1029/JB073i006p01959. Turcotte, D., Oxburgh, E., 1972. Mantle convection and the new global tectonics. Ann. Rev. Fluid Mech. 4, 33–66. Wegener, A., 1929. Die Entstehung der Kontinente und Ozeane, 4. Auflage. Friedrich Vieweg & Sohn, Braunschweig. Wilson, J.T., 1965. A new class of faults and their bearing on continental drift. Nature 207, 343–347. https://doi.org/ 10.1038/207343a0. Wilson, J.T., 1966. Did the Atlantic close and then re-open? Nature 211, 676–681.

C H A P T E R

2 The Physics and Origin of Plate Tectonics From Grains to Global Scales Elvira Mulyukovaa and David Bercovicib a

Department of Earth and Planetary Sciences, Northwestern University, Evanston, IL, United States b Department of Earth and Planetary Sciences, Yale University, New Haven, CT, United States

1. Introduction

his generation were concerned, the Vietnam War had been going on forever and would never end. The second author entered graduate school under less than propitious circumstances during the Ronald Reagan Era (when many of us were convinced we could never have a worse president) and published his first paper the year the first author was born. That paper had nothing to do with plate tectonics. The generation of plate tectonics was not even on the second author’s radar until after receiving his PhD, and then it never left his radar again. The first author moved from Murmansk to Norway the same year the second author left Hawaii for Connecticut, which was also the year George W. Bush was elected the US President (when many of us were certain we could never have a worse president). One of us was a young teenager at the time. One of us was not. Years later, the authors crossed paths at a small conference in the Kongsberg Silver Mines in Norway in the summer of 2013, shared a bond of not taking themselves (or anyone else) seriously, and began working together officially in June 2015

1.1 In the beginning… Normally such a section heading calls for a grandiose opening statement about the subject matter, for example, how “plate tectonics is one of the grand unifying theories of the physics of the Earth,” etc. But this beginning is about a personal reflection, and is also about an ending, or to quote T.S. Eliot What we call the beginning is often the end. And to make an end is to make a beginning.The end is where we start from.

The theory of plate tectonics is typically assigned a birth year of 1968 (although it had been brewing long before that), but 1968 would hardly be called the year of plate tectonics. In that year, both Martin Luther King, Jr. and Robert F. Kennedy were assassinated, and Richard Nixon was elected the US President (and many of us were convinced we could never have a worse president). The second author was 8 years old, and as far as he and

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2. The Physics and Origin of Plate Tectonics From Grains to Global Scales

for the next 6 years. (Is there really any need to finish the parenthetical joke about Presidents?) However, at the moment of this writing, summer of 2021, the authors have, in a sense, parted ways, and so this chapter is our first contribution from different institutions. It includes, in the context of decades of research, most of our work and thinking, and understanding and misunderstandings together over the last half decade and then some. But as such, it is both a milestone and a road map, and is in every way, a beginning, not an end.

1.2 Ok, but seriously… Plate tectonics is one of the grand unifying theories of the physics of the Earth, and its inception in the 1960s revolutionized all the Earth sciences. The mathematical theory of plate tectonics (McKenzie and Parker, 1967; Morgan, 1968) has provided the framework with which to explain the Earth’s geological activity associated with earthquakes, volcanic activity, and mountain building, as well as surface processes controlling the evolution of the Earth’s atmosphere and climate (Berner, 2004; Walker et al., 1981) and the maturation of hydrocarbons (McKenzie, 1981). The forces driving plate tectonics have long been recognized as arising from mantle convection, linking Earth’s surface motion to the dynamics of its deep interior and its thermochemical evolution. At the heart of our search for understanding how and why Earth, alone among the known terrestrial planets, has plate tectonics, is determining the unique physical conditions that allow mantle convection at depth to look like plate tectonics at the surface. The necessary physics lies in the complexities of rock deformation. The plate tectonic style of mantle convection involves severe deformation of the top convective boundary layer, that is, lithosphere, to form narrow plate boundaries separating largely

undeformed lithosphere that comprise plate interiors. However, plate-boundary formation is itself a paradox. The ductile strength of mantle rocks is strongly temperature dependent and when accounting for this effect alone, the cold lithosphere is too strong to deform by convective forcing (Solomatov, 1995). In principle, no terrestrial planet should have its lithosphere deforming as rapidly as the vigorously convecting mantle. Most of the terrestrial planets in our solar system seem to obey this common-sense rule, all except Earth. Some physical mechanism on the Earth necessarily counters or neutralizes the thermal stiffening of the lithosphere, a mechanism unique to the conditions of our planet, or even possibly responsible for these conditions on our planet. The occurrence of plate tectonics thus requires a mechanism for localized weakening to form plate boundaries with which to mobilize the surface (Bercovici, 1993, 1995a, 2003; Bercovici et al., 2015b; Kaula, 1980; Mont
esi, 2013; Mulyukova and Bercovici, 2019b; Tackley, 2000a, b). Plate boundaries span the entire depth of the lithosphere, and thus require strain localization across a range of rheological regimes: from brittle faulting in shallow regions to ductile shear at depth. The strength of the lithosphere peaks in its cold ductile region around 40–80 km depth (e.g., Karato, 2008; Kohlstedt et al., 1995, pp. 338–358) and thus presents the main impediment to shear localization and the formation of tectonic plate boundaries. Many modern plate boundaries appear to form over preexisting weak zones (Crameri et al., 2020; Gurnis et al., 2000; Hall et al., 2003; Lebrun et al., 2003; Stern, 2004; Stern and Bloomer, 1992; Tommasi et al., 2009; Toth and Gurnis, 1998), which implies that the mechanism responsible for lithospheric weakening is long lived and allows for the inheritance and reactivation of older plate boundaries. The occurrence of strain localization requires a self-weakening positive feedback mechanism,

1 Introduction

whereby deformation causes weakening, which leads to increased deformation, further weakening, and so on. Lithosphere and crustal rocks can weaken through a number of phenomena, such as an increase in temperature, the presence of fluids, and the changes in microstructure. Frictional heating in a deforming rock and the associated strain localization due to thermal weakening allows for some strain localization in the lithosphere. However, the weakening effects of shear heating are mitigated by thermal diffusion, which rapidly erases the warm weak regions over a few million years (Bercovici, 1998; Bercovici and Karato, 2003; Bercovici et al., 2015b; Lachenbruch and Sass, 1980). In addition, the accelerated grain growth at higher temperatures and the resulting increase in viscosity further curbs the weakening effect of frictional heating (Foley, 2018; Kameyama et al., 1997). The presence of fluids, such as water, in the lithosphere may induce weakening through lubrication of plate boundaries at subduction zones or by reducing friction through increasing pore pressure (Bercovici, 1998; Gerya et al., 2008). While the rheological effects of fluids may be long lived, because of relatively slow diffusion of hydrogen in minerals, they are restricted to shallow lithospheric depths (i.e., the upper tens of kilometers), since it is prohibitively difficult to deliver water to the deepest regions of the lithosphere (e.g., Korenaga, 2017). Microstructural influences on rock strength include viscous anisotropy, for example, by crystal preferred orientation, or CPO (Durham and Goetze, 1977; Hansen et al., 2012b; Zhang and Karato, 1995), mineral grain-size reduction through dynamic recrystallization (DRX), microcracking, or other forms of damage (see Bercovici et al., 2015b; Mulyukova and Bercovici, 2019b, for recent reviews). Lithospheric weakening through changes in microfabric can be persistent, since, once deformation ceases, erasing the microstructure (e.g., randomizing crystal orientations in the

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case of CPO, or grain growth in the case of grain-size reduction) is much slower than convective mantle overturn, thus allowing for inheritance and reactivation of lithospheric weak zones (Bercovici and Ricard, 2014; Tommasi et al., 2009). In areas where deeper portions of the lithosphere are exposed at the Earth’s surface, such as in ophiolites (Hansen et al., 2013; Jin et al., 1998; Kohlstedt and Weathers, 1980; Linckens et al., 2011; Skemer et al., 2010; Warren and Hirth, 2006), severe deformation is coincident with highly reduced grain size, as is evident in mylonites and ultramylonites (Furusho and Kanagawa, 1999; White et al., 1980). Strain weakening by grain-size reduction is also commonly seen to accompany rock deformation in experimental studies (De Bresser et al., 1998; Green and Radcliffe, 1972; Hansen et al., 2012a; Karato et al., 1980). CPO microfabric is observed in lithospheric shear zones as well, but is restricted to the coarse-grained and less deformed parts of the rock (Ebert et al., 2007; Linckens et al., 2011). Thus, the rheological feedbacks due to CPO development are probably more important in the earlier stages of moderate rock deformation (Durham and Goetze, 1977; Ebert et al., 2007; Hansen et al., 2012a, b; Zhang and Karato, 1995), while the effect of grain-size reduction dominates for extreme deformations evident in mylonites. The mechanisms responsible for mylonitization and grain-size reduction are thought to play a key role in lithospheric shear localization, and grain-damage theory provides a theoretical framework for understanding these mechanisms. Specifically, the theory describes the positive feedback between rock deformation, grain-size reduction, and resultant mechanical weakening, which together may lead to lithospheric shear localization, and, ultimately, plate-boundary formation (Austin and Evans, 2007; Bercovici and Ricard, 2005, 2012, 2014, 2016; Rozel et al., 2011). In what follows, we

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2. The Physics and Origin of Plate Tectonics From Grains to Global Scales

outline the physics of grain-damage theory, its applications to some leading questions in the origin and operation of plate tectonics, and lastly to the recent and future directions.

The conventional notion of viscosity μ is defined such that τ ¼ 2μe_ , in which case (1) yields: μ¼

2. Grain-damage physics 2.1 Grain damage in monominerallic materials A rock’s rheological behavior is determined by the creep mechanisms by which it deforms under deviatoric stress. In the deeper ductile portion of the lithosphere, rocks deform predominantly by diffusion and dislocation creep. The dominant creep mechanism is commonly assumed to be the one that induces fastest strain, or is most efficient at releasing stress; in this case, given only diffusion and dislocation creep, and assuming isotropic behavior, this leads to a composite rheological law in which the strain rates of each creep mechanism are summed in series:

 B Rm e_ ¼ Aτn1 + m τ ¼ Aτn1 1 + Fm τ (1) R R where e_ and τ are the strain rate and stress tensors, respectively; τ2 ¼ 12 τ : τ is the second invariant of the stress tensor; B and A are the diffusion and dislocation creep compliances; m and n are grain size and the stress exponents, respectively; and R is the mean grain size of a rock sample (see Table 1 for typical parameter values). Moreover, we have introduced the rheological field boundary grain size
1=m B RF ¼ (2) Aτn1 at which the strain rates associated with the two creep mechanisms are equal. At a given stress, grains that are smaller than RF deform predominantly by diffusion creep, and grains larger than RF deform by dislocation creep.

τ1n
 Rm F 2A 1 + m R

(3)

When diffusion creep is dominant (i.e., for R ≪ RF), μ is sensitive to grain size R, and grain-size reduction leads to softening. When dislocation creep is dominant, (R ≫ RF) viscosity is grainsize insensitive, but is stress dependent for n6¼1; for creep in rocks, n > 1 and thus an increase in stress also leads to softening. A rock’s viscosity is also profoundly affected by temperature, which enters through the rheological compliances B and A (see Table 1). Cooling by a few hundred degrees Celsius can induce orders of magnitude increase in viscosity, depending on the activation energies of the different creep mechanisms. The peak strength in the ductile lithosphere, and thus the main bottleneck to lithospheric deformation and plateboundary formation, falls within the temperatures 800–1400 K (Kohlstedt et al., 1995). That the lithosphere is the coolest layer of the mantle also makes it the strongest, and to a large extent the challenge of plate-boundary generation is undoing so-called thermal stiffening. Convective mantle cooling acts as an engine that supplies mechanical work to deform the lithosphere. For our composite rheological law (1), the rate at which deformational work is done to a rock is

m  RF Ψ ¼ e_ : τ ¼ 2Aτn+1 1 + (4) R There are two ways in which deformational work can be deposited: as irrecoverable energy, in which work is dissipated as heat and contributes to entropy production, and as recoverable energy, in which work is stored as new grain boundary (or surface) energy upon the formation of new grains and subgrains (Austin and Evans, 2007;

9

2 Grain-damage physics

TABLE 1

Material and model properties.

Property

Symbol

Value/definition

Dimension

Gas constant

RG

8.3144598

J K1 mol1

Shear modulus

G

70

GPa

Length of Burgers vector

b

0.50

nm

Surface tension

γ

1

J m2

ϕi

ϕ1 ¼ 0.4, ϕ2 ¼ 0.6

η

3ϕ1ϕ2  0.72

Phase-volume fractiona a

Phase distribution function Dislocation creep

b

e_ disl ¼ Aτn

Activation energy

Edisl

530

kJ mol1

Prefactor

A0

1.1  105

MPans1

Stress exponent

n

3.5

Compliance

A

Þ A0 exp ð REdisl GT

MPans1

Diffusion creepb

e_ diff ¼ Brm τ

Activation energy

Ediff

300

kJ mol1

Prefactor

B0

13.6

μmm MPa1 s1

Grain-size exponent

m

3

Compliance

B

Þ B0 exp ð REdiff GT

μmm MPa1 s1

Activation energy

EG

200

kJ mol1

Prefactor

G0

2  104

μmp s1

Exponent

p

2

Grain-growth rate

GG

G0 exp ð REGGTÞ

Exponent

q

4

Interface coarsening rate

GI

q qp GG p ðμmÞ 250

Grain growthc

μmp s1

Interface coarseningd

a

μmq s1

Phase distribution function from Bercovici and Ricard (2012), assuming a peridotite mixture with volume fractions of 40% and 60% pyroxene and olivine, respectively. b Our model values for rheological parameters are representative of experimentally determined olivine creep laws (Hirth and Kohlstedt, 2003; Karato and Wu, 1993). c Olivine grain-growth law from Karato (1989) after Kameyama et al. (1997); note that the value for EG has been debated (e.g., Evans et al., 2001). d Interface coarsening law from Bercovici and Ricard (2013), based on the analysis done in Bercovici and Ricard (2012).

10

2. The Physics and Origin of Plate Tectonics From Grains to Global Scales

Bercovici and Ricard, 2005; Ricard and Bercovici, 2009). The splitting of grains into new smaller grains is part of the process of DRX, which is accommodated by the migration and accumulation of dislocations to form new grain boundaries, which eventually leads to a reduction in average grain size. Storing mechanical work as new surface energy on grain boundaries is referred to as grain damage, since it weakens the material through grain reduction, provided the material is deforming through a grain-size-sensitive (GSS) mechanism such as diffusion creep. While deformational work acts to reduce grain size by damage, it is counteracted by the process of coarsening or grain growth, or the tendency toward minimum net grainboundary surface energy. Specifically, grain growth occurs because smaller grains have greater boundary curvature and surface tension than larger grains and are effectively squeezed to higher pressure and hence internal energy (i.e., chemical potential); this pressure difference, or equivalently energy contrast, causes diffusive transfer of mass from the smaller to the larger grains, inducing a net increase in average grain size. This process of normal grain growth is also known as healing, since it undoes some of the mechanical damage and stiffens the material while it is dominated by diffusion creep. Grain size thus evolves through the competition between grain growth and grain reduction, or between healing and damage. More generally, grain evolution occurs through changes in the grain-size distribution and the interaction between grain populations of different sizes (see Hillert, 1965; Ricard and Bercovici, 2009). To simplify the analysis of grain-size evolution, Rozel et al. (2011) approximated the grain-size distribution with a self-similar log-normal function, which retains the same shape through time such that its mean, variance, and amplitude are uniquely determined by one characteristic grain-size R (see Ricard and Bercovici, 2009, for the full analysis of grain-size evolution).

The resulting theoretical model of grain-size evolution in a continuum is derived from a nonequilibrium thermodynamic framework (de Groot and Mazur, 1984), wherein fluxes of mass driven by forces associated with grain growth and grain damage always satisfy positivity of entropy production, that is, the second law of thermodynamics (Ricard and Bercovici, 2009); when reduced to a model with a selfsimilar log-normal grain-size distribution, this leads to a grain-size evolution law: dR G 2λf R2 Aτn+1 ¼ p1  dt pR 3γ

(5)

The first term on the right-hand side of Eq. (5) describes grain growth, where G is a temper ature-dependent coarsening coefficient and p is a grain growth exponent. The second term on the right-hand side of Eq. (5) describes grain-size reduction by damage, where γ is the grain-boundary surface tension (see Table 1), 2 λ ¼ exp ð5σ2 Þ is specific to the log-normal grain-size distribution (Rozel et al., 2011, with dimensionless variance σ typically set to 0.8), and 2fAτn+1 is the amount of mechanical work that goes toward creating new grain boundary area and energy. New grain boundaries are created through DRX, in which dislocations merge to form subgrains and eventually new grains. DRX can only occur in grains, where the strain is predominantly accommodated by dislocation creep. Therefore, out of the total mechanical work Ψ from Eq. (4), only the work done by dislocation creep (2Aτn+1) can be used to form new grains (Austin and Evans, 2007; Rozel et al., 2011), and out of this work only a fraction f, called the damage partitioning fraction, goes toward creating new grain boundary area and energy. When no deformational work is done on the system (i.e., Ψ ¼ 0), the standard static grain growth relation is recovered (Evans et al., 2001), and for which the range of values for G and p has been determined experimentally (see Table 1).

2 Grain-damage physics

Grain size reaches a dynamical equilibrium when the rates of coarsening and damage are in balance. As inferred by Eq. (5), the steadystate grain size is a function of the deformational work rate and can therefore serve as a paleowattmeter (Austin and Evans, 2007; Rozel et al., 2011): that is, the measured grain size can be translated into work rate, based on the theoretical considerations above. Analogously, the observed grain size can be translated into the driving stress, thus serving as a paleopiezometer, based on the empirically deduced correlation between grain size and stress ( Jung and Karato, 2001; Karato et al., 1980; Post, 1977; Van der Wal et al., 1993). Furthermore, comparing the observed steady-state grain size and that predicted by the grain-damage model at different stress and temperature conditions can be used to determine the range of possible values for f (Mulyukova and Bercovici, 2017; Rozel et al., 2011). The work fraction f is found to be temperature dependent and, starting off roughly at 0.1 for coldest temperatures of 800 K, decreases several orders of magnitude as temperature increases to 1600 K, depending on the assumed activation energies for grain growth and rheological compliances, among others. As already noted, shear localization by grain damage requires a positive feedback, whereby grain-size reduction induces weakening, which focuses deformation, which in turn accelerates grain-size reduction, and so on. However, in monomineralic materials, grain-size reduction occurs during dislocation creep, while the necessary weakening due to GSS viscosity occurs during diffusion creep. That these two creep mechanisms are somewhat exclusive (except over a narrow range of grain sizes near the rheological field boundary, e.g., by dislocation accommodated grain-boundary sliding; see Hansen et al., 2011; Hirth and Kohlstedt, 1995) precludes the coexistence of grain reduction and self-weakening (e.g., De Bresser et al., 2001), and hence the localizing feedback leading to mylonites. However, actual lithospheric rocks

11

consist of at least two phases (olivine and pyroxene), and the presence of secondary phases is known to have a significant effect on grain-size evolution and the resulting material strength (Herwegh et al., 2005, 2011; Linckens et al., 2011, 2015). Moreover, field studies suggest that mylonitization preferentially occurs in the polymineralic domains, especially for upper mantle peridotites (Skemer et al., 2010; Warren and Hirth, 2006). To that end, Bercovici and Ricard (2012) developed the two-phase grain-damage theory to infer how the interaction between phases facilitates grain reduction, inhibits healing, and enhances shear localization.

2.2 Grain damage in polymineralic materials The two dominant mineral phases in the lithosphere are olivine and pyroxene, with effectively trace minerals remaining; in essence olivine comprises 60% of the lithosphere by volume and pyroxene most of the remaining 40%. Thus, we treat the polyminerallic lithosphere as a two-phase medium. The physical description of a grained two-phase continuum involves the volume fraction ϕi and the mean grain size Ri for each of the phases (where the subscript i ¼ 1 or 2 denotes the individual phases, for example, i ¼ 1 is for the secondary phase like pyroxene, and i ¼ 2 is for the primary phase like olivine), which are both functions of space and time. When the phases are unmixed, the interface that separates them is smooth, meaning that it has a large radius of curvature r (r is also referred to as interface roughness), and the interface area is minimized. When the phases are well mixed, in which one phase is well dispersed through the other phase, the interface is rougher and has a small radius of curvature r. The two-phase grain-damage theory tracks the coupled evolution of the interface roughness r and the grain size Ri of each of the phases i (see Bercovici and Ricard, 2012):

12

2. The Physics and Origin of Plate Tectonics From Grains to Global Scales

f r2 dr ηGI ¼ q1  I Ψ dt qr γI η

(6)

dRi Gi λR2 1 ¼ p1 Zi  i f G 2Ai τn+1 i Zi dt 3γ i pRi

(7)

where GI (Gi) and q ( p) are the rate and the exponent of interface (grain boundary) coarsening, respectively, γ I (γ i) is the interface (grain boundary) surface energy, respectively, and Zi is the Zener pinning factor, to be discussed shortly. The phase distribution function η ¼ 3ϕ1ϕ2 ensures that the interface area vanishes in the limit of ϕi ! 0 or 1 (Bercovici et al., 2001, Section 2.2). The partitioning fractions fI and fG determine how much of the deformational work is partitioned into new interface and grain boundary area, respectively (see also Bercovici and Ricard, 2016), where the work to produce more grain boundary area is restricted to dislocation creep (see Section 2.1). The work to produce more interface area is tapped from the phase-volume average of the total work rate (Ψ ¼ Σi ϕi Ψ i ), since any mode of deformation will cause interface distortion by stretching, rending or mixing. The range of possible values for fI and fG were determined in Bercovici and Ricard (2016) and Mulyukova and Bercovici

Pinning slows grain growth Single phase: coarsening

Low surface and internal energy

where λ* ¼ 0.87 depends on the assumed shape of the grain-size distribution (Bercovici and Ricard, 2012, Appendix F.4). The evolution laws for grain sizes in each phase given by Eq. (7) are similar to that for the single phase (Section 2), except that they are also affected by the interface between the phases in two ways. First, the interface obstructs grain-boundary migration and hence retards grain growth (Fig. 1A). Specifically, if a grain can only grow by wrapping itself around an

Pinning helps damage

Two phases: coarsening impeded by pinning

High surface and internal energy

(2017), by comparing the theoretically predicted steady-state grain sizes to the ones reported from experimental ( Jung and Karato, 2001; Karato et al., 1980; Linckens et al., 2014; Post, 1977; Van der Wal et al., 1993) and field data (e.g., Herwegh et al., 2011; Linckens et al., 2011, 2015) for different stress and temperature conditions. As before, the factor λ is specific to the log-normal grain-size distribution. The Zener pinning factor Zi in Eq. (7), which couples the grain size and the interface roughness evolution, is defined as (Bercovici and Ricard, 2012):
2 Ri * Zi ¼ 1  λ ð1  ϕi Þ (8) r

Low surface and internal energy

High surface and internal energy

Damage: work goes to high surface energy

High surface and internal energy

damage: less work needed for high surface energy

High surface and internal energy

(A)

(B)

FIG. 1 Cartoon of how Zener pinning distortion increases the mean curvature and surface and internal energy of large grains, and thus impedes grain growth driven by energy contrasts (A) and facilitates grain-boundary damage (B).

2 Grain-damage physics

obstacle (i.e., a grain of a different phase) or by wedging itself between two grains of opposite phase, then the associated grain boundary distortion involves an increase in grain-boundary area, which requires additional surface energy, and therefore no longer becomes energetically favorable. Obstruction of grain growth by the interface is also known as Zener pinning: for small Ri/r, pinning is negligible (Zi  1), but as Ri/r becomes larger and Zi approaches 0, as indicated by Eq. (8), pinning impedes coarsening (first term in Eq. 7), as is classically inferred for Zener pinning theory (e.g., Hillert, 1965, 1988; Manohar et al., 1998; Smith, 1948). The second effect of Zener pinning on grainsize evolution, as shown by Bercovici and Ricard (2012), is that when Zi decreases, the grain-size reduction by damage gets amplified. In particular, because of grain-boundary distortion by pinning bodies, it requires less energy to split grains in a two-phase medium, than in the single-phase one (Fig. 1B). In total, Eqs. (6), (7) show that in a polymineralic medium, grain damage is expressed in two forms, grain-boundary damage (i.e., reduction in Ri) and interface damage (i.e., reduction in r). Specifically, a fraction of work can be stored both as grain-boundary energy within a given phase (as in the monomineralic case), as well as interface energy between phases, through rending, stretching, and stirring of the interface when the rock deforms. The interface between phases gets distorted and its radius of curvature r decreases, while its area and energy increase at the cost of deformational work. Importantly, work can be transformed into interface damage regardless of creep mechanism, since such damage only requires that the medium deforms, one way or another. Thus, the roughness of the interface r, or the size of the pinning bodies, can shrink to sizes below the field boundary grain size, into the GSS diffusion creep regime. Shrinking pinning bodies drive down the grain size with them, through the dual Zener pinning effects, that is, enhancing damage by DRX even for small grains and

13

suppressing grain growth (Eq. 7). Interface damage thus allows for grain-size reduction and GSS rheology to coexist, providing a positive feedback mechanism for self-weakening of polymineralic rocks. The deformational work rate in Eqs. (6), (7) uses similar rheological relations to the composite rheology (Eq. 1), but for each phase i: τi e_ i ¼ Ai τni + Bi m (9) Ri where Ai and Bi are again the rheological compliances for dislocation and diffusion creep, respectively, e_ i and τi are the square root of the second invariant of the strain rate and stress tensors, respectively, and Ri is the mean grain size, with the grain-size distribution assumed to be log-normal in each of the phases. If the phases at a given point are assumed to have no relative motion, then their velocities and hence strain rates e_ i are the same, which is thus equivalent to the homogeneous strain approximation. If there is relative motion of phases due to phase mixing, this assumption is less valid, and discussed here. Grain size partially governs the rheological properties of the rock, as it determines the creep regime (Eq. 2) and the rock’s viscosity in the GSS regime. Thus, the microphysics of grain-size evolution affects the macroscopic processes of rock deformation. At the same time, the grainsize evolution laws (in single-phase [Eq. 5] as well as two-phase [Eqs. 6, 7] materials) depend on physical conditions (such as stress and temperature) that are governed by continuum-scale processes. In a full system of equations that describe rock deformation, the microscale and macroscale processes are connected by coupling the governing continuum equations, including the conservation of mass, momentum, and energy, through the rheological relations, to the grain-size evolution laws (for monomineralic [Eq. 5] or polymineralic [Eqs. 6, 7] materials). The complete system of coupled equations can be applied to geodynamic models featuring grain damage (see examples in Section 3).

14

2. The Physics and Origin of Plate Tectonics From Grains to Global Scales

2.3 Grain mixing and hysteresis The effectiveness of two-phase damage, especially the dual roles of Zener pinning, is controlled by the degree to which the different phases are interspersed or mixed with each other at the grain scale. The process of phase mixing, and its effect on rock rheology and shear localization, has been the focus of numerous recent experimental and theoretical studies (Bercovici and Mulyukova, 2018, 2021; Bercovici and Ricard, 2016; Bercovici and Skemer, 2017; Cross and Skemer, 2017; Tasaka et al., 2017a, b, 2020; Wiesman et al., 2018). At low pressures (i.e., 0 is a damping parameter, and set k ¼ k + 1. In step 2, the definition of the Jacobian leads to different nonlinear solvers. For example, in a Newton-based scheme, we seek a correction δX such that F(Xk + δX) ¼ 0, which is approximated via a Taylor expansion FðXk + δXÞ  FðXk Þ+JðXk ÞδX + OðδX2 Þ, which when dropping higher-order terms results in a Jacobian given by ∂Fi ðXk Þ , ∂xj

ih i δv

h vi

¼  FFp : |fflfflfflffl{zfflfflfflffl}|ffl{zffl} |{z}

JðXk ÞδX ¼ FðXk Þ,

JðXk Þ ¼ rX FðXk Þ ¼

where we have explicitly indicated that the shear viscosity is a function of the velocity and pressure. If a used in conjunction with a strain-rate dependent flow law (e.g., dislocation creep), or a viscoplastic rheology (e.g., DruckerPrager), the resulting correction associated with the nonlinear problem, here expressed only in terms of (v, p) is given by

(25)

which is the functional derivative of the nonlinear residual F with respect to X in the direction Xk. Other definitions of J are possible including the Picard linearization in which all terms involving the gradient of F are dropped from Eq. (25). The precise choice of J will affect how quickly (in terms of iterations) the nonlinear solver converges to jjF(Xk)jj < ε. Considering the viscous flow problem in Eq. (1) the nonlinear residuals are given by 
 F v ¼ r  ηðv, pÞ rv + rvT  rp + ρg, (26) F p ¼ r  v,

Jvv

Jvp

D

0

J

δp

δX

(27)

F

Note the pressure-dependent viscosity results (in general) in a nonsymmetric saddle point problem as the (1, 2) block will not be the transpose of the (2, 1) block, even if the spatial discretization resulted in a symmetric saddle point problem in the linear case. When the flow law does not include a pressure dependence, Jvp ¼ G. Using the flow laws described in Section 2.3, direct application of Newton-type methods has been demonstrated, in general (i.e., for arbitrary pressure dependence), to result in ill-posedness, resulting in a nonuniqueness of the velocity and pressure solution (Spiegelman et al., 2016). A number of recent works have proposed a variety of method to resolve this important issue including: restoring symmetry of the operator in Eq. (27) (Fraters et al., 2019); regularizing the viscoplastic flow law (Duretz et al., 2021), and using a dual-primal formulation which treats the stress as an independent variable during the Newton linearization (Rudi et al., 2020). The dual-primal formulation has been demonstrated to be robust and efficient for von-Mises materials (Eq. 12 with φ ¼ 0); however, its applicability for Drucker-Prager plasticity has not been evaluated.

4.4 Free-surface evolution Modeling a free surface requires both a geometric representation of the interface and a technique to enforce the free-surface constraints

557

4 Model components and their challenges

(i.e., the boundary condition σn ¼ 0). Using a body-fitted FE method with a mesh which that conforms to the free surface, both requirements are trivially achieved. In a nonbody-fitted method, such as FD, different approaches have been utilized in the geodynamic community. The most extensively used technique to circumvent this issue, usually with FD methods, is the “sticky-air” method (Crameri et al., 2012; Schmeling et al., 2008). In this approach, a layer of thickness Hair with viscosity ηair and density ρ ¼ 0 is introduced above the interface defining the boundary between rock and air (Γ fs). The union of the “air” domain (Ωair) and rock domain defines a larger domain Ωsa ¼ Ω [ Ωair with an exterior boundary which we denote by ∂Ωsa. The free-surface interface is embedded within, that is, Γ fs \ ∂Ωsa ¼ Ø. The conservation laws from Section 2 are applied within Ωair. Suitable boundary conditions are required on the segment ∂Ωsa n ∂Ω. These boundary conditions are chosen to yield the best approximation of the free-surface boundary condition. While attractive from many implementation perspectives, controlling the approximation error associated with this approximate freesurface boundary condition is complicated as it depends on Hair, ηair and the boundary conditions applied along ∂Ωsa n ∂Ω. How these model parameters and choices influence the approximation error have been previously studied (Crameri et al., 2012). Nevertheless, the stickyair approximation does not converge in the manner usually expected, that is, under mesh refinement. For example, if the thickness of the air layer and boundary conditions are fixed, then the air viscosity controls the approximation error associated with σn ¼ 0. In practice, one usually expects that under mesh refinement, that is, when the spatial discretization h decreases, the approximation solution vh, ph converges (at some rate) to the true solution, lim ðvh , ph Þ ¼ ðv, pÞ: h!0

However, in the sticky-air models this is no longer true (Duretz et al., 2016), rather one has lim ðvh , ph Þ ¼ ðv, pÞ: h!0 ηair !0

Such convergence behavior mandates that if the solution quality is to improve as the mesh is refined (h ! 0), then one needs to choose the 0 0 sticky-air viscosity as ηair ¼ ηair (h/h0 )k, where ηair is a reference air viscosity defined on the coarsest mesh resolution h0 > h, and k is the usual order of accuracy of the flow problem (i.e., in the absence of the sticky-air approximation). The usefulness of the sticky-air approach is questionable given that convergence under mesh refinement requires ηair be reduced. Many alternatives for nonbody-fitted discretizations employing different interface capturing schemes exist: the ghost fluid method (Fedkiw et al., 1999), the immersed interface method (LeVeque and Li, 1994), CutFEM (Burman et al., 2015), and the finite cell method (Parvizian et al., 2007). Many of these methods are yet to be explored in the context of geodynamic simulations. One exception is presented in Duretz et al. (2016), where the authors extend the classic free-surface capturing technique from Harlow and Welch (1965) to the case of fluids with a variable viscosity. Including the temporal evolution of the free surface (Eq. 7) introduces new computational challenges. It is well documented that evolving the free surface in conjunction with buoyancydriven fluids with variable viscosity may require the use of unusually small time steps (Δt) for the time integration to remain numerically stable (Kaus et al., 2010). The “sloshing” instability (Fig. 4) stems from the use of an explicit time integrator to evolve Eq. (7) forward in time. If instead one considered using the A-stable, lowest order, backward difference formula (BDF) time integrator

558

22. Numerical Modeling of Subduction

FIG. 4 Illustration of the temporal instability associated with a gravity driven flow with a free surface in which a dense, more viscous layer (light gray) sinks through a less dense, less viscous fluid (dark gray). a) Stable evolution using Δt ¼ 2500 year resulting in gradual topography buildup. b) Unstable evolution using Δt ¼ 5000 year. Here, the velocity field at the free surface alternates in direction from one-time step to the next, leading to a “sloshing” motion. Modified from Kaus, B.J.P., M€ uhlhaus, H., May, D.A., 2010. A stabilization algorithm for geodynamic numerical simulations with a free surface. Phys. Earth Planet. Inter. 181 (1–2), 12–20, Copyright 2010, Elsevier.

xtk +Δt ¼ xtk + Δtvtk +Δt ,

(28)

the sloshing instability would not occur for any size Δt. We note that the velocity at time tk + Δt is a function of xtk +Δt , hence the implicit time integration of the coordinates requires a full nonlinear solver for Y ¼ (v, p, x)—even if the flow problem is inherently linear. The monolithic nonlinear problem is thus J0 δY ¼ F0 , F0 ¼ (Fv, Fp, x xk  ΔtZv). Here, ZNx Nv is either the identity or an injection operator which extracts the velocity at the free surface at each of the Nx coordinates. For a flow problem which was linear before introducing the free-surface evolution, the Jacobian J0 is

2

A

6 J0 ¼ 4 D ΔtZ

J013

G

3

7 J023 5, 032 Nx Np , I33 Nx Nx ,

0 032

I33 (29)

J013

N v N x

J023

N p N x

where ¼ rx F  , ¼ rx F  . Noting that the residual associated with the coordinates is linear in v, p, x and the presence of the identify in the (3, 3) slot, Eq. (29) can be reduced to the following (v, p) problem:

 0  00 A G J J ¼  13 ½ I33 1 ½ Δt Z 032  D 0 J023

0  J Z 012 ¼ J + Δt 13 : (30) J023 Z 0 v

p

559

5 Software

Solving Eq. (30) is challenging, and becomes increasingly difficult as Δt increases. Moreover, the Jacobian entries for J0 are complicated to write down analytically, especially J013 which involves the flow law. In short, the direct application of an implicit time integrator turned a linear problem into a nonsymmetric nonlinear problem—however, temporal stability is guaranteed. In Popov and Sobolev (2008), a cheaper approach was sought in which the mesh coordinates were updated using Eq. (28) during each Newton iteration associated with a problem given by Eq. (26). This approach can be thought of as a Picard linearization of the nonlinear term appearing in the reduced system (Eq. 30). Finding the balance between a time integrator which is practical (from the perspective of implementation effort and computational time required to solve the discrete flow problem), and defines a radius of stability allowing sufficiently large time steps to be used, has led to the development of numerous alternatives. The free-surface stabilization algorithm (FSSA) is one such approach which maintains a lowcomputational cost through utilization of an explicit time integrator, while remaining stable when large time steps are used by considering a perturbed form of Eq. (1) (Kaus et al., 2010). The radius of stability of the time integrator is enlarged by including a high-order approximation of the buoyancy forcing term (ρg) at time tk + Δt. The other advantages of the FSSA are that the problem remains linear (when linear flow laws are used) and the saddle point problem remains symmetric (if the discretization admits D ¼GT). While derived for variational formulations (problems expressed in terms of weak solutions), the FSSA method has been extended and applied to strong formulations (Duretz et al., 2011). The FSSA method is widely used as the stable time step, which can be used within a subduction simulation can be increased by up to two orders of magnitude compared to a simulation without the FSSA. Numerous other similar stabilization techniques have been

developed in the geodynamic modeling community (Andres-Martı´nez et al., 2015; Kramer et al., 2012; Rose et al., 2017). The need for enhanced stability of the time integrator used to advance the coordinates of the free surface is independent of the specific geometric representation and boundary condition capturing approach. However, certain choices may suppress temporal instabilities. For instance, sticky-air models using a value for ηair that is comparable to the background rock viscosity will not tend to exhibit the “sloshing” instability.

5. Software In the modern era of scientific computing, the concept and benefits of open-source software (OSS) is well established and a driver of innovation (von Krogh and von Hippel, 2006). The computational geodynamics community has embraced the ideals of OSS and many highquality geodynamic software packages are built from external open-source software libraries, as well as themselves being open-source and thus publicly available to the community (e.g., Hwang et al., 2017). In the following, we review several open-source packages, which are dedicated to geodynamics, as well as libraries dedicated to the discretization of PDEs and to the solution of the resulting discrete systems.

5.1 Geodynamic software Currently, there are no open-source software libraries which are dedicated to performing simulations of subduction. However, there are numerous open-source software packages specifically developed for simulating geodynamic processes. Here, we discuss several which have previously been used to model subduction. We restrict the presentation to software which uses an inf-sup stable discretizations for the Stokes problem. These software packages are summarized in Table 1.

TABLE 1 Nonexhaustive summary of geodynamic software projects used for subduction simulations. Project

Language

Discretization

HPC support

Other features

Subduction studies

Underworlda (Mansour et al., 2019; Moresi et al., 2007)

C, Python API

b0 for Finite elements; 1 – Stokes; 1 SUPG for temperature; structured mesh, quadrilateral (2D) and hexahedral (3D) elements; material point method (Lagrangian particles for lithology [C] and history variables [X k ])

MPI

Segregated solver; geometric multigrid for the viscous block A; supports a true free-surface BC (Eqs. 3, 7) and the “sticky-air” approximation

Compositional buoyancy, thermochemical buoyancy; time-dependent material domains; nonlinear flow laws; 2D, 3D; see Stegman et al. (2006), Sharples et al. (2016), Schellart and Strak (2021), and Strak and Schellart (2021)

pTatin3Db (May et al., 2014, 2015)

C, CUDA

b1 for Finite elements; 2 – Stokes; 1 SUPG or FV for temperature; structured mesh, hexahedral elements, 2D†, 3D; material pointmethod (Lagrangian particles for lithology [C] and history variables [X k ])

MPI, OpenMP, GPU

ALE; coupled solver with segregated preconditioner; Newton nonlinear solvers; hybrid AMG-geometric multigrid for A; supports a true free-surface BC (Eqs. 3, 7)

Thermochemical buoyancy; thermomechanical; time-dependent material domains; nonlinear flow laws; 2D; see Mao et al. (2017) and Hamai et al. (2018)

LaMEMc (Kaus et al., 2016)

C++/C

stagFD for Stokes; FD + marker-in-cell for temperature; Lagrangian particles for lithology (C) and history variables (X k ); 2D†, 3D

MPI

Coupled multigrid preconditioner; Newton nonlinear solvers; support for adjoints. Free-surface BC approximated using the “sticky-air” approach

Thermal and compositional buoyancy; supports a “sticky-air” approximation to the true free-surface BC (Eq. 3, 7). Thermomechanical; time-dependent material domains; nonlinear flow laws; 2D, 3D; see Yang et al. (2021), Pusok et al. (2018), Pusok and Stegman (2019), and Wang et al. (2021)

ASPECTd (Heister et al., 2017; Kronbichler et al., 2012)

C++

Finite elements; 2 –1 for Stokes; 2 SUPG or entropy viscosity for temperature and composition/lithology; quadrilateral (2D) and hexahedral (3D) elements;

MPI

ALE; dynamic/static mesh refinement (e.g., Fig. 3d); coupled solver with segregated preconditioner; Newton nonlinear solvers; AMG or geometric multigrid for A; physics

Thermal buoyancy, thermochemical buoyancy, compositional buoyancy; time-dependent material domains; nonlinear flow laws; 2D, 3D; see Glerum

structured quad/oct tree meshes; grid-based or particle-based representation of lithology (C) and history variables (X k ) Fluiditye (Davies et al., 2011; Piggott et al., 2008)

C++, C, Fortran90, Fortran77, Python API

Finite elements; 2 –1 for Stokes, 2 or FV for temperature; unstructured mesh, triangle (2D) and tetrahedral (3D) elements; lithology is represented as a volume fraction and solved using FV

MPI

support for compressibility, two-phase flow; supports a true freesurface BC (Eqs. 3, 7) and the “sticky-air” approximation

et al. (2018) and Holt and Condit (2021)

Anisotropic adaptive mesh refinement; segregated solver with AMG preconditioner for A; supports a true freesurface BC (Eqs. 3, 7) and the “sticky-air” approximation

Thermal buoyancy; thermomechanical; timedependent material domains; nonlinear flow laws; 2D; see Perrin et al. (2018) and Garel et al. (2014)

Information presented based on statements made in project-specific publications and may not reflect the current (development) status of each project. † indicates the method can define a 2D domain by using a mesh with one element in the third dimension. a https://underworld2.readthedocs.io. b https://bitbucket.org/ptatin/ptatin3d. c https://bitbucket.org/bkaus/lamem. d https://aspect.geodynamics.org. e https://fluidityproject.github.io.

562

22. Numerical Modeling of Subduction

5.2 PDE libraries 5.2.1 Automated finite elements FEniCSc (Alnæs et al., 2014, 2015; Logg and Wells, 2010; Logg et al., 2012) is computational platform for solving PDEs that enable users to rapidly translate scientific models into efficient finite element code. This is achieved via a domain-specific language and an expressive Python interface that allows users to define weak forms using syntax closely resembling the mathematics used to describe variational problems. FEnciCS provides a large family of function spaces including the usual continuous and discontinuous Lagrange elements, as well as more exotic spaces such as Raviart-Thomas (RT), Brezzi-Douglas-Marini (BDM), and Nedelec. Importantly, the choice of FE function space is decoupled from the definition the userdefined weak form. These features provide enormous generality with respect to the PDE which can be defined, and how the PDE can be approximated (discretized). FEniCS opens the door to an unlimited number of complex, coupled multiphysics problems which can be defined and solved with minimal effort on the part of the application programmer (cf. hand rolling your own finite element code). To assist application programmers in managing choices associated with the PDE components c

relevant to geodynamics, software tools built on top of FEniCS exist (Wilson et al., 2017). We also note that an extension to FEniCS exists, which provides support for Lagrangian particles, and this functionality has been applied to geodynamic simulations of Rayleigh-Taylor instabilities (Maljaars et al., 2021). Other automated FE packages similar in design and scope to FEniCS exist—for example, Firedraked (Rathgeber et al., 2016). Firedrake has been demonstrated to be suitable for large-scale parallel computations of 2D and 3D thermal convection in the Earth (Davies et al., 2022). 5.2.2 Discretization libraries A large number of discretization software libraries exist, primarily for finite elements. Several widely used packages include: deal.II,e libMesh,f MFEM,g Dune,h Feel++,i and FreeFEM.j To the authors knowledge, there are few open-source, general-purpose finite difference software packages which offer similar flexibility to FEniCS or Firedrake. One project that does provide a generic way to describe arbitrary PDEs and discretize them via finite differences is Scikit-fdiff.k Despite the lack general-purpose PDE-based FD frameworks, numerous generic, and high-performance finite difference stencil libraries exist, for example, Devito.l With the exception of deal.II, the high-quality FE libraries

See https://fenicsproject.org. As of 2019, FEniCS is no longer actively developed. FEniCSx is the name given to the latest development version of FEniCS, with the problem-solving environment now known as DOLFINx (https:// github.com/FEniCS/dolfinx). d

See https://www.firedrakeproject.org.

e

See https://www.dealii.org.

f

See https://libmesh.github.io.

g

See https://mfem.org.

h

See https://www.dune-project.org.

i

See http://www.feelpp.org.

j

See https://freefem.org.

k

See https://scikit-fdiff.readthedocs.io.

l

See https://www.devitoproject.org.

563

6 Future directions

listed have not been exploited in the geodynamics modeling community. In large part, this may be because the perceived method of choice to represent lithology/material composition is via Lagrangian particle methods, and the packages listed earlier (except deal.II) do not support such methods. That said, for the subduction model variants which do not require Lagrangian material domains, any of the FE packages listed earlier are appropriate to use should one wish to develop a new subduction simulator.

5.3 Linear algebra libraries Once discretized, the resulting linear (e.g., Eq. 23) or nonlinear (e.g., Eq. 27) equations need to be solved be some computational means. There are two major linear algebra software libraries, which natively support parallelism and are thus suitable for high-resolution 2D and 3D subduction simulations: Trilinosm and PETSc. The following discussion focuses on PETSc due to its wide-spread use in the geodynamics community. PETScn (Balay et al., 1997, 2021)—The Portable, Extensible Toolkit for Scientific Computation—provides a suite of data structures and methods for the scalable (parallel) solution of scientific applications modeled by PDEs. PETSc supports MPI, and GPUs (through CUDA, HIP, or OpenCL), as well as hybrid MPIGPU parallelism. Besides fundamental support of linear algebra, PETSc provides a wide range of Krylov methods, preconditioners (including domain-decomposition methods [e.g., Zampini, 2016], algebraic and geometric multigrid components for extreme-scale computations [May et al., 2016], specialized block factorizations for multiphysics problems [Brown et al., 2012]), and advanced nonlinear solvers (Brune et al., 2015). m n

See https://trilinos.github.io. See https://petsc.org.

PETSc also provides discretization support for meshes: DMDA describes structured grids suitable for finite difference, finite element methods; DMSTAG describes a structured grid object designed for staggered grid finite differences and/or FV methods; and DMPLEX (Lange et al., 2016) is a general mesh object suitable for unstructured mesh representation comprised of cell types given by triangles, quadrilaterals, tetrahedra, hexahedra, and prisms/wedges. PETSc also includes an API to define discrete operators for FV and finite element methods. Generic high-level discretization support for particle methods is provided by DMSWARM. Such is its versatility and usefulness, PETSc is employed as the linear algebra back-end for the geodynamic packages Underworld, pTatin3D, LaMEM, Terra FERMa, Fluidity, and the automated finite element packages Firedrake and DOLFINx.

6. Future directions Advancing the state of the art in subduction modeling can be enhanced in several areas outlined here. • Flow discretizations: Finite element discretization comparison studies should be enlarged to explore macroelement continuous Galerkin FE, DG, and HDG discretizations for variable viscosity Stokes flow. A rigorous comparison and assessment of the trade-offs between methods is difficult as it should encompass solution accuracy, conservation properties, and practical matters such as the performance of efficient solvers, which are required by large-scale (high-resolution) 3D subduction models.

564

22. Numerical Modeling of Subduction

Adaptively refined meshes have been shown to be incredibly useful in subduction studies. Currently, this approach is only available to finite element discretizations. Adaptive meshes for stagFD have been developed in 2D (Gerya et al., 2013); however, the methodological developments required to use adaptive meshes in 3D with stagFD discretizations is lacking. • Subelement resolution: Multiscale methods should be investigated to better capture subelement material property variations. This may involve GMsFEM methods (Efendiev et al., 2013), homogenization techniques (Ma˚lqvist and Peterseim, 2020), or solving subcell problems with isotropic constitutive laws to define effective anisotropic constitutive behavior on the macrocell. • Enhanced solvers: There is a continual need for improving the robustness and efficiency of both the linear and nonlinear Stokes solvers used in subduction simulations. The efficiency of these solvers is of paramount importance as this ultimately dictates the time-to-solution of a subduction simulation. The suitability of multilevel domaindecomposition preconditioners have seemingly been unexplored for subduction, or geodynamic problems in general. Such methods represent practical alternatives to exact factorizations in terms of robustness; however, they possess much better parallel scalability and thus are more suitable for large-scale simulations. Methods such as finite element tearing and interconnecting— dual primal (FETI-DP), balancing domain decomposition by constraints (BDDC), and generalized eigenvalue in the overlap o

See https://www.khronos.org/opencl.

p

See https://github.com/ROCm-Developer-Tools/HIP.

q

See https://kokkos.org.

r

See https://julialang.org.

(GenEO) are suitable for solving A, Jvv (as required for a segregated solver), and also A,J as required for a coupled Stokes solver. Massively parallel implementations exist and are accessible via PETSc and thus should be evaluated for subduction problems (e.g., Jolivet et al., 2013; Zampini, 2016). We are at the dawn of the exascale computing era and the next generation of subduction simulations need to exploit such computational resources. Realizing the full potential of the next-generation exascale hardware is complicated, in part due to the trend to utilize onboard accelerators (GPUs). Such systems generally involve complex memory hierarchies, which are difficult for the application programmer to efficiently utilize manually, and moreover, hinder the development of performance portable code between different hardwares. New software tools and languages (e.g., OpenCL,o HIP,p Kokkos,q Juliar ) help mitigate this issue. The next-generation subduction simulations should be developed with awareness of such paradigms. Furthermore, changing trends in hardware require reassessment of current state-of-the-art discretizations and solvers in order to best utilize the available computational resources (e.g., R€ass et al., 2022). • Physics: Incorporating new physical processes will be essential to further the state of knowledge of subduction dynamics. Including the transport of melt and fluids (i.e., two-phase porous flow) is an essential component of subduction. Work in this direction has already commenced (Cagnioncle et al., 2007; Dannberg et al., 2019; Wilson et al., 2014). Performing two-phase

References

flow simulations over geological timescales (Myr) in a fully coupled and self-consistent manner requires: (i) suitable formulations of the governing equations (conservation of momentum, mass, energy, chemical species) which are stable across relevant parameter regimes (e.g., porosity (ϕ) within the range 0  ϕ < 1); (ii) advanced time integrators (e.g., implicit-explicit [IMEX] schemes) to efficiently deal with the wide range of time scales associated with the liquid velocity and solid velocity; and (iii) compatible flowtransport discretizations to accurately solve the solid-temperature and liquid-porosity/ concentrations systems. Furthermore, selfconsistent, robust 3D subduction models spanning across coseismic, interseismic, and geodological time scales need to be developed (e.g., in the spirit of van Dinther et al., 2013; van Zelst et al., 2019). • Predictive capabilities: The subduction community at large, presently does not have predictive capabilities comparable with the atmospheric/climate or physical oceanography communities. To enhance the predictive capabilities of subduction simulators, techniques allowing for quantifying solution variability as a function of model parameterizations (e.g., adjoint sensitivity, reduced-order models), data assimilation and in general, uncertainty quantification need to be incorporated into the next generation of open-source subduction simulation frameworks. While many of the methodological developments or software implementations exist in both the geodynamics community (Bocher et al., 2016; Bunge et al., 2003; Hu et al., 2018; Li et al., 2017; Liu et al., 2008; Ortega-Gelabert et al., 2020) and the applied mathematics community (Brunton et al., 2016; Bui-Thanh et al., 2008; O’LearyRoseberry et al., 2022; Rudy et al., 2017; Villa et al., 2021), they are not currently incorporated within any open-source community framework for simulating subduction.

565

In closing, we have summarized the governing equations commonly used to describe physics-based models of the long-term evolution of the Earth’s interior, and presented an overview of different discretizations, which can be used to obtain numerical solutions of such models. Some of the inherent complexities associated with obtaining accurate, stable, and thus reliable numerical solutions have been presented, as well as state-of-the-art modeling techniques to circumvent them. While we have centered our presentation of numerical modeling around subduction, the content presented here is directly applicable to other long-term geodynamic processes.

Acknowledgments I am indebted to conversations with my friends and colleagues (in alphabetical order) Jed Brown, Alice Gabriel, Taras Gerya, Mike Gurnis, Anthony Jourdon, Richard Katz, Boris Kaus, Laetitia Le Pourhiet, and Marc Spiegelman, who have directly and indirectly influenced the contents of this chapter.

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He, Y., Puckett, E.G., Billen, M.I., 2017. A discontinuous Galerkin method with a bound preserving limiter for the advection of non-diffusive fields in solid Earth geodynamics. Phys. Earth Planet. Inter. 263, 23–37. Heister, T., Dannberg, J., Gassm€ oller, R., Bangerth, W., 2017. High accuracy mantle convection simulation through modern numerical methods-II: realistic models and problems. Geophys. J. Int. 210 (2), 833–851. https://doi.org/ 10.1093/gji/ggx195. Hillebrand, B., Thieulot, C., Geenen, T., Van Den Berg, A.P., Spakman, W., 2014. Using the level set method in geodynamical modeling of multi-material flows and Earth’s free surface. Solid Earth 5 (2), 1087–1098. Holt, A.F., Condit, C.B., 2021. Slab temperature evolution over the lifetime of a subduction zone. Geochem. Geophys. Geosyst. 22 (6). e2020GC009476. Hu, J., Liu, L., Zhou, Q., 2018. Reproducing past subduction and mantle flow using high-resolution global convection models. Earth Planet. Phys. 2 (3), 189–207. Hwang, L., Fish, A., Soito, L., Smith, M., Kellogg, L.H., 2017. Software and the scientist: coding and citation practices in geodynamics. Earth Space Sci. 4 (11), 670–680. https://doi.org/10.1002/2016EA000225. Ismail-Zadeh, A., Fucugauchi, J.U., Kijko, A., Takeuchi, K., Zaliapin, I., 2014. Extreme Natural Hazards, Disaster Risks and Societal Implications. vol. 1 Cambridge University Press. Jolivet, P., Hecht, F., Nataf, F., Prud’Homme, C., 2013. Scalable domain decomposition preconditioners for heterogeneous elliptic problems. In: SC ’13. Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, Association for Computing Machinery, New York, NY. Kaus, B.J.P., M€ uhlhaus, H., May, D.A., 2010. A stabilization algorithm for geodynamic numerical simulations with a free surface. Phys. Earth Planet. Inter. 181 (1–2), 12–20. Kaus, B., Popov, A.A., Baumann, T., Pusok, A., Bauville, A., Fernandez, N., Collignon, M., 2016. Forward and inverse modelling of lithospheric deformation on geological timescales. In: Proceedings of NIC Symposium, vol. 48. John von Neumann Institute for Computing (NIC), NIC Series. Kramer, S.C., Wilson, C.R., Davies, D.R., 2012. An implicit free surface algorithm for geodynamical simulations. Phys. Earth Planet. Inter. 194, 25–37. Kronbichler, M., Heister, T., Bangerth, W., 2012. High accuracy mantle convection simulation through modern numerical methods. Geophys. J. Int. 191, 12–29. https://doi.org/10.1111/j.1365-246X.2012.05609.x. Lange, M., Mitchell, L., Knepley, M.G., Gorman, G.J., 2016. Efficient mesh management in Firedrake using PETSC DMPLEX. SIAM J. Sci. Comput. 38 (5), S143–S155. Lehmann, R.S., Luka´cova´-Medvid’ova´, M., Kaus, B.J.P., Popov, A.A., 2016. Comparison of continuous and discontinuous Galerkin approaches for variable-viscosity

Stokes flow. ZAMM-J. Appl. Math. Mech./Zeitschrift f€ ur Angewandte Mathematik und Mechanik 96 (6), 733–746. LeVeque, R.J., Li, Z., 1994. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31 (4), 1019–1044. Li, Z.-H., Ribe, N.M., 2012. Dynamics of free subduction from 3-D boundary element modeling. J. Geophys. Res. Solid Earth 117 (B06408), 1–18. Li, J., Widlund, O.B., 2007. On the use of inexact subdomain solvers for BDDC algorithms. Comput. Methods Appl. Mech. Eng. 196 (8), 1415–1428. Li, D., Gurnis, M., Stadler, G., 2017. Towards adjoint-based inversion of time-dependent mantle convection with nonlinear viscosity. Geophys. J. Int. 209 (1), 86–105. Liu, L., Spasojevic, S., Gurnis, M., 2008. Reconstructing Farallon plate subduction beneath North America back to the Late Cretaceous. Science 322 (5903), 934–938. Logg, A., Wells, G.N., 2010. DOLFIN: automated finite element computing. ACM Trans. Math. Softw. (TOMS) 37 (2), 1–28. Logg, A., Mardal, K.-A., Wells, G., 2012. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. vol. 84 Springer Science & Business Media. Louis-Napoleon, A., Gerbault, M., Bonometti, T., Thieulot, C., Martin, R., Vanderhaeghe, O., 2020. 3-D numerical modelling of crustal polydiapirs with volume-of-fluid methods. Geophys. J. Int. 222 (1), 474–506. Maljaars, J.M., Labeur, R.J., Trask, N., Sulsky, D., 2019. Conservative, high-order particle-mesh scheme with applications to advection-dominated flows. Comput. Methods Appl. Mech. Eng. 348, 443–465. Maljaars, J.M., Richardson, C.N., Sime, N., 2021. LEoPart: a particle library for FEniCS. Comput. Math. Appl. 81, 289–315. Ma˚lqvist, A., Peterseim, D., 2020. Numerical Homogenization by Localized Orthogonal Decomposition. SIAM. Mansour, J., Giordani, J., Moresi, L., Beucher, R., Kaluza, O., Velic, M., Farrington, R., Quenette, S., Beall, A., 2019. underworldcode/underworld2: v2.8.1b., https://doi. org/10.5281/zenodo.3384283. Mao, X., Gurnis, M., May, D.A., 2017. Subduction initiation with vertical lithospheric heterogeneities and new fault formation. Geophys. Res. Lett. 44 (22), 11–349. May, D.A., Moresi, L., 2008. Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics. Phys. Earth Planet. Inter. 171 (1–4), 33–47. May, D.A., Brown, J., Le Pourhiet, L., 2014. pTatin3D: highperformance methods for long-term lithospheric dynamics. In: SC’14: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 274–284.

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C H A P T E R

23 Literate, Reusable, Geodynamic Modeling Louis Moresi Research School of Earth Sciences, The Australian National University, Canberra, ACT, Australia

1. Introduction I find it remarkable that we can write mathematical expressions that are able to describe the natural world in exquisite detail. Even more remarkable, perhaps, is that the same mathematical expressions can be solved using algorithms that manipulate finite, numerical quantities. Those algorithms can be executed laboriously with pencil and paper or by manipulating binary logic in an electronic computer. Geodynamics is a discipline that has wholeheartedly embraced this form of computation as a means of understanding a slowly evolving planet which we can observe in detail only in the present and whose history we know only at the surface. It seems quite natural to hope that our computational models should, like other computational systems, grow in complexity and capability through the cumulative efforts of researchers over time. The open-source movement has shown us the way, and in contrast to Isaac Newton’s often quoted comment to Newton (1675): “… if I have seen further, it is by standing on the shoulders of giants.”

Dynamics of Plate Tectonics and Mantle Convection https://doi.org/10.1016/B978-0-323-85733-8.00010-X

we actually stand on the collective shoulders of every single one of the many, hardworking individuals who have made a small contribution to our field. As somebody whose fervently believes in this process and has spent many years on the development of the tools and improving the practices of open computational science, there are some important lessons that I would like to share in this chapter.

2. A very brief history of computational geodynamics The Earth sciences is a discipline that underwent a complete rejuvenation with the development of the theory of plate tectonics following a post–World War II explosion in observational data from the oceans and from a global effort in seismic instrumentation. This rejuvenation coincided with the availability of digital computing to civilian researchers and the globally focused Earth science community, particularly in geodynamics, has advanced in parallel with advances in computation, and has contributed novel

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Copyright # 2023 Elsevier Inc. All rights reserved.

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algorithms in modeling and data analysis arising from the particular challenges of our discipline. Early computational geophysical research was published mainly as supplementary information in research papers with the focus on explaining algorithms, benchmarking, and resolution tests, but with no expectations around public sharing of source code or explicit discussion of licenses. Typically, very little value was given to the specific implementation of algorithms and the writing of research code (from scratch) was considered the appropriate apprenticeship for a PhD in computational geoscience. Code sharing became more important as the algorithmic complexity exceeded the scope of a single PhD. Perhaps one of the primary drivers for this shift was the adoption of the finite element method in many research groups (e.g., Bathe, 2008; Hughes, 1987; Zienkiewicz et al., 2013). Finite element codes, while offering great flexibility in mesh geometry and resolution, the ability to handle interfaces and jumps in material properties, a generic treatment of constitutive behavior and a common approach to discretizing different systems of equations, are burdened by an inevitable complexity of implementation that is difficult to reproduce from scratch and on demand. Extending this complexity with the need to operate in a parallel, highperformance environment has driven the development of open-source, community codes initially from the ground up. The example that I am most familiar with is the CITCOM code, a project that I initiated at Caltech (Moresi and Solomatov, 1995) and released as an opensource project that was adopted and expanded by the community to include parallel and spherical versions (Shijie Zhong et al., 1998; Zhong et al., 2000) and grew to become very widely used in the community to the point that the community then felt the need to take over the support and development of the code. In the early 2000s, a number of initiatives established open-source software development as a core activity in the geosciences and were

funded to promulgate good software engineering practices in coding as well as encouraging the adoption of source code revision control and regression/unit testing. These included the geoframeworks initiative and the Computational Infrastructure in Geodynamics (www. geodynamics.org/) organization in the USA, and AuScope (www.auscope.org.au) in Australia. The numerical methods for global geodynamics have typically been quite specialized and not well handled by commercial engineering codes. Models need to be capable of handling very large material deformation (fluid mechanics) while keeping track of localized deformation and post-failure material evolution (solid mechanics) which is one of the reasons that stand-alone software for geodynamics continues to exist. However, much of the generic, parallel, numerical underpinnings of these specialized packages is outsourced to open, community frameworks, most notably PETSc which provides well-tested, parallel data management, linear and nonlinear algebra packages, and an extensive library of finite element machinery (Balay et al., 1997).

3. From reproducibility to reusability There are a number of ways in which scientific research results can be positively repeated (reproduced, replicated) and the following definitions are widely understood, if not universal. For my work to be reproducible, it means that if you follow my detailed instructions, and use the same experimental setup as me, then you will obtain the same answer as me. This follows from the most elementary principle, that any scientific result that is published has already been repeated by the author sufficiently to expect it to be reproducible by anyone, and that the presence of experimental error has been considered and accounted for in the way that the results are presented. A more demanding form of positive repeatability is the replication of research

3 From reproducibility to reusability

results. To replicate my results, it should be possible to use any equivalent experimental setup, not specifically the one I used, and still obtain the same answer. Neither of these two forms of repeatable experiments demand that you and I agree on the interpretation of the experiments, only that they can be positively repeated by either one of us. You are also free to disagree with my assertion that the results are meaningful, useful, or interesting! It is also worth remembering that repeating the work in this way does not automatically make it straightforward to expand upon the ideas in the research and be in a position to modify and build upon the experiment. This is the concept of reuse of research results and it requires more effort than reproducibility or replicability. Before we discuss this any further, we should review how reproducibility and replicability work in the context of computational science.

3.1 Reproducible computations The ability to repeat routine tasks without error is one of the reasons we use computers in the first place so, at first, reproducibility of computational results would seem to be a solved problem with, perhaps, a few minor, technical loose ends needing to be sorted out. In practice, anyone who relies upon a complex computational workflow for their research will appreciate how difficult it can be to orchestrate all the software packages and their dependencies, and to ensure that, over time, required updates to packages that maintain compatibility with changes to operating systems or data archives, do not introduce small but important changes to the results. Even “freezing” dependencies are not guaranteed to make the workflow transportable from one machine to another which is required for reproducibility. The advent of real-world virtualization (e.g., as envisaged by Popek and Goldberg, 1974) makes it possible to create an exact snapshot of a working environment in which a computational workflow

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can be run. Lightweight solutions such as software “containers” (e.g., Merkel, 2014) make it straightforward to build and distribute software environments that are specific to a given workflow and to bundle everything necessary to reproduce the work on any machine that can support the container. There are container options that are optimized to run in high-performance computing environments (e.g., Kurtzer et al., 2017) making reproducible high-performance computing a possibility. Not every computational result is deterministic as some algorithms require small random perturbations to seed the calculation, extreme sensitivity to initial conditions can produce different outcomes from ostensibly identical starting conditions, and order dependence often occurs in parallel solution algorithms. Under these conditions, as in a laboratory setting, reproducibility occurs when results repeatedly fall within the error analysis of the experiment. Even with technology on our side, making an element of computational research reproducible still requires concerted effort on the part of a researcher to release their source code, document and disclose each step of their workflow, and a cultural shift in the community that respects and rewards that effort (LeVeque, 2009; LeVeque et al., 2012).

3.2 Replication of computational research Traditionally, replicability implies that a research result is robust even if it is conducted in a different laboratory closely following the same recipe. The computational equivalent of replicability requires a little more thought because exact reproducibility through virtualization eliminates any systematic bias in an individual laboratory that replicability is supposed to address. We can reframe the reproducibility discussion to require not exact repeatability of an experiment time-after-time, but a statistical repeatability in the face of random perturbations. If the system exhibits bifurcations or is degenerated

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with respect to different symmetric states, then this will become apparent by exploring the perturbed state of the system and would be considered as part of the replication challenge. Any scientific discovery which is dependent upon some detail of the computational implementation is non-physical and we would rightly view it with deep skepticism. The next step in replication is to ensure that the research is expressed in terms of robust abstractions that describe the problem in a way that is independent of the details of the numerics, so that different implementations are broadly equivalent. We already use abstract descriptions of the implementation when we write a mathematical representation, but, if this is the language we are going to use to describe our models, then there is significant work on the part of the community software developers to help translate between mathematics and the numerical implementation (more on this below).

3.3 What more does reuse require? Reproducibility and replicability are both focused on “keeping researchers honest,” and ensuring openness and transparency in the way computational research is disseminated. This does not really guarantee that the research can be used as the foundation for new work, and the focus on a frozen state of computational software may even work against anyone trying to take a model and modify it. For example, in the original implementation of the Underworld software suite (Moresi et al., 2007), we enforced model reproducibility through a deterministic, machine readable (xml) user interface. This made re-running and sharing a model trivial, but the xml files themselves were fragile and modification and extension of the snapshot required considerable expertise. Is there a way to resolve the tension between reproducibility/replicability and the freedom to innovate that is required for models to be used as building blocks for further discovery? One possibility is to adopt ideas from literate

programming (Knuth, 1984), in which the description of an algorithm and the realization exist as a single document (or program) and thus are always perfectly synchronized (also see Schwab et al., 2000). Literate programming as a philosophy has evolved beyond Knuth’s specific implementation to include a looser integration of software and documentation in the form of “notebooks” (e.g., Granger and Perez, 2021; Kluyver et al., 2016; Perez and Granger, 2007). In notebook systems, documents with embedded code can be executed though there is no versioning for compilers or dependencies and very little structure is imposed to ensure strict reproducibility. In notebook-style literate programs, playful interactivity encourages reuse and re-purposing of code, and same virtual environments that can be used to ensure strict reproducibility of individual workflows can also launch an interactive session. It is not an accident that some of the most helpful reference points for geodynamics are still the foundation papers that laid out mathematically how a dynamic planet behaves (e.g., Turcotte and Oxburgh, 1972; McKenzie and Weiss, 1975; McKenzie, 1977; Hager and O’Connell, 1981). The computational models have not been reused but the mathematical formulations have their own worth and are timeless. In an appropriately abstract mathematical formulation, and appropriately framed research results, we can also imagine replicating a simple model with a more general, complex one as a means of validating the latter, and we can and do use results computed in 2D as complements to 3D models which are more demanding of computational resources. What remains in our quest for model reuse is a straightforward connection between the abstract and enduring (mathematical) representation of the model problem and the implementation that can be used by somebody with expertise in geodynamics or tectonics with a reasonable working knowledge of numerical methods.

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4 Mathematical choices

4. Mathematical choices If somebody wants to reproduce and build upon a model, they need to be able to disassemble it, take a step back, and set off along a new path. When we publish our work, we probably spend a considerable amount of time finding the simplest, most compact means of expressing ourselves because the simple conceptual, mathematical representation is the easiest to follow and extend. The same principle should apply to the computational model, and the best way to ensure that happens is if the conceptual, mathematical description can quickly be translated to a computational one. Of course, there is no one best way to solve every mathematical problem, and even very general tools such as the finite element method require careful tailoring for each new situation. Suppose we start with a simple example: the conservation principle for heat energy. The total heat present in a given volume can only change if heat is brought in across the boundary or generated within the volume. The rate of change of heat within the volume is governed by the net heat flux across the boundary and the rate of internal production of heat energy. In a textbook, this is often sketched for a small unit volume (see Fig. 1) in which the change of heat (or fluid) content of the volume is intuitively related to the difference between the flow in and the flow out. Fourier’s law connects the heat flux across a boundary to the temperature gradient perpendicular to the boundary (through an empirical material constant known as the thermal conductivity), and the heat energy is related to temperature through the heat capacity of the material. From a computation perspective, we can imagine linking a number of these conceptual containers together (as in the diagram shown in Fig. 2). The fluxes across the walls of the buckets (or cells/elements in a computational mesh) are now related to the gradients in the values within neighboring cells. This discrete representation of the problem becomes one in which we need to solve for a finite set of values, {Pi} in the domain.

FIG. 1 In this diagram, we see an idealized bucket with water flowing in and water flowing out. The change in the level of water in the bucket is a result of the net flux into the volume.

What we have just formulated in words is more typically expressed as a partial differential equation (PDE) as allow the representative volume of material to becoming infinitesimally small. For example, the conservation of heat energy is written: ρCp

∂T ¼ rðkrTÞ + hðx, tÞ ∂t

(1)

Here, ρ is the density, Cp is the heat capacity, T is temperature, k is thermal conductivity, and h is the heat production rate that can vary in space and time. The term k r T is the heat flux from Fourier’s law written in a way that is independent of the number of dimensions. The rate of change of temperature (left hand side) is determined by the gradients of fluxes and sources (right hand side). Partial differential equations describe the balance between gradients in different directions, to rates of change in time and they are typically very difficult to solve exactly unless we find a way to make significant simplifying assumptions. Once we have expressed the balance of fluxes and sources to express how a quantity changes, we have created an abstract mathematic form from which we can find solutions or to transform it to a problem we already understand. On the other hand, simplifying the mathematics is not a guarantee that we will have an easier time writing a computational representation.

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23. Literate, Reusable, Geodynamic Modeling

FIG. 2 Suppose we take the individual bucket from Fig. 1 and connect it to some neighbors. The flow out of one bucket is now caught by a neighbor. Intuitively, if the level of liquid between two neighbors is larger, the flow between them will be faster. This is a physical analog of Eq. (1).

For example, suppose we now consider the heat transport problem for a moving fluid which we need to do in mantle convection. Now Eq. (1) has to be written to include the effect of the fluid velocity, u as follows (neglecting physical constants): ∂T + ðu  rÞT ¼ r2 T + hðx, tÞ ∂t

(2)

The ðu  rÞ operator is the gradient in the direction of the velocity field at a point and so (u  r)T is the difference in the heat flux carried into and out of an elementary volume by u. This term, which is simple to describe, turns out to be quite complicated to implement numerically in because it introduces nonuniformity and an orientation to the flow of information which is hard to treat on a spatially regular grid (Fig. 2), leading to instabilities and inaccuracies. These become harder to deal with wherever the transport terms dominate the solution and specialized remedies are needed that often depend on knowing details of the underlying computational implementation (e.g., Brooks and Hughes, 1982; Smolarkiewicz and Margolin, 1998; Spiegelman and Katz, 2006). Exactly equivalent to (2) is the so-called Lagrangian form of the transport terms, usually written: DT ¼ r2 T + hðx, tÞ Dt

(3)

where D/Dt represents a time derivative from the point of view of an observer moving with the fluid. The use of this form is just a clarification of the fact that, when moving with the fluid,

the local perspective is identical to that of the static diffusion Eq. (1). Many different methods exist that exploit this particular simplification (usually at the expense of some complicating trade-off in other areas of the code) but one of the most widely adopted are the particle-in-cell methods that use the Lagrangian form only for quantities that are dominated by transport such as composition and history variables (e.g., Gassm€ oller et al., 2019; Gerya and Yuen, 2003; Moresi et al., 2003; with a comparison of Lagrangian methods in the last of these). Mathematically equivalent, the Eulerian form (2) and the Lagrangian form (3) imply very different forms of numerical representation. In the Eulerian form, we have additional terms in the inversion that change the character of the system of equations. In the Lagrangian form, those terms are absorbed in operations associated with moving the mesh points and updating computing derivatives on a deformed mesh. For those of us who develop, use, and distribute computational algorithms, the requirements for replicability require us to document very clearly the methods that are used in our modeling software, but also to provide abstractions that allow a typical user to quickly explore whether there are any aspects of their conclusions that are artifacts of the implementation.

5. An example: Underworld models We would like modeling tools that are close to the mathematical description but which also encourage users to make use of the most robust,

5 An example: Underworld models

efficient, and accurate algorithms available for any given problem. We prefer to keep unimportant details of the implementation at arms length, allow researchers to use models that are suited to their scientific problems (certainly not the other way around!), help them to avoid obvious algorithmic pitfalls, and make the use of the software as uncomplicated as we reasonably can. We developed the Underworld code (Beucher et al., 2019; Mansour et al., 2020; Moresi et al., 2007) with those ideas in mind, and we made a number of design choices: • A strong separation between the implementation layer and the user interaction layer. We use PETSc (Balay et al., 1997) to provide parallel data management, discretization, nonlinear solvers, etc. The user interface is in python. • Templated operators that map the low-level discrete, parallel constructs to objects at the user level. • A geodynamics/geology focused object model for the high-level (python) user interface which is process based rather than mathematical and preselects default algorithms and solvers. • A collection of generic, symbolic operations on parallel data structures that generalize the template mathematical expressions and support complete operator overloading and most of the operations available in sympy (Meurer et al., 2017). • Automatic differentiation and manipulation of these symbolic expressions to assist in the construction of model derivatives needed for nonlinear behavior and data assimilation. • Lagrangian and Eulerian data/variables that are equivalent to each other and can be interchanged and composed in building constitutive models, boundary conditions and initial conditions, and in analysis of results.

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• Self-reflectivity in models that allows the model to update itself as it runs (admittedly, this does make reproducibility more challenging). When we started development, we were aware that these ideas are not unique to our project but we also realized that the geodynamics problem, which is so dominated by historydependent rheology and post-failure deformation, poses some unique challenges. We should not play down the fact that the user community for geodynamics modeling software is very much focused on expanding knowledge in the geological sciences, and may not be well served by codes written by/for mathematicians in the engineering disciplines. Underworld is a hierarchical python code that interfaces to the PETSc data structures and solvers through the petsc4py interface. At the lowest level is the orchestration of the components of PETSc that support the Underworld object model: the mesh-based data structures, the Lagrangian swarm variables, the solver objects that assemble the terms making up an equation alongside the methods for solving it. The symbolic “function” layer of Underworld is the glue connecting all of these components in a manner that is transparent at the user level. For example, mesh and swarm variables have a symbolic form that can be used to write complicated expressions through most of the common mathematical operators, including differentiation. The symbolic form is only ever evaluated when required (that is, lazy evaluation) which means that the expressions that are made up of Underworld functions can be used to define the structure of equations, manipulated to the form required by PETSc solvers, and evaluated exactly at the times and places that the solvers require. For example, the heat transport Eq. (1) is implemented by specifying the unknowns, and the flux terms, to a generic solver template something like this:

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23. Literate, Reusable, Geodynamic Modeling

# X and T are sympy coordinate vector and unknown mesh variable. # Source term, coefficients may include mesh/swarm variables in mathematical expressions grad_T = sympy.derive_by_array(T, X) source_term = h flux_term = kappa * grad_T # template code to build petsc layer from source / flux ...

which keeps the implementation very close to the mathematical expression. To add the transport term is surprisingly simple: we add a Lagrangian history term that we use to keep track of past values of T at their previous locations. The Underworld function interface takes care of the projection of the unstructured information to the mesh locations subject to boundary conditions. In this formulation, the rate of change of T is a simple expression (and it is not too hard to see how to make a higher-order equivalent of this form).

## T, mesh variable, unknown ## T_star, swarm (history) variable grad_T = sympy.derive_by_array(T, solver.X).transpose() grad_T_star = sympy.derive_by_array(T_st ar, solver.X).transpose() # Lagrangian D/Dt term D_Dt = (T.fn - T_star.fn) / delta_t # flux term (Crank-Nicholson time-averaged) flux_term = 0.5 * (grad_T + grad_T_star) * kappa

There are other advantages in this kind of symbolic approach including the fact that

nonlinear solvers that require the computation of Jacobian derivatives are much less error prone if we can generate them automatically (something like this): # Jacobians are J_00 = sympy.derive_by_array (source_term, T) J_01 = sympy.derive_by_array(s ource_term, grad_T) J_10 = sympy.derive_by_array(flux_term, T) J_11 = sympy.derive_by_array(flux_term, grad_T) # Pass source, flux, J_xx to PETSc solvers ...

The hierarchical structure of Underworld means that users do not need to remember how to construct these expressions for standard problems as there are higher-level layers that define terms for specific applications —in this case: heat flux, diffusivity, temperature gradient, and so on. The same symbolic approach allows us to separate the problem description from the details of the mesh geometry. For example, the following Underworld code fragment (from the setup of a convection model) is expressed in a way that is independent of the coordinate system and the number of spatial dimensions.

References

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... # Define some values Rayleigh_number = ... A0 = ... B0 = ... my_mesh = MeshFromGmshFile(filename="my_gmsh_file.msh" , ... ) velocity = MeshVariable('u', my_mesh, ... ) pressure = MeshVariable('p', my_mesh, ... ) temperature = MeshVariable('T', my_mesh, ... ) stokes = Stokes(my_mesh, velocityField=velocity, pressureField=pressure, ... ) stokes.add_dirichlet_bc( values = (0.0,), ["Upper","Lower"] , dofs=(1,)) stokes.viscosity = A0 * sympy.exp(-B0 * temperature.fn) stokes.body_force = Rayleigh_number * temperature.fn * my_mesh.vertical_vector stokes.petsc_options["snes_type"] = "newtonls" stokes.solve() ...

The Underworld team advocate creating codes with strong, high-level abstractions, particularly if these can be kept closely aligned to the mathematical description that goes along with the results in a publication. This way, published results are imbued with the semantic content of the mathematical expression and this may provide a mechanism for searching through collections of models at a higher level than simply cataloguing algorithms, mesh dimensions, or checksums.

6. Discussion My argument for model reuse is that we should develop the ability to share computational ideas as quickly and freely as we do with mathematical descriptions of processes. Perhaps the most straightforward approach to this is to make the distance between the abstract mathematical form and models that can be derived from this. Symbolic and abstract computational representations are a valuable stepping stone in meeting that goal.

The use of automatic code generation and just-intime compilation means that this generality does not come at the cost of slower, less efficient code. Or, perhaps, it is fairer to say that if there is a speed penalty, it is more than offset by the speed that models can be created, modified, and so become reusable.

References Balay, S., Gropp, W.D., McInnes, L.C., Smith, B.F., 1997. Efficient management of parallelism in object oriented numerical software libraries. In: Arge, E., Bruaset, A.M., Langtangen, H.P. (Eds.), Modern Software Tools in Scientific Computing. Birkh€auser Press, pp. 163–202, https:// doi.org/10.1007/978-1-4612-1986-6_8. Bathe, K.-J., 2008. Finite element method. In: Wah, B.W. (Ed.), Wiley Encyclopedia of Computer Science and Engineering. John Wiley & Sons, Inc., Hoboken, NJ, USA, p. ecse159, https://doi.org/10.1002/9780470050118.ecse159. Beucher, R., Moresi, L., Giordani, J., Mansour, J., Sandiford, D., Farrington, R., et al., 2019. UWGeodynamics: a teaching and research tool for numerical geodynamic modelling. J. Open Source Softw. https://doi.org/ 10.21105/joss.01136.

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Brooks, A.N., Hughes, T.J.R., 1982. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32 (1–3), 199–259. https://doi.org/10.1016/00457825(82)90071-8. Gassm€ oller, R., Lokavarapu, H., Bangerth, W., Puckett, E.G., 2019. Evaluating the accuracy of hybrid finite element/ particle-in-cell methods for modelling incompressible stokes flow. Geophys. J. Int. 219 (3), 1915–1938. https:// doi.org/10.1093/gji/ggz405. Gerya, T.V., Yuen, D.A., 2003. Characteristics-based markerin-cell method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties. Phys. Earth Planet. Inter. 140 (4), 293–318. https://doi.org/10.1016/j.pepi.2003.09.006. Granger, B.E., Perez, F., 2021. Jupyter: thinking and storytelling with code and data. Comput. Sci. Eng. 23 (2), 7–14. https://doi.org/10.1109/MCSE.2021.3059263. Hager, B.H., O’Connell, R.J., 1981. A simple global model of plate dynamics and mantle convection. J. Geophys. Res. Solid Earth 86 (B6), 4843–4867. https://doi.org/10. 1029/JB086iB06p04843. Hughes, T.J.R., 1987. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1. Dr.). Prentice Hall, Englewood Cliffs, N.J. Kluyver, T., Ragan-Kelley, B., Perez, F., Bussonnier, M., Frederic, J., Hamrick, J., et al., 2016. Jupyter notebooks—a publishing format for reproducible computational workflows. In: Positioning and Power in Academic Publishing: Players, Agents and Agendas, p. 4, https://doi.org/10.3233/978-1-61499-649-1-87. Knuth, D.E., 1984. Literate programming. Comput. J. 27 (2), 97–111. https://doi.org/10.1093/comjnl/27.2.97. Kurtzer, G.M., Sochat, V., Bauer, M.W., 2017. Singularity: scientific containers for mobility of compute. PLoS One 12 (5), e0177459. https://doi.org/10.1371/journal.pone.0177459. LeVeque, R.J., 2009. Python tools for reproducible research on hyperbolic problems. Comput. Sci. Eng. 11 (1), 19–27. https://doi.org/10.1109/MCSE.2009.13. LeVeque, R.J., Mitchell, I.M., Stodden, V., 2012. Reproducible research for scientific computing: tools and strategies for changing the culture. Comput. Sci. Eng. 14 (4), 13–17. https://doi.org/10.1109/MCSE.2012.38. Mansour, J., Giordani, J., Moresi, L., Beucher, R., Kaluza, O., Velic, M., et al., 2020. Underworld2: python geodynamics modelling for desktop, HPC and cloud. J. Open Source Softw. 5 (47), 1797. https://doi.org/10.21105/joss.01797. McKenzie, D., 1977. Surface deformation, gravity anomalies and convection. Geophys. J. Int. 48 (2), 211–238. https:// doi.org/10.1111/j.1365-246X.1977.tb01297.x. McKenzie, D., Weiss, N., 1975. Speculations on the thermal and tectonic history of the earth. Geophys. J. Roy. Astron. Soc. 42 (1), 131–174. https://doi.org/10.1111/j.1365246X.1975.tb05855.x.

Merkel, D., 2014. Docker: lightweight linux containers for consistent development and deployment. Linux J. 2014 (239).  ´k, O., Meurer, A., Smith, C.P., Paprocki, M., Certı Kirpichev, S.B., Rocklin, M., et al., 2017. SymPy: symbolic computing in Python. PeerJ Comput. Sci. 3, e103. https:// doi.org/10.7717/peerj-cs.103. Moresi, L.-N., Solomatov, V.S., 1995. Numerical investigation of 2D convection with extremely large viscosity variations. Phys. Fluids 7 (9), 2154–2162. https://doi.org/10. 1063/1.868465. Moresi, L., Dufour, F., M€ uhlhaus, H.-B., 2003. A Lagrangian integration point finite element method for large deformation modeling of viscoelastic geomaterials. J. Comput. Phys. 184 (2), 476–497. https://doi.org/10.1016/S00219991(02)00031-1. Moresi, L., Quenette, S., Lemiale, V., Meriaux, C., Appelbe, B., M€ uhlhaus, H.-B., 2007. Computational approaches to studying non-linear dynamics of the crust and mantle. Phys. Earth Planet. Inter. 163 (1), 69–82. https://doi.org/10.1016/j.pepi.2007.06.009. Newton, I., 1675, February 5. Letter to Robert Hooke. https:// digitallibrary.hsp.org/index.php/Detail/objects/9792. Perez, F., Granger, B.E., 2007. IPython: a system for interactive scientific computing. Comput. Sci. Eng. 9 (3), 21–29. https://doi.org/10.1109/MCSE.2007.53. Popek, G.J., Goldberg, R.P., 1974. Formal requirements for virtualizable third generation architectures. Commun. ACM 17 (7), 412–421. https://doi.org/10.1145/361011.361073. Schwab, M., Karrenbach, N., Claerbout, J., 2000. Making scientific computations reproducible. Comput. Sci. Eng. 2 (6), 61–67. https://doi.org/10.1109/5992.881708. Smolarkiewicz, P.K., Margolin, L.G., 1998. MPDATA: a finitedifference solver for geophysical flows. J. Comput. Phys. 140 (2), 459–480. https://doi.org/10.1006/jcph.1998.5901. Spiegelman, M., Katz, R.F., 2006. A semi-Lagrangian CrankNicolson algorithm for the numerical solution of advection-diffusion problems. Geochem. Geophys. Geosyst. 7 (4). https://doi.org/10.1029/2005GC001073. Turcotte, D.L., Oxburgh, E.R., 1972. Mantle convection and the new global tectonics. Annu. Rev. Fluid Mech. 4 (1), 33–66. https://doi.org/10.1017/S0022112067001880. Zhong, S., Gurnis, M., Moresi, L., 1998. Role of faults, nonlinear rheology, and viscosity structure in generating plates from instantaneous mantle flow models. J. Geophys. Res. Solid Earth 103 (B7), 15255–15268. https:// doi.org/10.1029/98JB00605. Zhong, S., Zuber, M.T., Moresi, L., Gurnis, M., 2000. Role of temperature-dependent viscosity and surface plates in spherical shell models of mantle convection. J. Geophys. Res. Solid Earth 105 (B5), 11063–11082. https://doi.org/ 10.1029/2000JB900003. Zienkiewicz, O.C., Taylor, R.L., Zhu, J.Z., 2013. The Finite Element Method: Its Basis and Fundamentals, seventh ed. Elsevier, Butterworth-Heinemann, Amsterdam.

C H A P T E R

24 Perspectives on Planetary Tectonics Doris Breuer DLR, Institute of Planetary Research, Berlin, Germany

A comparison of our Earth’s direct neighbors, the so-called terrestrial planets, reveals some differences despite their similar internal structure with a basaltic crust, a silicate mantle, and an iron-rich core. Differences are in the tectonic and volcanic structures, the presence of a magnetic field, the surface temperatures, and the composition of the atmosphere (Table 1)—to name just a few. Why planets have evolved differently, and what internal and external processes are responsible, are thus key questions in comparative planetology. Ultimately, this goes back to mankind’s fundamental question of why Earth, the only planet that has life, is so special. An aspect often highlighted in the context of comparative planetology and one main focus of the present book is the planetary heat engine, which operates differently on the planets. Earth has an effective mechanism for transporting its heat to the surface through plate tectonics, where cold plates are subducted into the planet’s interior and hot mantle material flowing upward reaches rift zones near the surface. A mechanism helps to cool the interior and also the core, which in turn helps to generate a magnetic field. The subduction of cold plates has also the consequence that water and carbon stored in sediments by weathering processes are

Dynamics of Plate Tectonics and Mantle Convection https://doi.org/10.1016/B978-0-323-85733-8.00007-X

recycled into the interior, while they are released into the atmosphere by volcanism at volcanic arcs and ocean ridges. The so-called silicate– carbon cycle represents an integral part of plate tectonics and tends to stabilize the climate and the surface conditions. Because the solar activity has increased by 30% since the planets were formed, this balancing mechanism is particularly important and is considered one of the main reasons that life could evolve on Earth. The other terrestrial planets, notably Mars and Mercury, are single-plate planets operating in the so-called stagnant lid mode, in which convection occurs beneath a rigid lithosphere. Heat loss is relatively inefficient by conduction through the lid but can be enhanced by advective heat transport due to volcanism. In these planets, there is only a one-way transport of volatiles from the interior to the atmosphere. Longlived clement conditions on the surface are thus more difficult to maintain because the regulating effect by efficient crust and volatile recycling is missing, but is in principle possible, with carbonated sediments buried and sealed by lava flows. Venus present thermal heat engine is between these two end-member cases. Its surface has a stagnant lid, but locally some kind of surface mobilization or lithosphere delamination seems

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584 TABLE 1

24. Perspectives on Planetary Tectonics

Fundamental planetary parameters and observations.

Planet

Mercury

Venus

Earth

Mars

Massa (1024 kg)

0.3301

4.868

5.973

0.6418

2439

6052

6378

3396

5.43

5.24

5.51

3.93

Surface Temperature (K)

440

737

288

210

Surface pressure

–

93 bar

1 bar

0.0064 bar

Atmosphere composition (main components)

–

CO2 N2 SO2

N2 O2 Ar2

CO2 N2 Ar2

Magnetic field present

Yes

No

Yes

No

Magnetic field ancient

Yes

Not known

Yes

Yes

Tectonic regime

Stagnant lid

Squishy lid?

Plate tectonic

Stagnant lid

Equatorial Radiusa (km) a

3

Mean Density (kg/m ) a

a

Source: https://nssdc.gsfc.nasa.gov

to be taking place in the so-called coronae. These are oval-shaped volcano-tectonic features of various sizes which are likely formed by plume–lithospheric interaction. Some refer to this regime as squishy lid regime that forms as a consequence of lithosphere weakening due to volcanic intrusions. It is speculated that this dynamic regime was also active in early Earth history, before plate tectonics set in. Two general questions arise here: why do planetary heat engines function differently in the first place, and what is the consequence of different heat transport regimes on the thermal evolution of a planet? As for the first question, it should be noted that the stagnant lid convection is a more natural outcome when modeling the interior dynamics than plate tectonics, since a stiff layer, i.e., the stagnant lid, forms at the surface due to temperature-dependent viscosity and convection takes place below. As described in previous chapters in the book, the occurrence of plate tectonics requires more complex descriptions of the lithospheric rheology that also allow for “weakening” effects. In the context of the discussion about plate tectonics on large rocky exoplanets, the

influence of the planetary mass on the likelihood that a planet is in the plate tectonic regime has been investigated in the last years with the idea that the increase in planetary mass and radius also enhances the convective velocities and stresses necessary to “brake” plates. Here, the results diverge widely, i.e., some models predict that plate tectonics becomes more probable with increasing mass, while some suggest it is less probable or is even independent of planetary mass. The reason that the models show such different results is, among other things, due to different scalings for the stresses and velocities in which changes in parameters such as mantle thickness and viscosity enter. The estimate becomes even more uncertain when one considers that for large rocky exoplanets, the pressure in the interior increases significantly up to 1 TPa, because the pressure dependence of the viscosity, and of other material parameters, is not well known for those values. Looking at Mercury and Mars, which are both smaller than the Earth and are in stagnant lid regime, one could conclude that size matters and plate tectonics becomes more probable with increasing mass—at least in the range up to one

24 Perspectives on planetary tectonics

Earth radius. On the other hand, Venus, which has a similar size as the Earth, shows that the planetary mass or radius cannot be alone decisive, other factors must also play a role. Under discussion are the surface temperature and/or the existence of liquid water. In Venus, the 700 K surface temperature is much higher than on Earth and liquid water is not stable—in fact, extremely dry conditions are suspected at the surface and in the near-surface layers, which are not favorable for the formation and maintenance of plate tectonics. However, the surface conditions depend strongly on the atmosphere composition and pressure. Both, in turn, depend on thermal evolution in the planet’s interior and the associated volcanism and outgassing history. Accordingly, the tectonic regime may have changed during the planet’s evolution. Two main scenarios are currently being discussed for Venus. In one scenario, Venus was in a similar state to the Earth, with plate tectonics, moderate surface temperatures, and liquid water—Venus was habitable. In principle, habitability would have been possible to maintain with a stagnant lid or squishy lid regime, but only for a limited time, because the regulating effects would have been reduced and recycling of volatiles and carbonated sediments would only be achieved by the burial and sealing within lava flows. But only in its later evolution, a large amount of CO2 stored in carbonates at the surface was released due to impacts or a dynamic instability, leading to a climate dominated by a dense CO2 atmosphere and a high surface temperature, as we observe today. In an alternative scenario, Venus had a dense atmosphere and high surface temperatures shortly after its formation due to the outgassing of a magma ocean. This condition has only changed slightly during its evolution and Venus never had oceans or a stable hydrosphere. The heat transport mechanism in this case may have changed during the evolution from stagnant lid to catastrophic episodic lid. With the current observations we have of Venus and the existing

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thermo-chemical evolution models, none of the two scenarios can be disregarded. The present-day Venus has a young crust with an average age of 300–700 Ma. It is unclear whether the crust formed by a continuous resurfacing process or whether some kind of catastrophic surface renewal occurred—possibly in an episodic fashion. Active volcanism has also not yet been clearly identified, but there is evidence of surface material that appears to be little altered in the dense atmosphere, suggesting young basaltic material. Further information and observations are needed to decipher the present-day processes on Venus and, moreover, to understand how the planet’s tectonic behavior has changed over time in conjunction with atmospheric conditions. Future missions to Venus, such as the recently approved NASA VERITAS and DAVINCI+ missions and ESA’s EnVision mission, will hopefully help to better understand the present state of Venus and its evolution. Venus could also be a test case for the processes on early Earth. It should be noted that Venus is the least explored planet so far. Currently, there is an uncertainty in the core radius of plus or minus 300 km—far more than for the other planets, and it is not even certain whether the core is liquid or solid. Knowing the internal structure and having constraints on its thermal state, two of the goals of VERITAS and EnVision, will help to further constrain geodynamic models. In addition, especially for Venus with its high surface temperatures, coupling the atmosphere with the interior dynamics and tectonics, but also with the surface (accounting for weathering processes), will be essential. In addition to the comparison between Earth and Venus, also the view on the smaller terrestrial planets expands our knowledge about general geodynamic and tectonic processes. Mercury and Mars, both have, despite the common heat transport mechanism, different characteristic features. We can use some of these features to improve our understanding the

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24. Perspectives on Planetary Tectonics

thermal evolution of the planets in question, but for many other observations, there are still unanswered questions that concern their origins and as well as past and ongoing processes. Mercury, like the Earth, has a present-day magnetic field. This field is peculiar as it is very weak and has a dipole offset—both observations are difficult to explain with present dynamo models. There is also evidence of an early magnetic field about 3.9–3.7 Gyrs ago as suggested by remanently magnetized ancient crust measured by magnetometers on the NASA MESSENGER mission. It is unclear whether the magnetic field was present all the time, or whether there was an early dynamo which then diminished and was reactivated only later in the planet’s evolution. Having more information about the timing of an ancient dynamo can, however, be very important to constrain the thermal evolution of Mercury. This is because the heat transport of the mantle is crucial for the generation of a magnetic field. The more effectively the heat can be transported through the mantle, the more the core can cool, which in turn is necessary to generate and sustain convection and thus a magnetic field. In principle, there are two types of convection relevant for the dynamo, pure thermal convection (in which case there is a pure liquid core and the heat flux from the core must exceed a critical value) or chemical convection (in which case the core cools below the melting temperature and crystallizes—an effective mechanism to drive chemical convection and thus a dynamo). Important in considering the dynamic and thermal evolution of Mercury is its peculiar internal structure compared to the other terrestrial planets. Mercury has a large density, which points to a large core radius comprising more than 80% of the planetary radius—the other planets including the Earth are closer to 50%. Unlike Venus, the existence of a liquid outer core in Mercury is not debated. The large core, on the other hand, suggests a thin mantle and there is ongoing debate as to whether Mercury’s mantle

is convecting underneath its stagnant lid or whether the mantle may even be purely conductive. If the mantle is convecting, then the convection planform should likely be of small scale, of the order of the mantle thickness as expected for low Rayleigh number convection in thin spherical shells. An indication of the strength of past dynamics is provided by large impacts and associated volcanism that occurred several hundred million years after the large impact event. It has been shown that the convection in the underlying mantle is decisive for the time-delayed occurrence of volcanism after an impact. The volcanism with the observed larger time lag only occurs when convection was very sluggish at this early stage in its evolution—in contrast to a very dynamic mantle where the time lag would be smaller. A particularly tectonic feature on Mercury’s surface is the observed compressional structures called lobate scarps. These structures indicate a global contraction of less than 7 km since the end of the heavy bombardment, i.e., after about 3.7 Ga. The scarps provide important clues to the cooling behavior of the interior. For instance, efficient cooling and solidification of the core can make a large contribution to planetary contraction—up to 17 km if the entire iron-rich core froze. All three observations and constraints, i.e., present and early magnetic field, thin mantle in the stagnant lid regime, and a contraction rate of less than 7 km after 3.7 Ga currently support the following scenarios: Mercury, in its early history, had a dynamo powered by pure thermal convection. Later, the growth of an inner core may have begun. From this point on, the dynamo may have been additionally driven by latent heat and gravitational energy released by the growth of the inner core. These models suggest a small solid inner core and a late inner core growth could imply a period without an active dynamo in which the thermal dynamo would have ceased to operate and the chemical dynamo not yet have started. Alternatively, it is

24 Perspectives on planetary tectonics

also possible that the inner core grew early and that even the early magnetic field was generated by a chemical dynamo. This conjecture assumes a core with low concentrations of light elements and thus a comparatively high melting temperature. The inner core must have grown to a substantial size under these conditions before the heavy bombardment to be consistent with the observed low contraction. Such a model would predict a dynamo magnetic field for all or most of the planet’s evolution. Here, the ESA BepiColombo mission, which will arrive in Mercury in December 2025, will hopefully give us further constraints. Mars, even more so than Venus, is a candidate for a planet in which habitable conditions existed in its early evolution. Several observations indicate that water played an important role in its evolution. The morphology and composition of the Martian surface indicate the presence of liquid surface water, the main activity of which occurred between the end of the Noachian (3.9 Ga) and the beginning of the Hesperian (3.5 Ga). However, it is debated how warm the surface must have been to form these features and how long these warm conditions must have persisted. Also here, coupled models of the interior and atmosphere are essential. Taking into account early loss processes due to XUV radiation from the Sun, stagnant lid models considering melt production and outgassing show that the degassing of CO2 from the Martian mantle is less than 250 mbar, pressures only sufficient to stabilize liquid water transiently on the surface. The question remains as to how average surface temperatures above the freezing point of water could be maintained over an extended period of time. Thermal evolution models linking the interior to the atmosphere suggest that the climate on Mars has always been cold and moist, not warm and humid. Mars further exhibits the so-called crustal dichotomy, with an old heavily cratered southern hemisphere and a younger, less cratered

587

northern hemisphere. The two hemispheres differ in elevation by 1 to 3 km and in crustal thickness, with the thickness of the crust beneath the Northern lowlands being thinner than the crust beneath the Southern highlands—assuming that the crustal density is the same on both hemispheres. It is generally accepted that this crustal dichotomy is one of oldest features on Mars, with a formation time of earlier than 3.9 Ga or even as early as the first 50 Ma after the solar system formation. The origin of the crustal dichotomy is not well known and has been linked to internal processes such as single-degree convection and/or external processes such as impacts or as a result of inhomogeneous magma-ocean crystallization. None of the proposed formation mechanisms have been fully convincing to date and require future models that better link both mantle dynamics, impact processes, and melt formation with petrologic models. Interestingly, however, the crustal dichotomy shows important impact on thermal evolution and mantle dynamics as shown by 3D convection models that consider the variation of crustal thickness on top of a convecting mantle as a boundary condition. Due to the thermal insulating effect of the crust—crustal material usually has a lower thermal conductivity than mantle material—and the variation in crustal thickness, long-lasting magmatism is preferentially found in regions of thickened crust—as is observed in the major volcanic provinces of Tharsis and Elysium. The cause of the long-lasting volcanism over several billion years in a few regions was unexplained so far. The example of Mars shows also the important influence of the insulating crust especially for one-plate planets. Another significant contribution to a better understanding of the Martian interior is currently being made by NASA’s InSight mission, which is equipped with a broadband seismometer. It provides latest estimates of core size (1830  40 km—larger than previously assumed) and average crustal thickness (24–38 km for a two-layer crust or 39–72 km for a three-layer

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24. Perspectives on Planetary Tectonics

crust—currently preferred is the three-layer crust). The determination of the large core, for instance, helps us to draw further conclusions about the thermal and magnetic field evolution. The large size of the core means that it must have a relatively low density and thus many light elements to match the measured planetary mass and moment of inertia. This finding has further implications. The Martian core will still be completely liquid today—there are no thermal models where the core begins to freeze out, since many light elements also mean very low melting temperatures that cannot be exceeded. This would also be consistent with Mars not generating a dynamo today. The seismic data further indicate that the thermal lithosphere with values between 400 and 600 km is much thicker than that of the Earth. This is not surprising, since thicker lids are expected for stagnant lid planets. An open important question remains, however, how the observed recent volcanism can be

explained if the upper mantle is as cold as suggested by the seismic models. These examples show how mission observations and models can be used to draw conclusions about the evolution and workings of terrestrial planets. Upcoming missions and the development and improvement of models will certainly advance our knowledge in the future. Another prospect is to apply the models and the knowledge we gain from the planets of our solar system to rocky exoplanets. Currently, up to 5000 exoplanets have been discovered, of which about 47 are rocky exoplanets with a density larger than 5000 kg/m3 and smaller than 10 Earth masses. In the coming years, missions such as PLATO and ARIEL will continue to increase the number of planets found and further characterize their atmospheres. We will then be better able to understand how these planets function and eventually help in the search for habitable exoplanets.

Index

Note: Page numbers followed by f indicate figures and t indicate tables.

A Adjoint method, 249–250 Algebraic multigrid (AMG), 555 Ambizione program, 278 Analog models, 273, 274f Arabian–Nubian Shield (ANS), 310–311 Australian-Antarctic discordance (AAD), 224 Automatic relevance determination (ARD), 230

B Biogeodynamic coupling, 296–297 Brittle faulting, 466–467 Buoyancy-driven subduction models, 338–339

C Carbon recycling, 530 Cenozoic offshore record Izu–Bonin–Mariana (IBM) subduction zone, 364–366, 372f Puysegur–Fiordland subduction zone, 359–362 Tonga–Kermadec subduction zone, 366–367 Vanuatu and Hunter-Matthews subduction zone, 362–364 Clapeyron slope, 394 Code sharing, 574 Continental drift theory, 1 Continental rifting brittle faulting, 466–467 conceptual rift models, 466f force balance of, 467–469 geological features of, 462–465 lithospheric and mantle driving forces, 475 observations of rifts, 464f

overview of, 461–463f plate tectonic driving, 468 rifting and society, 472–475 from rift to mid-ocean ridge breakup and post-rift evolution, 470–472 concepts of rifted margin architecture, 469, 469f continent–ocean boundary, 469–470 faulting and mantle exhumation, 470 strain-induced weakening, 468 structural variability, 465–467 tectonic process, 459–460 timespan of, 460 Copernican Revolution, 192 Crustal dichotomy, 587

D Deformation boundary conditions, 541–542 boundary evolution, 543 nullspaces, 542 Deformation Mechanisms, Rheology and Tectonics (DRT) conference, 271–272 Dehydration, down-going slab, 531 Deserpentinization, 531–533 Diffuse oceanic plate boundaries (DOPBs), 84 deforming oceanic lithosphere, 95–97 evidence and analysis, 84–88 intra-oceanic plate deformation, 84 narrow oceanic plate boundaries, 84 Diffusion-driven grain boundary migration, 26 Dirichlet constraints, 541 Dislocation density piezometer, 25

589

DMSWARM, 563 Double subduction, 398 Dynamics subduction zones main driver of, 332–334 oceanic lithosphere at trench, 334–336 slab width, 336–337 Dynamic topography Amazon river drainage system, 224–225 computational approach, 245–250 data availability, 258 geodynamical models, 225 global representation of observational dataset ARD approach, 230–231 Gaussian process framework, 230–231 hierarchical Bayesian approach, 230 Matern covariance function, 230–231 power spectra, 232 long-wavelength residual topography, 240–241, 241f observational constraints, 242–245 Africa, 244–245 Australia, 244 North America, 243–244 observational estimates, 225–227, 228f observation-based dataset, 229–232 oceanic residual topography dataset shiptrack-derived measurements, 227–229 spot measurements, 227 ongoing uncertainties, 223–224 predictions from simulations of mantle flow

590 Dynamic topography (Continued) comparisons with observed geoid, 237–239 density and viscosity, 234–236 geological record, 232–234 inferred residual topography, 239f modeling approach and endmember cases, 234 power spectra for inferred residual topography, 233f synthetic predictions, 236–237 reconstructions into the geological past, 255–258 residual depth anomalies, 225 spatial pattern, 224 statistical theory of Gaussian process, 240 time-dependent global predictions, 250–255

E EarthByte research group, 429 Earth’s surface topography, 223 Earth’s tectonic evolution detrital zircons, 189–190 geodynamic regimes, 189 geological and geochemical evidence, 190 plate tectonics, 190–191 plutonic-squishy lid, 189 Earth systems science, 127–128, 128f Energetics of the solid earth gravitational energy release and viscous dissipation Earth’s mantle, 40–45 mantle energetics, 39–40 low-viscosity D" +plume +asthenosphere circuit, 45 mantle heat loss through the surface, 45–46 mantle’s “missing” energy supply, 48 non-hydrostatic internal deflections, 45 pattern and speeds, 56–58 plume-fed asthenosphere, 60–61 radioactive heat generation, 35 radioactive heat production in earth interior, 46–48 secular cooling of the mantle, 49–50 seismic observations, 36–39 upward return flow circuit lower mantle plumes, 58

Index

strong lateral flow in a shallow plume-fed asthenosphere, 59–60 strong lateral flow within the base of the D’’ layer, 58–59 Urey ratio, 48

F Faulting and mantle exhumation, 470 Feedbacks between internal and external dynamics changes in atmospheric CO2 and global surface temperature, 285–287, 286f dynamic interactions between continental collision, 276f effects on the evolution of life, 288 in extensional settings, 281–285 geological carbon cycle, 285–287 merging concepts toward an integrative earth system, 278–280 ongoing climate change, 288 From decompression melting to oceanic crust emplacement building the crust, 494–495 magma ascent and focusing, 493–494 magma generation, 491–493 From rift to mid-ocean ridge breakup and post-rift evolution, 470–472 concepts of rifted margin architecture, 469, 469f continent–ocean boundary, 469–470 faulting and mantle exhumation, 470

G Geochemical heterogeneity, 428 Geodynamic modeling buoyancy-driven subduction models, 338–339 flow law, 71–74 mantle convection simulations, 69–71 purpose of, 337–338 simulation set-up, 71 temporal evolution, 340–341 Geodynamic models MADS, 432 African, 445–447 evolution of, 444–445 LLSVP locations, 446t migration rates, 450, 451t

Pacific, 447–450 model plume characteristics, 431 conduit area and diameter, 440t estimated buoyancy flux, 437, 438t identification numbers and comparison of timings, 436t insights and remaining problems, 444 mantle structure and evolution, 444 paleomagnetic analyses from volcanic track, 441–444 temperature, 439t observations and analysis, 432–454 tectonic reconstructions, 453–454, 453t Geodynamics CITCOM code, 574 code sharing, 574 Earth science community, 573–574 mathematical choices, 577–578 numerical methods, 574 from reproducibility to reusability, 574–576 replication of computational research, 575–576 reproducible computations, 575 reuse model, 576 Underworld models hierarchical structure, 580 Lagrangian history term, 580 mathematical description, 581 PETSc data structures, 579 symbolic function layer, 579 GEOMAR, 529–530 Grain-damage model in deforming rock, 25 dislocation dynamics, 24–25 future aspects, 22–26 geometrically necessary dislocations (GNDs), 24–25 intragranular defects, 22–26 lithospheric deformation, 26–27 mixing and hysteresis, 14–17 monominerallic materials, 8–11 polymineralic materials, 11–13 Gravity potential energy (GPE), 277–278

H HeFesto, 395–396 Holmes model, 1–2

Index

I I3ELVIS geodynamic model, 277 Internal planetary feedbacks booting up plate tectonics, 150–151 boundary-layer interactions and plate-plume feedbacks, 139–146, 142f emergent plate tectonics, 152–153 mantle cooling trajectories, 134, 135f mantle Rayleigh number, 136f non-Newtonian upper mantle viscosity feedback, 138f plate tectonics, 129 plate tectonics-mantle dynamics, 146–150 self-sustaining feedbacks, 151 structural elements, 151 theory of plate tectonics, 150 thermal and deep water-cycling models, 132–134, 133f 3D spherical mantle convection model, 137, 137f Tozer feedback, 129, 130f International Panel on Climate Change (IPCC), 285 Izu–Bonin–Mariana (IBM) subduction zone, 364–366, 372f

K Kinematics subduction zones migration, 330 observations of, 332 velocities and reference frames, 331–332 velocity components, 330–331

L Large low shear velocity provinces (LLSVPs), 427–428 Large low velocity provinces (LLVPs), 235 Large-scale mantle flow, 398–399 Lithosphere-mantle interactions boundary conditions, 386–388, 386f geodynamics, 386 long-term, time-dependent, 399–400 observations and model design double subduction, 398 large-scale mantle flow, 398–399 slab breakoff, 397–398 slab width, 396–397 phase transitions, 393–396

recycling pathways and time along path, 400f stress dependence composite viscosity, 391–392 low-temperature plasticity, 390–391, 391f viscoelasticity, 392–393

M Magma-rich mid-ocean ridges, 500–502, 501f Mantle circulation models, 246 Mantle dynamics and mantle viscosity, 136–139 Mantle plumes and interactions Antarctic deformation and Pacific plate motion, 413f Copernican revolution, 407 future challenges, 421–422 in global geodynamics, 409–411 Hawaii–Emperor chain, 407–408, 408f hotspots fixed/moved, 411–412 hotspot tracks, 408 Iceland plume, 416, 418f initiation of subduction, 416–418 lithospheric uplift, 418–420 LLSVP margins, 414 plume-derived volcanism, 416 plume-induced plate rotations, 416–418, 419f plumes and continental breakup, 416 reference frames for plate motions, 412–413 thermal boundary layer (TBL), 409, 413–414 MARGINS community, 389 Mechanism and driving force of supercontinental formation analyses of the driving force, 208–211 prediction of future continental drift, 207–208 Mesoproterozoic-Neoproterozoic transitions biodiversity evolution, 297–300 biogeodynamics, 296–297 biological evolution, 300–303 Cryogenian, 313–314 enhanced erosion and weathering, 312–313 habitat formation and destruction, 313

591 large igneous provinces (LIPs), 313–314 marine carbonates, 311–312, 311f mobile surface environments, 296 nutrient supply, 310–311 plate tectonic revolution, 303–310 plate tectonics, 295–300, 315f single-lid to plate tectonics stimulated, 310–314 species richness, 298–299 strong tectonic–erosion, 313–314 Mid-ocean ridge (MOR) system, 484–485f axial morphology, 488f characteristics across spreading rates, 489f conduction and hydrothermal circulation, 483 cooling and building the oceanic lithosphere, 495–498 decompression melting to oceanic crust emplacement building the crust, 494–495 magma ascent and focusing, 493–494 magma generation, 491–493 history of, 485–491 large-scale ridge morphologies, 486–487 ridge axis morphologies, 487–491 seafloor record of interior dynamics, 502–504, 504f seafloor spreading rates, 484–485f seafloor through tectono-magmatic interactions detachment faults, 502 lithospheric thickness, 499–500 magma-rich mid-ocean ridges, 500–502, 501f quasi-amagmatic seafloor spreading, 502 upper mantle and lithosphere dynamics, 485 Modeled anomalously dense structures (MADS), 432 African, 445–447 evolution of, 444–445 LLSVP locations, 446t migration rates, 450, 451t Pacific, 447–450 Model plume characteristics, 431 conduit area and diameter, 440t

592 Model plume (Continued) estimated buoyancy flux, 437, 438t identification numbers and comparison of timings, 436t insights and remaining problems, 444 mantle structure and evolution, 444 paleomagnetic analyses from volcanic track, 441–444 temperature, 439t

N Neoproterozoic oxygenation event (NOE), 312 Neumann constraints, 541 Numerical convection model, 430–431 Numerical modeling of subduction compatible flow and transport, 552–553 enhanced solvers, 564 flow discretizations, 563 governing equations constitutive laws and coefficients, 543–544 deformation, 541–543 material domains and history variables, 545 thermal transport, 543 model definition, 540 open-source software (OSS) geodynamic software, 559–561, 560–561t linear algebra libraries, 563 PDE libraries, 562–563 PETSc, 563 physical processes, 564 physics-based models, 540 predictive capabilities, 565 realistic and accurate modeling, 541 solvers exact factorizations, 554–555 iterative methods, 555 linear flow solvers, 553–554 nonlinear flow solvers, 556 spatial discretization discontinuous Galerkin (dg) methods, 551 FE method, 548 flow, 548–551 inf-sup stable elements, 549–550 material regions, 551–552 stagFD discretization, 548 subelement resolution, 564

Index

three-dimensional subduction interface, 539–540, 540f Numerical simulation models description of, 215–217 NNR-MORVEL56 model, 216 reference viscosity, 215–216

O Oceanic lithosphere, 495–498 Onset of subduction, 116–118 Open-source software (OSS) geodynamic software, 559–561, 560–561t linear algebra libraries, 563 PDE libraries, 562–563 Ophicalcite, 530

P Pacific-Cocos-Nazca plate circuit, 97–99 PDE libraries, OSS automated finite elements, 562 discretization libraries, 562–563 Penrose Crust, 515–516 PERPLEX, 529–530 PETSc, 563 Planetary tectonics Mars, 587 Mercury, 586 mission observations and models, 588 parameters and observations, 584t stagnant lid mode, 583 terrestrial planets, 583 Venus, 585 weakening effects, 584 weathering process, 583 Planets’ evolution paradigm shift, 192 physics and numerical modeling, 191–192 Plate reconstruction models, 5 Plate subduction vs. plate bending and unbending, 532f Plate tectonics activation volume in the upper mantle, 78 in another planets, 191 automatic analysis of tectonics, 120 collapse of passive margins, 18–20 convection-tectonics system, 121 convective stress and yield stress, 77f

diffuse oceanic plate boundaries (DOPBs), 84 deforming oceanic lithosphere, 95–97 evidence and analysis, 84–88 intra-oceanic plate deformation, 84 narrow oceanic plate boundaries, 84 Earth’s tectonic evolution, 189–191 generation and onset, 17–18 geodynamic modeling, 69–74 grain-damage model in deforming rock, 25 dislocation dynamics, 24–25 geometrically necessary dislocations (GNDs), 24–25 intragranular defects, 22–26 lithospheric deformation, 26–27 mixing and hysteresis, 14–17 monominerallic materials, 8–11 polymineralic materials, 11–13 implications and model limitations, 79–80 inverse methods, 120 lithosphere strength, 69 mantle convection, 2 mechanisms of deformation, 92–93 memory localization, 119 oceanic lithosphere, 89–91 Pacific-Cocos-Nazca plate circuit, 97–99 parallel plate motion, 91–92 past of, 181–183 plates, climate, and planetary evolution, 21–22 poles of relative rotation across narrow plate, 94–95 poles of relative rotation between plates, 93–94 present of, 183 regime diagram, 74–78 rheology vs. initial conditions, 119 slab detachment, 20–21 spherical mantle convection simulations, 67–68 strength of the deep mantle, 68–69 surface mobility, 76f theory of, 2 viscosity and stress structures, 74 Plume-induced subduction initiation, 309

593

Index

Pseudo-plasticity context, 107–108 continental drift, 111–112 seafloor spreading, 112–114 temperature-dependent viscosity, 110–111 transform faults, 114–115 Puysegur–Fiordland subduction zone, 359–362

Q Quasireversibility (QRV) method, 249

R Residual topography, 224 RHEA, 387–388

S Seafloor record of interior dynamics, 502–504, 504f Seafloor through tectono-magmatic interactions detachment faults, 502 lithospheric thickness, 499–500 magma-rich mid-ocean ridges, 500–502, 501f quasi-amagmatic seafloor spreading, 502 Sequential data assimilation approach, 248 Serpentinization bend-fault, 530–531 bending-related faulting, 520–523 characteristics of, 513–514 crust–mantle boundary (CMB), 522f geochemical and biological consequences, 530–531 kinetics of, 514 mid-ocean ridge environment, 513, 515–516 multi-channel seismic (MCS) images, 520 peridotite–serpentine reaction, 511, 512f plate bending at trenches, 516–520 plate tectonics, 531 seafloor exposures, 513 slab fluid release, 519–520, 519f transform faults and fracture zones, 516 viscous slab bending and unbending, 533f water–rock reactions, 511

Slab breakoff, 397–398 Slab serpentinization, 530 Slab width, 396–397 Slip constraints, 541 Solvers exact factorizations, 554–555 iterative methods, 555 linear flow solvers, 553–554 nonlinear flow solvers, 556 Spatial discretization discontinuous Galerkin (dg) methods, 551 FE method, 548 flow, 548–551 inf-sup stable elements, 549–550 material regions, 551–552 stagFD discretization, 548 Stress dependence composite viscosity, 391–392, 391f low-temperature plasticity, 390–391, 391f rock deformation, 388 Subduction downwellings, 115–116 onset of, 116–118 Subduction initiation DSDP and ODP, 358 geological record, 357 mechanics of, 369–372 nucleation, 358–359 offshore Cenozoic record Izu–Bonin–Mariana (IBM) subduction zone, 364–366 Puysegur–Fiordland subduction zone, 359–362 Tonga–Kermadec subduction zone, 366–367 Vanuatu and Hunter-Matthews subduction zone, 362–364 onshore record, 367–369 Subduction zones dynamics main driver of, 332–334 slab width, 336–337 geodynamic modeling buoyancy-driven subduction models, 338–339 temporal evolution, 340–341 geology and geometry, 325–330 history of, 322–325 kinematics migration, 330 observations of, 332

velocities and reference frames, 331–332 velocity components, 330–331 South America and Scotia, 342–344, 344t Supercontinent cycle basal drag under continental plates, 211–212 continental growth, 198–199 cratonic lithosphere, 212 driving force of plate motion, 202–204 of supercontinental breakup, 204–207 of supercontinental formation, 207–211 dynamic interaction between mantle convection and continental drift, 200–202 numerical simulation of mantle convection, 199–200 unified mantle–core coupling system, 214f WSENS model, 213 super-Earths, 191

T Tectonic-convective modes compositional variations, 165 evolution of terrestrial bodies Earth, 168–169 heat pipe mode, 172 Mars, 172–173 terrestrial exoplanets, 173 Venus, 169–172 iso-chemical modes episodic-lid mode, 161 low-viscosity contrast mode, 161 mobile-lid mode, 161 ridge-only mode, 162 sluggish-lid mode, 161 stagnant-lid mode, 161 magmatism-induced modes heat pipe mode, 163 plutonic squishy lid/Plume-lid mode, 163 tectono-convective modes, 164f strain weakening and memory grain-size reduction, 167–168 shear heating, 168 texture (fabric) development, 168 yielding induced plate tectonics, 166–167

594 Tectonics of rocky planets and moons active tectonic regime, 184–185 episodic-lid regime, 186–187 heat-pipe regime, 185–186 mobile lid regime, 184f, 185 plutonic-squishy-lid regime, 187–188 ridge-only regime, 188 stagnant lid, 185 Tectono-magmatic style, 297 Thermochemically distinct regions, identification and evaluation, 431–432

Index

Thermo-mechanical numerical models, 273, 284f Tonga–Kermadec subduction zone, 366–367 Tozer feedback, 129, 130f

U Underworld models hierarchical structure, 580 Lagrangian history term, 580 mathematical description, 581

PETSc data structures, 579 symbolic function layer, 579

V Vanuatu and Hunter-Matthews subduction zone, 362–364 Venus, 585

W Whole Solid-Earth Numerical Simulation (WSENS), 213