Mantle Dynamics Following Supercontinent Formation by Philip J

Mantle dynamics following supercontinent formation

Philip J. Heron

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

Mantle dynamics following supercontinent formation

Philip J. Heron Doctor of Philosophy Graduate Department of Physics University of Toronto

2014

This thesis presents mantle convection numerical simulations of supercontinent formation. Approxi- mately 300 million years ago, through the large-scale subduction of oceanic sea floor, continental material amalgamated to form the supercontinent Pangea. For 100 million years after its formation, Pangea remained relatively stationary, and subduction of oceanic material featured on its margins. The present- day location of the continents is due to the rifting apart of Pangea, with supercontinent dispersal being characterized by increased volcanic activity linked to the generation of deep mantle plumes. The work presented here investigates the thermal evolution of mantle dynamics (e.g., mantle temperatures and sub-continental plumes) following the formation of a supercontinent. Specifically, continental insulation and continental margin subduction are analyzed. Continental material, as compared to oceanic material, inhibits heat flow from the mantle. Previous numerical simulations have shown that the formation of a stationary supercontinent would elevate sub- continental mantle temperatures due to the effect of continental insulation, leading to the break-up of the continent. By modelling a vigorously convecting mantle that features thermally and mechanically distinct continental and oceanic plates, this study shows the effect of continental insulation on the mantle to be minimal. However, the formation of a supercontinent results in sub-continental plume formation due to the re-positioning of subduction zones to the margins of the continent. Accordingly, it is demonstrated that continental insulation is not a significant factor in producing sub-supercontinent plumes but that subduction patterns control the location and timing of upwelling formation.

A theme throughout the thesis is an inquiry into why geodynamic studies would produce diﬀerent results. Mantle viscosity, Rayleigh number, continental size, continental insulation, and oceanic plate boundary evolution are explored in over 600 2D and over 20 3D numerical simulations to better understand how modelling method aﬀects conclusions on mantle convection studies. The results from this thesis show that the failure to model tectonic plates, a high vigour of convection, and a (pseudo) temperature-dependent viscosity would distort the role of mantle plumes, continent insulation, and subduction in the thermal evolution of mantle dynamics.

ii Dedication

To my parents, Ann and Ray.

iii Acknowledgements

Thank you to Prof Julian Lowman for all the time and eﬀort he has put into shaping my academic skill set. I’ve had the privilege of presenting work at conferences in Ottawa, San Francisco, Leeds, and in Massachusetts. For this, I am entirely grateful to Prof Lowman. Furthermore, my writing style has improved tremendously through Julian’s guidance (although I would now never use the word ‘tremendously’ (maybe ‘signiﬁcantly’ instead?)).

Through their honesty and thoughtfulness, my academic committee has made the PhD process very simple (although no less hard work). Thank you to Prof Stephen Morris and Prof Sabine Stanley. I am also grateful to Prof Stanley for the encouragement shown while I was her TA. Thanks to Prof Mathew Wells and Prof Dick Bailey who were part of my external committee. My external examiner, Prof Sam Butler, deserves credit for his excellent comments which helped improve the thesis. Keely O’Farrell and Sean Trim have always been on hand to offer help and encouragement (and to listen to a complaint about coding). I hope that one day we can publish our coffee table book, Numerical Modelling Mistakes. Robert Harrison and Ryan Vilim have helped with every presentation I have had to do. Thank you both for your excellent powerpoint skills and your 24/7 availability. A special thank you to the (past and present) Geophysics and Atmospheric graduates who have been great to be around (you know who you are). My graduate time in Toronto has been made a lot easier by the following people’s hard work and easy-going nature: Krystyna Biel, Crystal Liao, Teresa Baptista, Pierre Savaria, Jonathan Dursi, and all those at SciNet. I am also appreciative to Prof Tony Key for his excellent course on how to be an effective communicator, and to Becky Ghent, Adrian Lenardic, and Russ Pysklywec for taking an interest in my progress. This thesis is dedicated to my supportive parents. My family back home and my family in Ontario have been more caring and attentive to me than I have been to them over the past few years. Melissa has been a true collaborator with this work; she has heard every presentation and listened to every whinge. Thank you for your patience and encouragement.

iv Contents

List of Tables ix

List of Figures x

1 Introduction 1

2 Method 17

2.1 Introduction...... 17

2.2 Governingequations ...... 17

2.2.1 Approximations...... 18

2.2.2 Dimensionlessequations ...... 21

2.3 Numericalmodelling ...... 23

2.3.1 Massandmomentumequations...... 23

2.3.2 Energyequation ...... 28

2.4 Mantleviscosity ...... 28

2.4.1 Isoviscous and depth-dependent viscosity ...... 28

2.4.2 Temperature-dependentviscosity ...... 28

2.5 Force-balancemethod ...... 31

2.6 Time-dependentplatethickness ...... 33

2.7 Mantle temperatures and Rayleigh number ...... 33

2.8 Continentalinsulation ...... 35

2.9 Supercontinentmodelling ...... 38

2.10 Evolvingplategeometry ...... 41

2.11Models...... 43

v 3 The role of supercontinent thermal insulation and area in the formation of mantle plumes 44

3.1 Introduction...... 44

3.2 2DResults ...... 45

3.2.1 Initialcondition ...... 45

3.2.2 Continental coverage and mantle reversals ...... 46

3.3 3DResults ...... 48

3.3.1 Initialcondition ...... 48

3.3.2 Thermal response of the mantle after supercontinent formation ...... 50

3.3.3 Geothermsandmantletemperatures ...... 50

3.3.4 Non-insulating supercontinent ...... 53

3.3.5 Lower mantle viscosity and plume generation ...... 55

3.4 Discussion...... 55

3.4.1 Modelconsiderations...... 57

3.4.2 Mantlereversaltimeframe ...... 58

3.4.3 Continentalgeotherm ...... 59

3.5 Conclusion ...... 59

4 Plate mobility regimes and a re-evaluation of plate reversals 61

4.1 Introduction...... 61

4.2 Results...... 62

4.2.1 Mantlereversalsandmantleviscosity ...... 62

4.2.2 Parameter study: lithospheric cut-oﬀ temperature, TL ...... 64

4.2.3 Parameter study: thermal viscosity contrast, ∆ηT ...... 68

4.2.4 Parameter study: reference Rayleigh number, Ra0 ...... 71

4.2.5 Parameter study: aspect ratio and dimensionality study...... 74

4.3 Discussion...... 77

4.3.1 Platethickness ...... 80

4.3.2 Limitations ...... 81

4.3.3 AspectRatio ...... 82

4.3.4 Uniqueness ...... 84

4.4 Conclusion ...... 84

vi 5 The impact of Rayleigh number on the signiﬁcance of supercontinent insulation 86

5.1 Introduction...... 86

5.2 Comparing the vigour of mantle convection ...... 88

5.3 2DSupercontinentresults ...... 89

5.3.1 Initialcondition ...... 89

5.3.2 Isothermalcore-mantleboundary ...... 90

5.3.3 Insulatingcore-mantleboundary ...... 97

5.3.4 2D temperature increase due to insulation ...... 99

5.3.5 Average mantle temperatures and continental insulation...... 100

5.4 3DSupercontinentModels...... 103

5.4.1 3DSetup ...... 105

5.4.2 3DResults ...... 105

5.5 Discussion...... 107

5.5.1 Mantlepotentialtemperature ...... 108

5.5.2 Mantleheatingmode...... 108

5.5.3 Limitations ...... 110

5.6 Conclusion ...... 110

6 Inﬂuences on the positioning of mantle plumes following supercontinent formation 112

6.1 Introduction...... 112

6.2 Method ...... 113

6.3 2Dresults...... 116

6.3.1 Initial condition and supercontinent modelling ...... 116

6.3.2 Plume position as a function of subduction location ...... 117

6.4 3Dresults...... 120

6.4.1 3D initial condition and supercontinent modelling ...... 122

6.4.2 3D D100 T5...... 124

6.4.3 3D D30 T5 ...... 124

6.4.4 Non-insulating supercontinent ...... 126

6.4.5 Changingoceanicsubduction location ...... 126

6.5 Discussion...... 129

6.5.1 Viscosity proﬁle, continental coverage, and plume position...... 131

6.5.2 Plumegenerationzonesandsubduction ...... 132

vii 6.5.3 Sub-supercontinent isolation from subduction ...... 135 6.6 Conclusion ...... 135

7 Conclusion 136

Bibliography 141

viii List of Tables

2.1 Typical average properties for the variable in the Rayleighnumber ...... 35 2.2 Computational time for evolving and non-evolving oceanic plate boundary models . . . . 43

4.1 Benchmarking mantle parameters with Lowman et al. [2001]...... 63

5.1 Model parameters and initial condition properties for the isothermal basal boundary condition used in the Chapter 5 2Dstudy ...... 89 5.2 Input parameters and initial condition properties for models with the insulating basal boundary condition used in the Chapter 5 2Dstudy ...... 89

6.1 Input parameters for the 2D models in Chapter 6 ...... 115 6.2 Approximation of LIP position to continental margin subduction ...... 134

ix List of Figures

1.1 Thesupercontinentcycle...... 4

1.2 Present-day extent of large igneous provinces ...... 6

1.3 Present-daymantletomography...... 7

1.4 Circum-supercontinent subduction: Pangea and Rodinia ...... 8

1.5 The formation ofPangeathrough subduction ...... 9

1.6 Degree-1 and degree-2 mantle thermal structure ...... 12

1.7 PlumegenerationzonesandLLSVPs ...... 13

2.1 Flow chart of the calculation of temperature and velocityﬁeldsinMC3D ...... 24

2.2 Depth-dependent viscosity proﬁle used in Chapter 3 ...... 29

2.3 Depth-dependent viscosity proﬁles used in Chapters 4-6 ...... 30

2.4 Calculating the time-dependent plate thickness ...... 34

2.5 Temperature increase due to increasing continental insulation and increasing continental width ...... 36

2.6 Global paleogeographic reconstructions from 400-105Ma...... 39

2.7 An example of the evolution of oceanic plate boundaries ina3Dmodel ...... 40

2.8 Ageoftheoceanﬂoor ...... 42

3.1 Mantle reversals of supercontinent models: examples of no reversals for small continental coverage and sustained reversal for large continental coverage ...... 46

3.2 Subcontinental mantle ﬂow reversal results as a function of continental width and insulation for the 2D isoviscous convection study...... 47

3.3 3D initial temperature ﬁeld and model plate geometry...... 49

3.4 Plate geometry for pre- and post-supercontinent formation...... 49

x 3.5 Snapshots of temperature isosurfaces from a model with a supercontinent insulation factor of 0.25i...... 51

3.6 Average non-dimensional temperature as a function of depth from the model shown in Figure 3.5...... 52

3.7 Volume-averaged temperature beneath continents and oceans for the model presented in Figure 3.5...... 53

3.8 Snapshot of non-dimensional temperature ﬁeld from a model featuring a stationary su- percontinentwithanisothermalsurface...... 54

3.9 The two viscosity proﬁles as a function of mantle height used to model the generation of mantleplumes...... 55

3.10 Snapshots of deviation from the average horizontal temperature at depth 0.2d for two diﬀerentviscosityproﬁles...... 56

4.1 Mantle reversals (in an isoviscous model)...... 63

4.2 Plate velocity and heat ﬂux for models featuring mantle reversals and stagnant-lid tectonics 64

4.3 Stagnant-lid tectonics in a model featuring pressure- and geotherm-dependent viscosity . 65

4.4 Analysis of how the lithospheric cut-oﬀ temperature (TL) aﬀects plate mobility...... 66

4.5 Temperature snapshots of how changing the lithospheric cut-oﬀ temperature aﬀects mobility for a given internal heating rate...... 67

4.6 Analysis of how the thermal viscosity contrast (∆ηT ) aﬀects plate mobility...... 68

4.7 The volume-average and surface velocities as a function of average mantle temperature

for the ∆ηT parameterstudy...... 69

4.8 Temperature snapshots of how changing the thermal viscosity contrast aﬀects mobility at

agivenaveragemantletemperature...... 70

4.9 Analysis of how the reference Rayleigh number (Ra0) aﬀects plate mobility...... 72 4.10 The volume-average and surface velocities as a function of average mantle temperature

for the Ra0 parameterstudy...... 73 4.11 Temperature snapshots of how changing the reference Rayleigh number aﬀects mobility atagivenaveragemantletemperature...... 74

4.12 Comparing plate mobility as a function of internal heating rate/average mantle tempera- turefordiﬀerentaspectratiomodels...... 76

4.13 Temperature snapshots comparing Γ=1 (with one plate and reﬂective side-walls) and Γ=8 (with two plates and periodic side-walls) ...... 77

xi 4.14 Comparing plate mobility as a function of internal heating rate/average mantle tempera- turefordiﬀerentaspectratiomodels...... 78 4.15 Schematic tectonic regime diagram show the relationship between convective vigour and averagemantletemperature...... 79

4.16 Heating up and cooling down the mantle to change the tectonicregime...... 83

5.1 Plate thickness and average Rayleigh number relationship ...... 87 5.2 Geothermsfor2Dmodels ...... 90 5.3 2D supercontinent results: thermal ﬁelds for high and low Rayleigh number isothermal core-mantleboundarymodels ...... 91

5.4 2D supercontinent results: surface heat flux for high and low Rayleigh number isothermal core-mantleboundarymodels ...... 92 5.5 2D supercontinent results: time-series of volume-average temperatures beneath the oceans and continents for all isothermal core-mantle boundary models ...... 93 5.6 2D supercontinent results: volume-averaged temperature and basal and surface heat flux time-seriesathighandlowRayleighnumber ...... 94 5.7 2D supercontinent results: thermal fields for high and low Rayleigh number insulating core-mantleboundarymodels ...... 97

5.8 2D supercontinent results: time-series of volume-average temperatures beneath the oceans and continents for all insulating core-mantle boundary models...... 98 5.9 2D supercontinent results: heat generated per transit time as a function of average Rayleigh number for insulating core-mantle boundary models...... 98 5.10 2D supercontinent results: temperature increase solely due to insulation...... 100 5.11 2D supercontinent results: time-series of volume-average temperatures beneath the oceans

and continents for all isothermal core-mantle boundary models ...... 101 5.12 Mantle temperature deviation for 2D simulations ...... 102 5.13 Plate geometries: supercontinent formation ...... 103 5.14 3D supercontinent results: horizontal upper mantle ﬂow for low Rayleigh number models 104 5.15 3D supercontinent results: time-series of volume-average temperatures beneath the oceans and continents for the insulating core-mantle boundary model...... 106

5.16 3D supercontinent results: high Rayleigh number models...... 107

6.1 Paleo-subduction zone locations of the past 300Myr and the paleo-location of large igneous provincesfrom250Matopresent ...... 114

xii 6.2 Mantle geotherms and viscosity proﬁles for 2D models ...... 116 6.3 2D temperature snapshots of model D100 T5 showing the initial condition and location of sub-continental plumes with respect to subduction position in supercontinent models . 118 6.4 2D temperature snapshots for all models with supercontinent covering 50% (4.0d) of the surface...... 120 6.5 Plume position relative to continental margin location as a function of supercontinent coverage for models D100 T5, D30 T5, D100 T7, and D30 T7...... 121

6.6 3D initial condition and plate geometry for supercontinentformation ...... 122 6.7 Plate geometry, initial condition and temperature snapshots for 3D supercontinent models (using parameters given for model D100 T5) when continent coverage is 15%, 20%, 25% and 30% of the 4.25dx4.25d surface...... 123 6.8 Temperature field snapshots for 3D supercontinent models (using parameters given for model D30 T5)...... 125 6.9 3D temperature and surface heat flux snapshots for models featuring a non-insulating and aninsulatingsupercontinent...... 127 6.10 Insulating and non-insulating supercontinent temperature and heat flux time-series (model D100 T5) ...... 128 6.11 3D temperature snapshots comparing plume position for models featuring non-evolving and evolving oceanic plate boundaries 135Myr after supercontinent formation ...... 129

6.12 Temperature and heat ﬂux time-series for evolving and non-evolving oceanic plate boundaries (model D100 T5)...... 130 6.13 Thermal boundary layer analysis for high and low lower mantleviscosities ...... 133

7.1 Thesupercontinentcycle(revised) ...... 137

xiii Chapter 1

Introduction

The theory of plate tectonics describes the movement of Earth’s lithosphere, while the convective motion of the Earth’s mantle drives plates and therefore determines the present-day position of the continents. The twelve major tectonic plates that cover the Earth’s surface are comprised of ‘stiff’ lithospheric material (made up of the crust and part of the upper mantle) that moves over the more easily deformed asthenosphere [Schubert et al., 2001]. However, the movement of the plates is affected by the granitic continental crust being less dense than the mostly mafic oceanic crust [Fowler, 2005]. ‘Buoyant’ continental crust is significantly thicker (10-70km) than oceanic crust ( 7km), causing thin and dense oceanic ∼ material to subduct under continents at convergent continent-oceanic plate boundaries [Schubert et al., 2001; Fowler, 2005]. As a result, the age of the Earth’s surface is varied; some continental rocks are over 3.5Ga, whereas the oldest oceanic crust is less than 200Ma [Müller et al., 2008]. Continents are comprised of different pieces of buoyant lithosphere brought together by plate tectonic motion at sites of subduction of the oceanic lithosphere. Geological features formed by continent-continent collisions (e.g., mountain ranges, faulting) across Canada and America indicate that the North American continent consists of thirteen major cratons (old and stable continental lithosphere) amalgamated by plate tectonics [Hoffman, 1988]. Furthermore, similar fossils, flora and fauna on landmasses on either side of the Atlantic ocean indicate that North and South America were once attached to the African and Eu- ropean continents [Wilson, 1966]. These descriptions of the movement of the continents are a corollary of the theory of plate tectonics. However, the dynamic processes involved in plate tectonic motion, and its relation to the thermal evolution of the mantle, are still being debated.

At the University of Toronto in 1963, John Tuzo Wilson added a pivotal concept to the then peripheral theory of plate tectonics. Wilson [1963] suggested that the Hawaiian island volcanoes were created

1 Chapter 1. Introduction 2 by the north-west shifting of the Pacific tectonic plate over a fixed mantle hotspot. From that bold proposition and its implications, the plate tectonic theory began to generate more interest, and previous work supporting the theory was brought to the forefront of Earth Science research [e.g., Wegener, 1924; Holmes, 1931; Du Toit, 1937]. In 1966, based on evidence in the fossil record and the dating of vestiges of ancient volcanoes, Wilson proposed a cycle describing the opening and closing of oceanic basins, and therefore a method of amalgamating continental material (into a supercontinent) that would be subsequently dispersed (e.g., into the present-day continental configuration). Wilson [1966] outlined a four stage ‘Wilson cycle’ (as it was later known): the dispersal (or rifting) of a continent; continental drift, sea-floor spreading, and the formation of oceanic basins; new subduction initiation and the subsequent closure of oceanic basins through oceanic lithosphere subduction; and continent-continent collision and closure of the oceanic basin. The Wilson cycle has since been expanded upon and is often simply referred to as ‘the supercontinent cycle’. Over the past 50 years, geologists and geophysicists have progressed the theory of plate tectonics. However, the mechanisms involved in supercontinent formation and dispersal are still divisive. In 2014, the general form of the supercontinent cycle is comprised of four parts:

1. Continental material aggregates over a large downwelling to form a supercontinent [Santosh et al., 2009].

2. The formation of an almost stationary supercontinent [Scotese, 2001] generates subduction on its margins, with remnants of cold subducted material settling at the core-mantle boundary.

3. Thermal insulation by the continent traps underlying heat and the repositioning of subduction zones focuses thermal anomalies sub-supercontinent. A plume is formed beneath the supercontinent 50-100Myr after continental accretion [Li et al., 2003].

4. The supercontinent breaks up along pre-existing suture zones [Butler and Jarvis, 2004; Murphy et al., 2006, 2008], due to the lithosphere’s tensional yield stress being exceeded, and the continental fragments disperse. The timescale for the full cycle to be repeated is 200-400Myr [e.g., Zhong ∼ and Gurnis, 1993; Scotese, 2001; Yoshida and Santosh, 2011].

Figure 1.1 shows example thermal ﬁelds corresponding to the four parts of the supercontinent cycle. In order to better visualize the results presented in this thesis, 2D and 3D numerical simulations that correspond to each step in the cycle are shown alongside the schematic counterpart. Despite the advance- ments of our understanding of mantle convection, the roles of circum-supercontinent subduction (step 2, Figure 1.1) and continental thermal insulation (step 3, Figure 1.1) in the generation of sub-continental Chapter 1. Introduction 3

1) Supercontinent formation

2) Circum-supercontinent subduction

Figure 1.1 explanation overleaf. Chapter 1. Introduction 4

3) Continental insulation and plume generation

4) Supercontinent dispersal

Temperature 0.0 0.5 1.0

Figure 1.1: The supercontinent cycle. The figure shows a sketch of the widely agreed upon supercontinent formation and dispersal model. Thermal fields from 2D and 3D numerical calculations corresponding to each step of the cycle are shown for guidance. Colours of the thermal field correspond to the temperature key shown in (4). In 3D, the cold downwellings and warm upwelling are only shown in the interior of the model (for easier viewing), and the outline of the supercontinent plate is shown by the black lines on the surface of the model. In 2D, oceanic surfaces are indicated by dashed lines (arrow shows direction of plate motion) and continental surface is shown by a solid line (solid circles indicate no plate motion of the supercontinent). The cycle contains four parts: 1) supercontinent formation; 2) the generation of circum- supercontinent subduction; 3) continental thermal insulation and plume generation; and 4) supercontinent dispersal. In this study, supercontinent dispersal is not modelled. Chapter 1. Introduction 5 plumes (step 3, Figure 1.1) remain unclear. This thesis analyzes these key parts of the supercontinent cycle through the use of numerical models.

Geophysical observables are useful in understanding the supercontinent cycle. Past deep mantle plumes are thought to be manifested on the Earth’s surface by expansive areas of igneous material, erupted over relatively short geological timescales (e.g., large igneous provinces (LIPs)) [Burke and Torsvik, 2004]. Figure 1.2 shows twenty-three large igneous provinces (with deposition ages dating back to 251Ma) in their present-day location (with approximate eruption centres) [Torsvik et al., 2006]. The deep mantle origin of the large igneous provinces can be inferred from the generation of dyke swarms, surface uplift, and the geochemical signature of the erupted material [see Courtillot et al., 1999; Ernst et al., 2005]. Analyzing the rock record over Earth’s history shows little LIP activity during the amalgamation stage of the supercontinent cycle [e.g., Yale and Carpenter, 1998; Ernst et al., 2005; Ernst and Bleeker, 2010]. However, after a supercontinent has been formed for a period of time, the number of large igneous provinces increases on a global scale [e.g., Yale and Carpenter, 1998; Ernst et al., 2005; Ernst and Bleeker, 2010]. Below the surface, the thermal field of the present-day mantle may also hold information pertaining to supercontinent dynamics. Figure 1.3 shows horizontal cross-sections of a global seismic tomography model that depicts relative variations in shear velocity (with respect to the average) at 100km, 600km, 1000km and 2800km depth in the mantle [Kustowski et al., 2008]. Near the core- mantle boundary, anomalously warm material (characterized by slow shear wave velocities) is present beneath the Pacific and African plate, with the latter lying below the site of the last supercontinent Pangea. As a result of these present-day temperature anomalies, the mantle’s thermal and geoid profiles are characterized by a degree-2 harmonic structure.

Plate movement reconstructions (using paleomagnetism) and geological analysis of orogenesis (i.e., mountain building) also hold information pertaining to the supercontinent cycle. Studies analyzing the timing of continent-continent collisions and rifting sequences for the formation and dispersal of Pangea show the landmasses of Gondwana (the African, Antarctic, Indo-Australian and South American plates) and Laurasia (Eurasian and North American plate) colliding near the equator approximately 320Ma

[Smith et al., 1981; Hoﬀman, 1991; Scotese, 2001]. As a result of this collision, the Appalachian and Ural mountain belts were generated. Global plate reconstructions and analysis of volcanic arc lavas show Pangea to be ringed by subduction during the lifespan of the fully assembled supercontinent (Figure 1.4a) [Scotese, 2001]. The breakup of the supercontinent Pangea is thought to have occurred in two main stages: North America separating from the landmass 175Ma (starting the opening of the north Atlantic Ocean), followed by the dispersal of the Antarctic, Australian, Eurasian and South American continents between 140 and 100Ma [Smith et al., 1981; Hoﬀman, 1991; Scotese, 2001]. Chapter 1. Introduction 6

Figure 1.2: Present-day locations of large igneous provinces (green) and their eruption centres (dark green). Image and data from Torsvik et al. [2006] (with permission). The twenty-three large igneous provinces (with approximate age) are given as: CR, Columbia River Basalt (15Ma); AF, Afar Flood Basin (31Ma); GI, Greenland/Iceland (54Ma); DT, Deccan Traps (65Ma); SL, Sierra Leone Ridge (73Ma); MM, Madagascar (84Ma); BR, Broken Ridge (95Ma); WA, Wallaby Plateau (96Ma); HE, Hess Ridge (99Ma); CK, Central Kerguelen Plateau (100Ma); NA, Nauru Basalt (111Ma); SK, South Kerguelen Plateau (114Ma); RT, Rajhamahal Traps (118Ma); OP, Ontong Java Plateau (121Ma); MP, Manihiki Plateau (123Ma); MR, Maud Ridge (125Ma); PE, Parana-Etendeka (132Ma); BU, Bunbury Basalt (132Ma); MG, Magellan Ridge (145Ma); SR, Shatsky Ridge (147Ma); KR, Karroo Basalt (182Ma); CP, Central Atlantic Magmatic Province (CAMP) (200Ma); ST, Siberian Traps (251Ma). Chapter 1. Introduction 7

Figure 1.3: Horizontal cross-sections of a global seismic tomography model [Kustowski et al., 2008] showing relative variations (%) in shear velocity (with respect to the average) at 100km, 600km, 1000km and 2800km depth in the mantle. Slow velocities (warm material) are given as red while fast velocities (colder material) are given as blue. Figure is from Kustowski et al. [2008] (with permission). Chapter 1. Introduction 8

Figure 1.4: Circum-supercontinent subduction. Reconstructions of supercontinents (a) Pangea (195Ma) [Scotese, 2001], and (b) Rodinia (750Ma) [Li et al., 2008]. Image from Zhong et al. [2007] (with permission). Chapter 1. Introduction 9

340Ma

Eurasia

Rheic ocean

Gondwana

Approximate subduction zone direction

Figure 1.5: The formation of Pangea through subduction. Plate reconstruction during the formation of Pangea (340Ma). Triangles indicate subduction polarity (data taken from Scotese [2001] and approximately superimposed on this image). The Y-shaped arrangement of convergent plate boundaries, important in the aggregarion of continental material, is highlighted. The past location of the Rheic ocean is indicated. This ﬁgure uses a modiﬁed image from Blakely [2013] (with permission) and information from Scotese [2001] showing the location of convergent plate boundaries.

The supercontinent Rodinia formed with Laurentia (the North American craton) at its centre (Fig- ure 1.4b), and generated (amongst other mountain belts) the Grenville orogeny (including the Laurentian mountain range of Quebec) [Hoﬀman, 1991; Torsvik, 2003; Dalziel, 1991; Moores, 1991; Li et al., 2004, 2008]. By around 900Ma, Rodinia was fully assembled and subduction featured on its margins, similar to Pangea (Figure 1.4b) [Hoﬀman, 1991; Scotese, 2001]. Rodinia’s breakup began with India, Australia, East Antarctica and South China separating from Laurentia around 750Ma (150Myr after the full assembly of the supercontinent, a timescale comparable to the longevity of Pangea’s assembly) [Li et al., 2008].

The formation of Pangea and Rodinia have been attributed to large scale mantle downwellings amass- Chapter 1. Introduction 10 ing continental material (Figure 1.5) [Scotese, 2001]. Since plate tectonic theory’s early days, the closing of oceanic basins due to subduction has been linked to orogenesis and supercontinent growth [e.g., Wil- son, 1966; Dewey, 1969]. The process of ‘introversion’ [e.g., Murphy and Nance, 2003], where oceans that are interior to the supercontinent close to amass continental material, has been shown to have occurred in the formation of Pangea [Scotese, 2001]. The closing of the Iapetus and Rheic oceans (the latter through a sudden reversal in oceanic plate motion) (Figure 1.5) are believed to be fundamental in the introversion method of amalgamating the supercontinent Pangea [Murphy et al., 2006; Nance et al.,

2012]. After analyzing the topology within supercontinents, Santosh et al. [2009] proposed the large scale downwellings that amass continental material to be produced at a ‘Y-shaped’ plate boundary junction (Figure 1.5). This conﬁguration would promote stronger downwellings that could generate runaway subduction of oceanic material [Santosh et al., 2009]. As a result, subduction plays the key role in determining the location of a future supercontinent. If ‘introversion’ processes are dominant, then the Atlantic will act as the present-day versions of the Iapteus and Rheic oceans in the future formation of “Pangea Ultima” [Scotese, 2001]. However, if ‘extroversion’ subduction (the closing of an ocean exterior to the last supercontinent [e.g., Murphy and Nance, 2003]) is dominant, then the Paciﬁc will continue its reduction to form the supercontinent “Amasia” [e.g., Yoshida and Santosh, 2011].

Due to its relatively greater buoyancy, continental material remains on the Earth’s surface while oceanic plates are subducted. Furthermore, continental lithosphere inhibits heat loss from the Earth’s interior, relative to oceanic lithosphere, due to its thickness and the warmth of the radioactively enriched crust. Anderson [1982] first suggested that continental insulation could control the supercontinent cycle. As Pangea was relatively stable in one location for approximately 150Myr, Anderson [1982] proposed that continental insulation could have had a dramatic effect on the underlying mantle. Over a long timescale, the supercontinent would trap excess heat and cause uplift (through thermal expansion), partial melting of the mantle, and, ultimately, the dispersal of continental material [Anderson, 1982]. A large geoid high would be generated sub-supercontinent, similar to the present-day geoid profile over Africa, through the thermal expansion caused by the continental insulation. A similar hypothesis states that continental material would elevate mantle temperatures through the radioactively enriched crust warming the mantle below (the ‘thermal blanket’ effect) [e.g., Gurnis, 1988]. Recently, however, the ‘thermal blanket’ effect and continental insulation have been used more pseudonymously. For simplicity, the ‘thermal blanket’ effect is described in this study as the warming of the mantle by a continent, and therefore has the same definition as the term continental insulation.

Many numerical studies have shown that the combination of continental coverage and insulation can Chapter 1. Introduction 11 generate sub-supercontinental temperatures higher than sub-oceanic mantle material, suggesting that continental insulation acts as the main driver for supercontinent break-up [e.g., Gurnis, 1988; Zhong and Gurnis, 1993; Lowman and Jarvis, 1993, 1999; Yoshida et al., 1999; Phillips and Bunge, 2005; Coltice et al., 2009; Phillips et al., 2009; Phillips and Coltice, 2010; Yoshida, 2010a; Rolf et al., 2012]. Furthermore, a recent geochemical study into ancient lava samples from the Atlantic and Pacific oceans indicates increased mantle temperatures during the dispersal of the supercontinent Pangea [Brandl et al., 2013]. Through analyzing lava samples of the past 170Myr, Brandl et al. [2013] show that the post- supercontinent upper mantle beneath the Atlantic Ocean was 150K warmer than the present day values. By comparing mid-ocean ridges in the Atlantic and the Pacific (where the latter has samples that formed more than 2000km from the nearest continental ridge), Brandl et al. [2013] found that upper mantle temperatures in the Atlantic remained high for 60-70Myr (before returning to temperatures like those found beneath the Pacific Ocean). They attribute the temperature difference to continental insulation by the supercontinent Pangea.

The two large thermal anomalies beneath Africa and the Pacific (Figure 1.3) are thought to be mantle ‘super-plumes’ (or plume clusters), formed through processes related to the supercontinent cycle [e.g., Schubert et al., 2004; Zhong et al., 2007]. However, the generation of sub-supercontinental plumes that are of deep mantle origin cannot be easily explained by the mechanisms involved in the thermal blanket effect. Continental insulation may lead to partial melting in the upper mantle and subsequent rifting of a supercontinent, but the mechanism for generating a reversal in whole mantle flow (i.e., a super-downwelling forming a supercontinent being replaced by a mantle super-plume) is unclear. Aside from continental insulation, isolation from subduction [Lowman and Jarvis, 1996; Lowman and Gable, 1999], radiogenic heating at the core-mantle boundary by chemically distinct oceanic slabs [Maruyama et al., 2007; Senshu et al., 2009], and circum-supercontinent subduction [Zhong et al., 2007; Trubitsyn et al., 2008; O’Neill et al., 2009; Zhang et al., 2010; Heron and Lowman, 2010] have all been suggested to account for a sub-supercontinent reversal in mantle flow (with the latter mechanism gaining the most attention in the geophysics community).

In three-dimensional spherical shell mantle convection models with mobile lids, Zhong et al. [2007] presented two planform regimes for the Earth. When a supercontinent is absent, the mantle planform is characterized by a spherical harmonic degree-1 structure, with a major upwelling in one hemisphere and a major downwelling in the other [Zhong et al., 2007] (Figure 1.6a). Following the employment of a supercontinent above the downwelling, a degree-2 planform develops with two antipodal major upwellings (Figure 1.6b). Consequently, a degree-1 planform acts to form a supercontinent which, once fully assembled, changes the mantle planform to degree-2. Figure 1.6 shows the role of circum-supercontinent Chapter 1. Introduction 12

a) b)

Figure 1.6: Examples of (a) degree-1 and (b) degree-2 mantle thermal structure from geodynamic models featured in Zhong et al. [2007]. Thermal structures are plotted as isosurfaces of residual temperature with contour levels of -0.15 (blue for cold mantle) and 0.15 (yellow for hot mantle). (a) shows stable degree-1 mantle planform without a supercontinent. (b) shows mantle planform ﬁve transit times after a supercontinent is placed on the surface (in the northern hemisphere). Circum-supercontinent subduction separates distinct diametrically opposed upwellings. Images from Zhong et al. [2007] (with permission). Chapter 1. Introduction 13

Figure 1.7: Plume generation zones and LLSVPs (image and figure caption from Torsvik et al. [2006] with permission). The figure shows shear wave velocity anomalies (SMEAN model) at a depth of 2800km with LIPs restored to their original eruption locations (for more information see Torsvik et al. [2006]). Velocity anomalies (δVs) are expressed in percent. Blue denotes regions with high velocity and red with low velocity. The -1% contour (e.g., 1% slow) is shown in black. Chapter 1. Introduction 14 subduction in the modulation of mantle planform. The formation of a stationary supercontinent generates subduction on its edges which in turn generates a ‘super-plume’ that subsequently facilitates the dispersal of continental material (Figure 1.6b). Zhong et al. [2007] suggest that the Africa and Pacific antipodal super-plumes (the basis of the degree-2 structure of the present-day mantle) are a consequence of the supercontinent cycle. Therefore, Zhong et al. [2007] conclude that the mantle modulates between degree-1 and degree-2 mantle planform for supercontinent formation and dispersal, and that mantle plumes are generated by the formation of the supercontinent.

In addition to their apparent consistency with some geodynamic modelling of the supercontinent cycle, the two anomalously warm regions on the core-mantle boundary (Figure 1.3) have seismic characteristics that indicate more than just a thermal heterogeneity. Therefore, the sub-African and sub-Paciﬁc anomalies may not have a passive role in the supercontinent cycle as inferred by Zhong et al. [2007]. As shown in Figure 1.3, the two regions are characterized by low shear wave velocity values. As a result, they are widely known as large low shear velocity provinces (LLSVPs), and will be subsequently referred to as such throughout this thesis. The magnitude of the shear wave velocity anomaly is believed to be too large for a simple thermal feature [Karato and Karki, 2001; Brodholt et al., 2007]. Furthermore, the sharp lateral changes in shear wave velocity, and the ratio of lower mantle shear and compressional wave speeds, are also not indicative of purely thermal structures [Ritsema et al., 1998; Ni et al., 2002; To et al., 2005; Karato and Karki, 2001; Saltzer et al., 2001; Brodholt et al., 2007]. These seismic characteristics, alongside the anti-correlation of shear wave velocity and bulk-sound velocity for the LLSVPs [Ishii and Tromp, 1999; Masters et al., 2000; Trampert et al., 2004; Simmons et al., 2010; Della Mora et al., 2011; Koelemeijer et al., 2012], suggest sub-Africa and sub-Paciﬁc chemical and thermal heterogeneities.

Many studies have analyzed the role of chemical heterogeneities in mantle convection, focussing on whether LLSVPs represent long-lived structures that inﬂuence mantle dynamics [e.g., Tackley, 1998, 2002; McNamara and Zhong, 2005; Torsvik et al., 2006; Burke et al., 2008; Torsvik et al., 2008; Deschamps and Tackley, 2008, 2009; Davies and Davies, 2009; Schuberth et al., 2009; Simmons et al., 2010; Torsvik et al., 2010; Zhang et al., 2010; Tan et al., 2011; Davies et al., 2012; Steinberger and Torsvik, 2012;

Schuberth et al., 2012; Li and McNamara, 2013; Conrad et al., 2013; Davies and Goes, 2014]. Through correcting the present-day large igneous province locations (Figure 1.2) to the paleo-positions at the time of deposition, Torsvik et al. [2006] showed that the boundaries of the LLSVPs correlate with the projected origin of the deep mantle plumes associated with LIPs (Figure 1.7). Burke et al. [2008] speculated that the shape of the LLSVPs were such that their edges would facilitate the generation of mantle plumes (e.g., steep sides of the LLSVPs would create plume generations zones (PGZs), as outlined in Figure 1.7). Furthermore, the present-day location of diamond mines (which are typically in Chapter 1. Introduction 15 western Africa) also gives an indication on the behaviour of LLSVPs over time. The igneous kimberlite rock (which often contain diamonds) forms under high pressure at depth but can be transported to the surface by deep mantle plumes. Torsvik et al. [2010] showed the deep mantle projection of the paleo-position of kimberlite deposits to lie on the margins of the LLSVPs (e.g., the PGZ). Through the location and dating of kimberlite sites, Torsvik et al. [2010] proposed the plume generation zones must have been stable over long timescales. Furthermore, Torsvik et al. [2010] speculated that as LIPs and kimberlite deposits for the past 300Myr (and perhaps longer) correlate with the plume generation zones,

LLSVPs must be relatively ﬁxed in one location. A numerical study by Steinberger and Torsvik [2012] showed LLSVPs to control mantle dynamics, with circum-supercontinent subduction (e.g., Figure 1.4) interacting with plume generation zones (Figure 1.7) to generate the large igneous provinces (Figure 1.2) originating due to the formation of Pangea.

The hypothesis of stable LLSVPs has both global financial significance (e.g., determination of the possible locations of diamond mines) and implications for theories regarding the supercontinent cycle (i.e., that LLSVPs play a dominant role in mantle dynamics). However, recent thermochemical geodynamic models [e.g., Zhang et al., 2010; Tan et al., 2011; Li and McNamara, 2013] have shown difficulty in generating stable LLSVPs on timescales predicted by Torsvik et al. [2006]; Burke et al. [2008]; Torsvik et al. [2008, 2010] and Steinberger and Torsvik [2012]. A contrary hypothesis is that downwellings which reach the core-mantle boundary would sweep aside chemical piles [e.g., Tackley, 1998; Kellogg et al., 1999; Jellinek and Manga, 2002; McNamara and Zhong, 2005], and that the current shape of LLSVPs is due to the Earth’s subduction history (e.g., Figure 1.4) moulding chemical piles beneath upwelling regions of the Earth (e.g., Figure 1.6b) [McNamara and Zhong, 2005; Bull et al., 2009]. However, super-plumes generated beneath the centre of supercontinents (as shown in Figure 1.6), or on top of thermochemical piles [e.g., Deschamps et al., 2011], do not correlate with the paleo-position pattern of large igneous province locations (Figure 1.7). Furthermore, previous geodynamic simulations that disregard LLSVPs (e.g., isochemical models) show plumes forming away from downwelling regions and under the centre of supercontinental material (not in-keeping with Figure 1.7) [e.g., Schubert et al., 2004; McNamara and Zhong, 2005; Zhong et al., 2007; Santosh et al., 2009; Zhang et al., 2010; Heron and Lowman, 2010; Yoshida and Santosh, 2011; Rolf et al., 2012]. Therefore, the question remains as to what role subduction and LLSVPs play in the generation of mantle plumes.

This thesis focuses on three main topics related to the supercontinent cycle: continental insulation (Chapters 3 and 5), the generation of sub-continental plumes (Chapter 6), and how modelling method affects conclusions on mantle convection studies (Chapter 3-6). Discussing why geodynamic studies produce different results (e.g., meta-geodynamics) is crucial for validating the conclusions of the well Chapter 1. Introduction 16 established research field examining supercontinent formation and dispersal. This study analyzes what role continental insulation and subduction play in the generation of mantle plumes, and what ingredients in the model characteristics and setup are the most important in the study of the supercontinent cycle. Chapter 2

Method

2.1 Introduction

This chapter describes the methodology of modelling mantle convection using a numerical code (MC3D) initially developed by Gable [1989] and subsequently modiﬁed over the past two decades [e.g., Nettleﬁeld, 2005; Gait, 2007; Stein and Lowman, 2010; O’Farrell, 2013]. I have expanded MC3D to include attributes pertaining to modelling supercontinent formation and analysing the thermal evolution of the mantle. These new features in MC3D are described in sections 2.4 (geotherm-dependent viscosity), 2.6 (time- dependent plate thickness), 2.8 (continental thermal insulation), 2.9 (a stationary supercontinent) and 2.10 (a stationary supercontinent with evolving oceanic plate boundaries). All computations and coding for this thesis were conducted on the GPC supercomputer at the HPC SciNet consortium [Loken et al.,

2010].

2.2 Governing equations

The hydrodynamic equations governing mantle convection can be derived from the basic principles of conservation of mass, momentum and energy [Chandrasekhar, 1961; Jarvis and Peltier, 1989]. In a Cartesian geometry, the conservation of mass (the continuity equation),

∂ρ ∂ + (ρvi)=0, (2.1) ∂t ∂xi

expresses the fluid density ρ as a function of time t and position xi (where i = 1, 2, 3) and the fluid velocity vi as a function of time and position. Applying Newton’s second law of motion to an infinitesimal fluid

17 Chapter 2. Method 18 element and balancing the body and surface forces yields the conservation of momentum (the Navier- Stokes) equations, ∂vi ∂vi ∂Φ 1 ∂τij + vj +2εijkΩj vk = + , (2.2) ∂t ∂xj ∂xi ρ ∂xj where Ωj is the angular velocity of the rotating ﬂuid; εijk is the Levi-Civita tensor; Φ the apparent gravitational potential in the rotating frame of reference, i.e.,

1 Φ= U + Ω r 2, (2.3) 2| × | with U as the gravitational potential; and τij is the stress tensor,

2 τij = Pδij +2ηe˙ij ηδij e˙kk (2.4) − − 3

where P is the pressure, η is the Newtonian dynamic viscosity, δij the Kronecker Delta ande ˙ij is the strain rate tensor, 1 ∂vi ∂vj e˙ij = + . (2.5) 2 ∂xj ∂xi

The conservation of energy, known as the heat equation, is

∂T αT ∂P ∂T ∂T ∂ ∂T ∂vi ρCp + vi = k + χ + τij (2.6) ∂t − ρCp ∂t ∂xi − ∂xi S ∂xj ∂xi ∂xj

where α is the coeﬃcient of thermal expansion; Cp is the speciﬁc heat at constant pressure; χ the rate of internal heat generation per unit volume and k the thermal conductivity.

2.2.1 Approximations

For this study, the Boussinesq approximation is implemented [Boussinesq, 1897]. Any signiﬁcant temporal variations in ρ occur on timescales small enough to be considered irrelevant for convection processes (e.g., due to seismic eﬀects), and the Boussinesq approximation takes ρ as independent of pressure and only weakly temperature-dependent. Density variations are only taken into consideration in the body force term that drives convection and density is treated as constant elsewhere. Cp and g are also taken as constant.

The conservation of mass, momentum and energy are completed by an equation of state,

ρ = ρ [1 α(T T )], (2.7) 0 − − 0 Chapter 2. Method 19

where T0 is a reference temperature and ρ0 is a reference density deﬁned by ρ0=ρ(T0). As density is to be only weakly dependent on temperature, for the equation of state to hold then it must be that

α∆T << 1, (2.8) where ∆T is the maximum value of (T T ). For the Earth, α∆T is estimated to be approximately − 0 6 10−2 [e.g., Parise et al., 1990; Hofmeister, 1999]. Density variations in the conservation of mass and × energy equations, therefore, can be found to be negligible and the ﬂuid treated as incompressible.

Hydrostatic compression, where density varies considerably due to the pressure of overburden, is signiﬁcant for deep ﬂuid layers. The scale height, HT , is the layer thickness where the approximation that assumes density is not dependent on pressure breaks down:

C H = P . (2.9) T gα

7 For the whole mantle, HT 10 m. A condition to justify the application of the Boussinesq approximation ∼ is d << 1, (2.10) HT where d is the depth of the layer. However, for whole mantle convection in the Earth this condition still holds, but is at the threshold (as d

The viscous generation of heat in the energy equation is associated with the term τij (∂vi/∂xj), which can be shown to be negligible if the height of the convecting layer d is much less than the temperature scale height (HT ). The adiabatic temperature gradient, (∂T/∂xi)S, and the term containing the temporal derivative of pressure in the energy equation are also of the order d/HT and therefore negligible by the Boussinesq approximation.

A further simplification of the governing equations is to assume that the mantle material deforms by diffusion creep, where atoms migrate by the movement of adjacent vacancies (in contrast to dislocation creep, where slip in the crystalline lattice results in the breaking and reforming of individual bonds of neighbouring atoms) [Gordon, 1967]. Diffusion creep allows for a Newtonian flow, where dislocation creep results in a more nonlinear viscous rheology [e.g., Gordon, 1967]. The stress tensor (Eqn. 2.4) gives the relation of the strain rates to Newtonian dynamic viscosity.

Applying these assumptions, the governing equations are simpliﬁed before being non-dimensionalized. Chapter 2. Method 20

Due to the the ﬂuid being incompressible, the conservation of mass becomes

V = 0, (2.11) ∇ · which simpliﬁes the stress tensor (Eqn. 2.4) to

τ = P I +2ηe˙, (2.12) − where I is the identity tensor. The conservation of momentum can be re-written as

∂V ρ ρ [ + V V + 2(Ω V)] = P + ρ U + 0 Ω r 2 + η V + ( V)T (2.13) 0 ∂t · ∇ × − ∇ ∇ 2 ∇| × | ∇ · ∇ ∇ where U = g = -gˆz. Decomposing the pressure term into its hydrostatic (P ) and non-hydrostatic ∇ 0 components (P ), alongside the equation of state (Eqn. 2.7), allows for the gradient of the hydrostatic pressure ﬁeld (e P = -ρ gzˆ) to further simplify the conservation of momentum, ∇ 0 0

∂V ρ ρ [ + V V + 2(Ω V)] = P P ρgzˆ + 0 Ω r 2 + η V + ( V)T (2.14) 0 ∂t · ∇ × −∇ − ∇ 0 − 2 ∇| × | ∇ · ∇ ∇ e so

∂V ρ ρ + V V + 2(Ω V) = P +ρ gα(T T )ˆz+ 0 Ω r 2 + η V + ( V)T . (2.15) 0 ∂t · ∇ × −∇ 0 − 0 2 ∇| × | ∇· ∇ ∇ e Furthermore, due to the rotation of the ﬂuid body the centrifugal force can be included in a modiﬁed pressure term [Landau and Lifshitz, 1998],

ρ P = P 0 Ω r 2, (2.16) − 2 | × | b e which is substituted into Eqn 2.15 (and incorporated into 2.12). The conservation of momentum thus becomes

∂V ρ + V V + 2(Ω V) = P + ρ gα(T T )ˆz + η V + ( V)T . (2.17) 0 ∂t · ∇ × − ∇ 0 − 0 ∇ · ∇ ∇ b

Eliminating the terms in the energy equation (2.6) of order d/HT generates

∂T ∂T ∂ ∂T ρ0Cp + vi = k + χ (2.18) ∂t ∂xi ∂xi ∂xi Chapter 2. Method 21

∂T ∂T ∂ ∂T χ + vi = κ + , (2.19) → ∂t ∂xi ∂xi ∂xi ρ0CP where κ is the variable thermal diﬀusivity (κ=k/ρ0Cp). In the modelling of thermally and mechanically distinct plates, the thermal diﬀusivity in this study varies (see section 2.8).

2.2.2 Dimensionless equations

In order to analyze the terms in the equations with regard to their signiﬁcance for modelling mantle convection, it is helpful to convert the governing equations into dimensionless variables. Denoting a

2 ′ non-dimensional quantity by a prime, time can be scaled by the thermal diﬀusion time (t = (d /κ0)t ′ (where d is the depth of the mantle), the variable thermal diﬀusivity by κ= κ0κ , velocity by v = ′ ′ (κ0/d)v , dynamic viscosity by a reference viscosity (η = η0η ), the dimensionless pressure and stress

2 2 tensor by η0κ0/d (or ρ0ν0κ0/d where ν = η/ρ is the kinematic viscosity) and the temperature scale by T T =T ′∆T (where ∆T is the non-adiabatic temperature diﬀerence across the mantle model). The − 0 angular velocity can be written Ω=Ωˆz. Dropping the primes, the governing equations can be written in dimensionless form as κ ρ 0 V =0 V = 0 (2.20) 0 d2 ∇ · → ∇ · for the conservation of mass.

After simplifying Eqn. 2.17 with Eqn. 2.12, the following is attained:

κ2 ∂V κ2 κ η κ ρ 0 + 0 V V + 2Ω 0 (ˆz V) = 0 0 τ + ρ gαT ∆T zˆ (2.21) 0 d3 ∂t d3 · ∇ d × d3 ∇ · 0

κ ρ ∂V d2 ρ gαT ∆T d3 0 0 + V V + 2Ω (ˆz V) = τ + 0 zˆ (2.22) → η ∂t · ∇ κ × ∇ · κ η 0 0 0 0 Accordingly, 1 ∂V 1 2Ωd2 + V V + (ˆz V)= τ + RaT zˆ (2.23) → P r ∂t · ∇ P r κ × ∇ · 0 for the conservation of momentum. Ra and P r are the Rayleigh number and the Prandtl number, respectively.

The energy equation becomes

κ0∆T ∂T κ0 ∆T ∂T ∆T ∂ ∂T χ 2 + vi = κ0 2 κ + (2.24) d ∂t d d ∂xi d ∂xi ∂xi ρ0CP

∂T ∂T ∂ ∂T d2χ + vi = κ + (2.25) → ∂t ∂xi ∂xi ∂xi k0∆T Chapter 2. Method 22

∂T = (κ T ) V T + H (2.26) → ∂t ∇ · ∇ − · ∇ with k0=κ0ρ0CP the reference thermal conductivity (where H is the non-dimensional internal heating rate, and κ the non-dimensional variable thermal diﬀusivity). Expanding the V T term in 2.26 and · ∇ substituting the continuity equation (2.20), the energy equation becomes

∂T = (κ T ) (VT )+ H. (2.27) ∂t ∇ · ∇ − ∇ ·

Solutions to the above equations are determined by the Rayleigh number Ra, the Prandtl number P r and the non-dimensional internal heating rate H. The Rayleigh number [Chandrasekhar, 1961],

αg∆T d3 Ra = , (2.28) κ0ν0 is a measure of the relative importance of the buoyancy forces ( αg∆T) over the dissipative forces ∼ (νκ/d3), and is an indicator of the vigour of convection. The non-dimensional internal heating rate is the ratio of the Rayleigh number for non-dimensional internal heating RaH [Griﬃths, 1986] and the Ra, thus Ra H = H , (2.29) Ra where αgχd5 RaH = . (2.30) k0κ0ν0

The Prandtl number, ν P r = 0 , (2.31) κ0 is the ratio of the kinematic viscosity and thermal diffusivity of the fluid and therefore the relative importance of the diffusion of momentum and heat. The Prandtl number for the mantle is estimated as being as high as 1023 [Jarvis and Peltier, 1989] (where ν 1017m2s−1 and κ 10−6m2s−1), which 0∼ 0∼ indicates that the mantle disperses momentum very quickly and is not effective in transmitting heat. In comparison, the Prandtl number for the atmosphere is 0.75 (ν=1.5x10−5m2s−1 and κ=2.0x10−5m2s−1, ∼ [White, 2006]), indicating that momentum is dispersed slower than heat is transmitted. Pr 1023 is ∼ approximated as being of infinite value, which eliminates the inertial force term in the Navier-Stokes equation. Although the Coriolis force in the mantle is large (i.e., the multiplying factor in the third term of Eqn 2.23 (2Ωd2/κ) is of the order of 1015 [Schubert et al., 2001]), it is still small in comparison to the Chapter 2. Method 23

Prandtl number. Therefore, the Coriolis force term is also eliminated from Eqn 2.23 to give

τ = RaT z.ˆ (2.32) ∇ · −

Expanding the stress tensor gives the the equation of motion in its non-dimensional form as

η V + ( V)T P = RaT z,ˆ (2.33) ∇ · ∇ ∇ − ∇ − completing the set of governing non-dimensional equations for mantle convection. Speciﬁcally, Eqn’s 2.20, 2.27, and 2.33.

2.3 Numerical modelling

Mantle convection is modelled using the hybrid finite-difference spectral method code MC3D [Gable, 1989; Gable et al., 1991] to solve the dimensionless equations of mass (2.20), momentum (2.33) and energy (2.27) conservation for an infinite Prandtl number Boussinesq fluid in a Cartesian geometry. Figure 2.1 shows a simplified flow chart indicating how calculations are evolved. In summary, the partial differential equations for the mass and momentum equations are transformed into the spectral domain by taking the Fourier transform of the temperature (T(t0)), and velocity and stress fields are determined in spectral space [Gable, 1989]. The equations are simplified to obtain the velocity and stress fields in poloidal and toroidal space and finite differences are used to solve for the vertical gradients of the spectral coefficients [Gable, 1989; Gait, 2007]. After applying an inverse Fourier transform, the velocity field solution is used in the energy equation to advance the calculation in time (∆t), generating a new temperature field (T(t0+∆t)) (requiring a forward finite difference scheme integrated over the volume of each computational cell) whereupon the cycle begins again (Figure 2.1).

2.3.1 Mass and momentum equations

A spectral method is implemented for solving the partial diﬀerential equations for mass and momentum. In the numerical code MC3D, the temperature, velocity and stress ﬁelds are expressed in the (general) form ∞ ∞ − G(x,y,z)= gmn(z)e i(kmx+lny) (2.34) mX=0 nX=0

mn where the g are the complex Fourier coeﬃcients; km=2πm/Ax and ln=2πn/Ay the horizontal wavenum- bers, and Ax and Ay the non-dimensional solution domain length and width [Gable et al., 1991]. After Chapter 2. Method 24

T ∆t (time 0 stepping) Temperature field Forward finite Fourier volume transform Energy equation Mass (2.20) (2.27) and momentum equations (2.33) Inverse Finite Fourier difference transform approximation Velocity field

Figure 2.1: Flow chart of the calculation of temperature and velocity in MC3D. For more detailed explanation, see text. Chapter 2. Method 25 simpliﬁcation, the conservation of mass equation can be written as

∞ ∞ d mn mn mn V =0 v ikmv ilnv =0, (2.35) ∇ · → dz z − x − y mX=0 nX=0 where vx, vy and vz are the z-dependent Fourier coefficients of the velocity field components. If the distinct values of k, l and v determined by a unique m and n is assumed to be implicit (along with v on z), then the notation of equation 2.35 can be simplified so

∞ ∞ d mn mn mn d v ikmv ilnv =0 vz ikvx ilvy = 0 (2.36) dz z − x − y → dz − − mX=0 nX=0

(where this simpliﬁcation is used for the rest of the section).

Substituting the general form (equation 2.34) into the stress tensor (2.12) generates six unique components:

d 2ηikvx P η(ilvx + ikvy) η( vx ikvz)  − − − dz −  d τij = Pδij +2ηe˙ij = η(ilv + ikv ) 2ηilv P η( v ilv ) . (2.37) −  x y y dz y z   − − − −   d d d   η( dz vx ikvz) η( dz vy ilvz) 2η dz vz P   − − − 

Applying the general form to the momentum equation (where F is the buoyancy force of equation 2.32) d ikτxx ilτxy + τxz 0  − − dz    τ = F ikτ ilτ + d τ = 0 , (2.38) ∇ · →  xy yy dz yz     − −     d     ikτxz ilτyz + dz τzz   fz   − −    permits further simpliﬁcation to obtain a set of ordinary diﬀerential equations in spectral space. First,

d τxx τzz = ( 2ηikvx P ) (2η vz P ) (2.39) − − − − dz −

can be used with equation 2.36 to obtain a relation for τ xx

τxx = 2ηikvx 2η(ikvx + ilvy)+ τzz = 4ηikvx 2ηilvy + τzz. (2.40) − − − −

Following the same method, a relation for τ yy can also be found

τyy = 4ηilvy 2ηikvx + τzz. (2.41) − − Chapter 2. Method 26

d Concentrating on the x-component of equation 2.38, an expression for dz τxz can be obtained

d 2 τxz = ikτxx + ilτxy = ik( 4ηikvx 2ηilvy + τzz)+ ilτxy =4ηk vx +2ηklvy + ikτzz + ilτxy, (2.42) dz − −

and further simpliﬁed using τxy from equation 2.37:

d 2 2 2 τxz =4ηk vx +2ηklvy + ikτzz + il( η(ilvx + ikvy)) = (4k + l )ηvx +3ηklvy + ikτzz. (2.43) dz −

d Similarly, an expression for dz τyz can be obtained through rearranging the y-component of equation

2.38 and substituting τyy from equation 2.41

d τ = (4l2 + k2)ηv +3ηklv + ilτ . (2.44) dz yz y x zz

Accordingly, a set of ordinary differential equations for the z-derivatives of the Fourier coefficients of the velocity and stress spectral coefficients is found:

−1 −1 vx 0 0 η ik η 0 0 vx 0    −      v 0 0 η−1il 0 η−1 0 v 0  y     y       −              d  vz   ik il 0 0 00   vz   0    =     +   . (2.45) dz    2 2       τxz   η(4k + l ) 3ηkl 0 0 0 ik   τxz   0                   τ   3ηkl η(k2 +4l2)0 0 0 il   τ   0   yz     yz                     τzz   0 0 0 ik il 0   τzz   fz         

To reduce the computation time for solving the set of ordinary differential equation, variables are transformed into poloidal (P ) and toroidal (T ) components [Gable, 1989; Gait, 2007]. A vector field B can be defined as toroidal when, for a given scalar s,

B = sz, (2.46) ∇× b generating zero horizontal divergence. A vector ﬁeld can be deﬁned as poloidal if

B = sz, (2.47) ∇×∇× b generating zero vertical vorticity [Backus, 1986]. Gable et al. [1991] defined the new stress and velocity Chapter 2. Method 27 components as k l v = v + v , (2.48) P L x L y k l vT = vx vy, (2.49) L − L k l τ = τ + τ , (2.50) P L xz L yz k l τT = τxz τyz, (2.51) L − L where L=√k2 + l2, which simplifies the set of ordinary differential equations to yield

−1 vP 0 L η 0 0 0 vP 0         iv L 0000 0 iv 0  z     z       −         2      d  τP   4ηL 0 0 L 0 0   τP   0    =     +   , (2.52) dz          iτzz   0 0 L 0 0 0   iτzz   ifz     −               v   0 0000 η−1   v   0   T     T               2       τT   0 0 0 0 ηL 0   τT   0         

(which is subsequently solved for each combination of m and n). Six boundary conditions are applied to generate a two-point boundary value problem [e.g., Gable, 1989; Gable et al., 1991; Gait, 2007]); there is no shear stress on the core-mantle boundary,

τP (z =0)= τT (z =0)=0, (2.53) or any vertical velocity at the surface and the base of the model,

Vz(z =0)= Vz(z =1)=0. (2.54)

The ﬁnal boundary condition is that the surface moves with plate-like horizontal velocity (explained in section 2.5). Toroidal ﬂow can only be generated by any strike-slip motion at the plate boundaries in 3D simulations.

A finite-difference approximation [e.g., Press, 1986] is used to solve equation 2.52. Using the previous time-step as an initial guess at the solution, each iteration calculates a correction using a relaxation method. MC3D typically finds a solution after a small number of iterations using the relaxation method as the Courant-Friedrichs-Levy condition [Courant et al., 1928] keeps ∆t low and therefore the initial guess is often a good approximation. Chapter 2. Method 28

2.3.2 Energy equation

The numerical code MC3D solves the energy equation by using a temporal ﬁnite diﬀerence scheme [Travis et al., 1991]

∂T Tn+1 Tn = (κ T ) (VT )+ H − ∆Θ { (κ T VT ) ndA + H∆Θ (2.55) ∂t ∇ · ∇ − ∇ · → ∆t ≈ h ∇ − · in b where ∆Θ is the volume of a single cell; ∆t the time step; n the time count number (t = n∆t); and dA the area of each cell face in the calculation. The solution to the ﬁnite diﬀerence calculation is a second order (O(∆t2)) approximation due to the application of a corrective factor implemented in MC3D [e.g., Travis et al., 1991; Gait, 2007].

2.4 Mantle viscosity

The studies in this thesis use three diﬀerent viscosity conditions; Chapter 3 uses an isoviscous viscosity and a depth-dependent viscosity, while Chapters 4, 5 and 6 implement a depth-dependent viscosity with a geotherm-dependent factor.

2.4.1 Isoviscous and depth-dependent viscosity

7 Isoviscous models that omit internal heating in Chapter 3 feature Rab =1 10 , and the non-dimensional × mantle viscosity, η′, is 1 from the base of the plates to the core-mantle boundary. The models that include a depth-dependent mantle viscosity (Chapter 3) are characterized by a factor of 36 increase in viscosity

7 from the upper to lower mantle (Figure 2.2) [Pysklywec and Mitrovica, 1997] and feature a Rab =5 10 × (based on upper mantle viscosity). To obtain plate-like surface motion and structure in the models of Chapter 3, a ﬁnite thickness lithosphere is prescribed a viscosity that is 1000 times greater than the mantle directly below (Figure 2.2). The models of Chapter 3 have no time-dependent viscosity, and implement ﬁxed plate thickness (for both oceanic and continental plates) throughout the calculations.

2.4.2 Temperature-dependent viscosity

In Chapters 4, 5 and 6, the non-dimensional mantle viscosity, η′, is calculated through the product ′ of the geotherm- and (background) depth-dependent viscosities (η = η ¯ ηD). Models have a time- T × dependent viscosity resulting from the use of the horizontally-averaged non-dimensional mantle temper- ¯ ature (geotherm), T (z), in the calculation of ηT¯(z): Chapter 2. Method 29

Figure 2.2: Depth-dependent viscosity proﬁle from Pysklywec and Mitrovica [1997] (used in Chapter 3 only). Note the plates are 1000 times more viscous than the mantle directly below (indicated by the dashed line). Chapter 2. Method 30

0.8

0.6

Height 0.4

0.2

0 0 1 2 10 10 10 Viscosity

Figure 2.3: Depth-dependent viscosity proﬁles used in Chapters 4-6. Red: ηD=100 as described in text; green: ηD=30 as described in text. Chapter 2. Method 31

−T¯ ηT¯ = ∆ηT , (2.56)

where ∆ηT is the non-dimensional viscosity contrast owing to temperature. In this work, the range of non-dimensional viscosity associated with the thermal viscosity contrast (∆ηT ) is from 1 at the surface to (1/∆ηT ). As a result, an increase in ∆ηT will decrease the average viscosity of the model.

In Chapters 4 and 5, the non-dimensional depth-dependent viscosity (ηD(z)) is characterized by a factor of 100 increase from the surface to core-mantle boundary (with a factor of 40 increase between 600km and 700km depth and a factor of 50 increase from 700km to the core-mantle boundary). In

Chapter 6, an additional model featuring a factor of 30 increase in depth-dependent viscosity from the surface to the core-mantle boundary is used to analyze the response of the calculations to changes in lower mantle viscosity. Figure 2.3 shows the ηD(z) proﬁles used in the 2D and 3D studies of Chapters 4, 5 and 6.

The viscosity fields implemented in Chapters 4, 5 and 6 have no lateral dependence and are therefore not fully temperature-dependent. However, the methodology used here, of implementing a geotherm- and (background) depth-dependent viscosity (motivated by the effect of pressure and phase changes on viscosity), captures the dynamics due to fluctuations in thermal boundary layer thickness and accompanying changes in the depths over which a given viscosity contrast occurs. Furthermore, this methodology models a time-dependent lithospheric thickness (section 2.6) which results in a spectrum of different plate tectonic regimes so that the findings compare well with results from models featuring pressure, temperature and stress-dependent rheologies [e.g., Moresi and Solomatov, 1995; Stein et al., 2004; O’Neill et al., 2009; van Hunen and van den Berg, 2008; Foley and Becker, 2009; Korenaga, 2010; Stein et al., 2013]. In addition, Stein and Hansen [2014] showed layered temperature-dependent viscosity convection to be a suitable approximation to the full temperature dependence. When comparing mantle convection models featuring layered geotherm-dependent viscosity (as used in Chapters 4, 5 and 6) and fully temperature-dependent viscosity, Stein and Hansen [2014] found only minor differences in flow characteristics (including Nusselt number, convection flow regime, lid thickness, stress, and dynamic topography).

2.5 Force-balance method

Tectonic plates are described as having nearly uniform motion in their interiors, with plates separated by narrow bands of high strain-rate [e.g., DeMets et al., 1944; Minister and Jordan, 1978]. In this thesis, all Chapter 2. Method 32 calculations feature ‘plate-like’ surface velocities obtained by modelling dynamically determined, time- dependent, horizontal velocity boundary conditions. Global plate velocity and stress ﬁelds that neither add nor subtract energy from the system are continuously updated using a force-balance method [Gable et al., 1991; King et al., 1992; Brandenburg and van Keken, 2007] that balances the buoyancy and viscous resistance forces acting on the plates due to their motion. The linearity of the momentum equation allows the superposition of the solutions for the buoyancy force of the mantle ﬂow and the mechanical boundary conditions. Plate velocities are determined through the calculation of shear stresses at the base of the plates. The stress of the viscous resistance associated with plate movement is balanced against the buoyancy forces, so that there is no net force at the base of each plate [Gable et al., 1991; Gait, 2007].

The calculation of dynamic plate velocities is given by

i j −1 i l Vkj = ( Ckl) ( τ ), (2.57)

i l where Vkj is the plate velocity (in direction j=1, 2 of plate k=1,2...N, for a model with N plates); τ is the integrated shear stress on the base of plate l (where i=13, 23 and denotes the components of the

i j shear stress due to buoyancy); and Ckl is the plate interaction coeﬃcient that gives the stress (in the i component) on plate l due to the motion (in the j direction) of plate k. In summary, C quantiﬁes the viscous resistance to moving the plates when buoyancy forces are not present (and is a function of the plate geometry (i.e., plate size and shape)), while τ is determined by the buoyancy forces at the base of the plate. Eqn. 2.57 implies

13 1 13 1 13 2 13 1 13 2 τ C11 C11 . . . C91 C91 V11       23τ 1 23C1 23C2 . . . 23C1 23C2 V    11 11 91 91   12               .   ......   .               .  =  ......   .  , (2.58)              .   ......   .               13 9   13 1 13 2 13 1 13 2     τ   C19 C19 . . . C99 C99   V91         23 9   23 1 23 2 23 1 23 2     τ   C19 C19 . . . C99 C99   V92        for a 3D mantle convection model featuring nine plates (where the plate interaction coefficient matrix is 18 18). In MC3D, the calculation of the plate velocities is acquired through an inversion of Eqn. 2.58 × so that the buoyancy driven forces are balanced by the viscous resistance forces. As there is no laterally varying density in the models, the momentum equation is linear. At every time step the flow field is calculated as a superposition solution of the flow driven by body forces (with zero surface velocity) and Chapter 2. Method 33

flow driven by moving plates (with no body forces) [Gable et al., 1991]. This force balance method of calculating plate velocities is consistent with a strong rigid plate uniformly distributing applied stresses and has been shown to yield model plate velocities and heat flux values in agreement with methods that utilize rheologically defined plates [e.g. King et al., 1992; Koglin Jr. et al., 2005; Stein et al., 2013].

2.6 Time-dependent plate thickness

A tectonic plate (i.e., the upper most region of the mantle known as the lithosphere) is comprised of the crust and part of the upper mantle. The Earth’s crust is a thin layer of distinctive composition overlying the ultramafic upper mantle, the base of the crust marks a chemical boundary between the layers, and is seismologically defined as the Mohorovi˘cićdiscontinuity. The lithosphere-asthenosphere boundary is defined by a mechanical boundary where the rheology, or flow, of the ultra-mafic mantle changes. Over long timescales the lithosphere appears to behave as a rigid shell, whereas the asthenosphere behaves as a highly viscous fluid. Through seismic observations, this mechanical boundary is conventionally defined as the depth at which the mantle temperature passes 1350oC [McKenzie and Bickle, 1988], and ∼ delineates the base of the lithosphere (and therefore the tectonic plates).

In Chapters 4-6 (when geotherm- and depth-dependent viscosity is implemented), the lithosphere- asthenosphere transition is modelled as a mechanical boundary delineated by temperature, above which the mantle is sufficiently cool to behave in a more stiff manner and below which the material deforms more readily. The plate thickness is determined by the depth at which the geotherm (laterally-averaged temperature) exceeds a given lithospheric cut-off temperature, TL (Figure 2.4). The average plate viscosity is 2 orders of magnitude greater than the viscosity of the upper mantle (a small difference ∼ as compared to the Earth but sufficient to model relatively stiff plates). This method generates thick plates for low Rayleigh number models and thin plates for models which are convecting vigourously

(Figure 2.4). At this time-dependent depth, deﬁned by temperature TL, the force-balance calculation is applied. As a result, the oceanic and continental plates have a spatially uniform but time-dependent thickness (continental plates are given distinct thermal and mechanical properties that generate mantle insulation). In section 4.2.2, the parameter TL is explored to analyze its eﬀect on plate mobility.

2.7 Mantle temperatures and Rayleigh number

Recently, for plane-layer convection models with an Earth-like Rayleigh number, O’Farrell and Lowman [2010] showed that either no internal heating or a degree of internal cooling is necessary to attain the Chapter 2. Method 34

Time-dependent plate thickness Thin plate 1.0

0.9 Thick 0.8 plate 0.7 0.6 0.5 High Ra Height 0.4 Low Ra

0.3 TL=0.35 0.2 0.1 0.0 0.2 0.4 0.6 0.8 1.0 Temperature

Figure 2.4: Non-dimensional temperature against mantle height for a high Rayleigh number (green) and a low Rayleigh number (red) model. The dashed black line shows the lithospheric temperature, TL, which is 0.35 in this example. The base of the plates is deﬁned where the geotherm reaches the TL value. This method generates thick plates for low Rayleigh number models and thin plates for models which are convecting vigourously. Chapter 2. Method 35

Parameter Description Value α Coeﬃcientofthermalexpansion 3 10−5K−1 g Gravitational acceleration 10ms× −2 ∆T Mantle temperature change 3000K d Mantle depth ∼2900km κ Thermal diﬀusivity 10−6m2s−1 ν Kinematic viscosity ∼1018m2s−1 χ Rate of internal heat generation per unit volume∼ 3.5x10−8Wm−3 k Thermal conductivity 4.3Wm−1K−1

Table 2.1: Typical average properties for the parameters in the Rayleigh number (k taken from Hofmeister [1999] and the remaining values from Schubert et al. [2001]). χ is calculated by the product of the mean − − radiogenic heat generation per unit mass (7.4×10 12Wkg 1 [Schubert et al., 2001]) and the average mantle − density (4.7×103kgm 3 [Schubert et al., 2001]). Applying these values to the Rayleigh number (Eqn. 2.28), and internal heating Rayleigh number (Eqn. 2.30), estimates an Earth-like Ra to be ∼2×107, with a non-dimensional heating rate (H) of ∼25. spherical shell-type geotherms that occur with terrestrial concentrations of inferred internal heating.

Accordingly, for supercontinent formation models in Chapters 3, 5 and 6 that feature an isothermal core-mantle boundary, no internal heating is speciﬁed (only the 2D isoviscous models of Chapter 3 use internal heating). For purely internally heated models (Chapter 5) that feature an insulating core-mantle boundary, H values are used which generate interior temperatures and plate mobility similar to their bottom heated counterparts.

Applying Earth-like values to the parameters of the Rayleigh number and internal heating Rayleigh number (e.g., using Eqns. 2.28 and 2.30, and Table 2.1) approximates the Ra of the Earth to be 107 ∼ with a non-dimensional internal heating rate (H) of 25. These values would produce an internal heating ∼ 9 Rayleigh number (RaH ) on the order of 10 . The majority of the models in this thesis do not feature ∼ internal heating, but attain spherical shell-type geotherms. Therefore, the Cartesian models presented here can be considered to be approximating Earth-like convective vigour if Ra is 107 or higher.

2.8 Continental insulation

Continental insulation is prescribed by limiting the ability of the continental plate to conduct heat delivered from below by the mantle. The thermal diffusivity of the continental region, κc, is reduced in comparison to the oceanic lithosphere, allowing for the oceanic plates to have the thermal conductivity of the mantle and for the continental plate to be a relatively greater insulator. It should be noted that distinct oceanic and continental plate thicknesses [e.g., Rolf et al., 2012] are not modelled. However, by prescribing an insulating diffusivity in the high viscosity continental material, it is possible to mimic the thermal blanketing effect of thick continental lithosphere.

Figure 2.5 shows the eﬀect of continental insulation on temperature changes in the sub-supercontinent Chapter 2. Method 36

Continental temperature increase due to insulation 10 0 0.25 0.5 0.75

4 Temperature increase (%) 2

0 20 25 30 35 40 45 50 55 Continental width (% of total surface)

Figure 2.5: Temperature increase solely due to the thermal blanketing effect for a suite of high Rayleigh number models with varying continental insulation and continental width (e.g., 2D simulations in Figure ∼ 7 5 1.1). The volume-average Rayleigh number is 10 , with ηP and ηT¯ set at 100 and 10 , respectively (with H=20 for all models). The continental insulation parameter i is specified as 0 (a perfect insulator), 0.25, 0.5, or 0.75 (where i is the ratio of the continental and mantle diffusivities, κc/κ). Temperature increase is measured 500Myr (∼8 mantle transits) after supercontinent formation and calculated as the non-dimensional difference in the volume-average temperature beneath an insulating and non-insulating continent. Chapter 2. Method 37 mantle for varying supercontinental coverage in 2D models with a volume-average Rayleigh number of 107 (as given by equation 5.1). The influence of four continental insulation values (i) is presented; one ∼ end-member case of a perfectly insulating supercontinent and three cases where the ratio of the thermal

1 1 3 diffusivity in the continental material to that of the oceans is /4, /2 or /4. To isolate the effect of a ‘thermal blanket’, the models are purely internally heated and feature an insulating core-mantle boundary (so that stirring by plume formation does not affect sub-continental temperatures). Supercontinent formation is accompanied by the appearance of extensive circum-continental subduction zones, leading to the formation of pools of cold material beneath the newly formed supercontinent (see Figure 1.1). The cessation of subduction at the continental suture and the warming of the cold material means that sub-supercontinent temperatures will increase regardless of any continental insulation in effect (i.e., a background warming occurs due to the mechanics of forming even a non-insulating super-plate with oceanic thickness and thermal properties [Heron and Lowman, 2010]). To take this into account, the plots in Figure 2.5 show the percentage temperature increase under the supercontinent solely due to continental insulation. This is achieved by subtracting the mean sub-continental temperature found in a model with a non-insulating continental plate (T¯nonins) from the mean sub-continental temperature in the case with an insulating continent (T¯ins). The temperatures are evaluated 500Myr after supercontinent formation (well in excess of the proposed maximum time-scale for a supercontinent cycle [Scotese, 2001; Yoshida and Santosh, 2011]) when any influence due to insulation should have been in effect for a significant period. To measure the effect of insulation, ∆T¯ins is analyzed where

T¯ins(500Myr) T¯nonins(500Myr) ∆T¯ins(500Myr)= − . (2.59) T¯nonins(500Myr)

All speciﬁed parameters in the non-insulating models analyzed with Eqn. 2.59 are identical to those in the insulating models (except for the continental insulation parameter). This isolates the true eﬀect of continental insulation.

From Figure 2.5 it can be seen that increasing the continental insulation increases the temperature under the supercontinent. Moreover, for all continental insulation parameters, i, an increase in continental coverage generates an increase in sub-supercontinent temperature, as shown in previous studies [Phillips and Coltice, 2010; Heron and Lowman, 2011; Rolf et al., 2012]. Figure 2.5 demonstrates the ability of the model continents to generate a thermal blanket eﬀect as produced by thick continent material (albeit with no distinction between the thicknesses of continental and oceanic plates). For instance, 500Myr after forming, a supercontinent that is a perfect insulator covering 50% of the surface will generate a 14% increase in sub-continental temperature relative to the initial condition temperature Chapter 2. Method 38

(58% of this total can be contributed to thermal insulation and 42% to the cessation of subduction at the continental suture and the warming of the cold material). Applying Eqn. 2.59, there is an 8.3% temperature increase, relative to a non-insulating sub-continental temperature, that is solely due to thermal insulation (Figure 2.5).

A continental diffusivity value of 0.25 of the oceanic value is chosen (unless otherwise specified) for the remainder of the thesis (the temperature increase shown in Figure 2.5 for this parameter is comparable to that observed in other recent studies of supercontinents in mantle convection models [Yoshida, 2013]). The effective thickness of the continental lithosphere for this continental diffusivity value can be interpreted by considering the thermal diffusion timescale

2 Lc tp = , (2.60) κc

which characterizes the time taken (tp) for a temperature change to diﬀuse a distance Lc (the continental lithosphere thickness). By setting κc = 0.25κ, the continent eﬀectively insulates as if its thickness were doubled.

2.9 Supercontinent modelling

Figure 2.6 shows global paleogeographic reconstructions from 400-105Ma (in 100Myr intervals), indi- ∼ cating the formation of an almost stationary supercontinent between 300-200Myr [Scotese, 2001]. As discussed in Chapter 1 (and shown in summary Figure 1.1), when analyzing the thermal evolution of the mantle post-supercontinent formation it is appropriate to model stationary continental material.

MC3D has been modified to model a stationary supercontinent and oceanic plate motion that neither adds nor subtracts energy from the mantle convection system. Requiring the supercontinent plate to have a velocity of zero generates a net flow in the system given the force-balance method (section 2.5). To counter the net flow resulting from the imposed stationarity of the continental plate, the overall difference in plate velocities is added to the mantle velocity at every grid point and every time step. The plate velocities and stresses, which are determined by the plate interaction and buoyancy forces, are then calculated by accounting for net flow generated from the influence of the stationary plate. Chapter 2. Method 39

a) b)

400Ma 300Ma c) d)

200Ma 105Ma

Figure 2.6: Global paleogeographic reconstructions of relative continental positions from: a) Early Devo- nian (400Ma); b) Pennsylvanian (300Ma); c) Early Jurassic (200Ma); d) and late Early Cretaceous (105Ma) (where Scotese [2001] and Blakely [2013] explain the method of calculating global paleographic maps). The supercontinent Pangea is inferred to be relatively stationary for 100Myr (between 300-200Ma [Scotese, 2001]) compared to the continental conﬁguration at 400Ma and 105Ma. Images from Blakely [2013] (with permission). Chapter 2. Method 40

a) 1 2 1 d) 1 2 1 2’ 4 3 4 3 5 3 3 Continent 5 6 7 6 2 1 1 2 0Myr 1 60.2Myr b) 1 2 1 e) 1 2 1 4 4 2’

3 5 3 3 3 5 6 1 2 2 1 40.5Myr 1 100.4Myr c) 1 2 1 f) 1 4 1 4 2’ 2’ 3 3 5 3 3 5 6 2 1 2 1 1 40.7Myr 1 4 180.1Myr

Figure 2.7: An example of the evolution of oceanic plate boundaries in a 3D model. A full description of the model parameters is given in Chapter 6. Plate conﬁgurations are shown for t=0Myr (a); 40.5Myr (b); 40.7Myr (c); 60.2Myr (one mantle transit time) (d); 100.4Myr (e); and 180.1Myr (f) after supercontinent formation. The model initially features seven oceanic plates (panel a) labelled in terms of their ‘age’ (1 being youngest and 7 being oldest). In (b), Plate 2 covers more than 25% of the surface and in accordance with the plate model rules outlined in the text, is subsequently fragmented into two plates of roughly equal size (c). Chapter 2. Method 41

2.10 Evolving plate geometry

In the majority of the models presented in this thesis, plate boundaries do not evolve following supercontinent formation. However, in Chapter 6 MC3D is coupled with a finite element mesh generation code (LaGriT) to determine the motion of oceanic plate boundaries in a 3D model. Figure 2.7 shows the plate boundary evolution for the model (a full description of the model parameters is given in Chapter 6). The initial plate configuration at the time of supercontinent formation features seven oceanic plates and sixty-two distinct vertices on the plate boundaries (Figure 2.7a). The oceanic-oceanic plate boundaries move with a velocity equal to the area weighted mean of the adjacent plates [e.g., Gait, 2007; Gait et al., 2008; Stein and Lowman, 2010]. However, the continent is stationary (i.e., the supercontinent has no plate velocity and its boundaries do not move). Oceanic plate boundaries that migrate over the supercontinent perimeter are projected to the margin of the continent (so that the continental size and shape remains the same for the model duration (Figure 2.7a-f)). The same method of extrapolating plate boundary points that migrate over another plate also occurs for oceanic-oceanic plate interaction, with the ‘younger’ oceanic plate (given by the 1 (youngest) to 7 (oldest) plate numbering in Figure 2.7a) maintaining its shape as the ‘older’ oceanic plate is modified. The pseudo ‘plate age’ is generated based on the oceanic plate’s distance away from the supercontinent (Figure 2.7a). Furthermore, an oceanic plate is fractured if it becomes large enough that internal stresses would facilitate its breakup (here it is specified that when an oceanic plate covers over 25% of the model surface the plate should break arbitrarily). The implementation of this condition is apparent in Figure 2.7b-c where Plate 2 has fractured to make two plates of roughly equal size with the new boundary introduced along a line of high internal stress (due to sub-continental plume position). Modelling plate tectonics in this idealized way permits the analysis of first order processes relating to the thermal evolution of mantle temperatures post-supercontinent formation (e.g., how changing oceanic-oceanic subduction location affects sub-continental mantle dynamics).

Although the plate evolution method described here is simplistic when compared to the complex nature of Earth tectonics, fundamental features of oceanic plates are modelled. Figure 2.8 shows the age distribution of the present-day ocean ﬂoor (Figure 2.8). Oceanic plates are less buoyant than continental plates, and old oceanic lithosphere is the densest. As a result, the longevity of oceanic lithosphere is short and following its formation at oceanic ridges it does not remain on the Earth’s surface for periods much longer than two mantle transit times. In Figure 2.7, after one mantle transit time ( 60Myr, Figure 2.7d), ∼ the oceanic plate geometry has changed considerably and the three ‘oldest’ oceanic plates have been almost entirely subducted. As circum-supercontinent subduction exists throughout the calculation, most Chapter 2. Method 42

Figure 2.8: Age of the present-day ocean ﬂoor (from M¨uller et al. [2008], with permission). Chapter 2. Method 43

Model Resolution Time (Myr) Time (days) Non-evolving oceanic boundaries 426 426 129180 8 Evolving oceanic boundaries 426×426×129 180 225 × × Table 2.2: Computational time for evolving and non-evolving oceanic plate boundary models. A full description of the model parameters is given in Chapter 6. of the original oceanic material in Figure 2.7a is no longer present 100Myr after continental formation (as the oldest plates have a boundary with the continent and are undergoing subduction) (Figure 2.7e). Oceanic plate boundaries 180Myr after the formation of a supercontinent are dramatically diﬀerent to those featured in the initial geometry (in-keeping with Earth’s evolution, Figure 2.8). The ﬁndings of the supercontinent formation model featuring evolving oceanic plate geometries are presented in Chapter 6. Table 2.2 shows the numerical expense of the high Rayleigh number mantle convection calculations. The addition of evolving oceanic plates in this 3D model leads to a 2700% increase in computational run time.

2.11 Models

The following chapters present results from 2D and 3D mantle convection models that look at the thermal evolution of the mantle following the formation of a supercontinent. First, the role of continental size and insulation in the supercontinent is analyzed cycle using calculations featuring a depth-dependent viscosity and high Rayleigh number convection (Chapter 3). After implementing a more complex mantle viscosity law (Chapter 4), the relative importance of continental insulation is monitored when changing the mantle heating mode (Chapter 5). Finally, inﬂuences on the position of sub-continental mantle plumes generated in the supercontinent formation models are analyzed (Chapter 6). In summary, the work presented furthers the understanding of the many changes in mantle thermal structure that tran- spire post-supercontinent formation, and to quantify the relative importance of mantle heating mode, continental insulation, circum-supercontinent subduction, and lower mantle viscosity on supercontinent cycle dynamics (e.g., Figure 1.1). Chapter 3

The role of supercontinent thermal insulation and area in the formation of mantle plumes

3.1 Introduction

Continental insulation during the Mesozoic may offer an explanation as to why the mantle below the African plate, a former site of continental aggregation, is hotter than normal. Numerical modelling studies have shown that the formation of a supercontinent over a mantle downwelling can initiate a reorganization of mantle convection planform, resulting in sub-continental upwellings and anomalously high sub-continental temperatures. However, many numerical models omit modelling oceanic plates despite convection influenced by their presence significantly differing from convection in which plate-like surface motion is absent [Bunge and Richards, 1996; Zhong et al., 2000; Monnereau and Quéré, 2001]. In this chapter, the evolution of mantle dynamics is examined following supercontinent accretion along a convergent plate boundary. To isolate the dominant influence of continental aggregation on the mantle, different mechanical and thermal boundary conditions are implemented.

The models feature high Rayleigh numbers, stratiﬁed viscosities and oceanic plates, as discussed in

7 Chapter 2 (e.g., sections 2.4.1, 2.8 and 2.9). The isoviscous calculations implement Rab =1 10 , while × 7 all the depth-dependent viscosity models employ Rab = 5 10 (based on upper mantle viscosity) with × additional heating (when speciﬁed) prescribed by a uniform non-dimensional internal heating rate, H, of

44 Chapter 3. Continental insulation and the formation of mantle plumes 45

10. The dimensional thickness of the plates in the 2D and 3D studies are 98km and 135km, respectively (assuming a model depth of 2900km). Two-dimensional calculations are performed on grids with 705 × 177 nodes with three-dimensional calculations implementing 601 601 129 nodes. The models have × × periodic side walls and an isothermal free-slip bottom boundary. The insulation factor (i), as discussed in section 2.8, is the ratio of the thermal diﬀusivity in the continental material to that of the oceans. For example, 1.0 and 0.0 values of i denote an oceanic plate material and a purely insulating continent, respectively. Through decreasing the i value, the vertical heat ﬂow out of the plate is suppressed.

The work presented in this chapter follows closely to my peer-reviewed publication, Heron and Lowman [2011] (Heron P.J. and J.P. Lowman, 2011, The eﬀects of supercontinent size and thermal insulation on the formation of mantle plumes, Tectonophysics, 510, 28-38). After publication of this article, I developed an improved method of modelling continental insulation in comparison to that used in the published study. The ﬁndings from Heron and Lowman [2011] are the same as those presented in this section (and the two studies are compared throughout the chapter).

3.2 2D Results

First, the eﬀect of continental size and thermal insulation properties on the appearance of post supercontinent formation sub-continental plumes is investigated.

3.2.1 Initial condition

The initial conditions for all models are formed through modelling a two-plate system that produces a downwelling at the vertical midplane of the solution domain (through forcing a symmetrical solution about the midplane). Once the system reaches a statistically steady state (i.e., no long term heating or cooling trends are evident in the solution time series), the plate geometry is modified and the symmetric forcing removed. A continental plate (with a prescribed velocity of zero) is centred over the initial downwelling. Two oceanic plates are present on either side of the continent. The emplacement of the continental plate simulates the collision of two smaller continental plates at the site of the mantle downwelling. The mantle thermal field is advanced in time for cases with varying continental width and insulation parameters, and monitored for reversals of the sub-continental flow (i.e., the development of a sub-supercontinent upwelling where initially a downwelling had existed below the accreted continental material). All 2D models are analyzed over six mantle transit times (where a mantle transit time is defined as the time it takes a particle to traverse the mantle depth at the RMS velocity of the initial condition flow field). Six mantle transit times easily spans a period relevant to supercontinent breakup, as Chapter 3. Continental insulation and the formation of mantle plumes 46

a) No reversal 1.12 d

6.0 d b) Sustained reversal 1.34 d

6.0 d Figure 3.1: Temperature ﬁelds from supercontinent formation models: examples of (a) no reversals for small continental coverage and (b) sustained reversal for large continental coverage. Continental coverage is given as a function of mantle depth, d. Unsustained reversals can also occur (where a plume forms sub- supercontinent, but is not sustained under the continent for a signiﬁcant amount of time (e.g., 300Myr)).

one transit time for the Earth’s mantle is 60Myr [Zhong and Gurnis, 1993] and supercontinent dispersal ∼ is thought to occur 300-500Myr after formation [Condie, 1998; Senshu et al., 2009]. At the end of the ∼ six mantle transit times the models are classiﬁed under one of three categories: the sub-continental mantle featured a sustained reversal; a non-sustained reversal (a reversal which occurred and was not maintained beneath the supercontinent); or no reversal (no plume appeared beneath the supercontinent at any time). Figure 3.1 gives examples of a convection model with no reversal (Figure 3.1a) and a sustained reversal (Figure 3.1b).

3.2.2 Continental coverage and mantle reversals

The results for a model with a simple isoviscous mantle indicate that continental width is the most important factor for initiating sustained sub-continental mantle ﬂow reversals (Figure 3.2). The critical width found to produce a sustained reversal directly below a continent trapped between subduction zones at each of its margins was found to be only 128% of the mantle depth (d). Moreover, the insulation Chapter 3. Continental insulation and the formation of mantle plumes 47

0.0 0.1 Key: 0.2 0.3 No reversal 0.4 0.5 Unsustained 0.6 reversal 0.7 0.8 Sustained 0.9 reversal

Continental insulation, i 1.0 1.08 1.16 1.23 1.31 1.39 1.47 1.55 Continental width, d

Figure 3.2: Subcontinental mantle ﬂow reversal results as a function of continental width and insulation for the 2D isoviscous convection study. A mantle reversal features a plume forming beneath a supercontinent (where there was once a downwelling below the accreting continents). Red triangles, red circles and blue triangles indicate sustained, unsustained and no subcontinental ﬂow reversals, respectively. Each calculation was examined for six mantle transit times following supercontinent formation. Calculations are performed with a grid resolution of 705 × 177.

factor has little eﬀect on the overall mantle dynamics leading to plume formation (except in the narrow band of continental coverage in which the transition to mantle reversals occurs). Therefore, subduction has an important role in the production of mantle plumes in purely thermal convection. Due to the cessation of subduction at the continental suture, within 1.5 mantle transit times (tM ) the cold centred downwelling of the initial condition dissipates for all models. Downwellings move to the margins of the stationary continent, while thermal instabilities form at the bottom boundary layer. The disappearance of cool material descending from below the overlying continent allows the downwellings at the continental margin to trap the instabilities in the lower boundary layer so that an upwelling can form, generating a mantle reversal. For the band of unsustained reversals shown in Figure 3.2, the plumes move out from beneath the continent (despite the presence of thermal insulation). When no reversals are generated, the downwellings at the margins of the supercontinent act to push thermal instabilities out from under the continent. The space beneath the continent is not large enough to produce a sub-supercontinent plume.

As discussed in section 3.1, Heron and Lowman [2011] presented a suite of 2D models analyzing continental width and mantle reversals in supercontinent models featuring thermal insulation (prescribed by limited surface heat ﬂux via surface temperature manipulation). The models of isoviscous convection presented by Heron and Lowman [2011] are in excellent agreement with Figure 3.2, having a transition Chapter 3. Continental insulation and the formation of mantle plumes 48 to mantle reversals when continental coverage exceeds 1.28d of the surface. For models featuring depth- dependent viscosity, Heron and Lowman [2011] described thermal insulation as having a limited role in the production of mantle reversals in supercontinent formation simulations.

3.3 3D Results

Given the ﬁndings of the 2D modelling study, only a small number of 3D calculations are examined. These feature large solution domains and multiple oceanic plates surrounding a large model supercontinent. A 6 6 1 solution domain is modelled corresponding to a lateral extent equal to the surface area of Earth’s × × mid-mantle. Depth-dependent viscosity (as described in section 2.4.1) is implemented for the 3D study.

3.3.1 Initial condition

Figure 3.3 shows the temperature field used as an initial condition for the 3D calculations. The red and blue isosurfaces have non-dimensional temperature values of 0.7 and 0.375. The randomly chosen pattern of the fixed plate boundaries is given by the thick black lines on the surface (and are also represented in Figure 3.4a). Sidewalls are periodic boundaries and do not coincide with plate boundaries. The initial condition was obtained by projecting a 2D solution from a model with the same depth-dependent viscosity into the third dimension and specifying a plate geometry featuring nine plates (Figure 3.4a). This 3D model was then integrated forward in time for several mantle overturns until the system reached a statistically steady state. A suitable flow pattern mimicking the formation of Pangea (with Y-shaped subduction [Santosh et al., 2009] aggregating three large plates) was eventually obtained as the model naturally evolved. The resulting thermal field was then taken as the initial condition (Figure 3.3). The model supercontinent is formed through joining the three plates that converge at the large-scale model subduction zone (occurring along the yellow plate boundary in Fig 3.4). The plates are joined to form one continent and, at the time of continental aggregation, a new oceanic plate is also formed from the residual portion of the three plates. This configuration (e.g., producing one oceanic plate from the continental plates) allows for the formation of a supercontinent of comparable size to Pangea (covering 29% of the ∼ system surface area). The new plate boundary separating the supercontinent and the new oceanic plate is indicated by the dashed black line in Figure 3.3. Furthermore, by breaking the continent it ensures that oceanic plate 3 (Figure 3.4a) does not border the supercontinent on opposing margins (to aid circum- supercontinent subduction to form dynamically). The plate geometry featuring one supercontinent (brown colour) and seven oceanic plates (blue and purple colours) is given in Figure 3.4b. Chapter 3. Continental insulation and the formation of mantle plumes 49

Figure 3.3: Model plate geometry and initial temperature field used in the 3D study. The red and blue isosurfaces have temperature values of 0.7 and 0.375. The top and bottom 6% of the isosurfaces has been omitted to permit a view of the model interior. The horizontal slice shows the temperature field at a depth of 0.98d. The black lines on the surface show the locations of the specified plate boundaries, with the yellow line indicating the collisional boundary along which three plates are sutured to form the supercontinent. The dashed black line shows the boundary between the supercontinent and a new oceanic plate that is introduced at the time of continental aggregation. The models examined feature seven oceanic plates and a supercontinent covering 29% of the surface area. The grid resolution is 601 × 601 × 129.

a) b) 1 3 2 1

6 4 4 5

1 1 2 3

Reference velocity (proportional to vrms )

Figure 3.4: Plate geometry for (a) pre- and (b) post-supercontinent formation. (a) shows the initial plate geometry with plate velocities. Arrow length is proportional to plate velocity at the time of collision. Three continental plates aggregate over a downwelling. (b) sutured continents form one large supercontinent and an additional oceanic plate. The supercontinent covers 29% of the surface area. Chapter 3. Continental insulation and the formation of mantle plumes 50

3.3.2 Thermal response of the mantle after supercontinent formation

Figure 3.5 shows snapshots of temperature isosurfaces from the 3D calculation with a supercontinent

1 insulation factor of 0.25i. A continental diﬀusivity value of /4 of the oceanic value is chosen for the remainder of the study (the temperature increase for this parameter is comparable to that observed in other recent studies of supercontinents in mantle convection models [Yoshida, 2013]). Panels are shown at intervals of one mantle transit time following supercontinent formation. After one mantle transit time, downwellings have formed on some of the new supercontinent’s margins, while remnants of the initial downwelling remain at the base of the subcontinental boundary (marker A). The previous site of the arrival of cold lithospheric material at the base of the mantle exhibits the growth of hot thermal instabilities around 120Myr (2tM ) after supercontinent formation (shown by marker B). These instabilities produce a plume that has penetrated the upper mantle in panel 3, leading to a cluster of mantle plumes beneath the supercontinent after four mantle transit times (C). Despite the presence of the highly insulating supercontinent above, the plume cluster moves out from beneath the continent (D). These plumes are replaced by a secondary plume (E) under the supercontinent 5-6 mantle transit times after formation.

3.3.3 Geotherms and mantle temperatures

Laterally averaged temperature as a function of depth (i.e., the geotherm) is shown in Figure 3.6 (below the continent and oceans) at the time of continental formation (dashed curves) and four mantle transit times later (solid curves) for the supercontinent model shown in Figure 3.5. At both times subcontinental temperatures in the mid-mantle are comparable. Considerable heating is seen at the base of the sub- continental mantle, where subducted lithosphere from the initial condition has been replaced by mantle plumes. The oceanic geotherm throughout the mantle remains almost constant over time. The left inset of Figure 3.6 shows that from just below the surface down to just above the basal boundary the sub-continental temperature never exceeds the mean temperature of the oceans, despite the overlying supercontinent insulation. An analysis of pressure-temperature conditions of mantle xenoliths from oceans and continents [Santosh et al., 2009] show sub-oceanic geotherms to be of a higher temperature than continental geotherms, in agreement with the 3D ﬁndings (shown in right inset of Figure 3.6).

To compare this study with similar published work featuring a diﬀerent thermal insulation method, Figure 3.7 shows the time-series of the volume-average temperatures beneath the continent and oceanic plate post-supercontinent formation for the model shown in Figure 3.5 and for a highly insulating supercontinent from Heron and Lowman [2011]. The volume-average temperature beneath the continent Chapter 3. Continental insulation and the formation of mantle plumes 51

A B

E D

Figure 3.5: Snapshots of temperature isosurfaces from a model with a supercontinent featuring an insulation factor of 0.25i. Panels are shown at intervals of one mantle transit time following supercontinent formation. Fields are rendered as in Figure 3.3. The repositioning of downwellings to the margins of the supercontinent generates mantle plumes (and therefore a mantle reversal) sub-supercontinent. Chapter 3. Continental insulation and the formation of mantle plumes 52

1.0

0.8 0.98

0.96 0.6 0.94

0.92 Height 0.4 0.90 0.2 0.88 0.86 0.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Temperature

Figure 3.6: Average non-dimensional temperature as a function of depth from the model shown in Fig- ure 3.5. Red lines show the temperatures in subcontinental regions, blue for sub-oceanic regions. Dashed and solid lines correspond to the initial thermal ﬁeld and the thermal ﬁeld four mantle transit times following supercontinent formation, respectively. Chapter 3. Continental insulation and the formation of mantle plumes 53

0.545

0.54

0.535 Heron and Lowman (2011) 0.53 This study

0.525

0.52

0.515 Volume averaged temperature 0.5 0.0 40 120 200 280 360 Time, Myr

Figure 3.7: Volume-averaged temperature beneath continents and oceans for the model presented in Figure 3.5 and for a highly insulating supercontinent from [Heron and Lowman, 2011] over 6 mantle transit times. The volume-average temperature beneath the continent never exceeds that below the oceans for either insulation model (Figure 3.6), despite the formation of mantle plumes below the supercontinent.

never exceeds that of the oceans for both insulation models (Figure 3.7), despite the formation of mantle plumes below the supercontinent.

3.3.4 Non-insulating supercontinent

Figure 3.8 shows a model identical to that of Figure 3.5 but without any continental thermal insulation properties (using the initial condition shown in Figure 3.3). Even without the addition of thermally insulating properties, the lack of motion of a supercontinent (and the subsequent formation of downwellings at its margins) is suﬃcient to produce a mantle ﬂow reversal (marker F, Figure 3.8) within four mantle transit times of supercontinent formation [e.g., Lowman and Gable, 1999]. Chapter 3. Continental insulation and the formation of mantle plumes 54

Figure 3.8: Snapshot of non-dimensional temperature ﬁeld from a model featuring a stationary supercontinent with an isothermal surface. Four mantle transit times have elapsed since supercontinent formation. Marker F indicates a subcontinental plume. Fields are rendered as in Figure 3.3. Chapter 3. Continental insulation and the formation of mantle plumes 55

Case 1 Case 2 Height

Viscosity

Figure 3.9: The two viscosity proﬁles as a function of mantle height used to model the generation of mantle plumes. Case 1 (red) shows a factor of 36 increase in the lower mantle (as used for the previous 3D results in this study), while case 2 (blue) shows a decreased viscosity near the core-mantle boundary. Case 1 is pressure-dependent while case 2 is pressure- and geotherm-dependent [e.g. Gait et al., 2008].

3.3.5 Lower mantle viscosity and plume generation

The Earth’s mantle viscosity is strongly temperature dependent. Here, a viscosity profile with a decrease in viscosity near the hot core-mantle boundary is implemented in order to assess its effect on the generation of deep mantle plumes. Figure 3.9 shows the new viscosity profile (case 2) as compared to the profile used earlier in this study (case 1).

Figure 3.10 shows snapshots of upper mantle variations in temperature for both viscosity cases shown in Figure 3.9. The average mantle temperature at a depth of 0.2d is subtracted from the temperature ﬁeld in order to highlight the plumes generated in the upper mantle. For case 1 (i.e., Figure 3.5), upper mantle thermal anomalies indicate the appearance of plumes below the supercontinent 140-150Myr after formation. Case 2, with a low-viscosity lower thermal boundary layer, allows upper mantle plumes to form faster than in case 1 (Figure 3.10). This result is in-keeping with previous studies on plume formation timescales following continental aggregation [Zhang et al., 2010].

3.4 Discussion

The results of the three-dimensional modelling support the conclusions of the 2D study and show that thermal insulation is not an imposing factor in producing sub-continental plumes. Moreover, a supercontinent does little to elevate mean subcontinental temperatures. However, the importance of circum- Chapter 3. Continental insulation and the formation of mantle plumes 56

Case 1 Case 2

130Myr 0.5

140Myr 0.0 Temperature variation (%) 150Myr -0.5

Figure 3.10: Snapshots of deviation from the average horizontal temperature at depth 0.2d (580km) for two different viscosity profiles. Timings are converted from diffusion time based on a full mantle transit − taking 60Myr, implying an upper mantle viscosity of 9.0x1016 m2s 1. The grey circles indicate the first plumes penetrating the upper mantle for each viscosity profile. Chapter 3. Continental insulation and the formation of mantle plumes 57 supercontinent subduction in creating an environment conducive to the growth of sub-supercontinental thermal instabilities is in agreement with previous mantle convection models [Gurnis, 1988; Zhong and Gurnis, 1993; Lowman and Jarvis, 1993; Zhong et al., 2007]. The insignificant influence on mantle flow of the highly insulating supercontinent is suggested by the supercontinental plume (Figure 3.5, marker D). The plume moves where the mantle flow dictates, as driven by the subduction pattern, rather than staying in a region dominated by continental insulation.

3.4.1 Model considerations

The motivation for employing 130km thick viscous plates (in 3D calculations) is ultimately determined ∼ by the fact that the model employs uniform thickness continents and oceanic plates. It is prudent, therefore, to model the thickest continents possible while modelling plates that are comparable in depth to the thickness of the cold upper thermal boundary layer. Modelling continents demands a thick lithosphere but modelling oceanic plates as thick as the continents is undesirable. As the average Rayleigh numbers in the calculations presented here are a little lower (perhaps a factor of 3) than that ∼ inferred for the Earth, the model oceanic plate thickness is increased in proportion to the relatively thick thermal boundary layer.

Heat loss from the oceanic plates modelled here is dominated by conductive cooling of the viscous oceanic lithosphere. However, advection allows hot underlying mantle material to reach the surface of the calculations at diverging plate boundaries. Hot material moves away from the simulated model oceanic ridges and cools in a similar manner to the heat lost from plate cooling models. By specifying a thick oceanic lithosphere the vertical advection of heat under the old oceans is inhibited. In contrast, thin continental lithosphere allows vertical advection to take heat closer to the surface where it would escape more easily. However, by specifying that the continents are thermal insulators, an eﬀectively thicker continental lithosphere is implicitly modelled by increasing the diﬀusion times across the supercontinent (as shown in right inset of Figure 3.6, and discussed in 2.8).

The simplicity of the model, particularly the use of a plane-layer geometry, means that constraining the Earth’s mantle geotherm or its heat budget is beyond the scope of this study. Instead, the aim is to show that the formation of a supercontinent is insufficient to produce significant heating of the subcontinental mantle on the timescale relevant to supercontinental longevity (e.g., up to 150 Myr). One can suggest the difference between these results and those of several previous studies [e.g. Gurnis, 1988; Zhong and Gurnis, 1993; Trubitsyn and Rykov, 1995; Yoshida et al., 1999; Honda et al., 2000; Phillips and Bunge, 2005, 2007; Zhong et al., 2007; Trubitsyn et al., 2008; Phillips et al., 2009; Coltice Chapter 3. Continental insulation and the formation of mantle plumes 58 et al., 2009; O’Neill et al., 2009; Yoshida, 2010a,b; Phillips and Coltice, 2010; Zhang et al., 2010; Yoshida and Santosh, 2011] is explained by the fact the model presented here includes oceanic plates. Oceanic plates mitigate the rate of oceanic heat loss in comparison to the heat flow obtained by modelling oceans with free-slip boundaries [Stein and Lowman, 2010]. The latter may therefore exaggerate the contrast between continents and oceans in a global scale model.

3.4.2 Mantle reversal time frame

Zhong and Gurnis [1993] state that the supercontinent cycle occurs with a period of 300-500Myr. More recent studies suggest that the formation of a plume beneath the supercontinent Pangea occurred 50- ∼ 100Myr after its formation [Li et al., 2003; Maruyama et al., 2007], with rifting and volcanism associated with the dispersal of Pangea indicating the timescale of supercontinent assemblage to be 200Myr ∼ [Condie, 1998]. The results presented here indicate that, following the cessation of subduction associated with the site of supercontinental suturing, at least two mantle transits are required for vigorous plumes to form. During that period, effects due to continental thermal insulation are inconsequential. Analyzing horizontal thermal anomalies at a depth of 0.2d, mantle plume heads appear at transition zone depths ( 580km) about 150Myr after supercontinent formation (Figure 3.10). The model supercontinent is ∼ therefore unaffected by underlying plumes for at least 150Myr (c.f. Figure 3.5), similar to the timescales suggested for the existence of Pangea [Scotese, 2001; Condie, 1998; Maruyama et al., 2007; Yoshida and Santosh, 2011]. The arrival time of plumes beneath the supercontinent can be seen to reduce by the specification of a low viscosity zone near the core-mantle boundary (Figure 3.10).

The results presented here identify the onset of plume generation with subducting cold material [Gurnis, 1988; Zhong and Gurnis, 1993; Lowman and Jarvis, 1993], where the location of the rising thermal anomalies is coupled to the subduction pattern produced by convergence at the supercontinent margins [e.g. Zhong et al., 2007]. Prolonged subduction of cold, oceanic plates preceding supercontinent formation is believed to generate a chemically distinct slab graveyard at the base of the mantle, which has been suggested to act as ‘fuel’ for mantle plumes [Maruyama et al., 2007; Santosh, 2010; Yoshida and Santosh, 2011] and dictate the position of plumes through ‘plume generation zones’ (PGZ) [Burke et al., 2008]. The model presented here is not able to take into consideration these chemical heterogeneities. However, the calculations do model remnant slab material drawing a particularly high heat flow from an isothermal core-mantle boundary that leads to rapid plume formation. Although the findings clearly show subduction patterns dictating the position of plumes, the inclusion of distinct mechanical or chemical oceanic slabs could have the affect of reducing the timing of plume formation below a supercontinent. Chapter 3. Continental insulation and the formation of mantle plumes 59

To take into consideration the effect of temperature on viscosity, the viscosity profile of the reference case was modified to include a region of low-viscosity ( 9 times less viscous than previous calculations) ∼ near the core-mantle boundary. The lower viscosity thermal boundary at the base of the models again generates thermal anomalies that penetrate the upper mantle in the same location as plumes obtained with the simple depth-dependent viscosity profile, though 10Myr sooner (Figure 3.10). The relatively ∼ low viscosity lower layer will generate higher velocities at the base of the mantle (when compared to the relatively higher viscosity lower mantle models), making it easier (and quicker) for instabilities to merge to generate plumes. Therefore, the application of depth- and geotherm-dependent viscosity is added to future models (Chapter 4), where a more complex viscosity profile is shown to affect mantle dynamics (Figure 3.10).

3.4.3 Continental geotherm

Previous studies incorporating insulating continents found sub-continental temperatures to be higher than those below oceans [Gurnis, 1988; Phillips et al., 2009; Yoshida, 2010b; Phillips and Coltice, 2010].

However, an analysis of pressure-temperature conditions of mantle xenoliths from oceans and continents [Boyd, 1973; Kramers, 1977; Sen et al., 1993; Santosh et al., 2009] shows sub-oceanic geotherms to be of a higher temperature than continental geotherms, in agreement with these 3D model ﬁndings (Figure 3.6).

The geotherm in Figure 3.6 successfully emulates the distinct temperature profiles inferred for oceanic and continental lithosphere [Boyd, 1973; Kramers, 1977; Sen et al., 1993; Santosh et al., 2009]. However, it should be noted that while this study incorporates the affects of convective heat transport, like many simple conduction models for lithospheric geotherms, the model used here neglects the influence of phase and mineralogical changes on heat flow [Sinha and Butler, 2009].

3.5 Conclusion

The generation of active mantle upwellings below a model continent has been observed in numerous past studies [e.g. Gurnis, 1988; Zhong and Gurnis, 1993; Lowman and Jarvis, 1993, 1996; Yoshida et al., 1999; Lowman and Gable, 1999; Zhong et al., 2007; O’Neill et al., 2009; Senshu et al., 2009; Yoshida, 2010b; Zhang et al., 2010; Heron and Lowman, 2010; Yoshida and Santosh, 2011]. To determine how the contrast between continental and oceanic regions might change when oceanic plates are modelled (as plates affect oceanic heat flow differently than a free-slip surface [Monnereau and Quéré, 2001; Lowman, 2011]), this study incorporates thermally and mechanically distinct continental and oceanic plates that cover the entire surface of the model. By modelling oceanic plates, as well as a supercontinent, the thermal contrast Chapter 3. Continental insulation and the formation of mantle plumes 60 between subcontinental and suboceanic regions essentially vanishes (Figure 3.7). However, the formation of a supercontinent results in sub-continental plume formation due to the re-positioning of subduction zones (Figure 3.5). Accordingly, it is demonstrated that continental insulation is not a significant factor in producing sub-supercontinental mantle reversals but that subduction patterns control the location and timing of upwelling formation (Figure 3.2 and 3.8). Chapter 4

Plate mobility regimes and a re-evaluation of plate reversals

4.1 Introduction

The fundamental mechanics that initiate and sustain plate motion on Earth are still not well understood [Davies, 1992; Hamilton, 1998; de Wit, 1998; Griﬃn et al., 2003; Davies, 2006; Stern, 2007]. Rapid changes in plate motion direction over short geological timescales might initiate or result from signiﬁcant changes in the thermal evolution of the mantle. For example, the opening and closing of the Rheic Ocean [Nance et al., 2012] through a reversal in plate motion is thought to have played a fundamental role in the process leading to the formation of supercontinent Pangea [Nance et al., 2010]. In addition, high mantle temperatures during the early Earth [Nisbet et al., 1993; Abbott et al., 1994; Grove and Parman, 2004; van Thienen et al., 2004; Labrosse and Jaupart, 2007] may have contributed to the past existence of plate tectonic regimes (e.g., stagnant-lid and episodic subduction tectonics) not seen in modern plate tectonics [Brown, 2006; O’Neill et al., 2007a,b; van Hunen and van den Berg, 2008; Condie, 2008; Halla et al., 2009; Sizova et al., 2010; Gerya, 2011; van Hunen and Moyen, 2012; Gerya, 2014]. This work may also have implications for exoplanets. The likelihood of plate tectonics on super-Earths is a much studied, and highly controversial, area of geodynamics [e.g., Valencia et al., 2006, 2007; O’Neill and Lenardic, 2007; O’Neill et al., 2009; Valencia and O’Connell, 2009; Korenaga, 2010; van Heck and Tackley, 2011; Stein et al., 2013].

In this chapter, a geotherm- and purely depth-dependent viscosity is implemented in a series of mantle convection models (with increasing mantle temperature) to identify the controlling factors on

61 Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 62 plate mobility and mantle ﬂow reversals. A time-dependent plate thickness is modelled by deﬁning the base of a plate as being temperature-dependent, similar to the mechanical boundary that marks the transition of the lithosphere to the asthenosphere in the Earth. The models presented here build upon the previous chapter in order to explore the role of modelling method in mantle convection studies. The methodology for implementing the time-dependent plate thickness and geotherm-dependent viscosity is described in Chapter 2 (sections 2.6 and 2.4.2, respectively).

4.2 Results

The effect of time-dependent plate thickness and geotherm- and depth-dependent viscosity is explored in relation to the generation of plate mobility. First, a direct comparison between results obtained using an isoviscous rheology and the geotherm-dependent viscosity model is analyzed in section 4.2.1, followed by a parameter study using the new modelling method (examining the effect of the lithospheric cut-off temperature (section 4.2.2), thermal viscosity contrast (section 4.2.3) and reference Rayleigh number (section 4.2.4) on plate mobility). The results from section 4.2.4 are used to study how aspect ratio and side-wall mechanics influence mantle convection models (section 4.2.5). Finally, 3D simulations are presented to help understand the effect of model geometry on plate mobility (section 4.2.5).

4.2.1 Mantle reversals and mantle viscosity

In mantle convection models featuring plates with a fixed thickness, Lowman et al. [2001] showed that plate motion in bimodally heated models (i.e., featuring both internal and basal heating) is characterized by changes in the direction of the mantle circulation, coinciding with a change in polarity of the plate velocity (defined as episodic mantle reversals). This result was subsequently found in numerous other studies that feature similar mantle viscosity and internal heating parameters (but different solution domain geometries) [e.g., Lowman et al., 2003, 2004; Monnereau and Quéré, 2001; King et al., 2002; Ghias and Jarvis, 2004; Koglin Jr. et al., 2005]. Figure 4.1 depicts temperature and horizontal velocity from a study with an isoviscous mantle reproduced from Lowman et al. [2001] that features mantle reversals (with Table 4.1 showing how this reproduced model is benchmarked against the original Lowman et al. [2001] study). Figure 4.2a depicts the plate velocity and heat flux (surface and basal) for Figure 4.1. The change in polarity of the plate velocity is shown by markers A and C in Figure 4.2a (corresponding to Figure 4.1a and Figure 4.1c, respectively), with a period of quiescence denoted by marker B (Figure 4.2a and corresponding to Figure 4.1b). In the single plate, aspect ratio (Γ) 1 study, a full mantle reversal cycle occurs on a period of about 180Myr (based on comparing mantle transit velocities and in-keeping Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 63

a) b) c)

Temperature 0.0 0.5 1.0

d) e) f)

Velocity Velocity Velocity -4500 0 4500 -1500 0 1500 -4500 0 4500

Figure 4.1: Temperature (a-c) and horizontal velocity (d-f) snapshots from an isoviscous model. This study is the same as that presented by Lowman et al. [2001]: the model has an aspect ratio (Γ) of 1, the upper mantle Ra is 1.5×107 and the non-dimensional internal heating rate is 15. Plates are 1000 times more viscous than the upper mantle and have a thickness of 0.05d. Panels (a) and (d) correspond to the same time, as do (b) and (e), and (c) and (f). The ﬁgure shows the reversal of plate motion: (a) and (d) depict the plate moving to the left, (c) and (f) show the plates moving to the right. The middle panels show a period of relative low velocity as the mantle ﬂow reversal is in a transition.

Model Resolution q¯surf T¯ Lowman et al, 2001 200 200 33.34 0.7983 This study 200×200 33.34 0.7983 × Table 4.1: Benchmarking the updated MC3D numerical code (this study) with the results of the Lowman et al. [2001] study. The model is of an isoviscous mantle convection simulation with reflective side-walls. The one plate (1000 times more viscous than the upper mantle) is 0.05d thick (where d is the depth of the mantle). The Rayleigh number is given as 1.5×107, and a non-dimensional internal heating rate of H=15 is specified.q ¯surf is the time average surface heat flux and T¯ the time-averaged mantle temperature (when the model is in a statistically steady thermal state). with [Lowman et al., 2001]). Lowman et al. [2001] state that internal heating (given as H=15 in this model) is important in the role of episodic mantle reversals because it generates a build-up of heat in the interiors of wide convection cells close to mantle downwellings. In Figure 4.3, time-dependent plate thickness and depth- and geotherm-dependent viscosity are Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 64

a) b) 8000 8000 Surface heat flux 50 A 50

Plate velocity Plate Basal heat flux Plate velocity Plate 40 40 Plate velocity B 0 30 0 30 Heat Flux Heat Flux 20 20 C 10 -8000 10 -8000 0 60 120 180 0 60 120 180 Time (Myr) Time (Myr)

Figure 4.2: Plate velocity and (surface and basal) heat flux for models featuring mantle reversals and stagnant-lid tectonics. (a) shows the mantle reversal cycles for the Lowman et al. [2001] study (blue) and the fluctuating surface (red) and basal heat fluxes (green). A mantle reversal cycle is ∼180Myr in this figure. Markers A, B and C correspond to the snapshots shown in Figure 4.1a, 4.1b, and 4.1c, respectively. (b) shows the stagnant-lid plate tectonics and the low surface heat flux for a study using the same surface Ra and internal heating rate specified by the Lowman et al. [2001] model, but featuring pressure- and geotherm-dependent viscosity (Figure 4.3). The basal heat flux has a negative value as the mantle has become so warm that heat is transferred across the base the of the box.

applied to a model with an internal heating rate of 15. A reference Rayleigh number (Ra0) is speciﬁed so that the Rayleigh number at the base of the mantle (when T=1) is 105, with the pressure and thermal viscosity contrast given as 102 and 105, respectively. The application of geotherm-dependent viscosity and the time-dependent plate thickness in this convection model (with high internal heating) does not generate episodic mantle reversals, despite the similar heating conditions to the model shown in Figure 4.1. Instead, the plate velocity reduces to almost zero (Figure 4.2b), characteristic of a stagnant- lid tectonic regime. The average mantle temperature increases rapidly as the plate velocity decreases. No episodic features exist once the mantle temperature becomes stable.

Figures 4.2b and 4.3 indicate that episodic mantle reversals due to the build-up of heat in the upper mantle [e.g., Lowman et al., 2001, 2003, 2004; Monnereau and Quéré, 2001; King et al., 2002; Ghias and Jarvis, 2004; Koglin Jr. et al., 2005] may be a characteristic of the model setup (namely the viscosity and temporally fixed plate thickness). The following sections explore the features of the mantle convection model that can produce a transition from mobile-lid to stagnant-lid tectonics (or generate a mantle reversal). Section 4.2.2 analyzes how the lithospheric cut-off temperature affects plate mobility.

4.2.2 Parameter study: lithospheric cut-oﬀ temperature, TL

Plate mobility (M) is deﬁned as the ratio of plate (surface) to volume averaged rms velocity [Tackley, 2000]. The classiﬁcation of a stagnant-lid regime occurs once the plate mobility ratio becomes lower Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 65

0.0 0.5 1.0

Temperature

Figure 4.3: Temperature snapshot of the steady-state mantle in a stagnant-lid tectonic regime. An eﬀort has been made to match the parameters of this model with those used in the isoviscous convection study of Lowman et al. [2001] but with depth- and geotherm-dependent viscosity. The aspect ratio and internal heating rate (Γ=1 and H=15) are kept the same. However, a temperature-dependent and a depth-dependent viscosity contrast of 105 and 100 (respectively) are applied in a model with a reference Rayleigh number of 400. The more complex rheology generates a stagnant-lid tectonic regime, with the average non-dimensional temperature in the box being greater than 1. The lithospheric cut-oﬀ temperature in this case is 0.4 and generates a plate thickness of 0.035d. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 66

(d) TL= 0.4 (a) 10000 10 Vrms Temperature 0.4 1000 Vsurface L Mobile

T Sluggish Temp 0.3 100 Stagnant 1 0.2 Velocity 10 0 5 10 15 20 1 Internal Heating (H) 0.1 0.1 0.1 1 10 Internal Heating (H) (e) (b) TL= 0.3 10000 10 Mobile Vrms

1.0 Temperature 1000 Vsurface 0.1 Sluggish Temp 100 Mobility (M) 0.01 Stagnant 1

0.001 Velocity 10 Lith T=0.4 0.0001 Lith T=0.3 1 Lith T=0.2 0.1 1 10 0.1 0.1 Internal Heating (H) 0.1 1 10 Internal Heating (H) (c) (f) TL= 0.2 Mobile 10000 10 1.0 Vrms 1000 Vsurface Temperature 0.1 Sluggish Temp 0.01 100 Mobility (M) Stagnant 1

0.001 Lith T=0.4 Velocity 10 0.0001 Lith T=0.3 Lith T=0.2 1 0.4 0.6 0.8 1.0 1.3 0.1 0.1 Volume average Temperature (T) 0.1 1 10 Internal Heating (H)

Figure 4.4: Analysis of how the lithospheric cut-off temperature (TL) affects plate mobility (M). In this study the only model parameters that change are H and TL. ∆ηT , ∆ηP and Ra0 are kept constant (at 5 10 , 100 and 400, respectively). Plate mobility (M), vrms, the plate velocity vsurf and the average mantle temperature (Temp) are calculated when the model has reached steady state. (a) shows at what internal heating rates a model will change tectonic regime for a given TL value. (b) shows the relationship between mobility and internal heating rate. (c) shows the relationship between mobility and average temperature. (d) - (f) show the volume-average and surface velocities as a function of internal heating rate for (d) TL=0.4, (e) TL=0.3 and (f) TL=0.2. The red shaded area denotes the transition from mobile-lid to sluggish-lid tectonics. The overall trend is that increasing the lithospheric cut-off temperature increases mobility at a given internal heating rate. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 67

Temperature H=8 0.0 0.275 0.550 0.875 1.10

a) b) c)

TL=0.2 TL=0.3 TL=0.4 Stagnant Sluggish Mobile

Figure 4.5: Temperature snapshots showing how changing the lithospheric cut-oﬀ temperature aﬀects mobility for a given internal heating rate (H=8). (a) shows a snapshot of the thermally steady stagnant-lid tectonic regime (M=0.0057) when TL=0.2, (b) sluggish-lid tectonic regime (M=0.16) when TL=0.3 and (c) mobile-lid tectonic regime (M=0.86) when TL=0.4. Model parameters of ∆ηT , ∆ηP and Ra0 are kept constant (at 105, 100 and 400, respectively).

than 0.01 [Stein et al., 2013], indicating that the plate (or plates) are moving at a velocity that is less than 1% of the average mantle velocity. If the mobility ratio (M) is greater than 0.01 but less than 0.5, the tectonic regime is described to be ‘slugglish’ [Stein et al., 2013], characterized by more episodic and delamination-style subduction. However, if M exceeds 0.5 the tectonic regime is said to be ‘mobile’ [Stein et al., 2013], characterized by complete upper thermal boundary layer subduction. As a reference, the estimated mobility ratio for the present-day Earth is approximately 2 [Kanamori and Brodsky, 2004].

In this section, the eﬀect of changing the lithospheric cut-oﬀ temperature, TL (Figure 2.4), is analyzed in terms of M, the mobility ratio. The parameters for these calculations feature a thermal viscosity

5 contrast of ∆ηT =10 , a pressure viscosity contrast of ∆ηP =100, and a reference Rayleigh number of Ra0=400. Figure 4.4 presents the ﬁndings from changing TL and the internal heating rate, H. In Figure 4.4a, the transition from mobile to sluggish/stagnant-lid tectonics occurs at lower internal heating rates as TL is decreased. This indicates that colder plates generate stagnant-lids more easily than thick lithospheric plates. A TL value of 0.4 produces stagnant-lid tectonics when the internal heating rate is 15 or higher (Fig 4.4b), or once the non-dimensional average mantle temperature is greater than 1.2 (Fig 4.4c). The red shaded regions (dashed vertical line) of Fig 4.4d-f indicate the transition from mobile-lid to sluggish/stagnant-lid plate tectonics as a function of heating rate. The trend for lowering

TL is that this transition from mobile-lid tectonics occurs at lower internal heating rates, illustrated by temperature snapshots of all three TL values at the same internal heating rate (H=8) in Figure 4.5. At this H value, stagnant-lid, sluggish-lid and mobile-lid tectonics can be found when TL=0.2, 0.3, and 0.4, Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 68

(a) (c) Mobile 1.0

T 10 6 η Mobile 0.1 Sluggish

SluggishSluggish Stagnant 5 0.01

10 Mobility (M) Mobile Stagnant Stagnant 4 0.001 6 10 η T=10 5 0.0001 η T=10 0 5 10 15 20 η =10 4 Internal Heating (H) T 0.1 1.0 10 Internal Heating (H) (b) (d) Mobile 1.0

T 6 η 10 0.1

Sluggish MobileMobile Sluggish Stagnant 5 0.01

10 Mobility (M) Stagnant Stagnant 4 0.001 6 10 η T=10 Sluggish 5 0.0001 η T=10 0.5 0.7 0.9 1.1 1.3 η =10 4 Volume average Temperature (T) T 0.4 0.6 0.8 1.0 1.4 Volume average Temperature (T)

Figure 4.6: Analysis of how the thermal viscosity contrast ∆ηT aﬀects plate mobility (M). In this study the only model parameters that change are H and ∆ηT . TL, ∆ηP and Ra0 are kept constant (at 0.35, 100 and 400, respectively). Plate mobility (M), vrms, the plate velocity vsurf and the average mantle temperature (Temp) are calculated when the model has reached steady state. (a) shows at what internal heating rate a model will change tectonic regime for a given ∆ηT value. (b) shows at what average mantle temperature a model changes tectonic regime for a given ∆ηT value. (c) shows the relationship between mobility and internal heating rate, and (d) between mobility and average temperature.

respectively. A low TL value produces a large viscosity contrast between the stiﬀ plates and the bulk convecting mantle. This allows the plates to detach from the mantle to produce a low plate mobility.

When TL is high, the viscosity contrast between the plates and the upper mantle is small, and a higher plate mobility is registered. Through increasing the internal heating value, H, the interior temperature of the model is increased (along with the vigour of convection in the upper mantle). An increase in H increases the viscosity contrast between the plates and the upper mantle, decreasing the plate mobility

(when TL is kept constant).

4.2.3 Parameter study: thermal viscosity contrast, ∆ηT

In this section, the thermal viscosity contrast (∆ηT ) is varied with internal heating rate. The lithospheric cut-oﬀ temperature (TL) is set at 0.35 for this study, while the reference Rayleigh number is kept at

400, as in the previous section. The TL value is based on choosing a non-dimensional temperature high enough to be in-keeping with Earth-like tectonics (the lithospheric-asthenosphere boundary is believed to occur once the mantle temperature passes 1350oC [McKenzie and Bickle, 1988]), but also low enough ∼ to possibly produce observable stagnant-lid convection [e.g., Moresi and Solomatov, 1995; Stein et al., Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 69

(a) 6 ηT=10 10000 Vrms 1000 Vsurface

100

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T) (b) 5 ηT=10 10000 Vrms 1000 Vsurface

100

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T) (c) 4 ηT=10 10000 Vrms 1000 Vsurface

100

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T)

Figure 4.7: The volume-average and surface velocities as a function of average mantle temperature are 6 5 4 shown for (a) ∆ηT =10 , (b) ∆ηT =10 and (c) ∆ηT =10 (with model parameters as Figure 4.6). The red shaded area denotes the transition from mobile lid tectonics to a sluggish plate regime. The overall trend is that increasing the thermal viscosity contrast decreases mobility at a given average mantle temperature. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 70

Temperature Tav ~1.00 0.0 0.275 0.550 0.875 1.10

a) b) c)

4 5 η =10 6 η T=10 η T=10 T M=0.43 M=0.089 M=0.00035 (H=10) (H=12) (H=16)

Figure 4.8: Temperature snapshots of how changing the thermal viscosity contrast aﬀects mobility at a given average mantle temperature (Tav∼1.00). (a) shows a temperature snapshot of the thermally steady 4 5 sluggish-lid tectonic regime when ∆ηT =10 , (b) sluggish-lid tectonic regime when ∆ηT =10 and (c) 6 stagnant-lid tectonic regime when ∆ηT =10 . Model parameters of TL, ∆ηP and Ra0 are kept constant (at 0.35, 100 and 400, respectively).

2004; O’Neill et al., 2009; van Hunen and van den Berg, 2008; Foley and Becker, 2009; Korenaga, 2010;

Stein et al., 2013; Stein and Hansen, 2014]. Viscosity is modelled in MC3D (section 2.4.2) so that an increase in the thermal viscosity contrast decreases the average non-dimensional viscosity in the model. Therefore, the average Rayleigh number (and the vigour of convection) will increase if the thermal viscosity contrast is increased (when all other mantle parameters remain ﬁxed).

Previous studies have shown that mantle convection models that feature strongly temperature- dependent viscosity can develop stagnant-lid tectonics [Christensen, 1984; Solomatov, 1995]. The cold, viscous plates can decouple from a vigourously convecting mantle so that the plate mobility reduces to zero if the temperature-dependent viscosity is high [Nataf and Richter, 1982; Morris and Canright, 1984;

4 Nataf, 1991]. This study looks at ∆ηT between the range 10 (e.g., ‘weakly’ temperature dependent) and 106 (e.g., ‘strongly’ temperature dependent). When H is less than 10, increasing internal heating rates across the range of thermal viscosity contrast considered shows decreasing plate mobility for higher ∆ηT values (Figures 4.6a and 4.6c). At higher internal heating rates, the relationship between

∆ηT and tectonic regime (e.g., mobility) is unclear (Figures 4.6a and 4.6c). However, comparing the volume average temperatures shows the transition from mobile to sluggish-lid and sluggish to stagnant- lid occurs at lower temperatures for higher ∆ηT values (Figures 4.6b and 4.6d). By modifying ∆ηT , the average Rayleigh number of the model changes, which impacts the eﬀect of the internal heating

6 rate. The high internal heating rates with strong temperature-dependent viscosity (∆ηT =10 ) do not eﬀectively heat the vigorously convecting mantle as greatly as a high internal heating rate with weak

4 temperature-dependent viscosity (∆ηT =10 ) (Figure 4.6). Due to the ambiguity in comparing internal Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 71 heating rates between models with diﬀerent average Rayleigh numbers, plate mobility is henceforth discussed in terms of volume average temperature. Figures 4.6b and 4.6d show the decoupling of the plate and the mantle occurs at lower temperatures for the high ∆ηT values. This feature may be explained by the greater range of viscosities modelled at high ∆ηT that increases the viscosity contrast between the stiﬀ plates and the deformable mantle interior at relatively cool mantle temperatures, which in turn generates stagnant-lid tectonics.

The implementation of a geotherm-dependent viscosity generates the tectonic regimes and the mantle decoupling mechanisms that has been shown in many models featuring fully temperature-dependent viscosity [e.g. Christensen, 1984; Nataf and Richter, 1982; Morris and Canright, 1984; Nataf, 1991; Solomatov, 1995]. Figure 4.7 shows that the overall trend is that increasing the thermal viscosity contrast decreases mobility at a given average mantle temperature. Figure 4.8 compares the thermally

4 6 steady-state temperature snapshots of mantle convection models for the ∆ηT range 10 -10 around the volume average non-dimensional temperature of 1.00. Figure 4.8 shows that as the thermal viscosity ∼ contrast is increased, the plate mobility is decreased.

4.2.4 Parameter study: reference Rayleigh number, Ra0

The reference Rayleigh number, Ra0, determines how vigorously a layer is convecting at a chosen depth and temperature. In this study, Ra0 is calculated at the surface, where T=0 and η=1 (the place in the model where the viscosity is highest). In this section, the eﬀect of a high (Ra0=250000), medium

(Ra0=25000) and low (Ra0=2500) vigour of convection is analyzed in terms of plate mobility for changes in internal heating rate.

Figure 4.9a shows that the transition from mobile to sluggish-lid convection occurs near the same internal heating rate (H=8) for vigorous and non-vigorous convection, with sluggish-lid convection common at high internal heating rates. However, as section 4.2.3 showed (and previously in O’Farrell [2013]), the internal heating rate parameter is not as eﬀective in warming the interior when the mantle is vigorously convecting (Figure 4.9b). A better comparison of how these model parameters work is to analyze the mobility in terms of average mantle temperature. Figures 4.9b and 4.9d show that more vigorously convecting models can produce stagnant-lid tectonics at low temperatures in comparison to low Rayleigh number models. This is shown again in Figure 4.10, where the transition from mobile- to sluggish-lid tectonics occurs at lower mantle temperatures for high Rayleigh number models. For a given mantle temperature of 0.87, the plate mobility is shown to be reduced as the reference Rayleigh number is increased (Figure 4.11). Figure 4.11 further highlights that to obtain similar average mantle Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 72

(a) (c) 1.0 Mobile 2.5x10 5 0 Mobile Sluggish 0.1 Sluggish

Ra 4 2.5x10 Mobility (M) 0.01 Stagnant 3 Stagnant 2.5x10 0.001 5 Ra 0=2.5x10 4 0 5 10 15 20 25 30 0.0001 Ra 0=2.5x10 3 Internal Heating (H) Ra 0=2.5x10 0.1 1 10 Internal Heating (H)

(b) (d) 1.0 Mobile 5 2.5x100 0.1 Mobile Stagnant Sluggish Ra 4 0.01 2.5x10 Mobility (M) Stagnant Sluggish 5 3 0.001 Ra 0=2.5x10 2.5x10 4 Ra 0=2.5x10 0.0001 3 0.4 0.6 0.8 1.0 1.2 Ra 0=2.5x10 Volume average Temperature (T) 0.6 0.8 1.0 1.2 Volume average Temperature (T)

Figure 4.9: Analysis of how the reference Rayleigh number (Ra0) aﬀects plate mobility (M). In this study 5 the only model parameters that change are H and Ra0. ∆ηT ,∆ηP and TL are kept constant (at 10 , 100 and 0.35, respectively). Plate mobility (M), vrms, the plate velocity vsurf and the average mantle temperature (Temp) are calculated when the model has reached steady state. (a) shows at what internal heating rates a model changes tectonic regime for a given Ra0 value. (b) shows at what average mantle temperature a model will change tectonic regime for a given Ra0 value. (c) shows the relationship between mobility and internal heating rate, and (d) between mobility and average temperature. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 73

(a) 5 Ra 0=2.5x10 10000 1000 Vrms 100 Vsurface

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T) (b) 4 Ra 0=2.5x10 10000 Vrms 1000 Vsurface

100

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T) (c) Ra =2.5x10 3 10000 0 Vrms 1000 Vsurface

100

Velocity 10 1 0.1 0.4 0.6 0.8 1.0 1.4 Temperature (T)

Figure 4.10: The volume-average and surface velocities as a function of average mantle temperature are 5 4 3 shown for (a) Ra0=2.5×10 , (b) 2.5×10 and (c) 2.5×10 (with model parameters as Figure 4.9). The red shaded area denotes the transition from mobile lid tectonics. The overall trend is for mantle temperatures greater than 0.6, the plate mobility is reduced as the reference Rayleigh number is increased. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 74

Temperature Tav ~0.87 0.0 0.275 0.550 0.875 1.10

a) b) c)

Ra =250000 Ra 0=2500 Ra 0=25000 0 M=0.34 M=0.21 M=0.0080 (H=10) (H=14) (H=30)

Figure 4.11: Temperature snapshots of how changing the reference Rayleigh number aﬀects mobility at a given average mantle temperature (Tav∼0.87). (a) shows a temperature snapshot of the thermally steady sluggish-lid tectonic regime when Ra0 =2500, (b) sluggish-lid tectonic regime when Ra0 =25000 and (c) stagnant-lid tectonic regime when Ra0 =250000. Model parameters of TL, ∆ηP and ∆ηT are kept constant (at 0.35, 100 and 105, respectively).

temperatures, a low Rayleigh number sluggish-lid model requires H=10 while a high Rayleigh number simulation implements H=30 (and yet yields stagnant-lid tectonics). The analysis of internal heating rate and Rayleigh number is discussed further in section 5.3.3. The specification of high Rayleigh number convection lowers the critical average mantle temperature needed to reduce the plate velocity to zero (Figure 4.9d). However, as the mantle is convecting vigorously (producing high surface heat flow), it is difficult to produce the warm average mantle temperatures required for the mantle to decouple from the plates and to reduce mobility. For a fixed internal heating rate, an increase in Ra0 therefore increases surface mobility (Figure 4.9c).

All of the calculations shown thus far in this chapter have been simple convection studies using aspect ratio 1 (Γ=1) models featuring reflective sidewalls [e.g., Stein et al., 2013]. The ability of mantle temperatures to generate stagnant-lid tectonics is apparent across all of the models shown. Many studies have shown that aspect ratio and model geometry affect mantle temperatures [e.g., Schmalzl et al., 2004; Butler and Jarvis, 2004; O’Farrell and Lowman, 2010; O’Farrell et al., 2013; O’Farrell, 2013]. Therefore, the following section analyzes the effect on plate mobility of changing the reference Rayleigh number in large aspect ratio 2D calculations and 3D simulations (where none of the models use reflective sidewalls).

4.2.5 Parameter study: aspect ratio and dimensionality study

The application of reﬂective sidewalls in Cartesian geometry mantle convection models locks the position of upwellings and downwellings to the sides of the solution domain. In this section, the reﬂective sidewalls are replaced by a periodic boundary (where material passes from one side of the box to the other), the Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 75 number of plates is doubled to two, and the aspect ratio increased from Γ=1 to Γ=8. The models presented here are more consistent with the supercontinent studies shown in Chapter 3.

In Figure 4.12, the plate mobility for the low Rayleigh number (Ra0=2500) study shown in section 4.2.4 (Γ=1) is compared to the mobility for the same parameters in models featuring a larger aspect ratio (Γ=8). The transitions to diﬀerent tectonic regimes occur at higher internal heating (and average mantle temperature) values once periodic boundaries are introduced and the aspect ratio and plate size are increased. Mantle temperatures for corresponding internal heating rates are similar in the two suites of models. Figure 4.13a-b shows the most dramatic diﬀerence occurs when H=20, as the Γ=1 model has already transitioned into a stagnant-lid regime while sluggish-lid tectonics prevails at Γ=8 (despite similar mantle temperatures in the two models).

A comparison of plate mobility at low, medium and high convective vigour (Ra0=2500, 25000, and 250000, respectively) when H=0 and H=20 is shown in Figure 4.14 for the two aspect ratio cases. For all Rayleigh numbers, increasing aspect ratio and plate size while removing reﬂective sidewalls generates a more mobile tectonic regime at comparable average mantle temperatures (Figure 4.14b). Figure 4.13c-d shows the comparison of the diﬀerent Γ models when the model is convecting vigorously (Ra0=250000). Although the tectonics in the two studies are in the same regime (sluggish-lid), the mobility of the Γ=8 model is three times greater than the Γ=1 model (despite having the same average mantle temperature).

In a study analyzing transitions to hard turbulence in mantle convection models, Schmalzl et al. [2004] showed that 2D and 3D studies have comparable mantle dynamics (i.e., vrms and Nusselt number) for infinite Prandtl number parameters. The Schmalzl et al. [2004] study, however, did not model tectonic plates which has been shown to influence the heat flow of mantle convection models [Monnereau and Quéré, 2001]. Figure 4.14 presents temperature and plate mobility results from a 3D simulation with dimensions of 6 6 1 featuring nine tectonic plates and periodic sidewalls [e.g., Heron and Lowman, × × 2014]. The model parameters in this 3D simulation are the same as used for the Ra0=250000 model of section 4.2.4. The impact of the high internal heating rate is minimal for the vigorously convecting model with a large surface area (and nine plates). Heat is more easily transported away at the multiple divergent plate boundaries and the mantle remains cool. As a result, the high internal heating rate produces only a marginally warmer mantle and a slightly reduced plate mobility (still mobile-lid) than when no internal heating is prescribed (Figure 4.14). Analyzing the effect of dimensionality for a H=20 model (with the same mantle parameters) shows increased mantle temperature and reduced plate mobility in 2D as compared to 3D (Figure 4.14). For a high Rayleigh number 3D mantle convection model to produce stagnant-lid tectonics, an internal heating rate is required that is greater than that used for a similar 2D simulation. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 76

0 10

−1 10

−2

Mobility 10

−3 10 2D aspect ratio 8 2D aspect ratio 1 Mobile−Sluggish Transition −4 Sluggish−Stagnant Transition 10 0 5 10 15 20 25 30 H (a)

0 10

−1 10

−2

Mobility 10

−3 10 2D aspect ratio 8 2D aspect ratio 1 Mobile−Sluggish Transition −4 Sluggish−Stagnant Transition 10 0.4 0.6 0.8 1 1.2 T (b)

Figure 4.12: Comparing plate mobility as a function of (a) internal heating rate and (b) average mantle temperature for different aspect ratio models (Γ=1 and Γ=8). The model parameters are the same for each 3 5 suite of results: Ra0, TL,∆ηT and ∆ηT are kept constant (at 2.5×10 , 0.35, 100 and 10 , respectively). The Γ=8 model is a two plate system with periodic sidewalls while Γ=1 has one plate with reflective sidewalls prescribed. Overall, increasing Γ increases plate mobility for similar internal heating rates. The average mantle temperatures behave differently at high aspect ratio. At similar temperatures, the higher aspect ratio models yield more mobile plate tectonics. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 77

Temperature H=20 0.0 0.275 0.550 0.875 1.10 Γ=1 Γ=8 a) b)

M=0.001 M=0.05 c) d)

M=0.06 M=0.2 Figure 4.13: Temperature snapshots comparing Γ=1 (with one plate and reﬂective side-walls (panels (a) and (c))) and Γ=8 (with two plates and periodic side-walls (panels (b) and (d))). (a) and (b) have 5 model parameters: Ra0=2500, H=20, TL=0.35, ∆ηT =10 and ∆ηP =100. (c) and (d) have the same parameters except for a higher reference Rayleigh number (Ra0=250000). Removing the reﬂective sidewalls and increasing the aspect ratio for both sets of models increases the plate mobility, despite the average mantle temperature being similar (Tav (a)=1.09, (b)=1.09), (c)=0.83, and (d)=0.85.

4.3 Discussion

Paleozoic reconstructions [Scotese, 2001] show the creation of the Rheic Ocean around 460Ma followed by its contraction 90Myr later, and its eventual closing to form Pangea 280Ma. In geodynamic models, episodic mantle reversals due to the build-up of heat in the upper mantle have been found to reverse plate motion on similar timescales [e.g., Lowman et al., 2001, 2003, 2004; Monnereau and Quéré, 2001; King et al., 2002; Ghias and Jarvis, 2004; Koglin Jr. et al., 2005]. However, these previous studies did not include geotherm- or temperature-dependent viscosity, the incorporation of which generates a different spectrum of plate tectonic regimes [e.g., Moresi and Solomatov, 1995; Stein et al., 2004; O’Neill et al., 2009; van Hunen and van den Berg, 2008; Foley and Becker, 2009; Korenaga, 2010; Lowman, 2011]. Through applying a geotherm- and depth-dependent viscosity in models with Rayleigh number and internal heating rates similar to isoviscous studies that exhibited episodic mantle reversals, stagnant- lid plate tectonics can be obtained (Figure 4.2-4.3). This study indicates that the generation of episodic mantle reversals could be a characteristic of the model parameters (in particular, the mantle viscosity). Therefore, the reason for the rapid change in plate direction that generated the opening and closing of the Rheic Ocean (amalgamating the supercontinent Pangea) remains elusive.

The range of non-dimensional viscosity associated with the thermal viscosity contrast (∆ηT ) is from

1 (at the surface) to (1/∆ηT ). Therefore, increasing the thermal viscosity contrast increases the average Rayleigh number (if all other parameters are kept constant) through decreasing the average viscosity in the model. In the mantle convection models of section 4.2.3, increasing the thermal viscosity contrast Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 78

0 10

−1 10

−2

Mobility 10 2D aspect ratio 1, Ra0=2500 2D aspect ratio 8, Ra0=2500 2D aspect ratio 1, Ra0=25000 2D aspect ratio 8, Ra0=25000 −3 2D aspect ratio 1, Ra0=250000 10 2D aspect ratio 8, Ra0=250000 3D aspect ratio 6, Ra0=250000 Mobile−Sluggish Transition −4 Sluggish−Stagnant Transition 10 0 5 10 15 20 H (a)

0 10

−1 10

−2

Figure 4.14: Comparing plate mobility as a function of (a) internal heating rate and (b) average mantle temperature for different aspect ratio models (Γ=1 and Γ=8). The model parameters are the same for each 3 5 suite of results: Ra0, TL,∆ηP and ∆ηT are kept constant (at 2.5×10 , 0.35, 100 and 10 , respectively). The Γ=8 model is a two plate system with periodic sidewalls while Γ=1 has one plate with reflective sidewalls prescribed. Overall, increasing Γ increases plate mobility at similar internal heating rates. The average mantle temperatures behave differently at high aspect ratio. At similar temperatures, the higher aspect ratio models generate more mobile plate tectonics. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 79

Sluggish- or stagnant-lid tectonics

Mobile-lid tectonics temperature Earth Increasing mantle Increasing

Increasing convective vigour

Figure 4.15: Schematic tectonic regime diagram show the relationship between convective vigour and average mantle temperature from the parameter study of section 4.2. An increase in convective vigour is deﬁned here as an increase in the thermal viscosity contrast and/or increasing the reference Rayleigh number. Sluggish- or stagnant-lid plate tectonics occurs at lower mantle temperatures for a vigorously convective mantle. This schematic only takes into consideration the mantle in a post-critical Rayleigh number regime.

facilitates a decoupling of the plates from the mantle and enables the generation of stagnant-lid tectonics (Figures 4.6-4.8) [e.g., Nataf and Richter, 1982; Morris and Canright, 1984; Nataf, 1991]. Furthermore, the reference Rayleigh (Ra0) number in section 4.2.4 acts as a gauge for the vigour of mantle convection. Increasing the reference Rayleigh number increases the effective Rayleigh number of a model (if all other parameters are kept constant). Figures 4.9-4.11 show that for models with similar interior temperatures, plate mobility will be less for mantle convection models that are convecting vigourously. Similar to the mechanism of generating stagnant-lid tectonics with increasing thermal viscosity contrast (e.g., Fig- ure 4.6), the viscous plate decouples from the vigorously convecting mantle at lower mantle temperatures for high Rayleigh number models than low Rayleigh number models (Figure 4.9). High internal heating rates are required to warm mantle temperatures for high Rayleigh number models, in comparison to low Rayleigh number models (Figure 4.6 and Figure 4.9), as a result of the reduced effectiveness of H to raise mantle temperatures when the mantle is vigorously convecting (as discussed further in section 5.3.3). Fundamentally, the effect of increasing H is to increase mantle temperatures. However, as internal heating rates do not warm the mantle efficiently at high Rayleigh numbers [e.g., Sotin and Labrosse, 1999; Butler and Peltier, 2000; Sinha and Butler, 2009; O’Farrell, 2013], H becomes difficult to compare across models with different mantle parameters. Therefore, to understand the effect on plate mobility of changing ∆ηT or Ra0 (and therefore changing the vigour of convection), the volume-average temperature of the models is analyzed to permit a comprehensive comparison to planetary interiors. For example, a high Rayleigh number mantle convection model is in stagnant-lid regime when H=30 (Figure 4.9a) but a low Rayleigh number model is in the same regime when H=20 (Figure 4.9a). Intuitively, this would Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 80 imply that the high Rayleigh number model would have a warmer mantle interior than the low Rayleigh number model. However, this is not the case (Figure 4.9d and Figure 4.6d). Therefore, the relationship between the thermal viscosity contrast and/or reference Rayleigh number and plate mobility is linked more with average mantle temperatures than the increase in internal heating rates (Figures 4.6 and 4.9). Comparing interior temperatures of mantle convection models shows that plate mobility is less when the mantle is vigourously convecting (e.g., when the thermal viscosity contrast and/or reference Rayleigh number is comparatively high) (Figure 4.6-4.11). The interaction of the plates and the upper mantle may offer an explanation as to why this occurs. In models featuring geotherm-dependent viscosity, as mantle temperature increases the viscosity contrast between the plates and the upper mantle also increases. For models featuring high ∆ηT or Ra0, the plates can decouple from the mantle to reduce the plate mobility at lower temperatures than when ∆ηT or Ra0 is lower.

Figure 4.15 generalizes the relationship between plate mobility and convective vigour after applying geotherm- and depth-dependent viscosity (with time-dependent plate thickness). Sluggish- or stagnant- lid plate tectonics occurs at lower mantle temperatures for a vigorously convecting mantle (Figures 4.6 and 4.9). In Figure 4.15, the Earth is characterized as vigorously convecting with mobile plate tectonics and a cool mantle temperature. According to this study, a warmer mantle temperature for an Earth with comparable convective vigour would reduce plate mobility and possibly generate a diﬀerent tectonic regime than seen today. Therefore, Figure 4.15 is in-keeping with the idea of stagnant-lid and episodic subduction tectonics existing [Brown, 2006; O’Neill et al., 2007a,b; van Hunen and van den Berg, 2008; Condie, 2008; Halla et al., 2009; Sizova et al., 2010; Gerya, 2011; van Hunen and Moyen, 2012; Gerya, 2014] in an early Earth setting with higher mantle temperatures than present day [Nisbet et al., 1993; Abbott et al., 1994; Grove and Parman, 2004; van Thienen et al., 2004; Labrosse and Jaupart, 2007].

In addition, the results indicate that further cooling of the mantle would ensure Earth plate tectonics would remain in a mobile-lid regime (if the mantle maintained the current vigour of convection).

4.3.1 Plate thickness

The base of a tectonic plate is defined by the lithosphere-asthenosphere boundary, where the flow of the ultra-mafic mantle changes from being relatively stiff (lithosphere) to being more readily deformable (asthenosphere). The lithospheric cut-off temperature (acting as the temperature with which the lithosphere-asthenosphere boundary is defined) is modified in section 4.2.2 to determine its effect on plate mobility. In this study, the higher the lithospheric cut-off temperature the more likely mobile plate tectonics is generated. If the lithospheric cut-off temperature is relatively low, plate tectonics would Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 81 be less likely (Figure 4.3). Due to the ambiguity of the total temperature change across the mantle (e.g., constraints on the core-mantle boundary temperature), it is difficult to obtain appropriate values of TL for the modern-day Earth. However, if the base of the lithosphere is conventionally defined as 1350oC [McKenzie and Bickle, 1988] and the super-adiabatic change in temperature across the mantle ∼ is approximately 3000K, then a lithospheric cut-off temperature of 0.45 would be sufficient to model the

Earth. The modelling method here indicates that models featuring TL=0.45 with appropriate mantle temperatures would generate a plate mobility in-keeping with present day values (Figure 4.4) [Kanamori and Brodsky, 2004].

4.3.2 Limitations

The viscosity model implemented here supports no lateral variance and is therefore not fully temperature- dependent. However, the methodology captures fluctuations in thermal boundary layer thickness and accompanying changes in the depths over which a given viscosity contrast occurs, generating a spectrum of different plate tectonic regimes that compares well with results from models featuring pressure, temperature and stress-dependent rheologies [e.g., Moresi and Solomatov, 1995; Stein et al., 2004; O’Neill et al., 2009; van Hunen and van den Berg, 2008; Foley and Becker, 2009; Korenaga, 2010; Stein et al., 2013]. In addition, Stein and Hansen [2014] showed layered temperature-dependent viscosity convection to be a suitable approximation to the full temperature dependence. Through comparing mantle convection models featuring layered geotherm-dependent viscosity (as used here) and fully temperature- dependent viscosity, Stein and Hansen [2014] found only minor differences in convective characteristics (including Nusselt number, convection flow regime, lid thickness, stress, and dynamic topography) for cases featuring the same heating modes and viscosity contrasts.

The strong temperature dependence of mantle viscosity [Weertmant, 1970; Karato and Wu, 1993] controls plate tectonics [e.g., Solomatov, 1995; Hansen and Yuen, 1993; Morris and Canright, 1984; Trompert and Hansen, 1998]. A high thermal viscosity contrast generates cold, stiﬀ plates which are resistant to the stress produced by mantle buoyancy forces. Why the Earth features mobile plate tectonics, while Venus does not, is often explained by the lithospheric rock having a yield stress that is not exceeded during a stagnant-lid regime [Fowler, 1993]. Once the yield stress is exceeded, viscoplastic failure would initiate subduction (and thus generate the mobile-lid regime seen on Earth) [Fowler, 1993]. Many mantle convection studies include a lithospheric yield stress in order to initiate plate tectonics at high thermal viscosity contrasts [e.g., van Heck and Tackley, 2011; Lenardic and Crowley, 2012; Weller and Lenardic, 2012; Stein et al., 2013]. Although the results presented here do not use a lithospheric yield Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 82 stress, in MC3D the force-balance method of calculating plate velocities has inherent (and permanent) weak zones deﬁned at the plate boundaries (which facilitates plate mobility). An issue with stress- dependent rheology models that use a lithospheric yield stress is that past sites of deformation ‘heal’ over short timescales [e.g., van Heck and Tackley, 2011; Lenardic and Crowley, 2012; Weller and Lenardic, 2012; Stein et al., 2013], rather than acquiring a history of lithospheric ‘damage’ pertaining to the generation weak zones [e.g., Bercovici and Ricard, 2014; Foley and Bercovici, 2014].

4.3.3 Aspect Ratio

This study implements small aspect ratio boxes with reflective sidewalls to make inferences about plate mobility [e.g., Stein et al., 2013]. Section 4.2.5 acknowledges the effect of this modelling method by increasing the aspect ratio and dimensionality of the 2D study. Overall, increasing the aspect ratio increases plate mobility for models featuring the same heating modes and viscosity contrasts with similar interior temperatures (Figure 4.12-4.14). The removal of the reflective sidewalls changes the focus of the mantle stress and in turn increases the plate mobility (which cools the mantle). Comparing a high Rayleigh number 3D model with H=20 to a 2D study with the same mantle parameters (but different number of plates) shows a great difference in plate mobility (Figure 4.14a). The increased number of plates in the 3D case may help to cool the mantle temperature (Figure 4.14b) and increase the plate mobility.

Assessing the likeliness of plate tectonics on super-Earths is beyond the scope of this study. However, rheological melting regimes and average temperatures for the extra-solar planet mantle would be essential in establishment of the possibility of plate tectonics on a given super-Earth. The modelling method used here indicates that planets with warmer interiors, but the same rheological qualities as Earth, would have lower plate mobilities. Similarly, an early Earth characterized by warmer mantle temperatures (but the same convective vigour) would be expected to have less plate mobility than today [e.g., Nisbet and Fowler, 1983].

Previous studies have shown that the choice of geometry affects mantle temperatures [O’Farrell and Lowman, 2010; O’Farrell, 2013], therefore it is important to take this into consideration as mantle temperatures play a large role in the mobility of plate tectonics. The effect of aspect ratio and model dimension is an area of study that requires further clarification if results from small-scale mantle convection models are to infer tectonic processes involved on super-Earths. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 83

a) d)

Temp 1.00

b) e) 0.75

0.50

0.25 c) f)

0.00

Figure 4.16: Heating up and cooling down the mantle to change the tectonic regime. Temperature snapshots depict the ‘uniqueness’ of a set of mantle parameters to generate a tectonic regime (isotherms given for the non-dimensional temperature values of 0, 0.25, 0.5, 0.75 and 1.0). Model parameters for this 5 ﬁgure are: Ra0=400, ∆ηT =10 , ∆ηP =100, and TL=0.30). (a)-(c) shows the cooling of a model from an initial stagnant lid regime (a), which was generated using the mantle parameters given and an internal heating rate of 16 (the interior of the model has temperatures in excess of 1.0). The internal heating rate is removed (H=0) and the model is allowed to cool. Panel (b) is presented at a non-dimensional time of t=0.02 after (a). (c) shows the steady-state temperature snapshot for the mobile-lid tectonic regime generated once the mantle has cooled. As the model cools, the tectonic regime changes from stagnant-lid to mobile-lid. (d)-(f) shows the cooling of a model from an initial mobile-lid regime (d), which was generated using the mantle parameters given and an internal heating rate of zero (as in (c)). An internal heating rate of H=16 is applied, and the mantle convection model is warmed. Panel (e) is presented at a non-dimensional time of t=0.02 after (d). (f) shows the steady-state temperature snapshot for the stagnant-lid tectonic regime generated once the mantle has suﬃciently warmed. As the model heats up, the tectonic regime changes from mobile-lid to stagnant-lid. Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 84

4.3.4 Uniqueness

Recently, Lenardic and Crowley [2012] presented a parameter study into tectonic regimes on super- Earths. By changing the initial condition of a mantle convection study and analyzing the tectonic regime generated, Lenardic and Crowley [2012] found that a strong history dependence exists in determining the plate mobility, and that multiple tectonic states can exist for a set of mantle parameters. Through changing plate yield stress, Weller and Lenardic [2012] also found the thermal evolution of a mantle to have an eﬀect on tectonic regime. However, in the study presented here a given set of mantle parameters can be treated as unique in generating a plate tectonic regime; changing the initial condition of the mantle convection models presented here will not change the overall plate mobility. An example of this is given in Figure 4.16, where a model is warmed and cooled to demonstrate that modelling in a forward or backward direction does not change the tectonic regime (contrary to the Lenardic and Crowley [2012] study). This diﬀerence in uniqueness of tectonic regime (for a given set of mantle parameters) may be due to the method of modelling plate tectonics (e.g., the force-balance method implemented here and the method of Lenardic and Crowley [2012] that utilizes a stress-dependent rheology). As discussed in section 4.3.2, MC3D has permanent weak zones at the plate boundaries that may facilitate plate motion. In Lenardic and Crowley [2012], for example, as the model evolves in time the areas where the lithospheric yield stress is exceeded has no permanent history of such deformation. As a result, the modelling method of Lenardic and Crowley [2012] and Weller and Lenardic [2012] requires persistent ‘breaking’ of the lithosphere which promotes non-uniqueness of the parameter space, whereas the force- balance method of MC3D obtains more unique tectonic regimes for a given set of mantle parameters.

4.4 Conclusion

A motivation for this study was to investigate whether mantle convection calculations featuring geotherm- and pressure-dependent viscosity could produce episodic plate velocity reversals seen in isoviscous mantle convection models [e.g., Lowman et al., 2001, 2003, 2004; Monnereau and Quéré, 2001; King et al., 2002; Ghias and Jarvis, 2004; Koglin Jr. et al., 2005]. However, the application of a geotherm-dependent mantle viscosity eliminated the production of episodic mantle reversals, but gave the model the ability to generate three tectonic regimes: mobile, sluggish, and stagnant lid tectonics (similar to studies featuring full temperature-dependent viscosity [e.g., Moresi and Solomatov, 1995; Stein et al., 2004; O’Neill et al., 2009; van Hunen and van den Berg, 2008; Foley and Becker, 2009; Korenaga, 2010; Stein et al., 2013; Stein and Hansen, 2014]). A better understanding of plate tectonic processes was attained through Chapter 4. Plate mobility regimes and a re-evaluation of plate reversals 85 analyzing the thickness of the lithosphere, the thermal viscosity contrast, and the reference Rayleigh number and their relation to mantle temperatures. For example, if the mantle is convecting vigorously, tectonic plates can decouple from the mantle to produce stagnant-lid tectonics when the interior temperature is relatively cool (Figure 4.9). Furthermore, aspect ratio, dimensionality, and the number of plates in mantle convection models featuring geotherm-dependent viscosity have an effect on plate mobility (Figure 4.12). A further analysis of modelling methods in mantle convection models is continued in the next section, analyzing the effect of Rayleigh number in supercontinent calculations. Chapter 5

The impact of Rayleigh number on the signiﬁcance of supercontinent insulation

5.1 Introduction

Continental lithosphere inhibits heat loss from the Earth’s interior, relative to oceanic lithosphere, due to its thickness and the warmth of the radioactively enriched crust. As a result, it has been suggested that insulation may cause the formation of reservoirs of heat accumulating beneath slow-moving continents [Jordan, 1975; Anderson, 1982; Pollack, 1986; Gurnis, 1988; Zhong and Gurnis, 1993; Anderson, 1994; Lowman and Jarvis, 1995, 1999; Lenardic et al., 2005; Phillips and Bunge, 2005; Coltice et al., 2007, 2009; Rolf et al., 2012]. However, numerical studies featuring thermally and mechanically distinct oceanic and continental plates fail to generate sub-supercontinental temperatures much greater than sub-oceanic temperatures over timescales relevant to supercontinent episodes [Heron and Lowman, 2010, 2011; Yoshida, 2013] (a result shown in Chapter 3 using models with depth-dependent viscosity and high Rayleigh number convection). Therefore, the signiﬁcance of continental insulation in such processes as the supercontinent cycle remains unsettled.

The major diﬀerences in the conclusions on the signiﬁcance of continental insulation can be attributed, in part, to the modelling methods employed. For example, the choice of the thermal boundary condition at the core-mantle boundary (e.g., modelling mantle convection driven purely by internal heating) has

86 Chapter 5. Rayleigh number and continental insulation 87

300 Isothermal CMB Fit 250 0.086

200 0.069

150 0.052

100 0.034 Plate thickness (km)

50 0.017 Non−dimensional plate thickness

0 0 0.5 1 1.5 2 2.5 3 Average Rayleigh number 7 x 10

Figure 5.1: Plot of average Rayleigh number against plate thickness (in km left, non-dimensional right) for 8 isothermal core-mantle boundary steady state models. The line of best ﬁt is given by the equation −0.242 δ=3380× Raavg and is shown by the dashed curve.

a substantial eﬀect on mantle dynamics. An insulating core-mantle boundary condition (e.g., where a model is not heated from below (qcmb=0) and is purely internally heated) fails to generate the deep active mantle plumes that are believed to have formed 50-100Myr after supercontinent formation [Li et al., 2003]. In addition, the spatial resolution required to resolve boundary features at high Rayleigh number is computationally expensive. As a result, 3D models with Earth-like convective vigour are not common.

In this chapter, the significance of continental insulation in calculations featuring different heating parameters are examined in both 2D and 3D Cartesian geometry numerical simulations (with the geotherm-dependent viscosity and time-dependent plate thickness methods of Chapter 4 implemented). The study presented here makes a quantitative comparison between high and low Rayleigh number mantle convection models with a focus on the effect of continental insulation when convective vigour is a variable. The results from this systematic study illustrate the major differences in mantle convection models with regards to the effect of continental insulation when different mantle heating modes are considered. Chapter 5. Rayleigh number and continental insulation 88

5.2 Comparing the vigour of mantle convection

Complex rheologies or viscosity laws in numerical models can make it difficult to gauge the vigour of convection. A reference Rayleigh number, Ra0, (calculated by using a reference viscosity value, η0, ata certain depth and temperature) does not necessarily give a real indication of how vigorously a model is convecting (unless average temperatures, viscosities, heat fluxes or thermal boundary layer thicknesses are also stated). Simulations featuring purely internally heated models are even more difficult to classify than those with purely basal or mixed-mode heating (with an isothermal core) because non-dimensional temperatures may not reach the temperature at which the reference viscosity and reference Rayleigh number is defined (e.g., a non-dimensional temperature value of 1.0). In a study comparing results obtained with high and low Rayleigh numbers it is important, therefore, to clearly define the vigour of mantle convection. In this chapter, the thermal boundary layer thickness is used as a gauge of Rayleigh number magnitude. Figure 5.1 shows plate thicknesses plotted against Raavg for eight statistically steady 2D models with no internal heating and with an isothermal base, each featuring a pair of equal size plates. The average Rayleigh number (Raavg) is defined as

1 Raavg = Ra (5.1) 0η′(z)

′ −1 where η (z) is the average of the inverse of the viscosity and Ra0 is the Rayleigh number at the surface, where T is 0 (η′=1). Finding a line of best ﬁt through the data for non-dimensional plate thickness (δ) and Raavg, yields − δ = 3380 Ra 0.242. (5.2) × avg

From the relation above, plate thicknesses in purely internally heated models can be used to describe the vigour of convection and estimate the eﬀective average Rayleigh number. Accordingly, the insulating core-mantle boundary models can be compared to isothermal core-mantle boundary models without the use of internal heating Rayleigh numbers. In all models in this study, TL, ∆ηT , and ηD(z) are kept constant (0.35, 105 and 100, respectively). For isothermal core-mantle boundary simulations, the desired vigour of convection is obtained by increasing Ra0. In the insulating core-mantle boundary cases, the internal heating rate and Ra0 are both changed in order to obtain a range of mantle temperatures and plate mobility values that are comparable to those in the isothermal core-mantle boundary models. The average Rayleigh number values for the insulating core-mantle boundary models are calculated a posteriori based on the plate thickness resulting from the speciﬁed values of H and Ra0 (Figure 5.1).

For the two-dimensional model study, four supercontinent formation simulations are examined for Chapter 5. Rayleigh number and continental insulation 89

Name Raavg Ra0 δ (km) vrms q¯surf M cmb1 9.8x104 2.5x103 0.070(203) 101 6 0.52 2.0 cmb2 4.8x105 1.0x104 0.047(137) 264 9 0.49 2.3 cmb3 1.7x106 2.5x104 0.038(109) 390 11 0.50 2.3 cmb4 2.0x107 2.5x105 0.018(51) 2000 23 0.54 2.3

Table 5.1: Model parameters and initial condition properties for the isothermal basal boundary condition used in the 2D study. δ is the non-dimensional plate thickness; vrms is the average magnitude of the mantle velocity;q ¯surf is the non-dimensional time-averaged surface heat ﬂux; is the non-dimensional volume-average temperature; and M refers to the mobility of the surface, and is given as the ratio of the plate velocity and vrms of the mantle. Raavg is the average Rayleigh number, while the reference Rayleigh ′ number (Ra0) is calculated at the surface of the model (where T=0 and η =1). All 2D models have an aspect ratio of 8 and a grid resolution of 2500×200.

Name Raavg Ra0 H δ (km) ∆T vrms q¯surf M ins1 1.2x104 1.75x103 5 0.085(247) 0.65 59 10.5 0.52 1.8 ins2 2.7x105 2.0x103 7.5 0.057(164) 0.72 115 15.8 0.56 2.1 ins3 3.0x106 2.5x104 15 0.028(81) 0.71 540 31 0.61 2.0 ins4 1.1x107 2.5x105 20 0.022(64) 0.61 1190 45 0.55 1.5

Table 5.2: Input parameters and initial condition properties for models with the insulating basal boundary condition used in the 2D study. Parameters are the same as Table 5.1 with H, the non-dimensional internal heating rate, and ∆T the maximum temperature variation in the mantle (for an isothermal basal boundary condition ∆T is 1). each core-mantle boundary condition, the model parameters are given in Tables 5.1 and 5.2. The plate thickness, surface heat ﬂux, and vrms of the models indicate a range of low to high Rayleigh numbers. The convective vigour of the models is also indicated in Figure 5.2 which shows the horizontally averaged temperatures (geotherms) of the models. The thickness of the plates (the depth at which the geotherm reaches the TL value of 0.35) can be seen to increase with decreasing convective vigour.

5.3 2D Supercontinent results

The eﬀect of Rayleigh number on continental insulation is investigated here through the study of temperature changes post-supercontinent formation. Timescales are converted by scaling 60Myr to one mantle transit time, tM (calculated by using the mantle vrms and the time required for a mantle particle to

1 travel the depth of the mantle). In this section, a continental diﬀusivity value of /4 of the oceanic value is used for the continental insulation parameter, i.

5.3.1 Initial condition

The initial conditions for all 2D models are formed through modelling a dynamic two-plate system. Snapshots of the evolution of these models are shown so that the convergent plate boundary appears on the vertical midplanes of the depicted temperature ﬁelds (0Myr panels in Figure 5.3). Once the two-plate Chapter 5. Rayleigh number and continental insulation 90

1 1 (a) (b) 0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 cmb1 ins1 cmb2 ins2 0.5 0.5 cmb3 ins3 Height cmb4 Height ins4 0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0 0 0.5 1 0 0.5 1 Temperature Temperature

Figure 5.2: Non-dimensional temperature against height above the base of the mantle for the initial conditions of the 2D isothermal (a) and insulating (b) core-mantle boundary models. The base of the plates is deﬁned where the geotherm reaches the TL value of 0.35.

system reaches a statistically steady state (i.e., no long term heating or cooling trends are evident in the solution), the plate geometry is modified. A continental plate (with a prescribed velocity of zero) is centred over the initial downwelling and two oceanic plates move and subduct on either side of the continent. The emplacement of the continental plate simulates the collision of two smaller continental plates at the site of the mantle downwelling. To explore the influence of the initial thermal field, two sets of 2D calculations are examined. In one case, insulation by dispersed continents is not considered and continental insulation is not specified until supercontinent formation occurs. (The initial thermal field is obtained by integrating the thermal field in time with the assumption that continental lithosphere is absent.) All 2D model data for these calculations are given in Table 5.1 and Table 5.2 and correspond to the time of the respective initial conditions. In an alternate set of experiments (section 5.3.5) insulation of the mantle is simulated by dispersed continents prior to the assembly of the supercontinent. The mean temperatures of the initial conditions therefore differ, the latter case features warmer initial conditions overall.

5.3.2 Isothermal core-mantle boundary

Figure 5.3a-b depicts temperature snapshots for low and high Rayleigh number models every 50Myr starting from the initial condition until 250Myr after continental aggregation. The temperature ﬁeld at Chapter 5. Rayleigh number and continental insulation 91

a) 0Myr

50Myr

100Myr

150Myr

200Myr

250Myr

500Myr

b) 0Myr

50Myr

100Myr

150Myr

200Myr

250Myr

500Myr Temperature 0.0 0.25 0.5 0.75 1.0 c)

Figure 5.3: 2D supercontinent results: thermal fields for (a) low (cmb1) and (b) high (cmb4) Rayleigh number isothermal core-mantle boundary models. Oceanic surfaces are indicated by dashed lines (small arrows show the direction of plate motion), continental surface is given by a solid line (solid circles indicate no plate motion of the supercontinent). The initial condition (0Myr), 50, 100, 150, 200, 250 and 500Myr temperature snapshots are shown with black contours at non-dimensional temperatures of 0, 0.25, 0.5, 0.75 and 1. Dimensional times are scaled to one mantle transit time being equal to 60Myr. Large black arrows indicate the strong horizontal upper mantle flow (that has the ability to reposition downwellings). (c) Sub- oceanic and sub-continental temperatures in the upper mantle (depth 0.1d) at the time of the formation of the supercontinent (black) and 250Myr after (green). Dashed lines correspond to cmb1, solid lines cmb4. In the high Rayleigh number case there is very little change in sub-oceanic temperature over time, and a slight warming in the sub-continental region. The low Rayleigh number case shows significant heating below the sub-oceanic and particularly the sub-continental regions. Chapter 5. Rayleigh number and continental insulation 92

a) 0.8 0.6 0.4 0.2 Heat flux Heat 0.0

Ocean Supercontinent Ocean

b) cmb1 Ocean Supercontinent Ocean

c) cmb4 Ocean Supercontinent Ocean

Temperature 0.0 0.25 0.5 0.75 1.0

Figure 5.4: 2D supercontinent results: (a) surface heat flux for (b) low and (c) high Rayleigh number convection with isothermal core-mantle boundary conditions. (a) surface heat flux from the oceanic and continental plates for cmb1 (blue) and cmb4 (red) 250Myr after supercontinent formation (and ∼100Myr after the generation of a sub-continental plume). (b) and (c) shows the thermal fields of cmb1 and cmb4, respectively, corresponding to the times in (a). Chapter 5. Rayleigh number and continental insulation 93

cmb1 cmb2 0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4 Temperature Temperature 0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr) cmb3 cmb4

0.6 0.6

0.5 0.5

0.4 0.4 Temperature Temperature 0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr)

Figure 5.5: Time-series of volume-average temperatures beneath the oceans (blue line) and continents (red line) for all isothermal core-mantle boundary models. The dashed black line shows the volume average temperature for the whole model. All parameters for the isothermal core-mantle boundary models are given in Table 5.1.

500Myr post-supercontinent formation is also shown to illustrate the effect of the prolonged emplacement of a thermal blanket effect on the sub-continental mantle. The differences in the convection in the models are apparent; much thinner thermal upwellings and downwellings exist for a high Ra vigorously convecting mantle. However, there are similarities in the mechanisms involved in the thermal evolution of the mantle post-supercontinent formation. A sub-continental plume arrives between 100-150Myr (1.5- 2.5 transit times) after supercontinent formation in both models (a timescale consistent with the results of Yoshida [2013]). A sub-continental plume is then present for the duration of the calculation (reduced continental heat flow is present in both models due to the prescribed thermal blanket effect (Figure 5.4)).

Figure 5.3c depicts sub-oceanic and sub-continental temperatures in the upper mantle at the time of the formation of the supercontinent and 250Myr later. The low Rayleigh number case shows signiﬁcant warming of the upper mantle beneath the oceanic and particularly the continental plates. However, despite the formation of a sub-continental plume and the trapping of heat due to continental insulation, the sub-continental upper mantle temperatures remain lower than those below much of the oceans in the low Rayleigh number cases at the time considered. In the higher Ra case, the sub-oceanic and sub-continental temperatures are similar for both the initial condition and after 250Myr.

Figure 5.5 shows the mean sub-oceanic, sub-continental, and average mantle temperatures for the Chapter 5. Rayleigh number and continental insulation 94

Key (a): Continent (surface) Ocean (surface) Continent (basal) Ocean (basal) Transit time ( τ) Transit time ( τ) 4 4 a) 0 1 2 3 0 1 2 3 b) 30 cmb1 30 20 25 10 20 0 15 -10 Heat flux Heat 10 -20 5 Contributions to Contributions 0 -30 0 0 50 100 150 200 250 heating sub-continental 50 100 150 200 250 Time (Myr) Time (Myr) Transit time ( τ) Transit time ( τ) c) 0 1 2 3 4 d) 0 1 2 3 4 30 cmb4 0.5 Horizontal warming 20 0.0 10 Horizontal cooling 0 -0.5 -10 -20 -1.0 Contributions to Contributions -30 Horizontal/vertical -1.5 heat flow sub-continent flow heat sub-continental heating sub-continental 0 50 100 150 200 250 0 50 100 150 200 250 Time (Myr) Time (Myr) Key Difference in basal Temporal Horizontal (b&c): and surface heat temperature gradient heat flux Solid: cmb1 Dashed: cmb4 flux sub-continent sub-continent sub-continent

Figure 5.6: Analysis of volume-averaged temperature and basal and surface heat flux time-series at high and low Rayleigh number. (a) Continental surface (black) and basal (red) heat flux and oceanic surface (cyan) and basal (blue) heat flux as a function of time for cmb1 (solid line) and cmb4 (dashed line) models. The orange shaded region indicates the plume formation period between 100-150Myr in both models (shown in Figure 5). (b) and (c) show contributions to the non-dimensional heat equation in the sub-continental region at low (b) and high (c) Rayleigh number. The purple line indicates the difference in the basal and surface heat flux under the supercontinent (vertical heat flux), green shows the temporal temperature gradient sub-continent (relative warming or cooling) and the brown curve gives the horizontal heat flux under the continent (negative values indicating heat flowing from the sub-continental region). (d) shows the ratio of the horizontal and vertical heat flow sub-continent (solid line shows cmb1 and dashed line cmb4). A negative ratio indicates more heat is leaving the sub-continental mantle horizontally than is arriving from the base of the mantle, while a positive ratio shows that horizontal heat flow (from below the sub-oceanic plates) actually warms the sub-continental mantle. A ratio less than -1 indicates that the sub-continental mantle is cooling overall. Chapter 5. Rayleigh number and continental insulation 95 four isothermal core-mantle boundary models (mean temperatures are calculated by taking the volume- average temperature from the surface to the base of the mantle beneath the oceanic and continental regions). Over supercontinental timescales, only the lowest Rayleigh number model exhibits sub- continental temperatures warmer than sub-oceanic material. For high Rayleigh number models, the sub-continental and sub-oceanic initial condition temperatures are much closer than for the lower Rayleigh number calculations. For the model with the most Earth-like convective vigour, cmb4, the sub-continental and sub-oceanic temperatures are almost equal over supercontinent timescales.

Figures 5.3 and 5.5 show that supercontinent formation in low Rayleigh number models warms the mantle in a fundamentally different way than in higher Rayleigh number cases. In low Rayleigh number models the build-up of heat due to thermal insulation and the appearance of large plume heads beneath a supercontinent generates a horizontal upper mantle flow strong enough to deflect downwellings (a feature not seen in higher Rayleigh number simulations).

Figure 5.6a analyzes the temperatures and basal and surface heat ﬂux time-series for high and low

Rayleigh number models on supercontinent timescales (<250Myr). In Figure 5.6a, continental surface heat flux can be seen to be low (and approximately constant) in both models (in keeping with Figure 5.4). Heat arriving from the base of the model from the sub-oceanic regions is also approximately constant for high and low Rayleigh numbers. At low Rayleigh number, basal heat flux under the continent decreases during the first 100Myr of supercontinent formation due to the termination of subduction at the continental suture and cessation of the delivery of cold material to the core-mantle boundary below. Approximately 100Myr after supercontinent formation, the repositioning of subduction on the edge of the supercontinent has generated cold downwellings that reach the base of the model (Figure 5.3a). The pinching of the thermal boundary layer as the large, cold downwellings arrive at the core-mantle boundary raises the basal heat flux between 100-150Myr, and a sub-continent plume is formed. In the subsequent 100Myr, the basal sub-continental heat flux in the low Rayleigh number model decreases but remains higher than the flux at the time of supercontinent formation. Between 0-100Myr, Figure 5.6a shows the heating mechanism in the high Rayleigh number case is similar to that in the low Rayleigh number case; the heat flux reduces due to the closure of the initial subduction zone. However, the arrival at the base of the mantle of smaller and thinner cold downwellings from the repositioned subduction zones fails to dramatically increase the basal heat flux in the manner seen in the low Rayleigh number case. In the high Rayleigh number model, the plume formation stage (between 100-150Myr) halts the decrease in basal continental heat flux (but does not increase heat flux as in cmb1). This is largely due to a smaller volume of cold material impacting the sub-continental thermal boundary layer. After the formation of the sub-supercontinental plumes, the basal heat flux beneath the continent decreases Chapter 5. Rayleigh number and continental insulation 96

(>150Myr).

Figure 5.6b-d analyses the horizontal and vertical heat flux contribution to the sub-continental warming for high and low Rayleigh number models. The magenta lines of Figure 5.6b-c shows the difference in surface and basal heat flux sub-supercontinent, while the brown lines show the flow of heat from the sub-continental region into the oceanic region (a negative value indicating heat moving horizontally from under the continent). The resulting sub-continent volume average of the temporal temperature gradient is given by the green line (positive values showing warming under the supercontinent). For the low Rayleigh number model heat is continually lost from under the supercontinent to the oceanic region (brown line, Figure 5.6b). The formation of the sub-continental plume generates a warming beneath the supercontinent after 100Myr. However, the behaviour in the high Rayleigh number model is different; the temperature gradient beneath the supercontinent is more variable throughout. The rate of sub-continental mantle warming increases from 50-150Myr, however the rate of warming reduces once a large plume is fully formed. The horizontal heat flow for the high Rayleigh number case shows heat drawn in from the sub-oceanic mantle during the plume formation stage, a feature not seen in low Rayleigh number models. Figure 5.6d identifies this by plotting the ratio of the horizontal and vertical heat flow (brown and magenta lines, respectively) sub-supercontinent. A negative ratio indicates more heat is leaving the sub-continental mantle horizontally than is arriving from the base of the mantle, while a positive ratio shows that horizontal heat flow (from below the sub-oceanic plates) actually warms the sub-continental mantle. A ratio less than -1 indicates that the sub-continental mantle is cooling overall. The low Rayleigh number case shows heat leaving the continental region for the entire period examined, while the more vigorously convecting model cmb4 includes periods where heat from the sub-oceanic regions contributes to warming the sub-continental mantle. The multiplicity and ‘pulsing’ nature of the plumes in the high Rayleigh number models is shown in Figure 5.6c by the high frequency fluctuations in the temperature gradient and horizontal heat flux sub-continent. In contrast to low Rayleigh number models, the decrease in lateral extent of the circum-supercontinent downwellings draws relatively mod- est amounts of heat into the sub-continental mantle (Figure 5.6c). In the low Rayleigh number model, no ‘pulsing’ plume features are found (Figure 5.3a, Figure 5.6a, or Figure 5.6b), but a large plume head dominates the volume of the sub-supercontinent mantle and therefore the average temperature calculation.

In all models, the mechanism that instigates the formation of the plumes (and therefore a return ﬂow sub-supercontinent) is the repositioning of subduction zones, rather than continental insulation (which has not had enough time to generate high temperatures). Chapter 5. Rayleigh number and continental insulation 97

a) 0Myr

ins1 250Myr

500Myr b) 0Myr

250Myr ins4

500Myr

1.0 c) 0.8 0.6 0.4 0.2

Temperature 0.0 Ocean Supercontinent Ocean

Figure 5.7: 2D model results: thermal fields for (a) low (ins1) and (b) high (ins4) Rayleigh number insulating core-mantle boundary models. Temperatures are as shown in Figure 5.3. Oceanic plates are indicated by dashed lines (small arrows show direction of plate motion), continental surface is shown by solid line (solid circles indicate no motion of the supercontinent). The initial condition (0Myr) and temperature field snapshots 250Myr and 500Myr after supercontinent formation are shown with black contours at non- dimensional temperatures of 0, 0.25, 0.5, 0.75 and 1. Dimensional times are scaled to one mantle transit time (60Myr). Large black arrows indicate the strong horizontal upper mantle flow. (c) Sub-oceanic and sub- continental temperatures in the upper mantle (depth 0.1d) at the time of the formation of the supercontinent (black) and 250Myr later (green). Dashed lines corresponds to model ins1 and solid lines to model ins4. In the high Rayleigh number case there is very little change in sub-oceanic temperature over time, and a slight warming in the sub-continental region. The low Rayleigh number case shows heating in the sub-oceanic and especially the sub-continental regions.

5.3.3 Insulating core-mantle boundary

Figure 5.7a-b depicts temperature snapshots for low (ins1) and high (ins4) Rayleigh number models with an insulating core-mantle boundary. The absence of active plumes below the supercontinent allows for analysis of the thermal blanket eﬀect in isolation. Heat does build-up beneath the supercontinent in both models, however the lower Rayleigh number model (ins1) shows the strong horizontal upper mantle ﬂow seen in cmb1 (Figure 5.3a). Figure 5.7c shows the temperatures in the upper mantle in both models. The lower Rayleigh number model case shows a substantial temperature increase beneath the supercontinent as well as an increase below the oceans. For the higher Rayleigh number case there is mild warming under the continent and a slight decrease in temperature under the oceanic plates after 250Myr. As in Figure 5.3c, Figure 5.7c illustrates that models with higher mantle convection vigour Chapter 5. Rayleigh number and continental insulation 98

ins1 ins2

0.6 0.6

0.5 0.5

0.4 0.4 Temperature Temperature

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr) ins3 ins4

0.6 0.6

0.5 0.5

0.4 0.4 Temperature Temperature

0.3 0.3 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr)

Figure 5.8: Time-series of volume-average temperatures beneath the oceans (blue line) and continents (red line) for all insulating core-mantle boundary models. Dashed black lines show the volume average temperature for the whole model.

Heat input (insulating core−mantle boundary) 0.1 25 ) γ

0.08 20

0.06 15

0.04 10

0.02 5 Non−dimensional internal heating rate (H) Non−dimensional heating per transit time ( 0 0 4 5 6 7 10 10 10 10 Average Rayleigh number (Ra ) avg

Figure 5.9: 2D model results: non-dimensional heat input into the mantle per transit time (γ) (red) and non-dimensional internal heating rate (H) (blue) as a function of average Rayleigh number for insulating core-mantle boundary models. The non-dimensional heating per transit time is calculated through the multiplication of the non-dimensional internal heating rate (H) and the diﬀusion times per transit time. All parameters for the insulating core-mantle boundary models are given in Table 5.2. Chapter 5. Rayleigh number and continental insulation 99 have almost the same sub-oceanic and sub-continental temperatures.

Figure 5.8 shows the sub-oceanic, sub-continental and average mantle temperatures for the four insulating core-mantle boundary models. As is the case with the isothermal core-mantle boundary models in Figure 5.5, sub-oceanic and sub-continental temperatures are very similar at high Rayleigh number, and the mantle beneath the continental plates is comparable to sub-oceanic temperatures on supercontinent cycle timescales.

The increased temperatures in the low Rayleigh number models are explained by the rate at which the mantle is heated. Given the insulating basal boundary condition, the heat input is entirely determined by the non-dimensional internal heating rate (H). The total heat input per unit volume per transit time for each case is tM γ = Hdt, (5.3) Z0 where H is unique to the model and tM is the non-dimensional time required to travel the depth of the mantle. The total heat input per unit volume (γ) per transit time decreases as the vigour of convection increases, despite the non-dimensional internal heating rate (H) increasing with Rayleigh number (Figure 5.9). In the insulating core-mantle boundary cases, the internal heating rate and Ra0 are changed in order to obtain mantle temperatures and plate mobility values that are comparable to the isothermal core-mantle boundary models. Nevertheless, the amount of heat being generated per transit time by internal heating is found to be five times greater in the low Rayleigh number model (ins1) in comparison to the high Rayleigh number case (ins4) (Figure 5.9). The effect of continental insulation at low Rayleigh numbers is therefore amplified by the excess heat generated per transit time in comparison to high Rayleigh number models.

5.3.4 2D temperature increase due to insulation

It is diﬃcult to interpret how continental insulation and Rayleigh number interact in the supercontinent cycle by only looking at the temperature increase beneath the supercontinent. For the suite of 2D models, Figure 5.10 shows the temperature increase solely due to continental insulation as a function of average Rayleigh number. This is achieved by subtracting the mean sub-continental temperature found in a model with a non-insulating continental plate (T¯nonins) from the mean sub-continental temperature in the case with an insulating continent (T¯ins). The temperatures are evaluated 250Myr after supercontinent formation (in keeping with supercontinent time-scales [Scotese, 2001; Yoshida and Santosh, 2011]): T¯ins(250Myr) T¯nonins(250Myr) ∆T¯ins(250Myr)= − (5.4) T¯nonins(250Myr) Chapter 5. Rayleigh number and continental insulation 100

18 Insulating CMB

16 Isothermal CMB

6 Temperature increase (%) 4

0 4 5 6 7 8 10 10 10 10 10 Average Rayleigh number

Figure 5.10: 2D modelling results: temperature increase solely due to insulation for both basal boundary conditions. The temperature increase for each simulation is evaluated 250Myr after supercontinent formation. Increasing the convective vigour of the model decreases the impact of continental insulation.

The overall trend is that increasing the Rayleigh number of a model decreases the eﬀect of continental insulation on sub-supercontinent temperatures.

5.3.5 Average mantle temperatures and continental insulation

The prevalent view in supercontinent modelling literature is that continental insulation drives up the temperature of the sub-continental mantle following continental aggregation. The assumption is that continental insulation has its greatest impact once a large, stationary supercontinent has been formed, and not when continents are dispersed [e.g. Cooper et al., 2004, 2006; Lee et al., 2005; Lenardic et al., 2005]. Forming a supercontinent, therefore, generates a warming of the overall mantle temperature by changing the eﬃciency of continental insulation. Figures 5.5 and 5.8 show the volume-average temperature of the mantle steadily increasing for all supercontinent simulations (with low Rayleigh number models showing a greater increase in temperature than high Rayleigh number models). However, a recent study by Lenardic et al. [2011] argues that the mantle has an overall temperature that is steady regardless of the conﬁguration of the oceanic and continental plates. Lenardic et al. [2011] propose that continental plates insulate throughout the thermal evolution of the mantle, but it is only during supercontinent episodes that a dichotomy between sub-oceanic and sub-continental mantle temperature can form.

The theory states that when continents are dispersed the eﬀect of the continental insulation is global (as the mantle is thermally well mixed). Once a supercontinent is formed, however, circum-continent subduction inhibits lateral mixing in the mantle, and the eﬀect of continental insulation is manifested Chapter 5. Rayleigh number and continental insulation 101

cmb2−cont cmb4−cont

0.65 0.65

0.6 0.6

0.55 0.55

Temperature 0.5 Temperature 0.5

0.45 0.45 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr) ins2−cont ins4−cont

0.65 0.65

0.6 0.6

0.55 0.55

Temperature 0.5 Temperature 0.5

0.45 0.45 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr)

Figure 5.11: Time-series of volume-average temperatures beneath the oceans (blue line) and continents (red line) for all isothermal core-mantle boundary models. Dashed black lines show the volume average temperature for the whole model. The parameters used here are given in Table 5.1 and 5.2.

locally. As a result, sub-continental mantle temperatures would increase (and sub-oceanic mantle temperatures would decrease), but the volume-average temperature would stay the same. To demonstrate their theory, Lenardic et al. [2011] manually prescribe a ‘subduction curtain’ to partially isolate the sub-continental mantle and inhibit thermal mixing post-supercontinent formation.

The thermal mixing theory of Lenardic et al. [2011] is addressed in this section by modelling continental insulation in our calculations prior to the formation of a supercontinent. A study of the thermal evolution of the mantle post-supercontinent formation is conducted with an initial condition that has an elevated, but steady, mantle temperature. The ins2, cmb2, ins4 and cmb4 models were used in order to analyze the eﬀect of insulation prior to supercontinent formation on sub-oceanic and sub-supercontinental mantle temperatures. Figure 5.11 shows the sub-oceanic, sub-continental and average mantle temperatures for the models featuring well mixed initial conditions with temperatures determined by a history of continental insulation (in accord with the study by Lenardic et al. [2011]). For all models, the average mantle temperature does not appear to change signiﬁcantly, while the isolated sub-continental material warms as the sub-oceanic material cools. However, over supercontinent assembly timescales sub-continental material does not become warmer than the sub-oceanic mantle.

Figure 5.12 shows the change in overall mantle temperature as a percentage of the initial condition temperature for both the initially elevated mantle temperature models (denoted with the suﬃx cont) and Chapter 5. Rayleigh number and continental insulation 102

Volume−average Temperature Volume−average Temperature 10 15 cmb2 (a) Ins2 (b) cmb2−cont Ins2−cont cmb4 Ins4 cmb4−cont Ins4−cont 10

0 Deviation from initial condition T (%) Deviation from initial condition T (%)

−5 −5 0 100 200 300 400 500 0 100 200 300 400 500 Time (Myr) Time (Myr)

Figure 5.12: Deviation of the average mantle temperature for models with and without elevated temperatures due to modelling continental insulation prior to supercontinent formation. Models with a history of continental insulation are denoted with the suﬃx cont. Chapter 5. Rayleigh number and continental insulation 103

a1 2 1 b 1 2 1 7 3 3 3 5 3 5

4 4 4 4 6 6 1 1 2 1 2 1

Figure 5.13: Plate geometry for imminent (a) and post- (b) supercontinent formation. The pre- supercontinent geometry features nine plates: six oceanic (various shades of blue) and three speciﬁed as continental (coloured in green and brown). Following the formation of the supercontinent (covering ∼30% of the model surface), a new oceanic plate (plate 7) is formed from a residual portion of the aggregated continental plates.

the non-elevated initial mantle temperature models. Lenardic et al. [2011] allow temperature deviations of 2-3% from the average mantle potential temperature in cases where the system is considered steady. Adopting this allowance, the simulations that include elevated mantle temperatures can be considered thermally steady (for both high and low Rayleigh number models, with either insulating or isothermal basal boundaries), in agreement with the ﬁndings of Lenardic et al. [2011]. In contrast, the models that do not take into account continental insulation prior to supercontinent formation show a mantle temperature increase (that is much greater for lower Rayleigh number cases).

5.4 3D Supercontinent Models

Given the ﬁndings of the 2D study, only a small number of 3D calculations are considered (focussing on the end-member models featuring low Rayleigh number convection with an insulating core-mantle boundary and high Rayleigh number convection with an isothermal core-mantle boundary). A 6 6 1 × × solution domain is modelled, corresponding to a lateral extent equal to the surface area of the Earth’s mid-mantle. The grid resolution is 601 601 129 for the high Rayleigh number model and 301 301 129 × × × × for the low Rayleigh number model. Sidewalls in the 3D models are periodic and plate velocities are dynamically calculated (plate boundaries do not evolve). Chapter 5. Rayleigh number and continental insulation 104

ins10Myr 125Myr 250Myr

Horizontal mantle flow

Temperature 0.0 0.25 0.50.75 1.0

Figure 5.14: 3D modelling results: low Rayleigh number (ins1). The initial condition (0Myr), and temperature field snapshots 125Myr, and 250Myr after supercontinent formation are shown. Top panels show the non-dimensional temperature isosurfaces of 0.1, 0.2, 0.3 and 0.4 (subducted oceanic lithosphere), with the top 16% of the model depth removed to allow for better viewing of the interior. Also shown are temperature field slices on the back vertical faces and the base of the model solution domain. The stencil of the plate geometries is shown on the top of the box (yellow and green lines indicate the sutured continental margins and the new plate boundary, respectively). Bottom panels show a horizontal slice of the thermal fields for ins1 at a depth of 660km. The temperature isosurfaces of the subduction zones (non-dimensional temperature 0.1 to 0.3) are shown for the upper mantle region (300 to 660km below the surface). For all panels, black contours at non-dimensional temperatures of 0, 0.25, 0.5, 0.75 and 1 are shown. Note significant warming in the upper mantle due to thermal insulation. Black arrows indicate horizontal upper mantle flow. Chapter 5. Rayleigh number and continental insulation 105

5.4.1 3D Setup

The initial condition for the supercontinent formation modelling in 3D follows similar criteria to the 2D study, insofar as a subduction zone brings continental material together above a thermally equibrilated system. An initial condition is obtained by projecting a 2D solution into the third dimension, and then specifying a plate geometry featuring nine plates (Figure 5.13a). The heating mode, Rayleigh number and parameters governing the viscous properties of the model are unchanged. The 3D model is integrated forward in time for several mantle overturns until the system reaches a statistically steady state with the new plate geometry (Figure 5.13a). Eventually, a suitable ﬂow pattern mimicking the formation of Pangea (with Y-shaped subduction [Santosh et al., 2009] occurring at the site of convergence of three large plates (see Figure 5.13a)) is formed as the model naturally evolves. The model supercontinent is assembled through suturing the three plates that converge at the Y-shaped model subduction zone, forming a single continent of comparable relative area to Pangea (covering 30% of the system surface). At ∼ the time of continental aggregation, a new oceanic plate is also created from the remaining portion of the three plates joined to form the supercontinent (plate 7, Figure 5.13b). The supercontinent is prescribed a

1 velocity of zero and given an insulation parameter (i) of /4 (as used in the 2D study). The two thermal boundary condition cases for the 3D study, ins1 and cmb4, have the same parameter properties as given in Tables 5.1 and 5.2. Given the findings of the previous section, the mean temperature of the initial thermal field has been shown to be not as important as Rayleigh number in determining the effectiveness of supercontinent insulation (Figure 5.11). Consequently, due to computational requirements, the study of 3D models focuses only on cases where continental insulation (resulting through specification of a low continental diffusivity) is not enacted until supercontinent formation occurs.

5.4.2 3D Results

Figure 5.14 shows snapshots of temperature isosurfaces from 3D calculations of ins1, and a horizontal temperature slice of the thermal ﬁeld in the upper mantle (660km) for the initial condition, 125Myr and

250Myr after supercontinent formation. The initial ‘Y-shaped’ subduction pattern is no longer present after 125Myr, but the isosurfaces show the formation of circum-supercontinent subduction. In keeping with the 2D results for the low Rayleigh number models, temperature builds-up due to continental insulation and generates a strong horizontal upper mantle flow that entrains passive upwelling flow below the supercontinent. After 250Myr, sub-continental heat has pushed the circum-supercontinent downwellings away from the continental margins. The horizontal temperature slice for ins1 (Figure 5.14) shows significant heating in the upper mantle. Figure 5.15a shows the time-series of the continental and Chapter 5. Rayleigh number and continental insulation 106

3D − ins1 3D − cmb4

0.55 0.55

0.5 0.5

0.45 0.45 Temperature Temperature

0.4 0.4 (a) (b) 0.35 0.35 0 100 200 0 100 200 Time (Myr) Time (Myr)

Figure 5.15: Time-series of volume-average temperatures beneath the oceans (blue line) and continents (red line) for: (a) ins1, insulating core-mantle boundary model and (b) cmb4, isothermal core-mantle boundary model. Dashed black line shows the volume average temperature for the whole model.

oceanic temperatures for ins1. Sub-continental temperatures exceed sub-oceanic temperatures after 150Myr due to continental insulation generating a build up of heat beneath the supercontinent, as shown in the 2D models with low Rayleigh number.

The 3D high Rayleigh number mantle convection model (cmb4) shows more complex features than the low Rayleigh number and 2D simulations. Figure 5.16 shows the dissipation of the Y-shaped downwelling and the positioning of subduction on the margins of the supercontinent (shown clearly by the dark blue features in the horizontal temperature slices of 60Myr and 200Myr). The re-location of the subduction pattern, and the associated cold material on the core-mantle boundary, acts to organize thermal instabilities and produce plumes that penetrate the sub-continental upper mantle within 50-100Myr of supercontinent formation (in keeping with geological estimations of plume formation beneath the supercontinent Pangea [Li et al., 2003; Maruyama et al., 2007]). Any remnants of the large-scale subduction that amassed the continents has dissipated from beneath the supercontinent after 200Myr. At this time a full sub-continental mantle ﬂow reversal can be said to have taken place and several mantle plumes have appeared in the upper mantle. Furthermore, the large-scale net horizontal upper-mantle ﬂow seen in the low Rayleigh number cases is absent for the high Rayleigh number model. The time-series of the cmb4 temperatures in Figure 5.15b shows a small temperature increase below the supercontinent, despite Chapter 5. Rayleigh number and continental insulation 107

Figure 5.16: 3D supercontinent results: high Rayleigh number model (cmb4). The initial condition (0Myr), 60Myr, and 200Myr temperature snapshots are shown with black contours at non-dimensional temperatures of 0, 0.25, 0.5, 0.75 and 1. The top layer shows the non-dimensional temperature isosurfaces of 0.25, 0.3, 0.35 (subducted oceanic lithosphere) and 0.8 (plumes), with the top 6% of the model removed to allow for better viewing of the interior (the temperature slice at the base of the model is 0.3d above the bottom of the box (i.e., not the core-mantle boundary)). The stencil of the plate geometries is shown on the top of the box (yellow and green lines indicate the sutured continental zone and the new plate boundary, respectively). Bottom panels show a horizontal slice of the thermal ﬁelds for cmb4 at a depth of 660km.

the formation of several plumes and the prescribed ‘thermal blanket’ eﬀect (and is thus consistent with the 2D high Rayleigh number results). On supercontinent timescales, the sub-continental temperatures do not exceed those below the oceanic plates.

5.5 Discussion

Some consensus has been reached amongst geodynamicists over the mechanisms involved in the supercontinent cycle. However, the role of continental insulation in this cycle remains unsettled. Results from 2D and 3D modeling presented here support the conclusion that continental insulation is not significant when a supercontinent is formed. In agreement with previous geodynamic modeling studies featuring insulating continents, this work finds that following supercontinent formation heat builds-up under a stationary supercontinent. However, the study shows that when heat is trapped sub-continentally it is not to an extent that would cause the sub-continental mantle to be substantially warmer than sub- Chapter 5. Rayleigh number and continental insulation 108 oceanic mantle temperatures, in agreement with the findings of Heron and Lowman [2010, 2011] and Yoshida [2013]. Moreover, the change in sub-continental temperature is strongly dependent on Rayleigh number. In models featuring heating from an isothermal base, the sub-continental warming occurs because circum-supercontinent subduction (and the remnants of the oceanic material subducted during the formation of the supercontinent) organizes thermal anomalies on the core-mantle boundary to produce upwellings [Lowman and Jarvis, 1999; Zhong et al., 2007]. These upwellings may subsequently act as drivers for supercontinent dispersal [Gurnis, 1988; Zhong and Gurnis, 1993; Trubitsyn and Rykov, 1995].

5.5.1 Mantle potential temperature

A recent study by Lenardic et al. [2011] presented the hypothesis that the mantle has an overall temperature that is steady regardless of the configuration of the oceanic and continental plates, and that supercontinental episodes coincide with inhibited thermal mixing between the sub-oceanic and sub- continental mantle regions. Lenardic et al. [2011] describe the mantle as having an average temperature that remains steady, with continental insulation playing an active role in the thermal evolution of the mantle regardless of whether the continents are in motion or assembled as a stationary supercontinent. The Lenardic et al. [2011] study examined the prevalent notion that continental insulation is quiescent until plate motion ceases and continental material aggregates. In a series of 2D models presented here, this thermal mixing theory was addressed by modelling insulating continents prior to the formation of a supercontinent. The calculations described in section 5.3.5 reproduce some of the findings of Lenardic et al. [2011], indicating that if continental insulation has a strong effect prior to supercontinent formation, then the oceanic and continental mantle would cool and warm (respectively) after continental aggregation (with the overall mantle temperature staying the same). However, in this study sub-continental temperatures still do not exceed sub-oceanic temperatures on the timescale of a supercontinent episode, despite taking into consideration an increased background temperature due to continental insulation.

5.5.2 Mantle heating mode

Previous studies have also looked at the role of heating mode in mantle convection models [e.g. Lowman and Jarvis, 1999; Phillips and Bunge, 2005, 2007]. The use of an insulating core-mantle boundary condition, and therefore no core heat flux, was found to generate a periodic supercontinent cycle in a spherical study featuring mobile continents [Phillips and Bunge, 2007]. However, adding core heating to the thermal budget generates strong mantle plumes that disrupt the periodicity of supercontinent formation [Phillips and Bunge, 2007]. In this study, it is shown that plumes may play an important role Chapter 5. Rayleigh number and continental insulation 109 in the supercontinent cycle but heating mode does not affect the importance of continental insulation. It is the mantle Rayleigh number that determines the strength of the ‘thermal blanket’ effect for models featuring either an isothermal or insulating core-mantle boundary (Figure 5.10).

The 2D and 3D studies both show a fundamentally diﬀerent thermal evolution for low and high

Rayleigh number mantle convection models. The low Rayleigh number models are consistent with the findings of several recent studies [e.g. Yoshida and Santosh, 2011; Rolf et al., 2012] which found that the thermal blanket effect warms the sub-supercontinent mantle to produce a large-scale horizontal upper mantle flow. Rolf et al. [2012] present sophisticated 3D spherical numerical simulations with oceanic plates generated by a temperature-and stress-dependent viscosity alongside compositionally and rheologically distinct mobile continents. The study showed strong time-dependence in sub-oceanic temperatures (indicating the importance of modelling oceanic plates) and that continental insulation played a major role in heating up underlying mantle during supercontinent assembly. However, the findings here indicate that the ‘thermal blanket’ effect seen in low Rayleigh number studies is amplified.

In a model featuring high Rayleigh number mantle convection but no oceanic plates, Yoshida and Santosh [2011] found that as warming sub-supercontinent occurs, the thick warm upper mantle region produces a large-scale flow that deflects mantle downwellings from the margins of the supercontinent, before penetrating down to the lower mantle. At the base of the mantle, the downwellings gather thermal instabilities to produce a return flow upwelling beneath the supercontinent. The strong horizontal upper mantle flow is also captured in the low Rayleigh number 2D and 3D results presented here (for both core- mantle boundary conditions), in accord with the supercontinent cycle mechanism described by Yoshida and Santosh [2011]. However, the continental insulation driven flow [e.g. Yoshida and Santosh, 2011; Rolf et al., 2012] is absent in convection models approaching Earth-like Rayleigh number with thermally and mechanically distinct oceanic and continental plates. Section 5.3.2 analyzes the differences in the heat flow of low and high Rayleigh number models with isothermal core-mantle boundary. During the plume generation period (between 100-150Myr), the broad downwellings of low Rayleigh simulations pinch the thermal boundary layer along much of the base of the mantle (Figure 5.3a) to generate an increase in sub-supercontinent basal heat flux (Figure 5.6a). Despite sub-continental heat being transported into the oceanic regions (Figure 5.6d), a sharp increase in sub-continental mantle temperature is exhibited, delivering more heat to the upper mantle to be trapped by the thermal blanket effect. At high Rayleigh number, the thinner downwellings do not generate a great intake in sub-supercontinent basal heat flux relative to the volume of the sub-continental mantle (Figure 5.6a). For insulating core-mantle boundary models featuring only internal heating, more heat is delivered to the mantle per transit time at low Rayleigh number than with a vigorously convecting mantle (Figure 5.9). Due to the change in the Chapter 5. Rayleigh number and continental insulation 110 advective and internal heating timescales, the thermal blanket effect is amplified.

5.5.3 Limitations

A limitation of the study is that the oceanic slabs do not remain more viscous than the ambient mantle once subducted. However, the models do capture the thermal influence of the onset of subduction at the continental margins, the separation of the sub-continental mantle and sub-oceanic regions by a subduction curtain [Lenardic et al., 2011] and the impact of the downwellings once they reach the relatively low viscosity thermal boundary layer at the base of the mantle. Consequently, it is not expected that the thermal evolution of these models is strongly influenced by the neglect of lateral viscosity variation. The most likely effect of including a completely temperature-dependent rheology would be on the timing of the events in the evolution of the model. For example, the slab sinking time and plume rise time will be affected when slab and plume viscosities differ from the surrounding mantle. Accordingly, the addition of a temperature-dependent rheology would be necessary to refine estimates of the timing of the events occurring in the subcontinental flow reversal sequence.

The use of temperature-dependent parameters in Cartesian geometry models requires an adjustment of heating parameters in order to obtain spherical shell-like geotherms [e.g. O’Farrell and Lowman, 2010; O’Farrell et al., 2013]. The agreement of the low Rayleigh number geotherms found in this study with those found in low Rayleigh number spherical models shows that the heating modes chosen emulate the temperatures found in a spherical shell geometry (Figure 5.2).

Although the evolution of plate boundaries has not been modelled, previous studies [e.g. Stein and Lowman, 2010] have shown that mantle temperatures rise with evolving plate boundaries as surface heat ﬂow becomes less eﬃcient when the deep mantle and the plate boundary positions become uncorrelated [Lowman et al., 2011]. If evolving plate boundaries were implemented here, warmer sub-oceanic mantle would likely result. Therefore, a reduced contrast in sub-oceanic and sub-continental temperature in high

Rayleigh number models would be promoted. Thus we do not expect evolving oceanic plate boundaries to aﬀect the conclusions regarding the minor importance of continental insulation during periods featuring assembled supercontinents. Nevertheless, realistic evolving plate boundaries will be implemented in the following chapter.

5.6 Conclusion

The inﬂuence of continental insulation on mantle temperatures and dynamics was examined using numerical models featuring a dynamically determined plate thickness and geotherm-dependent viscosity. Chapter 5. Rayleigh number and continental insulation 111

Through incorporating a region of reduced thermal diffusivity within a stationary continental plate (as compared to oceanic material), continental insulation is prescribed and a ‘thermal blanket’ effect is attained. In a series of models simulating supercontinent formation, the influence of continental insulation is seen to decrease as the vigour of convection is increased. Furthermore, in models approaching Earth- like Rayleigh number, it is difficult to obtain sub-continental temperatures in excess of sub-oceanic temperatures on timescales relevant to supercontinent episodes [e.g. Heron and Lowman, 2010, 2011; Yoshida, 2013], despite the ‘thermal blanket’ effect and the formation of plumes beneath the continent.

A recent study indicates that a reversal of mantle motion through the generation of plumes would be sufficient to disperse a supercontinent, despite sub-oceanic and sub-continental temperatures being comparable [Yoshida, 2013]. The work presented here is consistent with those findings. The formation of sub-supercontinental plumes are found to occur on timescales relevant to Pangea’s assembly, with the generation of subduction zones on the edges of the supercontinent shown to drive the genesis of the sub-continental mantle upwellings, rather than increased temperatures due to continental insulation. Using the results of Chapter 4 and Chapter 5, the next study applies geotherm- and depth-dependent viscosity in high Rayleigh number mantle convection calculations to analyze the effect of supercontinent formation and mantle viscosity on plume position (comparing the findings to the geological rock record). Chapter 6

Inﬂuences on the positioning of mantle plumes following supercontinent formation

6.1 Introduction

Several processes unfold during supercontinent formation, more than one of which might result in an elevation in subcontinental mantle temperatures through the generation of mantle plumes. Paleographic plate reconstructions and geochemical analysis of the rock record indicates that the formation of a supercontinent produces subduction on its margins (Figure 1.4), and correlates with an increase in large igneous provinces on a global scale (Figure 1.2) [e.g., Scotese, 2001; Yale and Carpenter, 1998; Ernst et al., 2005; Ernst and Bleeker, 2010]. Moreover, several numerical simulations of supercontinent formation and dispersal show that the genesis of subcontinental plumes follows the formation of subduction zones on the edges of the supercontinent [e.g., Zhong et al., 2007; Zhang et al., 2010; Steinberger and Torsvik, 2012]. Indeed, the results of Chapters 3 and 5 indicate continental insulation has a minimal role in the development of thermal anomalies sub-supercontinent. Here, the inﬂuence of the location of mantle downwellings on the position of subcontinental plumes is examined in detail.

Geodynamic modelling of the supercontinent cycle is often described in terms of a degree-1 thermal structure prior to continental assembly and degree-2 thermal structure post-supercontinent formation [e.g., Phillips and Bunge, 2005; Zhong et al., 2007; Zhang et al., 2010; Olsen et al., 2010], as shown

112 Chapter 6. Influences on the positioning of mantle plumes 113 in Figure 1.1 and Figure 1.6. However, the paleolocation of large igneous provinces over the past 300Myr (i.e., post-supercontinent formation, Figure 6.1) indicates a more complex thermal structure. The location of paleosubduction in Figure 6.1 shows the temporal evolution of Pangea’s margins, and the movement of continental material during the breakup of the supercontinent. Figure 6.1 shows that sub-continental plumes (inferred from the locations of large igneous provinces) did not develop under the centre of the supercontinent Pangea, or under the centre of the super-oceanic Paciﬁc plate, as isochemical geodynamic models would infer [e.g., Schubert et al., 2004; McNamara and Zhong, 2005; Zhong et al.,

2007; Santosh et al., 2009; Zhang et al., 2010; Heron and Lowman, 2010; Yoshida and Santosh, 2011; Rolf et al., 2012].

The paleoposition of large igneous provinces has previously been attributed to plume generation zones (PGZs) at the edges of large low shear velocity provinces (LLSVPs) situated beneath the present day locations of Africa and the Paciﬁc [Torsvik et al., 2006; Burke et al., 2008; Torsvik et al., 2008, 2010; Steinberger and Torsvik, 2012]. This hypothesis requires LLSVPs to be relatively stationary over 500Myr periods. However, thermo-chemical mantle convection models are often unable to maintain a ﬁxed location of LLSVPs over supercontinent timescales [e.g., Zhang et al., 2010; Tan et al., 2011; Li and McNamara, 2013]. Therefore, a secondary role of LLSVPs in the generation of sub-continental plumes has been suggested [Davies et al., 2012; Davies and Goes, 2014]. Here, the role of supercontinent margin subduction in the evolution of post-supercontinent mantle dynamics is examined.

The evolution of mantle dynamics after supercontinent accretion at a subduction zone is examined considering a range of continental coverage. 2D and 3D Cartesian geometry mantle convection simulations are presented featuring geotherm- and depth- dependent viscosity proﬁles (section 2.4.2) with thermally and mechanically distinct oceanic and continental plates (section 2.8 and section 2.9). Through changing the size of the supercontinent, geometrical factors involved in the generation of mantle plumes are analyzed considering purely thermal convection. The upper and lower mantle viscosity contrast is also changed to determine its relation to plume formation in simulations of vigorous mantle convection. In additional 3D models, the inﬂuence on the thermal evolution of the sub-continental mantle of continental thermal insulation and global subduction zone evolution is also investigated (e.g., section 2.10).

6.2 Method

Table 6.1 shows the viscosity parameters for the models featured in this study. The four models presented here feature changes in the depth-dependent viscosity contrast between the surface and the core-mantle Chapter 6. Influences on the positioning of mantle plumes 114

Figure 6.1: Subduction zones and large igneous province position. The positions of the subduction zones between 300Ma-present [Steinberger and Torsvik, 2012] are plotted alongside the positions of twenty-four large igneous provinces (LIPs) between 300-0Ma. The LIP positions have been corrected to their original paleo-position at the time of their deposition. The colour of the subduction zone and LIP correspond to the age at which it was present or ﬁrst formed, respectively. The outline of the present-day (0Ma) position of the continents is also shown as a guide. The subduction zone and LIP positions were determined by Steinberger and Torsvik [2012] and Torsvik et al. [2006], respectively. I am grateful to Bernhard Steinberger for providing the paleo-subduction data. Chapter 6. Influences on the positioning of mantle plumes 115

Model ηD ∆ηT q¯surf Ra0 TL vrms vp M D100 T5 100 105 23 2.5x105 0.35 2000 4600 2.3 0.54 D30 T5 30 105 23 7.0x104 0.35 3300 2600 0.8 0.71 D100 T7 100 107 23 8.0x103 0.40 1800 3500 1.9 0.60 D30 T7 30 107 23 1.5x103 0.45 2400 2800 1.2 0.69

Table 6.1: Input parameters and initial condition properties for the 2D models. ηD refers to the depth- dependent viscosity profile (Figure 2.3), and ∆ηT the thermal viscosity contrast. Models are named using the convention D* T* where D* is the increase in depth-dependence viscosity of the system (D*), and the power of the thermal viscosity contrast (e.g., ∆ηT =10*) determines T*.q ¯surf ,vrms, , TL, and M refer to the average surface heat flux, system root-mean-square velocity, volume-averaged mantle temperature, lithospheric cut-off temperature, and mobility ratio, respectively. The mobility ratio, M, is the ratio of surface plate velocity (vP ) to vrms. If M is > 0.5, a mobile tectonic regime exists. Ra0 is the Rayleigh ′ number at the surface, where T is 0 (η =1). Theq ¯surf values are matched (by changing the reference Rayleigh number, Ra0) and plate velocities vary by less than a factor of two for models with different viscosity parameters. The grid resolutions of the models are 2501×201 (D100 T5) or 2501×401 (D30 T5 and *T7).

boundary (ηD(z) varies from 100 (models D100*) to 30 (D30*)), and changes in the thermal viscosity

5 7 contrast, ∆ηT , from 10 (*T5) to 10 (*T5). Specifying this range, the response of the calculations to changes in viscosity is analyzed. Figure 2.3 shows the ηD(z) proﬁles used in the 2D and 3D study.

Chapter 5 showed that the effect of Rayleigh number in supercontinent models is important, specifically the effect of Ra on the ability of the mantle to warm through continental insulation. Consequently, high convective vigour is modelled in each calculation in this study. For example, the mantle parame-

5 ters of Model D100 T5 (Ra0=2.5x10 (Table 6.1)) generate a plate thickness of 0.018d, a time-averaged

7 non-dimensional heat ﬂux of 23, and an average Rayleigh number, Raavg, of 2.0x10 . In order to ensure that the vigour of convection in all models is high, the initial surface heat ﬂux values in each case are matched to Model D100 T5 (by adjusting the available parameters Ra0 and TL), while mobile plate tectonics is maintained (i.e., M>0.5) (Table 6.1). All the models presented in this section feature an isothermal core-mantle boundary, periodic side-walls, time-dependent plate thickness (section 2.6), geotherm- and depth-dependent viscosity (section 2.4.2), no internal heating (section 2.7), and thermally and mechanically distinct oceanic and continental material (section 2.8 and section 2.9).

Figure 6.2 shows the horizontally-averaged temperature (geotherm) and the non-dimensional viscosity proﬁle for the initial condition of the calculations in the 2D study. The thin surface boundary layer shown in the geotherms is indicative of a high vigour of convection (Figure 6.2a). Previous studies have shown

5 that mantle convection models featuring a highly temperature-dependent viscosity (e.g., ∆ηT >10 ) will decrease plate mobility [Christensen, 1984; Solomatov, 1995]. Therefore, the two depth-dependent viscosity proﬁles in this study act to reduce the eﬀect of the thermal viscosity contrast (Figure 6.2b). As a result, model D100 T5 has a much greater plate mobility and a lower interior temperature than Chapter 6. Influences on the positioning of mantle plumes 116

1 1

0.8 0.8

0.6 0.6 Height 0.4 Height 0.4

0.2 0.2

0 0 −6 −4 −2 0 0 0.2 0.4 0.6 0.8 1 10 10 10 10 Temperature Viscosity (a) (b)

Figure 6.2: Initial condition non-dimensional geotherms (a) and viscosity (η) (b) for 2D study (Model D100 T5, red; D30 T5, green; D100 T7, blue; D30 T7, purple). Dashed lines in (b) show the 440km and 660km layers (the mantle transition zone). Model parameters are given in Table 6.1.

D30 T5. In order to obtain mobile plate tectonics for the highly geotherm-dependent viscosities (*T7), the TL parameter is modiﬁed.

6.3 2D results

The eﬀect on mantle dynamics of mantle viscosity and supercontinent size is investigated here through studying the positions of new plumes post-supercontinent formation. Timescales are converted by scaling

60Myr to one mantle transit time.

6.3.1 Initial condition and supercontinent modelling

In the interest of clarity, the initial condition setup (which is similar to sections 3.2.1 and 5.3.1) will be briefly recapped here. The initial conditions for all 2D models are obtained through modelling a dynamic two-plate system. Snapshots of the evolution of these models are shown so that the convergent plate boundary appears on the vertical midplanes of the depicted temperature fields (see Figure 6.3a). Once the two-plate system reaches a statistically steady state (i.e., no long term heating or cooling trends are evident in the solution), the plate geometry is modified. A continental plate (with a prescribed velocity of zero) centred over the initial downwelling and two oceanic plates on either side (e.g., Figure 6.3b-d). The emplacement of the continental plate simulates the collision of two smaller continental plates at the site of the initial mantle downwelling. Insulation by dispersed continents is not considered and continental Chapter 6. Influences on the positioning of mantle plumes 117 insulation is not specified until supercontinent formation occurs. The supercontinent remains stationary (similar to the inferred history of Pangea’s evolution [Scotese, 2001]) for the duration of the model run.

6.3.2 Plume position as a function of subduction location

Figure 6.3 shows examples of calculating plume position as a function of subduction zone (convergent plate boundary) location for a range of D100 T5 models. For a supercontinent width of 0.8d (10% of the surface) (Figure 6.3b), plumes fail to form sub-continent (indicating that plumes only develop sub-continent after a critical supercontinent width has been exceeded [e.g., Lowman and Jarvis, 1993, 1995, 1996; Heron and Lowman, 2011]). Increasing the size of the supercontinent to 1.2d allows thermal instabilities to develop under the supercontinent, with the middle of the plume conduit reaching the upper mantle (e.g., z<0.23d (660km)) a distance 0.55d from the nearest subduction zone (e.g., the edge of the supercontinent) and therefore close to the plane of the continental suture (shown by the black dashed line in Figure 6.3c). Only the initial arrival locations of plumes are recorded (e.g., plumes forming in the ﬁrst 150Myr post-supercontinent formation), and their movement is not tracked. A three-plume system develops under a supercontinent covering 4.0d of the surface (Figure 6.3d), with the primary plume arrival located 1.25d away from the nearest subduction zone.

Figure 6.3 shows that for the model parameters in D100 T5, the size of the continent (and repositioning of subduction zones) has an effect on the number (and location) of the sub-continental plumes in models of supercontinent formation. However, changing the viscosity profile (e.g., ηD and ∆ηT ) of the models also has an effect on plume formation. Figure 6.4 shows the difference in plume locations and average mantle temperature across the range of 2D calculations featuring supercontinents covering 4.0d of the surface (after the supercontinent forms through continental collisions at an initial downwelling

5 (e.g., Figure 6.3a)). In this study, decreasing ηD from 100 to 30 for models featuring ∆ηT =10 , reduces the number of plumes that initially develop sub-continent (from three to two when comparing the results from Figure 6.4a (D100 T5) and Figure 6.4b (D30 T5)). Furthermore, Model D30 T5 features sub-continent upwellings that form closer to the subduction zones on the margins of the supercontinent, in comparison to the locations observed in Model D100 T5. Models D100 T7 (Figure 6.4c) and

7 D100 T5 (Figure 6.4d) show similar sub-continental plume locations. Therefore, increasing ∆ηT to 10 (Figure 6.4c-d) appears to negate any eﬀect of changing the depth-dependent viscosity (aside from the increase in average mantle temperature when ηD is reduced).

The relation between plume position and supercontinent margin location, as a function of supercontinent coverage, for Models D100 T5, D30 T5, D100 T7, and D30 T7 is summarized in Figure 6.5. Chapter 6. Influences on the positioning of mantle plumes 118

a) 8d d

Supercontinent b) Ocean 0.8d (10%) Ocean Temp 660km 1.0

0.5 c) 1.2d (15%)

A 660km 0.0

0.55 d d) 4.0d (50%) B CD 660km

0.6 d 0.5 d 1.25 d Figure 6.3: 2D temperature snapshots of model D100 T5 showing the initial condition and location of sub-continental plumes with respect to subduction position in supercontinent models. All 2D models feature a convergent plate boundary where continental material converges (analogous to the formation of Pangea [e.g. Santosh et al., 2009]), periodic sidewalls and an aspect ratio (Γ) 8 box. (a) Supercontinent formation initial condition (t = 0Myr) for model D100 T5. (b) Snapshot from model with a supercontinent covering 10% (0.8d) of the surface (t = 375Myr). Sub-continental plumes fail to develop under the supercontinent. (c) Snapshot from model with collided continents covering 15% (1.2d) of the surface (t = 150Myr). An initial sub-continental plume penetrates the upper mantle (660km) at a distance of 0.55d (plume A) from the nearest subduction zone location. (d) Snapshot of supercontinent covering 50% (4.0d) of the surface (t = 140Myr). The initial sub-continental plumes penetrate the upper mantle (660km) at distances of 0.6d (plume B), 1.25d (plume C) and 0.5d (plume D) from the nearest subduction position. Model parameters are given in Table 6.1. Chapter 6. Influences on the positioning of mantle plumes 119

Increasing continental coverage for D100 T5 models (Figure 6.5a) generates primary sub-continental plumes located further away from the edges of the supercontinent (and therefore closer to the prior location of initial subduction). As shown in Figure 6.3, the number of initial plumes generated sub- continent increases with growing continental coverage. A possible explanation as to why more plumes generate under the larger continents that feature the viscosity profile D100 T5, rather than any of the other viscosity profiles (Figure 6.5), is that the lower mantle viscosity retards velocities at the base of the mantle. The slower velocities lead to thermal instabilities on the core-mantle boundary that gather closer to the initial subduction location than in models featuring a diminished lower mantle viscosity. The interaction between the remnant flow field from the initial subduction and the flow from the new subduction combines to generate thermal gradients (to produce plumes). If the viscosity is relatively low at the base of the mantle, then plumes are quickly pushed out from beneath the centre of the supercontinent. However, if the viscosity in the lower mantle is relatively high, then the thermal instabilities are generated closer to the site of initial subduction due to the velocity of the flow not being strong enough to propel the instabilities away. At high Rayleigh number, there is enough space to produce plumes wherever a strong thermal gradient occurs. As a result, if the lower mantle velocities are relatively low, plumes can form both in close proximity to the initial subduction location and the margins of the supercontinent. However, if the lower mantle velocities are high, the plumes will only form close to the continental margins (due to the interaction between the initial subduction and the new subduction location) (Figure 6.5).

Figure 6.5a shows results from both aspect ratio (Γ) 8 (circles) and Γ=10 (triangles) D100 T5 models. As the aspect ratio was increased, a diﬀerent initial condition was produced (however, still in-keeping with the supercontinent formation set-up of Figure 6.3a). The positions of the primary plumes for both aspect ratio models are found to be consistent (Figure 6.5a). This ﬁnding indicates the presence of a sub-continent environment that is independent of the initial condition and solution domain aspect ratio.

For supercontinent coverage 1.6d and 2.4d in Model D30 T5 (Figure 6.5b), a sub-continental ≥ ≤ plume forms directly below the continental suture (if continental coverage is <1.6d, no sub-continental plume forms). However, for continental coverage >2.4d sub-continental plumes, in general, form closer to the supercontinent margins (as continental size increases), opposite to the trend found for D100 T5. The results of Figure 6.5a-b indicate that reducing the lower mantle viscosity could lock continental plumes to new circum-supercontinent subduction zones rather than promoting plume formation below the continental suture. Through increasing the thermal viscosity contrast (∆ηT ) the major diﬀerences between the two depth-dependent viscosity models essentially vanishes (Figure 6.5c-d). For continental coverage 2.8d, both Model D100 T7 (Figure 6.5c) and D30 T7 (Figure 6.5d) feature sub-continental ≥ Chapter 6. Influences on the positioning of mantle plumes 120

a) 4.0d (50%) 660km

b) 4.0d (50%) Temp 1.0 660km

0.5 c) 4.0d (50%) 660km 0.0 d) 4.0d (50%) 660km

Figure 6.4: 2D temperature snapshots for all models with supercontinent covering 50% (4.0d) of the surface. (a) Model D100 T5; (b) Model D30 T5; (c) Model D100 T7; (d) Model D30 T7. Model parameters are given in Table 6.1. Snapshots correspond to the arrival of the ﬁrst set of plumes sub-supercontinent (∼150My after supercontinent formation).

plume locations that ﬂuctuate between 0.8 and 1.0d in separation from the subduction at the edge of the supercontinent. This indicates that the repositioning of subduction zones to the supercontinent margins plays a strong role in the evolution of sub-continental dynamics, and highlights the independence of sub-continentnal plume positioning on continental suture location.

6.4 3D results

The effect on plume position of decreasing the degree of depth dependence of the viscosity profile is now considered in 3D calculations that use the parameters of D100 T5 (section 6.4.2) and D30 T5 (section 6.4.3). The effects of continental insulation (section 6.4.4) and evolving oceanic boundaries Chapter 6. Influences on the positioning of mantle plumes 121

Model D100 T5 Model D30 T5 2 2

1.5 1.5

1 1

0.5 0.5 iii iii 0 i ii Distance from subduction (d) Distance from subduction (d) 0 0 1 2 3 4 5 1 2 3 4 5 Supercontinent width (d) Supercontinent width (d) (a) (b)

Model D100 T7 Model D30 T7 2 2

1.5 1.5

1 1

0.5 0.5 i ii ii

Distance from subduction (d) 0 Distance from subduction (d) 0 1 2 3 4 5 1 2 3 4 5 Supercontinent width (d) Supercontinent width (d) (c) (d)

Figure 6.5: Plume position relative to continental margin location (circles) as a function of supercontinent coverage (with measurements given as in terms of model depth, d). Dashed line indicates the location of the continental suture. The location of the sub-continental plumes that ﬁrst arrive are interconnected by coloured lines. (a) Model D100 T5; (b) Model D30 T5; (c) Model D100 T7; (d) Model D30 T7. Lower-case roman numerals show the number of initial plumes generated sub-continent. Model parameters are given in Table 6.1. Triangles in (a) correspond to D100 T5 calculations in a Γ=10 box. Chapter 6. Influences on the positioning of mantle plumes 122

1 a)1 2 b) 4 5 3 3 B A

6 C 6 7 1 2 1

Figure 6.6: 3D initial condition and plate geometry for supercontinent formation. Continental collision is modelled along a Y-shaped conﬁguration of convergent plate boundaries (a geometry analogous to that existing during the accretion of Pangea [Santosh et al., 2009]). (a) Plate geometry pre-supercontinent formation (where plates A, B and C combine to form 30% of the model surface (note coverages of 15%, 20%, and 25% are also considered, see Figure 6.7a) surrounded by seven oceanic plates); (b) Temperature snapshot of the initial condition (for Model D100 T5 3D) showing the Y-shaped subduction pattern. The non-dimensional temperature isosurfaces of 0.25, 0.3, 0.35 (subducted oceanic lithosphere) and 0.8 (plumes) are shown, with the top 6% of the model removed to allow for better viewing of the interior. The stencil of the supercontinent plate geometry covering 30% of the surface is shown on the top of the box (these rendering parameters remain for all 3D models unless stated).

(section 6.4.5) on the generation of sub-continental thermal instabilities are also analyzed. Unless spec- iﬁed otherwise, a non-dimensional 4.25 4.25 1 solution domain is modelled. The grid resolution is × × 426 426 129 for the high Rayleigh number models. × ×

6.4.1 3D initial condition and supercontinent modelling

Although the setup of the initial condition is similar to sections 3.3.1 and 5.4.1, for the sake of clarity I will describe it again here. The initial condition for the supercontinent formation modelling in 3D follows similar criteria to the 2D study, insofar as plate motion convergence at a subduction zone brings continental material together above a thermally equibrilated system. An initial condition is obtained by projecting a 2D solution into the third dimension, and then specifying a plate geometry featuring nine plates. The Rayleigh number and parameters governing the viscous properties of the model are unchanged (Table 6.1). The 3D model is integrated forward in time (with the new plate geometry), for several mantle overturns, until the system reaches a statistically steady state. As the model naturally evolves, a suitable arrangement of subduction zones forms mimicking the plate geometry at the formation of Pangea, with a Y-shaped plate boundary conﬁguration [Santosh et al., 2009] occurring at the site of convergence of three large plates (A, B, and C, Figure 6.6a). For all 3D models, the model supercontinent Chapter 6. Influences on the positioning of mantle plumes 123

Figure 6.7: 3D Plate geometry (a) and temperature snapshots showing plume formation 60Myr (b) and 200Myr (c) after supercontinent formation (using parameters given for Model D100 T5) when continent coverage is (l-r) 15% (1.6d)2, 20% (1.9d)2, 25% (2.1d)2 and 30% (2.4d)2 of the 4.25dx4.25d surface.

is assembled through suturing the three plates that converge at the Y-shaped arrangement of convergent boundaries (Figure 6.6b). Through changing the area of designated continental plates A, B, and C, the supercontinent size is varied to cover 15% (1.6d)2, 20% (1.9d)2, 25% (2.1d)2 or 30% (2.4d)2 of the surface (Figure 6.7a). The sizes of oceanic plates 5 and 7 are modiﬁed based on the remaining portion of the three plates joined to form the supercontinent (Figure 6.6a and 6.7a). The supercontinent is prescribed

1 a velocity of zero and given an insulation parameter (i) of /4 (when speciﬁed). Chapter 6. Influences on the positioning of mantle plumes 124

6.4.2 3D D100 T5

Figure 6.7 shows the subduction and plume evolution of supercontinent formation models with varying continental coverage for case D100 T5, following from the initial condition shown in Figure 6.6b. Due to the forces acting on the plates and the cessation of subduction at the Y-shaped plate boundary, subduction moves to the edge of the supercontinent after its formation (Figure 6.6b), similar to the circum-Pangea downwellings associated with Permian plate tectonics (Figure 6.1). The primary sub- continental plumes generated for supercontinent covering more than 15% of the surface are generated close to the continental margins (Figure 6.7b). For a supercontinent covering 15% of the surface, the primary plumes form under oceanic material and are not impeded by the newly formed subduction on the continental margins (Figure 6.7b). Secondary sub-continental plumes for all supercontinent models in Figure 6.7c generate close to the region of the initial subduction location, in-keeping with the Model D100 T5 2D results of Figure 6.5a. Comparing the initial temperature snapshot of Figure 6.6b with 200Myr later (Figure 6.7c), the formation of a supercontinent and the repositioning of subduction zones at its margins have little eﬀect on sub-oceanic plumes and oceanic-oceanic subduction dynamics, despite the supercontinent size varying.

6.4.3 3D D30 T5

The four continental coverage scenarios of Figure 6.7a are analyzed for the mantle parameters of Model D30 T5. Supercontinent formation is modelled in the same way as Figure 6.6, with continental material amalgamating over a downwelling but for the case D30 T5. After 144Myr the mantle parameters of D30 T5 generates sub-continental plumes for even the smallest of the continental coverages (15%, Figure 6.8b), in keeping with the ﬁndings of the 2D study (and 3D D100 T5 models). Furthermore, increasing the continental coverage to 30% of the 4.25 4.25 surface (Figure 6.8e) generates a pair of × sub-continental plumes from close proximity to the circum-supercontinent subduction. The eﬀect of reducing the lower mantle viscosity locks in the generation of plumes to the supercontinent subduction pattern, as found in the 2D study (Figure 6.5b). Figure 6.8g-h shows the response of the mantle to amalgamation of a larger continent (covering 30% of a 5.25 5.25 surface) using the same supercontinent × formation setup. Despite the greater volume existing under this continent, sub-continental plumes form in close proximity to the circum-supercontinent subduction and not under the site of initial downwelling location (Figure 6.8g-h). Chapter 6. Influences on the positioning of mantle plumes 125

a) d) g) a)

b) e) h)

c) f) i)

Figure 6.8: Temperature ﬁeld snapshots for supercontinent models with mantle parameters for the case D30 T5 when continental coverage is 15% (1.6d)2 (a)-(c), 30% (2.4d)2 (d)-(f) of a 4.25dx4.25d surface, and 30% (2.9d)2 (g)-(i) of a 5.25dx5.25d surface (using the plate geometries shown in Figure 6.7a). The initial conditions for the models are given by snapshots (a), (d), and (g). The plume and subduction zone evolution is shown at 144Myr (b, e, and h) and 190Myr (c, f, and i) post-supercontinent formation. Chapter 6. Influences on the positioning of mantle plumes 126

6.4.4 Non-insulating supercontinent

The effect of continental insulation on plume position is analyzed in Figure 6.9. Comparing temperature snapshots in supercontinent formation models featuring the mantle parameters of D100 T5 and a stationary continent covering 30% of the surface, Figure 6.9a-b shows minor differences in mantle dynamics for models with and without continental insulation. Through modelling oceanic and continental plates that cover the surface of the solution domain, high heat flux is constrained to the plate boundaries (Figure 6.9c-d). Nevertheless, the effect of continental insulation on surface heat flux is highlighted when comparing a supercontinent prescribed with the thermal diffusivity of the oceanic material (κc=κ, Figure 6.9c) and one with a reduced oceanic thermal diffusivity (κc=0.25κ, Figure 6.9d). Inhibiting continental surface heat flux raises sub-supercontinent temperature and the volume-averaged temperature of the system (Figure 6.10a). The sub-continental build-up of heat does have an effect on circum-supercontinent and oceanic-oceanic subduction in the models, with downwellings being more prominent on the margins of the non-insulating supercontinent (Figure 6.9). However, the location of sub-continental and sub-oceanic mantle plumes are unaffected by continental insulation (Figure 6.9). Figure 6.10 shows a 50% reduction in continental surface heat flux for the model featuring an insulating supercontinent.

6.4.5 Changing oceanic subduction location

In this section a supercontinent formation model with evolving oceanic boundaries is compared to a model with fixed plate boundaries (the method employed to evolve oceanic plate boundaries is described in 2.10), where both models feature the mantle parameters of case D100 T5. Allowing the plate geometry to evolve has a dramatic effect on the shape and number of the oceanic plates, even after only one mantle transit time (Figure 2.7) [e.g., Stein and Lowman, 2010]. Figure 6.11 shows the difference in subduction zone location between the two models 135Myr after supercontinent formation. Despite some differences, the stationarity of the supercontinent results in subduction on the supercontinent margins and a sub- continental plumes in the same location for both models. Furthermore, sub-oceanic plume locations are also similar over the short timescales associated with supercontinent assembly, despite the evolution of the oceanic boundaries (Figure 6.11).

In the case studied, sub-continental temperatures of the insulating supercontinent with evolving oceanic boundaries are marginally higher than sub-continental temperatures with ﬁxed oceanic boundaries on supercontinent timescales (Figure 6.12a). The free evolution of the oceanic boundaries may generate greater subducted material on the supercontinent margin to draw in more sub-continental heat Chapter 6. Influences on the positioning of mantle plumes 127

a) b)

c)0.01 d)

qs 0.005

0.0 Figure 6.9: 3D temperature and surface heat flux snapshots for models featuring a non-insulating and an insulating supercontinent covering 30% (2.4d)2 of the surface using model D100 T5 parameters (in a 4.25dx4.25dxd box). (a) and (c) show the temperature and surface heat flux snapshots, respectively, 140Myr after supercontinent formation for a non-insulating continent. (b) and (d) show the temperature and surface heat flux snapshots, respectively, 140Myr after supercontinent formation for an insulating continent. Chapter 6. Influences on the positioning of mantle plumes 128

0.48

0.475

0.47

0.465

0.46

0.455

0.45

Non−dimensional temperature 0.445

0.44 0 20 40 60 80 100 120 140 160 180 Time (Myr) (a)

Heat flux 10

0 0 20 40 60 80 100 120 140 160 180 Time (Myr) (b)

Figure 6.10: Mean temperature (a) and heat ﬂux (b) time-series for an insulating (solid) and non- insulating (dashed) supercontinent (D100 T5). Temperature key: volume-average temperature (black); volume-average sub-continental temperature (red); volume-averaged sub-oceanic temperature (blue). Heat ﬂux key: continental surface (black); continental basal (red); ocean surface (cyan); ocean basal (blue). Chapter 6. Influences on the positioning of mantle plumes 129 a)1 b) 1 2 4 3 3 5

6 7 6

12 1 c) d) 2 1 1 4 2’ 3 3 5

1 1 2

Figure 6.11: 3D temperature snapshots comparing plume position for models featuring non-evolving (a-b) and evolving (c-d) oceanic plate boundaries 135Myr after a supercontinent formation (for Model D100 T5 with a supercontinent covering 30% of the 4.25d×4.25d surface). Plate geometry (a and c) and temperature snapshots (b and d) are compared for models started from the same initial condition (see Figure 6.6b).

from the core (red lines, Figure 6.12b), which can account for the increase in sub-continental temperature. A decrease in sub-oceanic temperature (Figure 6.12a) when oceanic plate size and shape are allowed to evolve can also be accounted for by a decrease in sub-oceanic basal heat ﬂux (blue lines, Figure 6.12b). However, more models (in larger solution domains) are required in order to qualitatively state the eﬀect of mobile oceanic plate boundaries on mantle temperatures.

6.5 Discussion

This study analyzes the eﬀect of changing both the lower mantle viscosity and the location of supercontinent margin subduction zones on the evolution of sub-supercontinent mantle dynamics. Previous studies have shown isochemical convection models to form deep mantle plumes away from downwelling loca- Chapter 6. Influences on the positioning of mantle plumes 130

0.48

0.475

0.47

0.465

0.46

0.455

0.45

Non−dimensional temperature 0.445

0.44 0 20 40 60 80 100 120 140 160 180 Time (Myr) (a)

Heat flux 10

0 0 20 40 60 80 100 120 140 160 180 Time (Myr) (b)

Figure 6.12: Mean temperature (a) and heat flux (b) time-series for an insulating supercontinent with non-evolving oceanic plates (solid) and an insulating supercontinent with evolving oceanic plates (dashed). Temperature key: volume-average temperature (black); volume-average sub-continental temperature (red); volume-averaged sub-oceanic temperature (blue). Heat flux key: continental surface (black); continental basal (red); ocean surface (cyan); ocean basal (blue). The time-series in this figure correspond to the calculations that produced the snapshots shown in Figure 6.11. Chapter 6. Influences on the positioning of mantle plumes 131 tions [e.g., Schubert et al., 2004; McNamara and Zhong, 2005]. The results here show that isochemical mantle convection models can produce sub-continental plumes close to the subduction on the margins of a supercontinent (in-keeping with the large igneous province observables of Figure 6.1).

6.5.1 Viscosity proﬁle, continental coverage, and plume position

Results from 2D and 3D modelling presented here show that changing the viscosity structure for mantle convection simulations (with similar surface heat ﬂux) can determine the position (and number) of sub-continental plumes penetrating the upper mantle post-supercontinent formation (Figure 6.5, 6.7, and 6.8). Furthermore, the sub-continental plume locations for the four viscosity proﬁles (Figure 6.2b) show varying degrees of dependence on the location of continent margin subduction post-supercontinent formation (Figure 6.5).

For 2D models featuring a high contrast in the geotherm-dependent viscosity (*T7) with continental coverage >1.2d (e.g., >15% coverage), plume positions strongly depart from the pre-supercontinent formation location of subduction (Figure 6.5c-d). For continental coverage >2.8d, primary and secondary plume locations appear to be locked to positions between 0.8d and 1.0d from continental margin subduction in a two plume system and therefore showing no dependence on the initial subduction location (Figure 6.5c-d). 2D and 3D D30 T5 models show similar results for the same continental coverage, with primary plume formation resonating between 0.6d and 0.8d from continent margin subduction in the 2D study (Figure 6.5b).

The viscosity profile featuring the weakest geotherm-dependence (D100 T5) shows the greatest depar- ture from sub-continental primary plumes being locked to continental margin subduction (Figure 6.5a). The high viscosity lower mantle featured in Model D100 T5 is the least affected by the influence of the geotherm, which may affect the mantle dynamics and produce plumes further away from a continental margin subduction zone than in the other models (which feature lower viscosities near the base of the mantle) (Figure 6.13). Cold downwellings in the lower mantle are broad in Model D100 T5 due to the relatively high viscosity in the lower mantle. After the cessation of subduction at the continental suture, the thermal boundary layer becomes stagnant (Figure 6.13a). The arrival of the broad downwelling formed at the edge of the supercontinent pinches the thermal boundary layer to drive heat flow into the base of the mantle. The cold slab produces a thickening of the thermal boundary layer away from the downwelling region (Figure 6.13b) and generates thermal instabilities (and plumes). For models with a lower viscosity in the lower mantle, subducted material would not form features as broad. The impact of material from the continental margin downwelling on the stagnant thermal boundary layer Chapter 6. Influences on the positioning of mantle plumes 132 would still produce thermal instabilities (and eventual plumes) due to generation of enhanced horizontal temperature gradients. However, the thickening of the thermal boundary layer would occur closer to the new subduction location as a result of the smaller wavelength features permitted by the lower viscosity (Figure 6.13c-d). Nevertheless, for all viscosity profiles that feature more than one sub-continental plume and a large supercontinent (e.g., coverage greater than 2.8d), there is no exact correlation with plume location and the location of continental suture at the pre-collision subduction site.

6.5.2 Plume generation zones and subduction

Recently, Steinberger and Torsvik [2012] presented thermo-chemical mantle convection models featuring kinematic plate velocities (inferred from plate reconstruction models for the past 300Myr). They found the margins of thermo-chemical domes (e.g., LLSVPs) correlate strongly with the generation locations of plumes. The role of subduction on plume generation was also addressed. Steinberger and Torsvik [2012] modelled a coupled plate tectonic and mantle dynamic system in which the location of subduction affects where plumes are formed on the margins of thermochemical piles. Although Steinberger and Torsvik [2012] found that downwelling influenced the shape of thermochemical piles, it was noted that subduction history prior to the formation of Pangea would be needed to comment on LLSVPs being relatively fixed on the core-mantle boundary [e.g., Torsvik et al., 2006; Burke et al., 2008; Torsvik et al., 2008, 2010]. Other geodynamic simulations featuring thermo-chemical piles have also found the location of LLSVPs to be greatly influenced by subduction [e.g., Zhang et al., 2010; Tan et al., 2011; Li and McNamara, 2013], complimenting the widely held notion that downwellings which reach the core-mantle boundary would sweep aside chemical piles [e.g., Tackley, 1998; Kellogg et al., 1999; Jellinek and Manga, 2002; McNamara and Zhong, 2005].

For models featuring both kinematic plate velocities and thermo-chemical piles, it is diﬃcult to determine which is more dominant: LLSVPs in the generation of plumes or subduction in the lateral movement of LLSVPs. However, the study presented here establishes a link between subduction location and the generation of plumes in supercontinent models with dynamic plate velocities that do not feature thermo-chemical piles. If LLSVPs are believed to only play a secondary role in mantle dynamics [Davies et al., 2012; Davies and Goes, 2014], then the past (and future) locations of large igneous provinces can be, in part, explained by subduction zones and mantle viscosity. Furthermore, Austermann et al. [2014] recently conducted a statistical analysis of the correlation between large igneous provinces and lower mantle seismic structure. Through using Monte-Carlo based statistical tests, Austermann et al. [2014] tested whether plumes are preferentially generated at the margins of LLSVPs. Their results show that Chapter 6. Influences on the positioning of mantle plumes 133

(a) Cessation of initial subduction and stagnation of thermal boundary layer in a high viscosity lower mantle model

Thermal boundary layer

(b) Continental margin subduction producing plume from the boundary layer in a high viscosity lower mantle model

Thermal boundary layer

Thermal boundary layer (d) Continental margin subduction producing plume from the boundary layer in a low viscosity lower mantle model

Thermal boundary layer

Figure 6.13: Thermal boundary layer analysis for high (a-b) and low (c-d) lower mantle viscosities. High lower mantle viscosity broadens the features of mantle dynamics. Broad downwellings arriving on the core- mantle boundary both pinch and thicken the thermal boundary layer in the immediate distance and further away from the slab, respectively (b). The generation of thermal instabilities produces plumes from the thermal boundary layer. The same process occurs for a low lower mantle viscosity, however the mantle features are not as broad. Therefore, the wavelength of the thickening occurs at a shorter distance from the new slab. The high viscosity lower mantle has slower mantle velocities than the lower viscosity lower mantle, as indicated by the arrows on the downwelling slabs. Chapter 6. Influences on the positioning of mantle plumes 134

LIP Symbol Age Lat Long LIP distance from subduction CAMP CP 200Ma 2.5 341.9 2400km (0.83d) KarooRidge KR 200Ma -44.6 2.8 ∼3300km (1.1d) BunburyBasalts BU 132Ma -55.3 81.6 ∼3100km (1.1d) ∼

Table 6.2: Approximation of LIP position to continental margin subduction. The three LIPs analyzed are the primary plumes in a sub-continental region post-supercontinent formation (and after 250Ma). Latitude (lat) and longitude (long) values are taken from Torsvik et al. [2008] and give the location of reconstructed plume centres. LIP distance from nearest subduction is an approximation as outlined in the text.

large igneous provinces correlate with the margins and the interior of LLSVPs, with the two correlations being indistinguishable. Therefore, Austermann et al. [2014] argues the deﬁnition of ‘plume generation zones’ from the LLSVP margins [e.g., Torsvik et al., 2006; Burke et al., 2008; Torsvik et al., 2008, 2010] to be premature.

Attempting to apply the results presented here to the LIP positions of Figure 6.1 requires approximations to the interaction between continental margin subduction location and plume generation that is beyond the scope of the study. If thermo-chemical piles on the core-mantle boundary do not play a role in the formation of sub-continental plumes then subduction on its own may be a driving force for gathering lower mantle thermal instabilities. However, it is difficult to quantify which paleo-subduction zone would interact with the mantle flow to produce any given LIP (e.g., Figure 6.1) due to the uncer- tainty of plume rise time and slab sinking rate. This is particularly the case with LIPs such as Skagerrak (297Ma) and the Siberian Traps (251Ma), which formed too early after 300Myr to be influenced by the circum-supercontinent subduction shown in Figure 6.1. The role of continental dispersal (and therefore the evolution of continental margin subduction) in the generation of mantle plumes is not featured in this study. However, a dispersing supercontinent would have an impact on mantle dynamics. As a result, it is difficult to analyze plumes in the numerical models with LIPs that are not the primary thermal anomaly in a region. As a result, only the Central Atlantic Magmatic Province (200Ma), Karoo Ridge (182Ma), and Bunbury Basalts (132Ma) LIPs can be analyzed from Figure 6.1. Table 6.2 gives the approximate values of the nearest continental margin subduction zone relative to the plume centre location (with the subduction locations and plume positions taken from Figure 6.1). All three of these plumes form within 0.8d (2320km) and 1.1d (3190km) of the nearest continental margin. Comparing this first-order analysis with the results of Figure 6.5 shows Models D100 T7 and D30 T7 as the viscosity profiles most relatable to the post-Pangean dynamics (e.g., plumes form at a distance between 0.7d and 1.0d for a supercontinent with a large width). Chapter 6. Influences on the positioning of mantle plumes 135

6.5.3 Sub-supercontinent isolation from subduction

Modelling the supercontinent cycle requires the formation of circum-supercontinent subduction (Fig- ure 6.1) and an insulating continent. However, the influence of the insulating supercontinent is secondary to subduction in the generation of sub-continental plumes (Figure 6.9). As a result, the formation of a large oceanic plate ringed by subduction (e.g., the modern-day Pacific) would also generate sub-plate plumes (e.g., the large igneous provinces and hot spots of the Pacific) [e.g., Heron and Lowman, 2010]. On timescales relevant to sub-continental plumes, sub-continental mantle dynamics is not affected by changes in the locations of the oceanic plate boundaries (Figure 6.11). The isolation from subduction under the supercontinent creates a micro-environment unique to the supercontinent cycle. Therefore, a relevant initial condition that models continental amalgamation is also important for robust conclusions (Figure 1.1). Furthermore, in cases featuring plate boundary evolution, sub-oceanic plume positions do not show much lateral movement despite the repositioning of subduction zones between the evolving and non-evolving models (Figure 6.11), a result previously shown by Lowman et al. [2008].

6.6 Conclusion

This study analyzes the role of subduction and mantle viscosity in the generation of sub-continental mantle plumes in supercontinent formation models. Mantle viscosity aﬀects the plate mobility of a mantle convection model which in turn can change the relationship between supercontinent margin subduction and sub-continental plume position. Models which feature relatively high viscosities in the lower mantle may generate long wavelength mantle dynamics in the sub-supercontinent micro-environment. However, for all mantle viscosity models sub-continental plumes appear to show some dependence on the location of supercontinent margin subduction (once a critical continent size has been reached). The results of the 2D and 3D studies show that the formation of subduction zones at the margins of a supercontinent has a profound eﬀect on mantle dynamics, and may help to explain how the sites of previous (and future) large igneous provinces were (or will be) determined. Chapter 7

Conclusion

After studying continental insulation (Chapter 3 and 5), the generation of sub-continental plumes (Chap- ter 6), and how modelling method affects conclusions regarding mantle convection studies (Chapter 3-6), I propose a refined mechanism for the formation and dispersal of supercontinents. Figure 7.1 depicts the updated supercontinent cycle taking into account the main results from this thesis. Continental insulation is not a significant factor in affecting mantle dynamics (e.g., sub-continental plumes and elevated sub-continental mantle temperatures) (Chapter 3 and 5). However, subduction patterns control the location and timing of mantle upwellings (Chapter 3, 5, and 6). For a mantle with a viscosity that is highly temperature-dependent, sub-continental plumes form near the margins of the supercontinent (in close proximity to the circum-supercontinent subduction) (Chapter 6) (Figure 7.1).

The mantle heating mode can affect conclusions made about the role of continental insulation and the position of sub-continental plumes. Chapter 3 shows how supercontinent formation models that feature thermally and mechanically distinct oceanic and continental plates can reduce the effect of continental insulation on mantle temperatures (a result that differs from previous studies that do not model oceanic plates [e.g., Gurnis, 1988; Zhong and Gurnis, 1993; Trubitsyn and Rykov, 1995; Yoshida et al., 1999; Honda et al., 2000; Phillips and Bunge, 2005, 2007; Zhong et al., 2007; Trubitsyn et al., 2008; Phillips et al., 2009; Coltice et al., 2009; O’Neill et al., 2009; Yoshida, 2010a,b; Phillips and Coltice, 2010; Zhang et al., 2010; Yoshida and Santosh, 2011]). Therefore, the introduction of tectonic plates that cover the whole surface of the solution domain is essential in modelling the supercontinent cycle. Chapter 5 showed how the vigour of convection can determine the effect of continental insulation on mantle temperatures. Although computationally expensive, the use of an Earth-like Rayleigh number is desirable for all future mantle convection models investigating Earth’s evolution. Furthermore, a

136 Chapter 7. Conclusion 137

Figure 7.1: The supercontinent cycle revised in light of the results from this thesis. A supercontinent is amassed through a super-downwelling. Subduction then forms on the margins of the continent, generating sub-continental plumes that are not beneath the site of the super-downwelling. The continental plumes facilitate the dispersal of the supercontinent. Continental insulation does not have a major role in the formation of sub-continental plumes. Chapter 7. Conclusion 138 pseudo-temperature-dependent viscosity has been shown to aﬀect the timing and location of mantle plumes (Chapter 6) as well as mantle temperatures (Chapter 4). The mantle viscosity used here resulted in mantle plumes forming close to subduction locations (a diﬀerent result to previous isochemical mantle convection models [e.g., Schubert et al., 2004; McNamara and Zhong, 2005; Zhong et al., 2007; Santosh et al., 2009; Zhang et al., 2010; Heron and Lowman, 2010; Yoshida and Santosh, 2011; Rolf et al., 2012]) (Chapter 6), showing some correlation with the paleo-positioning of deep mantle plumes (LIPs) found in the geologic record (Figure 6.1).

The use of Cartesian over spherical geometry is a shortcoming of the mantle convection models presented here [e.g., Butler and Jarvis, 2004]. The impact of geometry may have an effect on plume movement at the core-mantle boundary, due to a focussing of subduction on the basal thermal boundary layer from above. Furthermore, mantle temperatures may be significantly affected, which could change the plate tectonic regime (e.g., a spherical geometry would produce a cooler mantle and therefore more mobile plate tectonics than for the same parameters in a Cartesian solution domain). However, this study adheres to the previous work by O’Farrell and Lowman [2010] and O’Farrell et al. [2013], which out- lines the necessary steps required to help emulate spherical geometry temperatures in Cartesian mantle convection models. Although the method of modelling mantle viscosity has been developed throughout the chapters of this thesis, future work would benefit from a fully temperature-dependent viscosity. However, through comparing mantle convection models featuring layered geotherm-dependent viscosity and fully temperature-dependent viscosity, Stein and Hansen [2014] found only minor differences in flow characteristics (including Nusselt number, convection flow regime, lid thickness, stress, and dynamic topography). Nevertheless, the study would benefit from a fully temperature-dependent viscosity (e.g., in comparing downwelling and upwelling transit times).

Future work is necessary to further increase our understanding of the processes involved in the supercontinent cycle. A specialized study into the effect of slabs impacting on the core-mantle thermal boundary layer could lead to a better understanding of the role of circum-supercontinent subduction in the generation of mantle plumes. Tackley [2011] presented work on the dynamics of compositionally- stratified slabs reaching the core-mantle boundary, analyzing the formation of plumes and the separation of basalt and harzburgite from the subducted material. Recently, Quéréet al. [2013] analyzed the effects of mantle viscosity and Rayleigh number on the sinking rate of a slab. A combination of these two geodynamic studies, focussing on slabs impacting the core-mantle boundary while varying the viscosity and Rayleigh number, would help clarify the role of the thermal boundary layer at the core-mantle boundary in the supercontinent cycle. By systematically analyzing the role of viscosity and Rayleigh number, an analysis of the wavelength of thermal anomalies (in the thermal boundary layer) as generated Chapter 7. Conclusion 139 by the introduction of slabs may be possible. The results of an analysis may be compared with the geophysical observables of paleo-subduction and large igneous province location in order to improve geodynamic models.

A theme of this thesis has been to show that numerical simulation of mantle convection requires systematic investigation of the eﬀect of the input parameters in order to assess controls on the results. Complex geodynamic models that utilize every parameter and constraint available may generate geophysical features whose origin may be be diﬃcult to interpret. The increase of computational power and the rise of impressive spherical geometry codes may require a period of numerical introspection, with future work examining ‘meta-geodynamics’. An excellent example of this is the recent study by

Bello et al. [2014] which focussed on the use of plate motion (from paleo-reconstructions) as a kinematic surface boundary layer for mantle convection models. Through analyzing the errors introduced into the convective flow through plate motion and initial thermal conditions, Bello et al. [2014] showed an expo- nential growth in time for thermal evolution uncertainties related to forecasting and hindcasting. Their results found that such kinematic constraints on the models produced unrealistic convective structures over relatively short geodynamic timescales ( 100Myr) [Bello et al., 2014]. ∼ Furthermore, a better understanding of the role of the initial condition is important. The results here use the available geophysical and geological data to obtain an initial condition featuring a possible scenario consistent with supercontinent formation conditions. However, better knowledge of subduction locations during the formation of Pangea would help to recreate sub-supercontinent thermal conditions. A cross-field study using geological fieldwork and geodynamic modelling may help constrain subduction locations from earlier than 300Ma (as the the age of the surface features that pertain to pre-Pangean subduction, and the un-constrained corresponding paleo-longitude, currently make it difficult to analyze). Geological field samples combined with smaller-scale lithospheric geodynamic models [e.g., Gray and Pysklywec, 2013] may help to further constrain a pre-Pangea initial condition. Similarly, the evolution of the Rheic ocean has been highlighted as important in final stages of the formation of Pangea [Nance et al., 2010, 2012]. As a result, future work should be conducted on geodynamic modelling of the Rheic ocean evolution to generate a better understanding of the thermal state of the sub-supercontinent mantle.

Supercontinent dispersal is not modelled in this thesis. However, it would be desirable to determine locations where continental yield stress would be exceeded post-supercontinent formation (and how subsequent continental rifting would occur [e.g., Butler and Jarvis, 2004]). The evolving oceanic boundary model of Chapter 6 (and previous studies by Lowman et al. [2008] and Gli˘sovi´cet al. [2012]) showed mantle plumes to be long-lived structures. The movement of subduction zones during continental dispersal (e.g., Figure 6.1) interacting with mantle plumes would provide an added dimension of convective Chapter 7. Conclusion 140

ﬂow not featured in this thesis. However, the computational time required for evolving plate geometry models is expensive, especially as high Rayleigh number (and therefore high resolution) models are needed for robust conclusions. The results from this thesis show that mantle convection studies require the modelling of both oceanic and continental tectonic plates, a high vigour of convection, and a temperature- (or geotherm-) dependent viscosity, in order to adequately model the supercontinent cycle through mantle convection. Failure to include (at least) these model features distorts the role of mantle plumes, continent insulation, and subduction in the thermal evolution of mantle dynamics. Bibliography

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This section describes the copyright agreement for all material from the thesis that has been previously published.

Figure copyright

The following ﬁgures have a Licence Agreement between Philip J Heron and the publisher (as shown) provided by Copyright Clearance Center for reuse in a thesis. Figure 1.2: Torsvik et al. [2006] (Wiley). Figure 1.3: Kustowski et al. [2008] (Wiley). Figure 1.4: Zhong et al. [2007] (Elsevier). Figure 1.5: Blakely [2013] (R. Blakely). Figure 1.6: Zhong et al. [2007] (Elsevier). Figure 1.7: Torsvik et al. [2006] (Wiley). Figure 2.6: Blakely [2013] (R. Blakely). Figure 2.8: M¨uller et al. [2008] (Wiley). Figure 6.1: Steinberger and Torsvik [2012] (permission granted by B. Steinberger).

Chapter contribution

The work presented in Chapter 2 features material from the method sections of two peer-reviewed publications (Heron and Lowman [2011]: Heron P.J. and J.P. Lowman, 2011, The eﬀects of supercontinent size and thermal insulation on the formation of mantle plumes, Tectonophysics, 510, 28-38; and Heron and Lowman [2014]: Heron P.J. and J.P. Lowman, 2014, The impact of Rayleigh number on assessing the signiﬁcance of supercontinent insulation, Journal of Geophysical Research: Solid Earth, 10.1002/2013JB010484). All permissions required to reprint have been granted from the publishers.

The work presented in Chapter 3 follows closely to my peer-reviewed publication, Heron and Lowman [2011] (Heron P.J. and J.P. Lowman, 2011, The eﬀects of supercontinent size and thermal insulation on the formation of mantle plumes, Tectonophysics, 510, 28-38). After publication of this article, I developed an improved method of modelling continental insulation in comparison to that used in the published study. The ﬁndings from Heron and Lowman [2011] are the same as those presented in Chapter

159 BIBLIOGRAPHY 160

3. The manuscript by Trim et al. [2014] uses a method for modelling mantle convection which features a time-dependent plate thickness. This method is discussed in Chapter 2 and Chapter 4 of this thesis. The development of this method for modelling plate thickness is my contribution to the Trim et al. [2014] manuscript. The work presented in Chapter 5 is from a peer-reviewed publication (Heron and Lowman [2014]: Heron P.J. and J.P. Lowman, 2014, The impact of Rayleigh number on assessing the signiﬁcance of supercontinent insulation, Journal of Geophysical Research: Solid Earth, 10.1002/2013JB010484), where all permission to reprint has been granted from the publisher. My role in the paper was to conduct all of the modelling, and to research and write the manuscript. Together, Prof. Lowman and I discussed the results of the modelling and edited the manuscript into its published form. The work presented in Chapter 6 will be submitted to Journal of Geophysical Research: Solid Earth for publication (Heron P.J., J.P. Lowman and C. Stein, Inﬂuences on the positioning of mantle plumes following supercontinent formation). My role in the collaboration for this paper was to conduct all of the modelling, and to research and write the manuscript. The content has been presented at two conferences: AGU (2013) and CGU-CIG (2014). Furthermore, the work was delivered as an invited talk at the Global Modeling of the Deep Carbon Cycle Workshop (June 2014).