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On the Dynamics of Plate Tectonics: Multiple Solutions, the Influence of Water, and Thermal Evolution

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On the Dynamics of Plate Tectonics: Multiple Solutions, the Influence of Water, and Thermal Evolution On the Dynamics of Plate Tectonics: Multiple Solutions, the Influence of Water, and Thermal Evolution The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Crowley, John. 2012. On the Dynamics of Plate Tectonics: Multiple Solutions, the Influence of Water, and Thermal Evolution. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9385631 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA c 2012 John W. Crowley ! All rights reserved. Dissertation Advisor: Professor Richard J. O’Connell John W. Crowley On the Dynamics of Plate Tectonics: Multiple Solutions, the Influence of Water, and Thermal Evolution Abstract An analytic boundary layer model for thermal convection with a finite-strength plate and depth-dependent viscosity is developed. The model includes a specific energy balance for the lithosphere and accounts for coupling between the plate and underlying mantle. Mul- tiple solutions are possible with three solution branches representing three distinct modes of thermal convection. One branch corresponds to the classic boundary layer solution for active lid plate tectonics while two new branches represent solutions for sluggish lid convec- tion. The model is compared to numerical simulations with highly temperature dependent viscosity and is able to predict both the type of convection (active, sluggish, or stagnant lid) as well as the presence of single and multiple solution regimes. The existence of multiple solutions suggests that the mode of planetary convection may be history dependent. The dependence of mantle viscosity on temperature and water concentration is found to introduce a strong dynamic feedback with plate tectonics. A dimensionless parameter is defined to quantitatively evaluate the relative strength of this feedback and demonstrates that water and heat transport may be equally important in controlling present-day plate- mantle dynamics for the Earth. A simple parameterized evolution model illustrates the feedback and agrees well with our analytic results. This suggests that a simple relationship may exist between the rate of change of water concentration and the rate of change of temperature in the mantle. This study concludes by investigating the possibility of a magnetic field dynamo in early solar system planetesimals. The thermal evolution of planetesimals is modeled by considering melting, core formation, and the onset of mantle convection and then employing thermal boundary layer theory for stagnant lid convection (if possible) to determine the iii cooling rate of the body. We assess the presence, strength and duration of a dynamo for a range of planetesimal sizes and other parameters. We find that a minimum radius of O(500) km is required for a thermally driven dynamo of duration O(10) My. The dependence of the results on model parameters is made explicit through the derivation of an analytic solution. iv Contents Abstract iii Contents v List of Figures ................................. ix List of Tables ................................. xi Acknowledgements xii 1 Introduction 1 2 Simple convection scalings 7 2.1 Thermal convection in a fluid layer with mixed heating . 8 2.2 Stagnant lid convection . 20 2.3 Convection in an ‘open’ layer . 24 3 Thermal convection with strong plates 31 3.1 Introduction.................................... 31 3.2 Model ....................................... 37 3.2.1 Mechanicalenergy ............................ 37 3.2.2 Dissipationintheplate . 41 3.2.3 Dissipation in the fault zone . 43 3.2.4 Interior cell flow and lateral dissipation . 43 3.2.5 Cornerflowdissipation. 47 3.2.6 Energybalanceequations . 48 v 3.3 Results....................................... 53 3.3.1 Multiplesolutions ............................ 53 3.3.2 Thermalboundaryconditions . 55 3.3.3 Behavior with a weak plate . 57 3.3.4 Behaviorwithastrongplate . 58 3.3.5 Behavior with a low viscosity layer . 62 3.3.6 Sluggish-lid solutions in numerical simulations . ......... 66 3.3.7 Multiple states and hysteresis in numerical simulations........ 70 3.4 Discussion..................................... 76 3.4.1 Implications for the Earth and planets . 76 3.4.2 Limitations and extensions of the model . 79 3.5 Conclusions .................................... 81 4 On the relative influence of heat and water transport on planetary dynamics 84 4.1 Introduction.................................... 85 4.2 Amodelplanet.................................. 85 4.3 Relative strength of the thermal and water feedbacks . .......... 87 4.4 Relative strength of the feedbacks for present day Earth ........... 89 4.5 Time scales and relative rates of change . 91 4.6 Heat and water transport for an Earth-like planet . 93 4.7 Evolution of an Earth-like planet . 96 4.8 Discussion..................................... 100 4.9 Conclusions .................................... 105 5 Thermal evolution of early solar system planetesimals and the pos- sibility of sustained dynamos 107 5.1 Introduction.................................... 108 5.2 Model ....................................... 115 vi 5.2.1 TheInitialState ............................. 115 5.2.2 Thermalboundarylayermodel . 124 5.2.3 Conditionsforadynamo .. .. .. .. .. .. .. 133 5.3 An analytic solution for dynamo duration . 135 5.4 Results....................................... 140 5.5 Discussion..................................... 146 5.5.1 Conductive versus convective modeling . 146 5.5.2 The role of temperature-dependent viscosity . 148 5.5.3 Futurework................................ 149 5.6 Conclusions .................................... 149 6 Conclusions 152 A The delta function solution 158 A.1 Thedensityfunction .............................. 159 A.2 Thepropagatormethod ............................. 160 A.3 Solution for a layer with free-slip boundaries . .......... 163 A.4 Extended applications of the delta function solution . ........... 168 B Additional calculations for Chapter 3 172 B.1 Calculation of the boundary term for the lithosphere . .......... 172 B.2 Calculating lateral flow and dissipation in the mantle . ........... 174 C Additional calculations for Chapter 4 178 C.1 Model equations and parameters for numerical simulations.......... 178 C.2 A kinematic plate velocity for the thermal-water feedback .......... 179 D Additional calculations for Chapter 5 182 D.1 Onset of stagnant lid convection . 182 D.2 The effect of a low conductivity surface layer . 184 vii D.3 Effect of continuous accretion . 184 D.4 Effective specific heat as latent heat . 190 D.5 Scalinglaws.................................... 191 Bibliography 195 viii List of Figures Figure 2.1 Diagram for thermal convection in a fluid layer. .......... 8 Figure 2.2 Nonlinear relation between Nu, RaT , and RaH for thermal convection withmixedheating. ............................... 15 Figure 2.3 Comparison of mixed heating equation with the numerical simulations and scaling of Sotin and Labrosse (1999). 19 Figure 2.4 Thermal convection with depth dependent viscosity ......... 24 Figure 2.5 Numerical simulation run with a modified version of MC3D that demonstrates the ‘open’ layer convective flow. 25 Figure 2.6 Streamlines for flow in a convecting open layer for various aspect ratios usingthedeltafunctionsolution. 27 Figure 3.1 Geometry and model variables. 37 Figure3.2 Modelsforplatebending. 41 Figure 3.3 Flow structure of the model convection cell. .......... 44 Figure 3.4 Model results for an isoviscous mantle with a finite strength plate. 57 Figure 3.5 Results for an isoviscous mantle with a strong plate. ......... 59 Figure 3.6 Results with an asthenosphere and a weak plate. ......... 63 Figure 3.7 Results with an asthenosphere and a strong plate. .......... 64 Figure 3.8 Comparison of model with finite strength plates to numerical simula- tions......................................... 67 Figure 3.9 Convective mode as a function of plate strength and viscosity contrast in thermal convection simulations with highly temperature dependent viscosity. 72 ix Figure 4.1 Interaction of the thermal and water feedbacks during cooling and regassing. ..................................... 87 Figure 4.2 Characteristic thermal and volatile evolutions. ............ 97 Figure 4.3 Mantle viscosity, plate velocity, and relative strength number for cou- pled thermal-water evolutions. 101 Figure 5.1 Calculation of the maximum dynamo time using a simple energy bal- anceargument. .................................. 112 Figure 5.2 Stages of Planetesimal evolution. ........ 115 Figure 5.3 Schematic representation of mantle convection defining various lengths usedinthethermalevolutionmodel. 124 Figure 5.4 Assumed viscosity profile as a function of temperature and melt fraction.127 Figure 5.5 Thermal evolution of a planetesimal for the reference case and with a radiusof400km.................................. 130 Figure 5.6 Thermal evolution of a planetesimal for the reference case and with a radiusof1000km. ................................ 132 Figure5.7 Analyticcoreheatflux. 137 Figure 5.8 Dynamo duration as a function of planetesimal radius......... 139 Figure 5.9 Dynamo duration as a function of planetesimal radius......... 143 Figure 5.10 Dynamo duration and magnetic field strength using the different mag- neticfieldscalings.
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