Introduction to Numerical Geodynamic Modelling

Numerical Modelling Of Geodynamic Processes Was Predominantly The Domain Of High-level Mathematicians Experienced In Numerical And Computational Techniques. Now, For The First Time, Students And New Researchers In The Earth Sciences Can Learn The Basic Theory And Applications From A Single, Accessible Reference Text. Assuming Only Minimal Prerequisite Mathematical Training (simple Linear Algebra And Derivatives) The Author Provides A Solid Grounding In Basic Mathematical Theory And Techniques, Including Continuum Mechanics And Partial Differential Equations, Before Introducing Key Numerical And Modelling Methods. 8 Well-documented, State-of-the-art Visco-elasto-plastic, 2-d Models Are Then Presented, Which Allow Robust Modelling Of Key Dynamic Processes Such As Subduction, Lithospheric Extension, Collision, Slab Break-off, Intrusion Emplacement, Mantle Convection And Planetary Core Formation. Incorporating 47 Practical Exercises And 67 Matlab Examples (for Which Codes Are Available Online At Www.cambridge.org/gerya), This Textbook Provides A User-friendly Introduction For Graduate Courses Or Self-study, Encouraging Readers To Experiment With Geodynamic Models--provided By Publisher. Until Now, Numerical Modelling Of Geodynamic Processes Has Been The Domain Of Highly Trained Mathematicians With Long Experience Of Numerical And Computational Techniques. Now, For The First Time, Students And New Researchers In The Earth Sciences Can Learn The Basic Theory And Applications From A Single, Accessible Reference Text. Assuming Only Minimal Prerequisite Mathematical Training (simple Linear Algebra And Derivatives) The Author Provides A Solid Grounding In The Basic Mathematical Theory And Techniques, Including Continuum Mechanics And Partial Differential Equations, Before Introducing Key Numerical And Modelling Methods. Eight Well-documented And State-of-the-art Visco-elasto-plastic, 2d Models Are Then Presented, Which Allowrobustmodelling Of Key Dynamic Processes Such As Subduction, Lithospheric Extension, Collision, Slab Break-off, Intrusion Emplacement, Mantle Convection And Planetary Core Formation. Incorporating 47 Practical Exercises And 67matlabexamples (forwhich Codes Are Available Online At Www.cambridge.org/gerya) This Textbook Provides A Userfriendly Introduction For Graduate Courses Or Self-study, And Encourages Readers To Experiment With Geodynamic Models First Hand--provided By Publisher. Machine Generated Contents Note: Acknowledgements; Introduction; 1. The Continuity Equation; 2. Density And Gravity; 3. Numerical Solutions Of Partial Differential Equations; 4. Stress And Strain; 5. The Momentum Equation; 6. Viscous Rheology Of Rocks; 7. Numerical Solutions Of The Momentum And Continuity Equations; 8. The Advection Equation Marker-in-cell Method; 9. The Heat Conservation Equation; 10. Numerical Solution Of The Heat Conservation Equation; 11. 2-d Thermomechanical Code Structure; 12. Elasticity And Plasticity; 13. 2-d Implementation Of Visco-elastic-plastic Rheology; 14. The Multi-grid Method; 15. Programming Of 3-d Problems; 16. Numerical Benchmarks; 17. Design Of 2-d Numerical Geodynamic Models; Epilogue: Outlook; References; Index. Taras V. Gerya. Includes Bibliographical References (p. 326-339) And Index. Half-title 3 Title 5 Copyright 6 Contents 7 Acknowledgements 12 Introduction 15 What is this book? 15 What this book is not 15 Get started 15 Short history of geodynamics and numerical geodynamic modelling 18 A few words about programming and visualisation 21 Units 22 How to use this book 22 Programming exercises and homework 23 1 The continuity equation 25 1.1 Continuum – what is it? 25 1.2 Continuity equation 27 1.3 Eulerian and Lagrangian points – what is the difference? 28 1.4 Derivation of the Eulerian continuity equation 29 1.5 Derivation of the Lagrangian continuity equation 32 1.6 Comparing Eulerian and Lagrangian continuity equations. Advective transport term 34 1.7 Incompressible continuity equation 37 Analytical exercise 37 Programming exercise and homework 38 2 Density and gravity 39 2.1 Density of rocks and minerals. Equations of state 39 2.2 Gravity and gravitational potential 44 Analytical exercise 48 Programming exercises and homework 49 3 Numerical solutions of partial differential equations 51 3.1 Finite-difference method 51 3.2 Solving linear equations 57 3.3 Geometrical and global indexing of unknowns 61 Programming exercises and homework 62 4 Stress and strain 65 4.1 Stress 65 4.2 Strain and strain rate 70 Analytical exercise 73 Programming exercise and homework 74 5 The momentum equation 75 5.1 Momentum equation 75 5.2 Newtonian law of viscous friction 78 5.3 Navier–Stokes equation 79 5.4 Poisson equation 82 5.5 Stream function approach 83 Analytical exercise 85 Programming exercise and homework 85 6 Viscous rheology of rocks 87 6.1 Rock rheology 87 6.2 Effective viscosity 88 6.3 Non-Newtonian channel flow 93 Programming exercises and homework 94 7 Numerical solutions of the momentum and continuity equations 97 7.1 Grids 97 7.2 Discretisation of the equations 100 7.3 Conservative finite differences 101 7.4 Boundary conditions 106 7.5 Indexing of unknowns 109 Programming exercises and homework 115 8 The advection equation and marker-in-cell method 119 8.1 Advection equation 119 8.2 Eulerian advection methods 120 8.3 Marker-in-cell techniques 127 Programming exercises and homework 133 9 The heat conservation equation 137 9.1 Fouriers law of heat conduction 137 9.2 Heat conservation equation 138 9.3 Heat generation and consumption 141 9.4 Simplified temperature equations 142 9.5 Heat diffusion timescales 143 Analytical exercises 144 Programming exercises and homework 145 10 Numerical solution of the heat conservation equation 147 10.1 Explicit and implicit formulation of the temperature equation 147 10.2 Conservative finite differences 149 10.3 Advection of temperature with Eulerian methods 154 10.4 Advection of temperature with markers 155 10.5 Thermal boundary conditions 158 Programming exercises and homework 160 11 2D thermomechanical code structure 163 11.1 What do we expect from geodynamic codes? 163 11.2 Thermomechanical code structure 164 Step 1: Interpolation of scalar properties from markers to nodes 166 Step 2: Solving the momentum and continuity equations 168 Steps 4–7: Solving the temperature equation 170 11.3 Adding self-gravity and free surface 172 Programming exercise and homework 177 12 Elasticity and plasticity 179 12.1 Why care about elasticity and plasticity? 179 12.2 Elastic rheology 179 12.3 Rotation of elastic stresses 182 12.4 Maxwell visco-elastic rheology 186 12.5 Plastic rheology 187 12.6 Visco-elasto-plastic rheology 189 Analytical exercise 191 Programming exercises and homework 191 13 2D implementation of visco-elasto-plastic rheology 193 13.1 Viscous-like reformulation of visco-elasto-plasticity 193 13.2 Structure of visco-elasto-plastic thermomechanical code 194 Step 1: Defining an optimal computational time step 196 Step 2: Interpolation of scalar fields, vectors and tensor fields 198 Step 3: Solving the momentum and continuity equations 198 Step 5: Interpolation of stress changes from nodes to markers 200 Steps 6 and 7: Solving the temperature equation 202 Step 10: Rotation of stresses 203 13.3 Visco-elasto-plastic iterations 203 Programming exercises and homework 205 14 The multigrid method 207 14.1 Multigrid – what is it? 207 14.2 Solving the Poisson equation with multigrid 214 14.3 Solving Stokes and continuity equations with multigrid 219 Constant viscosity case 221 Adding variable viscosity 226 Programming exercises and homework 231 15 Programming of 3D problems 235 15.1 Why simply not always 3D? 235 15.2 3D staggered grid and discretisation of momentum, continuity, temperature and Poisson equations 236 15.3 Solving discretised 3D equations 245 Programming exercises and homework 253 16 Numerical benchmarks 255 16.1 Code benchmarking: why should we spend time on it? 255 16.2 Test 1. Rayleigh–Taylor instability benchmark 256 16.3 Test 2. Falling block benchmark 258 16.4 Test 3. Channel flow with a non-Newtonian rheology 260 16.5 Test 4. Non-steady temperature distribution in a Newtonian channel 261 16.6 Test 5. Couette flow with viscous heating 264 16.7 Test 6. Advection of sharp temperature fronts 267 16.8 Test 7. Channel flow with variable thermal conductivity 267 16.9 Test 8. Thermal convection with constant and variable viscosity 269 16.10 Test 9. Stress build-up in a visco-elastic Maxwell body 274 16.11 Test 10. Recovery of the original shape of an elastic slab 275 16.12 Test 11. Numerical sandbox benchmark 277 16.13 Possible further benchmarks 281 Programming exercises and homework 281 17 Design of 2D numerical geodynamic models 283 17.1 Warning message! 283 17.2 What is numerical modelling all about? 283 17.3 Material properties 284 17.4 Visco-elasto-plastic slab bending 285 17.5 Retreating oceanic subduction 290 17.6 Lithospheric extension 293 17.7 Continental collision 296 17.8 Slab breakoff 301 17.9 Intrusion emplacement into the crust 305 17.10 Mantle convection with phase changes 310 17.11 Deformation of self-gravitating planetary body 315 Programming exercise and homework 320 Epilogue: outlook 321 Where are we now? 321 Where to go further? 321 State-of-the-art overview 325 Efficient direct solvers 326 Parallelisation of numerical codes 327 Mesh refinement algorithms 327 Including complex realistic physics in numerical geodynamic models 328 3D visualisation challenges 331 Conceptual warning 332 Conclusion 332 Appendix: MATLAB program examples 333 Introduction 333 Chapter 1 333 Chapter 2 333 Chapter 3 333 Chapter 4 333 Chapter 5 334 Chapter 6 334 Chapter 7 334 Chapter 8 334 Chapter 9 335 Chapter 10 335 Chapter 11 335 Chapter 12 335 Chapter 13 336 Chapter 14 336 Chapter 15 337 Chapter 16 337 Chapter 17 339 References 340 Index 354 0521887542,9780521887540 Until now, numerical modelling of geodynamic processes has been the domain of highly trained mathematicians with long experience of numerical and computational techniques. Now, for the first time, students and new researchers in the Earth Sciences can learn the basic theory and applications from a single, accessible reference text. Assuming only minimal prerequisite mathematical training (simple linear algebra and derivatives) the author provides a solid grounding in the basic mathematical theory and techniques, including continuum mechanics and partial differential equations, before introducing key numerical and modelling methods. Eight well-documented and state-of-the-art visco-elasto-plastic, 2D models are then presented, which allow robust modelling of key dynamic processes such as subduction, lithospheric extension, collision, slab break-off, intrusion emplacement, mantle convection and planetary core formation. Incorporating 47 practical exercises and 67 MATLAB examples (for which codes are available online at www.cambridge.org/gerya) this textbook provides a user friendly introduction for graduate courses or self-study, and encourages readers to experiment with geodynamic models first hand Numerical modelling of geodynamic processes was predominantly the domain of high-level mathematicians experienced in numerical and computational techniques. Now, for the first time, students and new researchers in the Earth Sciences can learn the basic theory and applications from a single, accessible reference text. Assuming only minimal prerequisite mathematical training (simple linear algebra and derivatives) the author provides a solid grounding in basic mathematical theory and techniques, including continuum mechanics and partial differential equations, before introducing key numerical and modelling methods. 8 well-documented, state-ofthe-art visco-elasto-plastic, 2-D models are then presented, which allow robust modelling of key dynamic processes such as subduction, lithospheric extension, collision, slab break-off, intrusion emplacement, mantle convection and planetary core formation. Incorporating 47 practical exercises and 67 MATLAB examples (for which codes are available online at (http://www.cambridge.org/gerya) www.cambridge.org/gerya ), this textbook provides a user-friendly introduction for graduate courses or self-study, encouraging readers to experiment with geodynamic models.