About this book:
Have you ever thought that modelling of geological processes is an exciting topic but too difficult to enter because there is no introductory textbook on this subject? Yes? Then come the good news! Here is the textbook written for you to learn numerical geodynamic modelling from scratch. It does not require any preliminary knowledge besides simple linear algebra and derivatives. It provides a consistent basic background in continuum mechanics, partial differential equations, numerical methods and geodynamic modelling. It is illustrated with 47 practical exercises and 67 MATLAB examples as successive and successful stages on your learning path. In addition, several state-of-the-art, well-commented visco-elasto-plastic codes are provided to allow numerical modelling in two-dimensions of several key geodynamic processes such as subduction, lithospheric extension, collision, slab breakoff, intrusion emplacement, mantle convection and planetary core formation. Below are keywords and the book content.
Keywords for this book:
geodynamic models, models of tectonic plates, models of lithospheric processes, divergent plate model, convergent plate model, model of continental collision, subduction zone model, mantle convection model, igneous intrusion model, model of the core of Earth, rocks properties, Earth mantle properties, numerical modeling, applied numerical methods, finite differences, particle in cell, numerical code, numerical example, numerical analysis, numerical integration, numerical matlab codes, numerical tests
INTRODUCTIONTheory: What is this book? What this book is not? Get started. Seven golden rules for learning the subject. Short history of geodynamics and numerical geodynamic modelling. Few words about programming and visualization. Nine programming rules.
Exercises: Starting with MATLAB. Visualization exercise.
CHAPTER 1: THE CONTINUITY EQUATIONTheory: Definition of a geological media as a continuum. Field variables used for the representation of a continuum. Methods for definition of the field variables. Eulerian and Lagrangian points of view. Continuity equation in Eulerian and Lagrangian forms and their derivation. Advective transport term. Continuity equation for an incompressible fluid.
Exercises: Computing the divergence of velocity field in 2D.
CHAPTER 2: DENSITY AND GRAVITYTheory: Density of rocks and minerals. Thermal expansion and compressibility. Dependence of density on pressure and temperature. Equations of state. Poisson equation for gravitational potential and its derivation.
Exercises: Computing and visualising density, thermal expansion and compressibility.
CHAPTER 3: NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONSTheory: Analytical and numerical methods for solving partial differential equations. Using finite-differences to compute various derivatives. Eulerian and Lagrangian approaches. Transition from partial differential equations to systems of linear equations. Methods of solving large systems of linear equations: iterative methods (Jacobi iteration, Gauss-Seidel iteration), direct methods (Gaussian elimination). Indexing of unknowns in 1D and 2D.
Exercises: Numerical solutions of Poisson equation in 1D and 2D.
CHAPTER 4: STRESS AND STRAINTheory: Deformation and stresses. Definition of stress, strain and strain-rate tensors. Deviatoric stresses. Mean stress as a dynamic (non-lithostatic) pressure. Stress and strain rate invariants.
Exercises: Computing the strain rate tensor components in 2D from the material velocity fields.
CHAPTER 5: THE MOMENTUM EQUATIONTheory: Momentum equation. Viscosity and Newtonian law of viscous friction. Navier-Stokes equation for the motion of a viscous fluid. Stokes equation of slow laminar flow of highly viscous incompressible fluid and its application to geodynamics. Simplification of the Stokes equation in case of constant viscosity and its relation to the Poisson equation. Analytical example for channel flow. Stream function approach.
Exercises: Solving continuity and momentum equations for the case of constant viscosity with a stream function approach.
CHAPTER 6: VISCOUS RHEOLOGY OF ROCKSTheory: Solid-state creep of minerals and rocks as the major mechanism of deformation of the Earth’s interior. Dislocation and diffusion creep mechanisms. Rheological equations for minerals and rocks. Effective viscosity and it’s dependence on temperature, pressure, and strain rate. Formulation of the effective viscosity from empirical flow laws.
Exercises: Programming viscous rheological equations for computing effective viscosities from empirical flow laws.
CHAPTER 7: NUMERICAL SOLUTION OF THE MOMENTUM AND CONTINUITY EQUATIONSTheory: Types of numerical grids and their applicability for different differential equations. Staggered, half-staggered and non-staggered grids in one, two and three dimensions. Discretisation of the continuity and Stokes equations on a rectangular grid. Conservative and non-conservative discretisation schemes for Stokes equations. Mechanical boundary conditions and their numerical implementation. No slip and free slip conditions.
Exercises: Programming different mechanical boundary conditions. Solving continuity and momentum equations for the case of variable viscosity.
CHAPTER 8: THE ADVECTION EQUATION AND MARKER-IN-CELL METHODTheory: Advection equation. Solution methods for continuous and discontinuous variables. Eulerian schemes: upwind differences, higher order schemes, flux corrected transport (FCT). Lagrangian schemes: marker-in-cell method. Runge-Kutta advection schemes. Numerical interpolation schemes between markers and nodes.
Exercises: Programming of various advection schemes and markers
CHAPTER 9: HEAT CONSERVATION EQUATIONTheory: Fourier’s law of heat conduction. Heat conservation equation and its derivation. Radioactive, viscous and adiabatic heating and their relative importance. Heat conservation equation for the case of a constant thermal conductivity and its relation to the Poisson equation. Analytical examples: steady and non-steady temperature profiles in case of channel flow.
Exercises: Computing shear heating and adiabatic heating distribution for buoyancy driven flow.
CHAPTER 10: NUMERICAL SOLUTION OF THE HEAT CONSERVATION EQUATIONTheory: Discretisation of the heat conservation equation with finite differences. Conservative and non-conservative discretisation schemes. Explicit and implicit solution schemes of the heat conservation equation. Advective terms: upwind differences, numerical diffusion. Advection of temperature with markers. Subgrid diffusion. Thermal boundary conditions: constant temperature, constant heat flow, combined boundary conditions. Numerical implementation of thermal boundary conditions.
Exercises: Programming various thermal boundary conditions. Solving the heat conservation equation in the case of constant and variable thermal conductivity with explicit and implicit solution schemes. Advecting temperature with Eulerian schemes and markers.
CHAPTER 11: 2D THERMO-MECHANICAL CODE STRUCTURETheory: Principal steps of a coupled thermo-mechanical solution with finite differences and marker-in-cell techniques. Organisation of a thermo-mechanical code for the case of viscous, multi-component flows. Adding self-gravity. Handling free planetary surfaces with weak layer approach.
Exercises: Building a 2D thermo-mechanical code.
CHAPTER 12: ELASTICITY AND PLASTICITYTheory: Elastic rheology. Maxwell visco-elastic rheology. Rotation of stresses during advection. Plastic rheology. Plastic yielding criterion. Plastic flow potential. Plastic flow rule.
Exercises: Stress buildup/relaxation with a visco-elastic Maxwell rheology.
CHAPTER 13: 2-D IMPLEMENTATION OF VISCO‑ELASTO-PLASTICITYTheory: Numerical implementation of visco-elasto-plastic rheology. Organisation of a thermo-mechanical code in case of 2D, visco-elasto-plastic, multi-phase flows.
Exercises: Programming a 2D thermo-mechanical code with a visco-elasto-plastic rheology.
CHAPTER 14: THE MULTIGRID METHODTheory: Principles of multigrid method. Multigrid method for solving the Poisson equation in 2D. Coupled solving of momentum and continuity equations in 2D with multigrid for the cases with constant and variable viscosity.
Exercises: Programming of multigrid methods for solving Poisson equation and coupled solving of momentum and continuity equations in 2D.
CHAPTER 15: PROGRAMMING OF 3-D PROBLEMSTheory: Formulation of thermo-mechanical problems in 3-D and its numerical implementation. Multigrid method for solving temperature, Poisson, momentum and continuity equations in 3-D
Exercises: Programming of multigrid methods for temperature and Poisson equations and coupled solving of momentum and continuity equations in 3D.
CHAPTER 16: NUMERICAL BENCHMARKSTheory: Numerical benchmarks: testing of numerical codes for various problems. Examples of thermo-mechanical benchmarks.
Exercises: Programming of models for various numerical benchmarks.
CHAPTER 17: DESIGN OF 2-D NUMERICAL GEODYNAMIC MODELSTheory: Warning message! What numerical modeling is about? Rock properties for numerical geodynamic models. Designing of numerical models for different geodynamic processes: visco-elasto-plastic slab bending, retreating subduction, lithospheric extension, collision, slab detachment, intrusion emplacement, core formation. Comparison with natural examples.
Exercises: Designing numerical model for extension of the continental lithosphere.
EPILOGUE: OUTLOOKTheory: Where are we now? Where to go further? Current and future directions of numerical geodynamic modelling development: 3D, MPI, OpenMp, PETSC, AMR, FEM, FVM, GPU/Cell-based computing, interactive computing, realistic physics, visualization challenges etc.
Exercises: No more exercises and home works!
MATLAB PROGRAM EXAMPLES