1. Introduction ................................................. 1
1.1 What is a Model? ......................................... 1
1.2 Spatial Dimension of Geological Problems ................. 6
1.2.1 Reducing Spatial Dimensions ........................ 10
2. Plate Tectonics ............................................. 15
2.1 Historical Development .................................. 15
2.2 Working on a Spherical Surface .......................... 20
2.2.1 ... or is the Earth Flat After All? ............... 21
2.2.2 Geometry on a Sphere .............................. 23
2.2.3 Kinematics on a Sphere ............................ 25
2.2.4 Mechanics on a Sphere ............................. 25
2.3 Map Projections ......................................... 28
2.3.1 The Mercator Projection ........................... 29
2.3.2 National Grids .................................... 30
2.4 The Layered Structure of Earth .......................... 33
2.4.1 Crust and Lithosphere ............................. 35
2.4.2 The Lithospheric Plates ........................... 39
2.4.3 The Plate Boundaries .............................. 39
2.4.4 The Wilson Cycle .................................. 46
2.5 Problems ................................................ 47
3. Energetics: Heat and Temperature ............................ 51
3.1 Principles of Heat Conduction ........................... 52
3.1.1 The Heat Conduction Equation ...................... 52
3.1.2 The Laplace Equation .............................. 60
3.1.3 The Error Function ................................ 61
3.1.4 Time Scales of Diffusion .......................... 62
3.2 Principles of Heat Production ........................... 64
3.2.1 Radioactive Heat Production ....................... 66
3.2.2 Mechanical Heat Production ........................ 68
3.2.3 Chemical Heat Production .......................... 76
3.3 Principles of Heat Advection ............................ 80
3.3.1 Heat Advection by Magma ........................... 81
3.3.2 Heat Advection by Solid Rock ...................... 82
3.3.3 Heat Advection by Fluids .......................... 85
3.3.4 The Peclet Number ................................. 86
3.4 Heat in the Continental Lithosphere ..................... 87
3.4.1 Stable Geotherms: The Relevant Equation ........... 90
3.4.2 The Contribution of Radioactivity ................. 92
3.4.3 Geotherms with Heat Flow Boundary Condition ....... 93
3.4.4 Geotherms with Fixed Basal Boundary Condition ..... 98
3.5 Heat in the Oceanic Lithosphere ........................ 101
3.5.1 Aging Oceanic Lithosphere ........................ 102
3.5.2 Subduction Zones ................................. 106
3.6 Thermal Evolution of Intrusions ........................ 109
3.6.1 Step-shaped Temperature Distributions ............ 110
3.6.2 One-dimensional Intrusions ....................... 112
3.6.3 Two-dimensional Intrusions ....................... 118
3.6.4 Modeling Realistic Intrusions .................... 120
3.7 Selected Heat Transfer Problems ........................ 125
3.7.1 Periodic Temperature Fluctuations ................ 125
3.7.2 Folded Isotherms ................................. 126
3.7.3 Isotherms and Surface Topography ................. 128
3.7.4 Temperature Distribution Around Faults ........... 132
3.8 Problems ............................................... 134
4. Kinematics: Morphology and Deformation ..................... 139
4.1 Strain - The Basics .................................... 139
4.2 Reference Levels ....................................... 142
4.2.1 Sphere, Spheroid and Geoid ....................... 143
4.2.2 Lagrangian and Eulerian Reference Frames ......... 144
4.2.3 The Undeformed Reference Lithosphere ............. 147
4.3 Uplift and Exhumation .................................. 149
4.3.1 Definition of Uplift and Exhumation .............. 149
4.3.2 Vertical Motion in Active Orogens ................ 152
4.3.3 Exhumation Processes ............................. 153
4.3.4 Modeling Vertical Kinematics ..................... 154
4.4 Isostasy ............................................... 160
4.4.1 Hydrostatic Isostasy ............................. 161
4.4.2 Flexural Isostasy ................................ 170
4.5 Geomorphology .......................................... 178
4.5.1 Erosion Models on Orogenic Scale ................. 181
4.5.2 Short Range Transport ............................ 184
4.5.3 Long Range Transport: Drainages .................. 193
4.5.4 Discontinuous Landscape Formation ................ 200
4.5.5 The Shape of Volcanoes ........................... 200
4.5.6 Fractals ......................................... 202
4.6 Problems ............................................... 205
5. Mechanics: Force and Rheology .............................. 209
5.1 Stress - The Basics .................................... 210
5.1.1 The Stress Tensor ................................ 210
5.1.2 Deformation Laws ................................. 220
5.2 Rheology of the Lithosphere ............................ 233
5.2.1 Rheology of the Continental Lithosphere .......... 234
5.2.2 Rheology of the Oceanic Lithosphere .............. 242
5.3 Forces Applied to Lithospheric Plates .................. 245
5.3.1 Transmission Mechanisms .......................... 245
5.3.2 Forces in Oceanic Lithosphere .................... 253
5.3.3 Forces in Continental Plates ..................... 257
5.4 Problems .............................................. 260
6. Dynamic Processes .......................................... 265
6.1 Continents in Extension ................................ 265
6.1.1 Basin Subsidence Mechanisms ...................... 268
6.1.2 Basin Types ...................................... 269
6.1.3 Subsidence Analysis .............................. 271
6.1.4 Models of Continental Extension .................. 276
6.2 Continents in Collision ................................ 284
6.2.1 Thermal Evolution of Collisional Orogens ......... 285
6.2.2 Mechanical Description of Colliding Continents ... 294
6.2.3 Orogenic Wedges .................................. 308
6.3 Selected Geodynamic Processes .......................... 315
6.3.1 Flood Basalts and Mantle Plumes .................. 315
6.3.2 Delamination of the Mantle Lithosphere ........... 318
6.3.3 Low Pressure - High Temperature Metamorphism ..... 321
6.3.4 High Pressure Metamorphism ....................... 323
6.3.5 Tectonic Overpressure ............................ 326
6.3.6 Feedback and Episodicity ......................... 329
6.4 Problems ............................................... 334
7. Metamorphic Processes ...................................... 337
7.1 Introduction ........................................... 337
7.1.1 What Exactly are P-T- and P-T-t-D-Paths? ......... 338
7.1.2 The Metamorphic Facies ........................... 339
7.2 Basic Principles of Petrology .......................... 342
7.2.1 Equilibrium Information: Thermobarometry ......... 346
7.2.2 Non-equilibrium Information: Kinetics ............ 349
7.3 Classification of P-T-Paths ............................ 353
7.3.1 Slope and Curvature of P-T-Paths ................. 355
7.4 Interpretation of Р-T-t-D-Relationships in Orogens ..... 357
7.4.1 Interpreting Temporal Relationships .............. 359
7.4.2 Interpreting Spatial Relationships ............... 360
7.5 Problems ............................................... 364
A. Topographic Maps ........................................... 369
B. Mathematical Tools ......................................... 393
B.1 What is a Differential Equation? ....................... 393
B.1.1 Terminology Used in Differential Calculus ........ 395
B.2 The Finite Difference Method ........................... 399
B.2.1 Stability and Accuracy ........................... 401
B.2.2 Implicit and Explicit Finite Difference
Methods .......................................... 402
B.2.3 Approximation of the Transport Equation .......... 404
B.2.4 Grids and Boundary Conditions .................... 407
B.2.5 Dealing with Irregular Grid Boundaries ........... 408
B.2.6 Recommended Reading .............................. 409
B.3 Scalars, Vectors and Tensors ........................... 410
B.4 Using Fourier Series ................................... 413
B.5 Selected Numerical Tricks .............................. 416
B.5.1 Integrating Differential Equations ............... 416
B.5.2 Analytically Unsolvable Equations ................ 416
B.5.3 The Least Squares Method ......................... 417
B.5.4 Basic Statistical Parameters ..................... 419
B.6 Problems ............................................... 421
С. Maths Refresher ............................................ 423
D. Symbols and Units .......................................... 429
E. Answers to Problems ........................................ 439
F. Internet Addresses ......................................... 455
References .................................................... 463
Index ......................................................... 485
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