Global dynamics of the Earth : applications of viscoelastic relaxation theory to solid-earth and planetary geophysics

This Volume Opens Up New Perspectives On The Physics Of The Earth’s Interior And Planetary Bodies For Graduate Students And Researchers Working In The Fields Of Geophysics, Planetary Sciences And Geodesy. It Looks At Our Planet In An Integrated Fashion, Linking The Physics Of Its Interior To Geophysical And Geodetic Techniques That Record, Over A Broad Spectrum Of Spatial Wavelengths And Time Scales, The Ongoing Modifications In The Shape And Gravity Field Of The Planet. Basic Issues Related To The Rheological Properties Of The Earth And To Its Slow Deformation Are Considered, In Both Mathematical And Physical Terms, Within The Framework Of An Analytical Relaxation Theory. Fundamentals Of This Theory Are Developed In The First Two Chapters. Chapters 3-9 Deal With A Wide Range Of Applications, Ranging From Changes In The Earth’s Rotation To Post-seismic Deformation And From Sea-level Variations Induced By Post-glacial Rebound To Tidal Deformation Of Icy Moons Of The Solar System. This Second Edition Improves Substantially Our Formalism Implementing Compressibility In Viscoelastic Relaxation. Chapter 5 Now Contains New Developments In The Physics Of The Gravitational Effects Of Large Earthquakes At Subduction Zones, Made Possible By New Gravity Data From Space Missions. The New Chapter 9 Of This Second Edition On Deformation And Stresses Of Icy Moons Enlarges The Applications Of The Book To Planetology, Dealing With The Additional Complications In The Theory Of Viscoelastic Relaxation Introduced By The Shallow Low-viscosity Zones And Inviscid Water Layers Of The Moons Of Jupiter And Saturn. 1. Viscoelastic Relaxation Theory, Momentum And Poisson Equations -- 2. Incompressible And Compressible Analytical Viscoelastic Models -- 3. Rotational Dynamics Of Viscoelastic Planets Linear Theory -- 4. Tpw And J2 Induced By Ice-sheet Loading -- 5. Detection Of The Time-dependent Gravity Field And Global Change -- 6. Sea-level Changes -- 7. Tpw Driven By Subduction. Non-linear Rotation Theory -- 8. Post-seismic Deformation -- 9. Icy Moons -- Appendices -- Index. By Roberto Sabadini, Bert Vermeersen, Gabriele Cambiotti. Foreword to the Second Edition 6 Foreword to the First Edition 8 Acknowledgments 10 Contents 11 1 Viscoelastic Relaxation Theory, Momentum and Poisson Equations 15 1.1 Rheological Models 15 1.2 Mathematics 17 1.2.1 The Linear Maxwell Solid 21 1.2.2 Compressible and Incompressible Earth's Models 23 1.2.3 The Correspondence Principle 25 1.3 Expansion in Spherical Harmonics 27 1.3.1 Volume Changes and Surface Forces 29 1.3.2 Spheroidal and Toroidal Deformations 30 1.4 Spheroidal Deformations 32 1.5 Toroidal Deformations 34 1.6 Boundary Conditions 35 1.6.1 The Earth's Surface 35 1.6.2 Chemical Boundaries 39 1.6.3 Core-Mantle Boundary 40 1.7 Elastic and Viscoelastic Solutions 45 1.7.1 Load and Tidal Love Numbers 47 1.7.2 Application of the Correspondence Principle 48 1.8 The Relaxation Spectrum 50 1.8.1 Modal and Non-modal Contributions 55 1.9 The Complex Contour Integration 56 1.10 Point Sources 57 1.10.1 Point Loads 57 1.10.2 Fault Discontinuities 59 References 64 2 Incompressible and Compressible Analytical Viscoelastic Models 66 2.1 Analytical Solution 66 2.2 Green Functions for Incompressible and Compressible Stratified Viscoelastic Earth's Models 66 2.2.1 Core-Mantle Boundary (CMB) Matrix 67 2.2.2 Propagators and Fundamental Matrices 68 2.3 Layered Incompressible Models 70 2.4 Relaxation Times for Incompressible Earth's Models 75 2.5 The Self-compressed, Compressible Sphere 81 2.5.1 The Analytical Solution 83 2.5.2 The Relaxation Spectrum of the Self-compressed Compressible Sphere 87 2.5.3 The Compositional Modes 89 2.6 Viscoelastic Perturbations Due to Surface Loading 92 2.7 Toroidal Solution 94 2.8 Time Dependent Loading Love Numbers 95 References 97 3 Rotational Dynamics of Viscoelastic Planets: Linear Theory 99 3.1 Introduction to Earth's Rotation 99 3.1.1 Liouville Equations 102 3.2 MacCullagh's Formula 103 3.2.1 Inertia Perturbations Due to Changes in the Centrifugal Potential 106 3.3 Linearized Liouville Equations 109 3.4 The Concept of True Polar Wander (TPW) 111 3.4.1 Reference Frame 113 3.4.2 Adjustment of the Equatorial Bulge 114 3.5 Developments of Linearized Rotation Theories 115 3.5.1 Comparison Between Different Rotation Theories 120 3.5.2 Omission of the M0 Rotation Mode 121 3.5.3 Analytical Formula for the M0 Rotation Mode 124 3.5.4 Unification of the Different Approaches 126 3.6 Non-hydrostatic Bulge Contribution 127 3.7 Readjustment of the Rotational Bulge 130 3.8 Compressible and Incompressible Readjustment of the Equatorial Bulge 133 3.9 Long-Term Behavior of the Rotation Equation 137 3.9.1 Theory for Rotation Changes Due to Mantle Convection 139 3.10 Time-Dependent Inertia Due to Mantle Convection 144 3.10.1 TPW Simulations 146 3.11 Polar Wander on the Earth, Moon, Mars and Venus 150 References 156 4 TPW and 2 Induced by Ice-Sheet Loading 161 4.1 TPW and 2 from PGR 161 4.2 The Inference of Mantle Viscosity from TPW and 2 Data 163 4.3 Loading 165 4.4 Mantle Viscosity 168 4.4.1 Variations in Depth of the Two-Layer Mantle Viscosity Profile 179 4.4.2 Upper Mantle Viscosities Lower Than 1021Pa s 180 4.5 Ice Age Cycles and the Polar Wander Path: Lithospheric and Mantle Rheology 183 4.6 Ice Age True Polar Wander in a Compressible and Non-hydrostatic Earth 187 4.6.1 The Role of Mantle Heterogeneities 190 References 197 5 Detection of the Time-Dependent Gravity Field and Global Change 200 5.1 Changes in the Long-Wavelength Geoid Components from Satellite Laser Ranging Techniques 200 5.2 Trade-Off Between Lower Mantle Viscosity and Present-Day Mass Imbalance in Antarctica and Greenland 206 5.3 Time Dependent Gravity Field from the GRACE Space Mission: The Importance of PGR Models 213 5.3.1 Global Vertical and Horizontal Displacements from PGR 217 5.4 The 2004 Sumatran and 2011 Tohoku-Oki Giant Earthquakes 221 5.4.1 Modeling the 2004 Sumatran Earthquake 222 5.4.2 The GRACE Data 225 5.4.3 Constraining the 2004 Sumatran Earthquake 226 5.4.4 The 2011 Tohoku-Oki Earthquake: Gravitational Seismology 229 References 232 6 Sea-Level Changes 236 6.1 The Issue of Sea-Level Change, a Present-Day Concern 236 6.2 Sea-Level Variations, Geoid and Gravity Anomalies Due To Pleistocene Deglaciation 238 6.2.1 Mathematical Formulation 239 6.2.2 Sea-Level Variations, the Geoid and Free-Air Gravity Anomalies 242 6.3 Glacial Isostatic Adjustment (GIA) Versus Tectonic Processes: The Example of the Mediterranean Sea 246 6.4 Sea-Level Fluctuations Induced by Polar Wander 253 6.5 Sea-Level Changes Induced by Subduction 257 6.5.1 Sea-Level Variations, Geoid Anomalies and The Long-Wavelength Dynamic Topography 258 6.5.2 A Single Sinking Slab 260 6.5.3 A Distribution of Slabs 262 References 265 7 TPW Driven by Subduction: Non-linear Rotation Theory 268 7.1 Formulation of the Non-linear Rotation Problem 268 7.2 Polar Wander Velocity for a Distribution of Slabs 276 References 278 8 Post-seismic Deformation 279 8.1 Global Post-seismic Deformation 279 8.2 Post-seismic Deformation for Shallow Earthquakes 287 8.2.1 The Umbria-Marche (1997) Earthquake 287 8.2.2 The Irpinia (1980) Earthquake 294 References 299 9 Icy Moons 302 9.1 Diurnal and Non-synchronous Rotation (NSR) Stresses Acting on Europa's Surface 302 9.2 The Tidal Potential 305 9.3 The Interior of Europa 310 9.4 The Impulse Tidal Response of Europa 311 9.4.1 The Impulse Response of Interior Models with a Global Subsurface Ocean 311 9.4.2 Boundary Conditions 314 9.4.3 Application to Icy Moons I: Normal Modes 316 9.4.4 Application to Icy Moons II: Impulse Response to Tidal Forces 316 9.5 Radial Deformation at the Surface 319 9.6 Stresses at the Surface of Europa 322 9.6.1 Diurnal Stresses at the Surface 322 9.6.2 NSR Stresses at the Surface 327 9.7 Stress Patterns on Europa's Surface 331 9.8 Morphology of the Europa Icy Moon 338 References 341 Appendix A Dyads and Vector Identities 345 Appendix B Analytical Functions 349 Appendix C Icy Moons 354 Index 361 Annotation This volume opens up new perspectives on the physics of the Earth's interior and planetary bodies for graduate students and researchers working in the fields of geophysics, planetary sciences and geodesy. It looks at our planet in an integrated fashion, linking the physics of its interior to geophysical and geodetic techniques that record, over a broad spectrum of spatial wavelengths and time scales, the ongoing modifications in the shape and gravity field of the planet. Basic issues related to the rheological properties of the Earth and to its slow deformation are considered, in both mathematical and physical terms, within the framework of an analytical relaxation theory Front Matter....Pages i-xvi Viscoelastic Relaxation Theory, Momentum and Poisson Equations....Pages 1-51 Incompressible and Compressible Analytical Viscoelastic Models....Pages 53-85 Rotational Dynamics of Viscoelastic Planets: Linear Theory....Pages 87-148 TPW and \(\dot{J}_2\) Induced by Ice-Sheet Loading....Pages 149-187 Detection of the Time-Dependent Gravity Field and Global Change....Pages 189-224 Sea-Level Changes....Pages 225-256 TPW Driven by Subduction: Non-linear Rotation Theory....Pages 257-267 Post-seismic Deformation....Pages 269-291 Icy Moons....Pages 293-335 Back Matter....Pages 337-358