1. Geodynamics DayLecturerLectures 2BBTemperature in the mantle 3BBGoverning equations and approximate solutions 4CLBNumerical, analytical and laboratory models 5CLBPlates, slab and subduction 6CLBPlumes, hotspots,transition zone and CMB 9CLBGeological Constraints 10BBComposition and origin of the core 11BBGoverning equations and the geodynamo 12BBThermal and dynamical evolution of Earth's and planets

2. Numerical, Analytical and Laboratory Models Lecture 4: Geodynamics Carolina Lithgow-Bertelloni

3. FAULTS! Large range of Time- & Length-Scales Mass - Momentum- Energy - Non-linear What is right Constitutive Relation? [Tackley, 1999] Governing Equations

4. Approaches Static Processes Dynamic Processes Experimental - Laboratory Observational - Modeling Theoretical - Numerical Simulations Present Past

5. Problems in Mantle Geodynamics  Understanding Earth and Earth-like planets  Sources of energy: internal vs. basal heating  Constitutive law: How to make plates  Scales of flow: plates, plumes  Phase transitions and their effect  Layering: what happens to slabs  Heterogeneity: scales, nature, origin  Destruction of heterogeneity: mixing  Understanding Earth history  Present-Day  Gravity, Plate Motions (driving forces), Deformation  History  Past plate motions (driving forces), rearrangements  Thermal evolution  True Polar Wander  Geochemical variations

6. Plate Tectonics Mantle Convection [Zhao et al., 1997]

7. Mantle Convection and Plate Tectonics [Turcotte and Oxburgh, 1967]

8. Plumes [Whitehead and Luther, 1975]

9. How to construct a numerical model?  Numerical methods for PDE’s  Spectral, Finite element, Spectral element  Flexibility  Grids (geometry, adaptability)  Resolution  Material property contrasts  Speed!  Regional vs. Global  Boundary conditions  Resolution, Speed  Nature of problem  Inputs  Material properties (from mineral physics)    as a function of  Rheology (viscosity, but not only)  As a function  P dependence requires compressibility  Energy sources (from geochemistry, and …)  Rate of internal heating  Basal heating (heat flow coming out of the core)  Chemical Composition (from geochemistry in a broad sense)

10. Difficulties  Choice of rheological law (does it matter?)  Olivine rheology?  Making plates, asymmetric subduction  Lithosphere and mantle hard to treat together(Lagrangian vs Eulerian)  Full thermodynamics  Phase transitions (including melting)  Mixing  Tracer methods (substantial differences!)  Other methods better?  Characterizing mixing [from Louis Moresi]

11. Recent Work Mantle Circulation Model? Slabs and Plumes: regional models Geochemical heterogeneity [Farnetani et al., 2002] [Zhong et al., 2000] [Billen, 2004]

12. Making plates [Bercovici, 2003] [Tackley, 2000]

13. Dynamics and chemical heterogeneity [Xie and Tackley, PEPI, in press]

14. Why do experiments? Fluid dynamics is studied both theoretically and experimentally, and the results are described both mathematically and physically. The phenomena of fluid motion are governed by known laws of physics--conservation of mass, the laws of classical mechanics (Newton's laws of motion), and the laws of thermodynamics. These can be formulated as a set of nonlinear partial differential equations, and in principle one might hope to infer all the phenomena from these. In practice, this has not been possible; the mathematical theory is often difficult, and sometimes the equations have more than one solution, so that subtle considerations arise in deciding which one will actually apply. As a result, observations of fluid motion both in the laboratory and in nature are also essential for understanding the motion of fluids. Scaling analysis makes it possible to infer when two geometrically similar situations--of perhaps quite different size and involving different fluids will give rise to the same type of flow. Same Ra, ~ same Pr and you are in business.  For the Earth (why not just numerics?)  Benchmarking, reality check  Parameter Range (the higher the Ra #… the greater the resolution)  Large rheological variations  Thermochemical convection  Mixing  New physical phenomena?

15. Plumes and Entrainment [Jellinek and Manga, 2002]

16. Slabs and trench rollback [Kincaid and Griffiths, 2003]

17. FAULTS! Large range of Time- & Length-Scales Mass - Momentum- Energy - Non-linear What is right Constitutive Relation? [Tackley, 1999] Governing Equations

18. Instantaneous Flow Mantle Density Heterogeneity Model [Hager & O’Connell, 1979] -Induced Viscous Flow -Can be solved analytically For a spherical shell -Predict: Radial Stresses Dynamic topography Based on Geologic Information-Plate Motion History Seismic Tomography- Convert velocity to density [ Lithgow-Bertelloni and Richards, 1998] [ Masters and Bolton]

19. Geoid and Viscosity Structure [Forte and Mitrovica, 2001]

20. Plate Motions [Conrad and Lithgow-Bertelloni, JGR, in PRESS]

21. Anisotropy [Gaboret et al., 2003; see also Becker et al, 2003]

22. Deformation [Lithgow-Bertelloni and Guynn, 2004] Lithospheric Stress Field Contribution from Mantle Flow

23. Past, Present and Future What have we learned? -Mantle and Plates are an intimately coupled system -Deep mantle structure is important for the surface -Geological information provides quantitative constraints -Mixing is complicated! Where are we now? -Circulation models -Generation of plates with exotic rheologies -Making real subduction zones! -Modeling isotopic and petrological heterogeneity -Modeling of observations in simple contexts (complications) Where are we going? -Self-consistent modeling of mantle flow and lithospheric deformation -Connection to surface processes (sea-level; climate) -Understanding deep Earth structure and consequences (seismology via mineral physics) -Feedback between geodynamic models and tectonics