Geodynamics
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Lectures
Temperature in the mantle
Governing equations and approximate solutions
Numerical, analytical and laboratory models
Plates, slab and subduction
Plumes, hotspots,transition zone and CMB
Geological Constraints
Composition and origin of the core
Governing equations and the geodynamo
Thermal and dynamical evolution of Earth's and planets
Numerical, Analytical
and Laboratory Models
Lecture 4: Geodynamics
Carolina Lithgow-Bertelloni
Governing Equations
Momentum-

v i 

p  2


v i   v j
 v  f i
t
x j
x i x i 2 ijkl j
T
T
 2T
 vi
 2  H
t
x i
x i
Energy -

Mass
-

  v i 
 vi

t
x i
x i

Non-linear
What is right Constitutive Relation?
[Tackley, 1999]
FAULTS!
Large range of Time- & Length-Scales
Approaches
Observational - Modeling
Theoretical - Numerical Simulations
Experimental - Laboratory
Present
Past
Static Processes
Dynamic Processes
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
Plate Tectonics
Mantle Convection
[Zhao et al., 1997]
Mantle Convection and Plate Tectonics
[Turcotte and Oxburgh, 1967]
Plumes
[Whitehead and Luther, 1975]
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
(P,T, X)
Rheology (viscosity, but not only)
As a function
Ý)
(P,T, X,,
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)
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
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
[from Louis Moresi]
Recent Work
Mantle
Circulation
Model?
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
[Zhong et al., 2000]
Slabs and Plumes: regional models
Geochemical
heterogeneity
[Farnetani et al., 2002]
[Billen, 2004]
Making plates
[Tackley, 2000]
[Bercovici, 2003]
Dynamics and chemical heterogeneity
[Xie and Tackley, PEPI, in press]
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?
Plumes and Entrainment
[Jellinek and Manga, 2002]
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Slabs and trench rollback
[Kincaid and Griffiths, 2003]
Governing Equations
Momentum-

v i 

p  2


v i   v j
 v  f i
t
x j
x i x i 2 ijkl j
T
T
 2T
 vi
 2  H
t
x i
x i
Energy -

Mass
-

  v i 
 vi

t
x i
x i

Non-linear
What is right Constitutive Relation?
FAULTS!
Large range of Time- & Length-Scales
[Tackley, 1999]
Instantaneous Flow
Seismic Tomography- Convert velocity to density
Mantle Density Heterogeneity Model
 v  0
  T  gˆz  0
Ý
T   pI  2
 2 V  4G
[ Masters and Bolton]
Based on Geologic Information-Plate Motion History
[ Lithgow-Bertelloni and Richards, 1998]
-Induced Viscous Flow
-Can be solved analytically
For a spherical shell
-Predict: Radial Stresses
Dynamic topography
Geoid and Viscosity Structure
[Forte and Mitrovica, 2001]
Plate
Motions
[Conrad and Lithgow-Bertelloni, JGR, in PRESS]
Anisotropy
[Gaboret et al., 2003; see also Becker et al, 2003]
Deformation
Lithospheric Stress Field
Contribution from Mantle Flow
[Lithgow-Bertelloni and Guynn, 2004]
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