Transcript Numerical, Analytical and Laboratory Models
Geodynamics
6 9 10 11 12
Day
2 3 4 5
Lecturer
BB BB CLB CLB CLB CLB BB BB BB
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-
t
i
v j
x v j i
p
x i
2
x i
2
ijkl v j
f i
Energy -
T
t
v i
T
x i
2
x i T
2
H
t
v i
x i
x v i 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 (
P
,
T
,
X
, ) P dependence requires compressibility Rate of internal heating 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]
Mantle Circulation Model?
Recent Work
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]
[Bercovici, 2003] [Tackley, 2000]
Making plates
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.
phenomena from these the motion of fluids.
These can be formulated as a set of nonlinear partial differential equations, and in principle one might hope to infer all the
. 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
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-
t
i
v j
x v j i
p
x i
2
x i
2
ijkl v j
f i
Energy -
T
t
v i
T
x i
2
x i T
2
H
t
v i
x i
x v i i
Non-linear What is right Constitutive Relation?
[Tackley, 1999] FAULTS!
Large range of Time- & Length-Scales
Instantaneous Flow
Seismic Tomography Convert velocity to density Mantle Density Heterogeneity Model
v
T
0
g
ˆ 0
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]
[Gaboret et al., 2003; see also Becker et al, 2003]
Anisotropy
[Lithgow-Bertelloni and Guynn, 2004]
Deformation
Lithospheric Stress Field Contribution from Mantle Flow
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