Geodynamics

Day Lecturer Lectures2 BB Temperature in the mantle3 BB Governing equations and approximate solutions4 CLB Numerical, analytical and laboratory models5 CLB Plates, slab and subduction6 CLB Plumes, hotspots,transition zone and CMB9 CLB Geological Constraints10 BB Composition and origin of the core11 BB Governing equations and the geodynamo12 BB Thermal and dynamical evolution of Earth's and planets

Numerical, Analytical and Laboratory Models

Lecture 4: GeodynamicsCarolina Lithgow-Bertelloni

€

∂∂t

ρv i( ) + v j

∂ ρv i( )∂x j

= −∂p

∂x i

+∂ 2

∂x i2 η ijklv j( ) + f i

€

∂T

∂t+ v i

∂T

∂x i

= κ∂ 2T

∂x i2

+ H

€

∂ρ∂t

+ v i

∂ρ

∂x i

=∂ ρv i( )

∂x i

FAULTS! Large range of Time- & Length-Scales

Mass -

Momentum-

Energy -

Non-linearWhat is right Constitutive Relation?

[Tackley, 1999]

Governing Equations

Approaches

Static ProcessesDynamic Processes

Experimental - Laboratory

Observational - Modeling

Theoretical - Numerical Simulations

PresentPast

Problems in Mantle Geodynamics

Understanding Earth and Earth-like planetsSources of energy: internal vs. basal heatingConstitutive law: How to make platesScales of flow: plates, plumesPhase transitions and their effect

Layering: what happens to slabsHeterogeneity: scales, nature, originDestruction of heterogeneity: mixing

Understanding Earth historyPresent-Day

Gravity, Plate Motions (driving forces), DeformationHistory

Past plate motions (driving forces), rearrangementsThermal evolutionTrue Polar WanderGeochemical 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 elementFlexibility

Grids (geometry, adaptability)ResolutionMaterial property contrasts

Speed!

Regional vs. GlobalBoundary conditionsResolution, SpeedNature of problem

InputsMaterial properties (from mineral physics)

ρ as a function of

Rheology (viscosity, but not only)As a function

P dependence requires compressibilityEnergy sources (from geochemistry, and …)

Rate of internal heatingBasal heating (heat flow coming out of the core)

Chemical Composition (from geochemistry in a broad sense)€

(P,T,X,σ , ˙ ε )

€

(P,T,X)

DifficultiesChoice of rheological law (does it matter?)

Olivine rheology?Making plates, asymmetric subductionLithosphere and mantle hard to treat together(Lagrangian vs Eulerian)

Full thermodynamicsPhase transitions (including melting)

MixingTracer methods (substantial differences!)

Other methods better?Characterizing mixing

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

[from Louis Moresi]

Recent WorkMantle CirculationModel?

Slabs and Plumes: regional models

Geochemical heterogeneity

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

[Farnetani et al., 2002]

[Zhong et al., 2000]

[Billen, 2004]

Making plates

[Bercovici, 2003]

[Tackley, 2000]

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 checkParameter Range (the higher the Ra #… the greater the resolution)Large rheological variationsThermochemical convectionMixingNew physical phenomena?

Plumes and Entrainment

[Jellinek and Manga, 2002]

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Slabs and trench rollback

[Kincaid and Griffiths, 2003]

€

∂∂t

ρv i( ) + v j

∂ ρv i( )∂x j

= −∂p

∂x i

+∂ 2

∂x i2 η ijklv j( ) + f i

€

∂T

∂t+ v i

∂T

∂x i

= κ∂ 2T

∂x i2

+ H

€

∂ρ∂t

+ v i

∂ρ

∂x i

=∂ ρv i( )

∂x i

FAULTS! Large range of Time- & Length-Scales

Mass -

Momentum-

Energy -

Non-linearWhat is right Constitutive Relation?

[Tackley, 1999]

Governing Equations

Instantaneous Flow

Mantle Density Heterogeneity Model

∇ •r v =0

∇ •T +δρgˆ z =0

T =−pI +2η˙ ε

∇2V =4πGδρ

-Induced Viscous Flow

-Can be solved analyticallyFor a spherical shell

-Predict: Radial StressesDynamic topography

Based on Geologic Information-Plate Motion History

Seismic Tomography- Convert velocity to density

[ Lithgow-Bertelloni and Richards, 1998]

[ Masters and Bolton]

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

[Lithgow-Bertelloni and Guynn, 2004]

Lithospheric Stress FieldContribution from Mantle Flow

Past, Present and FutureWhat 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