Numerical, Analytical and Laboratory Models

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Geodynamics

Day

Lecturer

Lectures

2

BB

Temperature in the mantle

3

BB

Governing equations and approximate solutions

4

CLB

Numerical, analytical and laboratory models

5

CLB

Plates, slab and subduction

6

CLB

Plumes, hotspots,transition zone and CMB

9

CLB

Geological Constraints

10

BB

Composition and origin of the core

11

BB

Governing equations and the geodynamo

12

BB

Thermal and dynamical evolution of Earth's and planets

Numerical, Analytical

and Laboratory Models

Lecture 4: Geodynamics

Carolina Lithgow
-
Bertelloni

FAULTS!


Large range of Time
-

& Length
-
Scales

Mass
-

Momentum
-

Energy
-

Non
-
linear

What is right Constitutive Relation?

[
Tackley,

1999]

Governing Equations


Approaches

Static Processes

Dynamic Processes


Experimental
-

Laboratory

Observational
-

Modeling

Theoretical
-

Numerical Simulations

Present

Past

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


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)

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
]

Recent Work

Mantle

Circulation

Model?

Slabs and Plumes: regional models

Geochemical
heterogeneity

[
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 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]

Slabs and trench rollback

[
Kincaid and Griffiths,
2003]

FAULTS!


Large range of Time
-

& Length
-
Scales

Mass
-

Momentum
-

Energy
-

Non
-
linear

What is right Constitutive Relation?

[
Tackley,

1999]

Governing Equations


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
]

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 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