Dynamics and Composition of
the Mantle: From the Atomic to
the Global Scale
Day Lecturer
Lectures
2  CLB
Mantle Flow- Deformation; Fluid Mantle
2 CLB
Mantle Flow- Governing Equations; Theory and applications
Tutorial 2: Modelling the geoid and dynamic topography
3 CLB
Geoid and Dynamic Topography
3 CLB
Dynamics of Plate Motions
Tutorial 2: Modelling the geoid and dynamic topography (continued)
Earthquakes
Short time-scales
Volcanoes
Long Time-Scales
Mantle Convection
Topography
Result of Plate Tectonics
Stress, Strain and Plate Tectonics
What is Tectonic Activity?
For the Earth we mean the large-scale strains or
deformation that occur over geological time-scales. There
are other geodynamic processes over shorter time-scales,
but for now we will deal only with long-term processes.
What are the stresses (or forces) that cause this activity?
That is the major question we are trying to answer. What
are the magnitude, nature, and origin of the stresses
(forces) causing tectonic activity?
Types of Geologic Strain
Types of Geologic Strain
€
How do we define strain?
Measure of the change of shape and size of a body
l
l
Strain
ε=
∂

Stress
σ=
€
F
(Pressure)
A
Strain
ε=
dx

€
dx
l
x
x
Types of strain
= constant
High T
Strain
Elastic Strain
(recoverable)
creep
Slope = ε˙
Yield point
Non-recoverable
strain
€
Elastic deformation
Apply
Yielding occurs
Time
Release
Types of strain
Wha t a r e the different types of s t r ai ns?
I) Recoverable (elastic)
II) Non-Recoverable (permanent)
i. Brittle Failure
ii. Creep (ductile deformation)
Recoverable Strain- mostly associated with small amounts of deformation
Small stress ! " Small Strain
When Stress (or Force) is released snaps back to original position
Stress is proportional to amount of strain (Hooke’s L a w )
Non-Recoverable (anelastic) Strain- Associated with large deformations
Large Stresses applied slowly (small strain rate) or fast (large strain rate)
Stress is proportional to strain rate
Strain rate
Brittle
Low
T
€ ε˙
Fast
ε˙ =
dε
dt
Creep
High
Slow
T
ε˙
Quantifying types of strain
We need to relate stress to strain for both types of strain.
To do so we need the properties of the material (moduli)
Elastic Strain is related to the elastic or rigidity moduli (G). σ
G=
ε
Anelastic Strain to viscosity (η).
σ
η=
ε
What are some typical values for the solid Earth?
Let’s work it out
Strain and Strain Rate
Elastic ~ .2-.5%
Anelastic ~ 100%!
Strain rate ~ 10-15s-1
In the Earth clearly the permanent, anelastic deformation is
much greater than the elastic strain. But over what time scales?
We saw that what type of strain develops depends on the P
and T conditions and the strain rate. So what is an appropriate
characteristic time scale?
Maxwell Relaxation Time
i.e.
How long before anelastic strain becomes the dominant
mode of deformation?
Maxwell Relaxation Time
ε σG µ
τM = =
=
ε˙ σ
G
µ
time<< τM à Elastic Behavior
time>> τM à Fluid (creep) Behavior
Let’s get some characteristic values for the Earth:
Material
G (GPa)
(Pa s)
Na-Ca glass (250°C)
4 x 1011
25
16
Salt (200°C)
3 x 10
20
Ice (0°C)
1013
4
Upper Mantle (1300 °C) 1020
50
Mantle creep is clearl y the domin ant mechanism of
scales t > 1010 to 1011 s ~ 103 to 109 years >> M
M
(s)
20
2 x 106
17 days
2.5 x 103
90 minutes
2 x 109
300 yr
deformation for geological time
Maxwell Relaxation Time
Material properties strong function of temperature!
Age of the Universe
1 byr
crust
(500°K)
creep regime
1 myr
τ>>τ
M
1000 years
1 year
mantle
(1500°K)
Maxwell Relaxation Time
This pictures shows to first order that
1)Only hot planets (Tinterior > 1500 K) can be “active”
2)Crust and near-surface rocks cannot deform by creep on geological time scales
In the Earth we have – to first order - the simple picture – of plate tectonics
Cold brittle plates over hot creeping mantle
we%know?%
How do we know?
2ηk 4 πν
=
ρg
gλ
ρgτ
η=
Rebound
2k
τ=
Isostasy; Post-Glacial
€
Richmond%Gulf,%Hudson’s%Bay,%Canada%
2ηk 4 πν
τ=
=
ρg
gλ
ρgτ
η=
2k
Pel$er&(2
Mantle Convection
Mantle Convection
Boundary Layers
Plates
Mantle Convection
Continuous generation of
dynamical (thermal) +
geochemical (compositional) =
seismic heterogeneity
[including phase transitions (TZ!!)]
[Zhao et al., 1997]
Research Methods
Observational - Modeling
(a)
Theoretical - Numerical Simulations
Experimental - Laboratory
Present
Past
Static Processes
Dynamic Processes
Governing Equations
Momentum-
∂
( ρ v) + ∇ ⋅ (ρ v) = ∇ ⋅ σ + f
∂t
Energy -
€
∂T
+ v ⋅ ∇T = ∇ 2T + H
∂t
Mass -
∂ρ
+ ∇ ⋅ ( ρv) = 0
∂t
€
[Tackley, 1999]
Non-linear
What is right Constitutive Relation?
σ = − pI + 2ηe
How to solve?
— Numerical methods for PDE’s
— Finite Difference, Spectral, Finite element, Finite
Volume, etc.
— Flexibility
— Grids (geometry, adaptability)
— Resolution
— Material property contrasts
— Speed!
— Regional vs. Global
— Boundary conditions
— Resolution, Speed
ª Inputs
— Nature of problem
ª Material properties (from mineral physics)
ª α, κ, ρ
ª as a function of
ª Rheology (viscosity, but not only)
ª As a function
ª P dependence requires compressibility
ª Energy sources (from geochemistry, and …)
ª Rate of internal heating
ª Basal heating (heat flow coming out of the core)
ª Chemical Composition (from geochemistry in a broad sense)
Mantle convection Movies
http://www.gps.caltech.edu/~gurnis/Movies/movies-more.html
http://www.ipgp.jussieu.fr/~labrosse/movies.html
Approximations
-Infinite Prandtl # fluid: i.e. Inertial forces are not important
-Fluid is Incompressible, Newtonian
-Properties Homogeneous
2
∇p − η∇ v = f
f = δρg
δρ = αρ oΔT
∂T
2
+
v
⋅
∇T
=
κ
∇
T+H
€
€ ∂t
But suppose you€
know δρ ? €
€
Falling Sphere
v
Δρ
Buoyancy
3
4 πr Δρg
FB = gΔm =
3
Resistance
2
2 v
˙
FR = Aσ R = 4 πr ηε ≈ −4 πr η = −4 πrηv
r
Force Balance: FB+FR=0
1 r 2Δρg
v≈
3 η
Actual coefficient varies between 1/3 (inviscid) and 2/9 (solid)
Falling Sphere
v
2
v≈
1 r Δρg
3 η
Δρ
€
Object Plume head
Xenolith r
Δρ (kg/m3)
η (Pa s) v
500 km 30
1022
80 km/Ma
10 cm
100
90 km /week
200
How to Solve Stokes..
2
∇p − η∇ v = δρ g
2
∇ Φ = 4π Gδρ
Take each variable: [v r,vθ ,vϕ , τ rr , τ rθ , τ rϕ ,δp,δρ]
Expand using spherical harmonics, scalar:
€
and vector fields:
[Alterman et al., 1959; Takeuchi and Hasegawa, 1965; Kaula, 1975; Hager and O Connell, 1979; 1981]
Aside: Spherical Harmonics
The spherical harmonics are the angular portion of the solution to Laplace's
equation in spherical coordinates 2
∇ Φ=0
€
Spherical Harmonics
Continuing…..
m

m

m

m
, ϕ
vr = a (r )Y
Y
vθ = bm (r)Y,mθ + c m (r)Y,mϕ
vϕ = b (r )Y
m
, ϕ
m

m
, θ
− c (r )Y
1 ∂Ym
=
sinθ ∂ϕ
Y,mθ
€
∂Ym
=
∂θ
[So for example the continuity (mass conservation) equation]
m
m
−2a
(
+1)b


a˙m =
+
€
r
r
…. Substitute the expansions for velocity, stress, with those for pressure and density… We end with 6 coupled ODEs
…..We could
€ solve NUMERICALLY… but if we perform a simple variable substitution for each spherical harmonic coefficient
….. We get two nicely defined differential equations
[See Hager and O Connell, 1979 and 1981]
Propagator Matrices
l
dy
Computing
l l
l Mantle Flow
=A y +f
[Poloidal]
dλ
λ = ln(r /RE )
dt l
l l
[Toroidal]
=Bt
dλ
€via Propagator matrices
These equations can be solved
[See Gantmacher]
λ l
l
0
0
Al
λ0
y( λ) = P ( λ, λ )y +
∫ P(λ,ξ )f (ξ )dξ
We can now solve for velocities and stresses at any depth once P is known!
€
Boundary Conditions: Continuity of velocity and stresses at boundary interfaces
Use: Free-slip at CMB; Free-slip or no-slip at surface
Advantages & Disadvantages
-Spectral Solutions VERY FAST! (You ll see)
-Can change radial viscosity profile (explore effects of viscosity structure)
-Spherical Shell
-Predict observables (Geoid, Topography, Plate Motions, Flow (Anisotropy))
-Can explore compressibility (less than 10% effect at long wavelengths) -LACK OF RHEOLOGICAL COMPLEXITY
Lateral viscosity variations
Plate boundary rheology
-MUST ASSUME A DENSITY HETEROGENEITY
-No TIME DEPENDENCE
So now what?
Seismic Tomography- Convert velocity to density---- BUT HOW?!
Mantle Density Heterogeneity Model
[ Masters and Bolton]
Based on Geologic Information-Plate Motion History"
Depth = 1000 Km [ Lithgow-Bertelloni and Richards, 1998]
Velocity-Density Scaling
Birch s law
δv=aδρ
Factors=0.1-0.5 g s/km cm3
Velocity-Density Scaling
Rρ / S
$ δ ln ρ '
=&
)
% δ lnVS ( Depth
If due to lateral T variations:
Rρ / S
$ ∂ ln ρ ' $ ∂ lnVS '−1
=&
) &
)
% ∂T ( P % ∂T ( P
Attenuation (anelasticity)
decreases value significantly
Claim: Cannot be negative
Not so! (phase transformations)
$ ∂ ln ρ ' $ ∂ ln ρ '
$ ∂ ln ρ ' $ ∂n '
&
) =&
) +&
) & )
% ∂T ( P % ∂T ( P,n % ∂n ( P,T % ∂T ( P
Karato and Karki (2001)
the three prominent sets of peaks in αme near 410,
520, and 660 km depth. Assuming typical values ⟨ρ⟩ =
Velocity-Density Scaling
n
R
and Lithgow-Bertelloni
(2007)
Fig. 8. Stixrude
Relative variations
of density and shear wave
velocity ξ (red)
and the isomorphic contribution to ξ (blue) compared with the results
w
i
t
a
t
c
c
r
n
t
v
e
Predict Geoid
Best fitting viscosity structure
Lithosphere-10 * UM
Lower Mantle-50 * UM
Plate Motions
Earth’s Stress Field
Contributions: Mantle Stresses; Crustal Heterogeneity
[Reinecker, J., Heidbach, O. and Mueller, B., 2003]
(available online at www.world-stress-map.org)
Stress Field
LVC+TD0
Fit to observations (Variance Reduction)
Azimuth-59%
Regime-61%
Combined effect of crustal contribution and mantle flow
[Lithgow-Bertelloni and Guynn, 2004]