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Lecture 3. Global Lecture 3. Global models: Towards models: Towards
modeling plate tectonicsmodeling plate tectonics
Global surface observables
Major ingredients of plate tectonics
Linking mantle convection and lithospheric deformations
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How weak are the plate boundaries?
How weak is asthenosphere?
How to make plate boundaries?
Key questionsKey questions
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Spherical harmonicexpansion
Modified from websiteOf Svetlana Panasyuk
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Spherical HarmonicsSpherical Harmonics
Example Components for Degree (L) = 8
Zonal (m=0) Sectoral (m=L) Tesseral ( m=L/2)
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GeoidGeoid
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GeoidGeoid
Measured by modelling satellite Measured by modelling satellite orbits.orbits. Spherical harmonic representation, Spherical harmonic representation,
L=360.L=360.
From, http://www.vuw.ac.nz/scps-students/phys209/modules/mod8.htm
Range+/- 120 meters
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Derivative of geoid (continents)Derivative of geoid (continents) Measured over the oceans using Measured over the oceans using
satellite altimetry (higher resolution).satellite altimetry (higher resolution).
Free-Air GravityFree-Air Gravity
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Derivative of geoid (continents)Derivative of geoid (continents) Measured over the oceans using Measured over the oceans using
satellite altimetry (higher resolution).satellite altimetry (higher resolution).
Free-Air GravityFree-Air Gravity
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Geoid/Free-air Gravity Geoid/Free-air Gravity SpectraSpectra
L = 60-360 short wavelength.600-110 km. Red Spectrum
Dominated by signal at long
wavelengths
L = 2-3 very long wavelength > 13,000 km
L = 4-12 long wavelength10000-3000 km
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Dynamic TopographyDynamic Topography
Isostatically Compensated
Dynamically Supported
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Post-Glacial Rebound Post-Glacial Rebound (PGR)(PGR)
Glacial Isostatic Glacial Isostatic Adjustment (GIA).Adjustment (GIA). returning to returning to
isostatic isostatic equilibriumequilibrium..
Unloading of the Unloading of the surface as ice surface as ice melts (rapidly).melts (rapidly).
From:http://www.pgc.nrcan.gc.ca/geodyn/docs/rebound/glacial.html
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Plate MotionPlate Motion
Well-known for the present time.Well-known for the present time. Accuracy degrades for times further in Accuracy degrades for times further in
the past.the past.
Data: Argus & Gordon 1991 (NUVEL-NNR), Figure: T. Becker
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Summary of Surface Summary of Surface ObservationsObservations
Observation Quality Observation Quality . .
Post Glacial Rebound Post Glacial Rebound variable variable (center) (center)
Plate Motion Plate Motion good (recent)good (recent)
Geoid Geoid good (<100 good (<100 km)km)
Free-air GravityFree-air Gravity good (shallow)good (shallow)
Dynamic TopographyDynamic Topography poor (magnitude)poor (magnitude)
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Observed plate velocities in no-net-rotation (NNR) reference frame
Plate MotionPlate Motion
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… and observed net-rotation (NR) of the lithosphere
Based on analyses of seismic anisotropy Becker (2008) narrowed possible range of angular NR velocities down to 0.12-0.22 °/Myr
Plate MotionPlate Motion
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Ax
T
xDt
DTC ijij
iip
)(
ijeff
ij
i
j
j
iij Dt
D
Gx
v
x
v
2
1
2
1)(
2
1
01
i
i
x
v
Dt
DT
Dt
DP
K
Dt
Dvg
xx
P ii
j
ij
i
Full set of equations
energy
Full system of equations
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Ax
T
xDt
DTC IIII
iip
)(
ijeff
ij
i
j
j
iij Dt
D
Gx
v
x
v
2
1
2
1)(
2
1
01
i
i
x
v
Dt
DT
Dt
DP
K
Dt
DvgPT
xx
P ii
j
ij
i
),(
Full set of equations
mass
energy
Simplifying...
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ijeffi
j
j
iij x
v
x
v
2
1)(
2
1
Full set of equations
energy
0
i
i
x
v
0)(
ij
ij
i
gxxx
P
Simplified system of equations
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Full set of equations
energy
0
i
i
x
v
0),,())(( 321
ii
j
j
ieff
ji
gxxxx
v
x
v
xx
P
Stokes equations
Main tool to model mantle convection
Solution is most simple if viscosity depends only on depth
),()(0
lml
l
lm
ilmi YrVv
),()(0
lml
l
lmlm YrPgdrP
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Ax
T
xDt
DTC IIII
iip
)(
ijeff
ij
i
j
j
iij Dt
D
Gx
v
x
v
2
1
2
1)(
2
1
01
i
i
x
v
Dt
DT
Dt
DP
K
0),(
ij
ij
i
gTPxx
P
Full set of equations
mass
Simplifying...
))(1))((),((),( 00 TTrTrPTP
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Solving Stokes equations with FE code Terra
(Bunge et al.)
Mantle convection
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Ingredients of plate tectonics
Convection (FE code Terra)
Weak plate boundaries
Ricard and Vigny, 1989; Bercovici, 1993; Bird, 1998; Moresi and Solomatov, 1998;Tackley, 1998, Zhong et al, 1998; Trompert and Hansen, 1998;Gurnis et al., 2000….
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Ingredients of plate tectonics
Bercovici, 1993,1995, 1996, 1998, 2003; Tackley, 1998, 2000; Moresi and Solomatov, 1998; Zhong et al, 1998; Gurnis et al., 2000…
Generating plate boundaries
van Heck and Tackley, 2008
Incr
easi
ng y
ield
str
ess
Tendency: towards more realistic strongly non-linear rheology
Viscous rheology-only and emulation of brittle failure
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Solving Stokes equations with code Rhea
(adaptive mesh refinement)
Burstedde et al.,2008-2010
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Solving Stokes equations with code Rhea
Stadler et al., 2010
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They can not also reproduce realistic one-sided subduction and pure transform boundaries
Global models can not generate yet present-day plates and correctly reproduce plate motions
Point 1
They employ plastic (brittle) rheolgical models inconsistent with laboratory data
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Modeling deformation at plate Modeling deformation at plate boundariesboundaries
Sobolev and Babeyko, Geology 2005; Sobolev et al., 2006
(Sobolev et al., EPSL 2005, Petrunin and Sobolev, Geology 2006, PEPI, 2008).
Subduction and orogeny in Subduction and orogeny in AndesAndes
Dead Sea TransformDead Sea Transform
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Balance equations
Momentum: 0
Energy:
iji
j
i
i
g zx
qDUr
Dt x
1ˆ Elastic strain:
21
Viscous strain: 2
Plastic strain:
el vs plij ij ij ij
elij ij
vsij ij
eff
plij
ij
G
Q
Deformation mechanisms
Popov and Sobolev ( PEPI, 2008)
Mohr-Coulomb
„Realistic“ rheology
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Dislocation
Diffusion
Peierls
Three creep processes
( Kameyama et al. 1999)
Diffusion creep
Dislocation creep
Peierls creep
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Combining global and Combining global and lithospheric-scale models lithospheric-scale models
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Mantle and lithospheric codes are coupled through continuity of velocities and tractions at 300 km.
Lithospheric code (Finite Elements)
Mantle code(spectral or FEM)
Coupling mantle convection and lithospheric deformation
Sobolev, Popov and Steinberger, in preparation
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Below 300 km depth
Spectral method (Hager and O’Connell,1981) with radial viscosity and 3D density distributions based on subduction history (Steinberger, 2000)
Above 300 km depth
3D temperature from surface heat flow at continents and ocean ages in oceans, crustal structure from model crust2.0
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Mantle rheology
olivine rheology with water content as model parameter
Parameters in reference model by Hirth and Kohlstedt (2003) with n=3.5 +-0.3.
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Plate boundaries
Plate boundaries are defined as narrow zones with visco-plastic rheology where friction coefficient is model parameter
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Mantle code (spectral)
Lithospheric code (FEM)
Mantle and lithospheric codes are coupled through continuity of velocities and tractions at 300 km.
The model has free surface and 3D, strongly non-linear visco-elastic rheology with true plasticity (brittle failure) in upper 300km.
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Mesh for low-resolution model
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Benchmark test
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Our model vrs. model by Becker (2006) (CitcomS)
Misfit= = 0.19
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How weak are plate boundaries?
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Friction at boundaries 0.4 (Smax=600 MPa)
Effect of strength at plate boundaries
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Friction at boundaries 0.2 (Smax=300 MPa)
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Friction at boundaries 0.1 (Smax=150 MPa)
much too low velocities
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too low velocities
Friction at boundaries 0.05 (Smax=75 MPa)
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about right magnitudes of velocities
Friction at boundaries 0.02 (Smax= 30 MPa)
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too high velocities
Friction at boundaries 0.01 (Smax= 15 MPa)
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Strength (friction) at plate boundaries must be very low (<0.02), much lower than measured friction for any dry rock (>0.1)
Point 2
No high pressure fluid=no plate tectonics
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How weak/wet is asthenosphere?
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Mantle rheology
olivine rheology with water content as model parameter
Parameters in reference model by Hirth and Kohlstedt (2003) with n=3.5 +-0.3.
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Modeling scheme
For every trial rheology (water content in asthenosphere) we calculate plate velocities for different frictions at plate boundaries, until we get best fit of observed plate velocities in the NNR reference frame
Next, we look how well those optimized models actually fit observations
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Lithospheric net rotation
Be
cke
r (2
008
)
Hirth and Kohlstedt (1996)
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Lithospheric net rotation
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Lithospheric net rotation
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Misfit=
Plate-velocities misfit
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Plate-velocities misfit
Misfit=
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Plate velocities in NNR reference frame
Model
Tp=1300°C,
lith: dry olivine;
asth:1000 ppm H/Si in olivine, n=3.8
Plate bound. friction:
Subd. zones 0.01-0.03, other 0.05-0.15
misfit= 0.25
misfit=0.25 (0.36 previous best by Conrad and Lithgow-Bertelloni, 2004)
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The current views on the rheology and water content in the upper mantle are consistent with the observed plate velocities, if the stress exponent in the wet olivine rheology is pushed to the highest experimentally allowed values (3.7-3.8)
Point 3
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How to make plate boundaries?
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Mohr-Coulomb plasticity
Plasticity
Damage rheology: strain softenimng
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We do 1 Mln. yr. modeling runs without
predefined plate boundaries with damage
rheology leading to frictional softening from
some initial value to 0.02
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Strain rate distribution
August 2009 S.Sobolev, GFZ PotsdamSuzdal
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Major plate boundaries and realistic plate velocities can be generated in a “short-memory” model with the damage rheology
Point 4
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Distribution of dissipation rate