Geodynamics 4: MixingLouise Kellogg

University of California, Davis

Outline:

•Why care about mixing?

•Physics of mixing

•Mixing in the mantle

(from Harpp and White,2001, G-cubed)

Fine-scale variations in the Galapagos

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apag

os I

slan

ds

Global scale: mantle contains both well-mixed regions and heterogeneity

Fine scale heterogeneity

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Scales of heterogeneity in an exposed peridotite

Allègre and Turcotte, Nature (1986)

The scale of heterogeneity led Allègre and Turcotte (1986) to propose a ‘marble cake’

structure to the mantle

Image from epicurious.com

Allègre and Turcotte 1986

A. Levander et al. Tectonophysics 416 (2006) 167–185

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Stretching and folding

Molecular diffusion

Stretching and folding

Breakup

Stretching and folding

Starting point (figure from Ottino)

300,000 particlesStarting position

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Dynamics of mixing in a simplified mantle model

Some mixing scales

Length scale (meters)

Reyn

old

s N

um

ber

10-6 100 106 1012

10-20

10-10

1010

100

1020

turbulent

laminar

Astrophysicsinteriors of stars

Mechanical Engineeringcombustion

Atmosphericdispersion

OceanographyChemical engineering

chemical reactors

Physiologyblood vesselsBioengineering

aeration in bioreactors

Food engineeringblending additives

Polymer EngineeringGeophysicsmantle convection

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clip from fluid dynamics film series - 13:09

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Stretching and folding

Molecular diffusion

Stretching and folding

Breakup

Stretching and folding

Starting point (figure from Ottino)

Kellogg and Turcotte, 1987 EPSL

What about chemical diffusion?

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Allègre and Turcotte, Nature (1986) looked at timescales as a way of figuring out the scales

of heterogeneity

Some ways to analyze mixing

in models of the mantle

•Dispersal of heterogeneities (visually or using statistical methods)

•Computing derived isotopic signatures

Convective mixing and the fine structure of mantle heterogeneity, Peter Olson, David A. Yuen and Derick Balsiger

Physics of the Earth and Planetary Interiors, 36 (1984) 291—304

“Underresolved sampling leads to apparent homogeneity” - From Olson et al. 1984

Convective mixing and the fine structure of mantle heterogeneity, Peter Olson, David A. Yuen and Derick Balsiger

Physics of the Earth and Planetary Interiors, 36 (1984) 291—304

Mixing in 2-D with particles •Added at subduction zones •Removed at mid-ocean ridges

2900 km

670 km

Normalized viscosity

Dep

th

0 km

1 10 100

Hunt and Kellogg 2000

Constant viscosity

Pressure-dependent viscosity: smooth increase

Transition zone viscosity: Jump at 670 km

Hunt & Kellogg - effect of viscosity on mixing

viscosity

1 10 100

1 10 100

1 10 100

Initial location of particles

(Hunt and Kellogg model)

Similar to the kinematic mixing shown by Ottino

Badro et al. (Science, 2003,2004) show that Fe2+ in the two major phases in the deep lower mantle undergoes a transition from a high-spin to a low-spin state.

Lower mantle phases:Perovskite: (Fe,Mg)SiO3

Magnesiowüstite:(Fe,Mg)O

Possible effects of this transition?

•Thermal conductivity INCREASES (because low-spin Fe2+ is nearly translucent to thermal radiation.)

•Viscosity INCREASES (relating to a rise in the melting temperature of perovskite with increasing Mg content.)

•Hypothesis: Changes in these properties may inhibit thermal convection, creating a stagnant layer or layered regime in the lower mantle (Badro et al. (2003,2004)).

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Particle Distributions after 4 Ga

Starting particle distributionat steady-state conditions

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f=10

f=1

f=100

f=150

40

1

0.5

~161,000 Particles Total

~40,250 in lower right quadrant↑

Configurational Entropy

f=1

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

0 1 2 3 4 0 1 2 3 4

f=10 f=50

f=100 f=150

Calculating the configurational entropy of particles with starting positions above the increase in viscosity thermal conductivity - varies as function of the distribution of particles in space.

Method defined by Goltz and Bose (2004) and Turcotte (2001)

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time (Ga) time (Ga) time (Ga)

time (Ga) time (Ga)

1 10 50 100 150f

.5

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2

Mixing in a 3D spherical model of present-day mantle convectionPeter van Keken and Shijie Zhong

Ferrachat & Ricard, Mixing in 3-D plate driven flows

Chaotic trajectories occur even in steady-state flows

50 Poincare sections - Ferrachat & Ricard 2001

Lyapunov exponents estimated by tracking tracers: Both chaotic and laminar

mixing are observed Ferrachat & Ricard

2001

Computing isotopic signaturesEvolution of U-Pb and Sm-Nd systems in numerical models of mantle convection and plate tectonics

Shunxing Xie and Paul J. Tackley, J. Geophys. Research, 109, B11204, 2004

T

1 By 2 By 3 By

206/204 Pb

The role of viscosity contrasts

Mixing of heterogeneities in the mantle: Effect of viscosity differences

Michael Manga

GEOPHYSICAL RESEARCH LETTERSVOL. 23, NO. 4, PAGES 403-406, FEBRUARY 15, 1996

IsoviscousMore viscous

Less viscous

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