Einat LevEinat Lev (MIT)(MIT)

October 2006October 2006

Collaborators:MIT/WHOI – Brad Hager, Greg HirthMonash/VPAC – Louis Moresi, Patrick Sunter, Alan Lo and

many other helpful mates

Outline

• Why we care? (“Motivation”)

• What we know? (“Introduction”)

• How we do it? (“Methods”)

• What we find? (“Results”)

• What should you take home? (“summary”)

Motivation

deformation

fabricfabricrheology

• Main question in this talk –

What is the effect of “anisotropic viscosity” on flow models• Underlying motives:

– (Un)verify of a basic assumption in modeling– “Closing the loop” - self-consistency in models

relating anisotropy (e.g. seismic) and flow

Anisotropy

“Physical properties”– Seismic wave speeds– Electrical/Magnetic/Thermal Conductivity– Optical properties– Strength/viscosity

AnisotropyAnisotropy - - variation of a variation of a physical physical propertyproperty depending on the direction depending on the direction in which it is measuredin which it is measured

Potential Applications

• More realistic estimation of stresses and the implied deformation mechanisms

• Chemical mixing and entrainment rates • Modeling tectonics in settings with existing

fabric, on different scales:– Folding of layered media – Changes in tectonics (e.g. strike-slip becoming

transpression in NZ).– Boundaries of regions with different histories

• Salt dome dynamics

“Hard Coupling” - Pervious Work

Richter&Dali – Rayleigh-Benard instabilities

Chrsitensen – instructive study of PGR and thermal convection

Honda – equations for transverse isotropy

Mühlhaus,Moresi, Cada – PIC+FE; folding; convection…

Han & Wahr – PGR and anisotropic visco-elasticity

1978 1986 19971987 2000-2006

In parallel:• Studies in other fields, such as glaciology, biology and fluid crystals and composites – tools may be out there!• Many studies with “soft coupling” : flow fabric, using crystallographic models (D-Rex, VPSC, finite strain)

1993

Chastel et al – VPSC & flow

Anisotropy in rocks

• Lattice preferred orientation (LPO)

• Shape preferred orientation (SPO)

• Crystal vs. aggregate properties - averages

Jung&Karato, 2001

Wilson & Zhang (1996)

Basic concepts

• The simplest form of anisotropy - transversely isotropytransversely isotropy (like a stack of layers)

• Can be characterized by the ratio of the “normal” to “shear” viscosities, called δ

After Teargus (2003)

• For solid rocks - δ≈5, but when , for instance, melt it present, it may be >100 (Honda, 1986)

ηs ηN

Anisotropy in the Equations of Flow

• Anisotropy is part of the constitutive lawconstitutive law:

Isotropic part Anisotropic “correction”

• Reduced anisotropic constitutive law we use:

Developing anisotropic fabric – The Directors

• “Directors” come from liquid crystal physics• Treated as normal vectors to layers

• Directors and the viscosity tensor:

Developing a Fabric• Time evolution of directors by the velocity:

• Testing vs. lab experiments:

Anisotropy and Finite Elements

• General form of the FE Stiffness Matrix

Ciso Caniso

Where

ResultsSeveral test cases:

2. A “rock” under simple shear

3. Density-driven flows - Rayleigh-Taylor instabilities at different levels of sophistication

4. Thermal convection

5. 3D strike slip – time dependent crust/mantle coupling and the effect on the strain field under New Zealand

Simple Shear of an “Aggregate”

Added vorticity in the flow field:

A Sinking Cylinder

• Just look at the shapes…

Anisotropic background Anisotropic cylinder

Sinking Slab• Different behavior of a slab as it deflects

on a barrier (the 660km, the CMB…)

Anisotropic upper mantle Isotropic upper mantle

McNamara, van Keken & Karato, 2002

Flow around a sinking slab• The fate of a 3D sinking dense slab is

controlled by return flow, which is influenced by the fabric in the upper mantle

Anisotropic UM Isotropic UM

Lithospheric Delamination• Invoked to explain topography and

gravity anomalies, as well as large volcanic provinces

• In our tests – the effect of pre-existing fabric on the shape and growth rate of the instabilities and the patterns of fabric developed

After Zandt (2004)

Fabric Near a “Drip”• Using our tools we can look at developing

fabric during the flow• An upwelling near lithospheric delamination (e.g. Tanton

and Hager, 2000) - would have to interact with this strong fabric

Effect of Existing Fabric on Instability Growth Rate

Fastest growth rate

Slowest growth rate

Up/Downwellings at or near easy-pure-shear blocks

Slow growth - Up/Downwellings at easy-simple-shear blocks

time

(τ)

<V>~e t/τ

Diapirism• Compare the strain rate – In the anisotropic case, strain is localized along

the boundary between the layers

Anisotropic Lower layer Isotropic

Strain Rate 2nd Invariant

Velocity Field

light

dense

Thermal convection• Purely thermal convection for a

box heated from below

Isotropic MediumAnisotropic Medium

• Anisotropic case is stabilizes much faster, and maintains original up/down directions• Mühlhaus et al (2003) – Nusselt number higher for anisotropic cases, but for δ>10 it stays constant

A Strike-slip Plate Boundary with an Anisotropic Lower Crust

• The strength of the lower crust affects lithospheric coupling and the strain field

After time…

Two distinctly different end-members:

Or not? What would happen if we add compression?

Strong lower crust Weak lower crust

Summary

• We’ve shown the application of a simple and fast fabric development technique in geodynamics flows

• Adding anisotropy to the examples shown had very interesting effects

• Anisotropic viscosity may play an important role in many geodynamical settings and should not be ignored

• Viscous anisotropy of earth materials – more lab experiments? Maybe through geoid/tomography mismatches?

• Relating viscous and seismic anisotropy – do they always appear together? How do they scale?

• What would be the effect on three- dimensional, time-dependent flows? Or a full (or at least orthorhombic) fabric?

Some Burning Questions: