Scaling of viscous shear zones with depth dependent viscosity and power law stress

strain-rate dependence

James Moore and Barry Parsons

Introduction

What are the dominant physical mechanisms that govern localisation of shear at depth in a strike-slip regime?

Depth dependent viscosity Major control Shear zones of 3-7km for reasonable crustal parameters

Non-linear stress strain-rate relationship Also significant, but secondary

Thermomechanical coupling Further localisation consequence of a pre-existing narrow

shear zone Scaling relation for continental lithologies Viscosity structures that explain post-seismic

deformation at NAF

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

2D approximation for infinitely long strike-slip fault. Stokes flow for anti-plane conditions: Far field driving velocities Rigid lid moves as block motion

Model construction

Solution Domain

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Contours at 10% intervals, dashed for 50% Width of domain: At the base of the layer, shear is widely distributed:

Constant viscosity layer

90% of far field motion at 1.66d50% of far field motion at 0.56d

Depth DependentPlots

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Depth dependent viscosity

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Scaling of shear zone width with DDV

Force balance:

Simple scaling relation, valid for small z0.

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Non-linear, uniform properties

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Scaling of shear zone width with n

Horizontal derivative of the velocities is, in general, much greater than the vertical.

Simple scaling relation, valid for large n:

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Scaling of shear zone width with n

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Arrhenius law

Viscosity structure:

0th order Taylor expansion:

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Arrhenius viscosity structure

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Arrhenius velocity field

ArrheniusDepth Dependent

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Comparison of mechanisms

Material Parameters from Hirth & Kohlstedt (2003), Hirth et al. (2001)

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Combined scaling law

Depth dependent Effective z0 for Arrhenius Non-linear scaling

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Shear zone width for crustal lithologies

Crustal Lithologies

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Combined scaling law

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Recent observations

Yamasaki et al. 2014 require a region of low viscosity beneath the North Anatolian Fault to explain post-seismic transient deformation following the 1999 Izmit and Duzce earthquakes

Could this be the fingerprint of a zone of localised shear?

What are the viscosity structures from our model?

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Viscosity structures

Ab

solu

te V

iscositie

s

Relative Viscosities

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Conclusions

Depth dependence of viscosity produces narrow shear zones

Power law rheology also provides a strong control

Shear heating and further localisation of shear is a consequence of having a pre-existing narrow shear zone

Viscosity structures generated by shear heating and/or power law rheology are important for the dynamics of post-seismic deformation

Scaling law:

Introduction Model Depth dependentNon-linear Arrhenius CombinedViscosity StructuresConclusions

Fin

Thank you for listening

Moore and Parsons (submitted), Scaling of viscous shear zones with depth dependent viscosity and power law stress-strain rate dependence, Geophysical Journal International.

This work was supported by the Natural Environment Research Council through a studentship to James Moore, the Looking into the Continents from Space project (NE/K011006/1), and the Centre for the Observation and Modelling of Earthquakes, Volcanoes and Tectonics (COMET).

We thank Philip England for helpful discussions during the course of this work.

Scaling of shear zone width with DDV

Shear Heating

0.1<z0<0.2 shear heating leads to a decrease in shear zone width of 5-20%

For 30km crust with these values, you would already have a shear zone of 6-14km

Shear heating will further localise deformation in these zones to 5-13km Important, but secondary

factor

Constant viscosity would give much wider region of deformation, of the order of 50km

Linear ductile shear zones

Exponentially Depth Dependent Viscosity

Viscosity structure:

Governing equation:

Solution:

Rheological Parametersz0: e-folding lengthη0: viscosity coefficient

Variablesu: velocity

Constantsw: width of domain

Rheological Parametersz0: e-folding lengthη0: constant viscosity

Variablesu: velocity

Linear ductile shear zones

Arrhenius law Viscosity structure:

Thermal structure:

Governing equation:

Approximate solution may be obtained by Taylor expansion of RHS about z=1/2.

ConstantsR: gas constantte: elastic lid thickness / d

Rheological ParametersB: material constantQ: creep activation energybeta: Geotherm

VariablesT: temperatureη: viscosity

Linear ductile shear zones

Arrhenius law To a first order approximation this is equivalent to

an exponentially depth dependent viscosity with

Velocity profile at z=1 is accurately captured with this approximation

Extremely high viscosity gradients in the shallow crust cause further shear localisation for z <1/2.

Higher order approximation is in agreement with numerical results

Non-linear ductile shear zones

Uniform properties: Viscosity structure:

Governing equation:

Approximate solution assuming :

ConstantsJ2: Second invariant of strain tensorw: width of domain

Rheological Parametersn: power law

Variablesu: velocityη: viscosity

Non-linear ductile shear zones

Exponentially depth dependent viscosity:

Viscosity structure:

Governing equation:

Approximate solution assuming :

ConstantsJ2: Second invariant of strain tensorw: width of domain

Rheological Parametersn: power law

Variablesu: velocityη: viscosity

Additional Equations

1st order approximate Arrhenius solution

Previous work

Yuen et. al. [1978] analysed the 1-D problem to investigate the relationship between thermal, mechanical and rheological parameters that govern shear zone behavior Once a shear zone forms it will remain localised due to shear-stress

heating

Thatcher and England [1998] investigated the role of thermomechnical coupling, or shear heating in the more complex 2-D problem Broad range of behaviors but for reasonable parameter values shear zones

are narrow. Shear localisation driven by dissipative heating near the axis of the shear

zone causing reduction in temperature dependent viscosity

Takeuchi and Fialko [2012] used a time dependent earthquake cycle model Thermomechanical coupling with a temperature dependent power-law

rheology will localise shear

Do we need themomechanical coupling, or a power law rheology, to generate shear zone localisation?