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 Conclusions – 3-7km for crustal conditions etc. Results – scaling, DDV is major control (2-3 bullet points on here) What have I done, don’t worry about other people. Add solution figure in IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity 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 IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity 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 IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Depth dependent viscosity IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Scaling of shear zone width with DDV  Force balance:  Simple scaling relation, valid for small z 0. IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Non-linear, uniform properties IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity 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: IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Scaling of shear zone width with n IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Arrhenius law  Viscosity structure:  0 th order Taylor expansion: IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Arrhenius viscosity structure IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Arrhenius velocity field Arrhenius Depth Dependent IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Comparison of mechanisms  Contours at 10% of driving velocity Material Parameters from Hirth & Kohlstedt (2003), Hirth et al. (2001) IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Combined scaling law  Depth dependent  Effective z 0 for Arrhenius  Non-linear scaling IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Shear zone width for crustal lithologies  Reasonable parameter values for continental crust outlined in yellow  For 30km thick crust, expect shear zones of 3-7km IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Combined scaling law IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Recent observations  Yamasaki et al 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? IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

Viscosity structures IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity StructuresConclusions

 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: IntroductionModelDepth dependentNon-linearArrheniusCombinedViscosity 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<z 0 <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 Parameters z 0 : e-folding length η 0 : viscosity coefficient Variables u: velocity Rheological Parameters z 0 : e-folding length η 0 : viscosity coefficient Variables u: velocity Constants w: width of domain Rheological Parameters z 0 : e-folding length η 0 : constant viscosity Variables u: velocity Constants w: width of domain Rheological Parameters z 0 : e-folding length η 0 : constant viscosity Variables u: 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. Constants R: gas constant te: elastic lid thickness / d Rheological Parameters B: material constant Q: creep activation energy beta: Geotherm Variables T: temperature η: viscosity Constants R: gas constant te: elastic lid thickness / d Rheological Parameters B: material constant Q: creep activation energy beta: Geotherm Variables T: temperature η: 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 : Constants J 2 : Second invariant of strain tensor w: width of domain Rheological Parameters n: power law Variables u: velocity η: viscosity Constants J 2 : Second invariant of strain tensor w: width of domain Rheological Parameters n: power law Variables u: velocity η: viscosity

Non-linear ductile shear zones  Exponentially depth dependent viscosity:  Viscosity structure:  Governing equation:  Approximate solution assuming : Constants J 2 : Second invariant of strain tensor w: width of domain Rheological Parameters n: power law Variables u: velocity η: viscosity Constants J 2 : Second invariant of strain tensor w: width of domain Rheological Parameters n: power law Variables u: velocity η: viscosity

Additional Equations  1 st 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?