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Shear Senses and Viscous Dissipation of Layered Ductile Simple Shear Zones

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Abstract

Velocity profiles and shear heat profiles for inclined, layered Newtonian simple shear zones are considered. Reverse fault-like simple shear of the boundaries and upward net pressure gradient act together in such shear zones. As the velocity of the boundary increases, the point of highest velocity shifts from the lower layer of less viscosity into the upper layer. The shear heat profile shows a temperature peak inside the lower layer. For a more viscous upper layer, the point of highest velocity is located inside the upper layer and shifts towards the upper boundary of the shear zone. The shear heat profile shows a maximum temperature within the upper layer. Depending on the flow parameters of the two layers, the slip rate of the boundary, and the dip and thickness of the shear zone, a shear sense in reverse to the relative movement of the shear zone boundaries may develop. These models can decipher thermo-kinematics of layered shear zones in plate-scale hot orogens.

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Acknowledgments

Department of Science and Technology’s (New Delhi) satellite project: IR/S4/ESF-16/2009(G) supported SM. Thanks to two anonymous reviewers for critical comments, Eugenio Carminati for editorial handling and reviewing, Rakesh Biswas for discussions, and Priyanka Ganesh for assistance.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematical Sciences, University College, Cork, Ireland

    Kieran F. Mulchrone

  2. Department of Earth Sciences, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, Maharashtra, India

    Soumyajit Mukherjee

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  1. Kieran F. Mulchrone
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Correspondence to Soumyajit Mukherjee.

Appendix

Appendix

Solutions of Eq. (12): shear heat profile for layer-1 and 2:

$$ \begin{aligned} T_{1} \left( y \right) & = \frac{1}{{24k\left( {h_{1} + h_{2} } \right)\mu_{1} \mu_{2} \left( {h_{2} \mu_{1} + h_{1} \mu_{2} } \right)^{2} }}\left( { - G_{2}^{2} h_{2}^{4} \left( { - y + h_{1} + h_{2} } \right)\mu_{1} (h_{2}^{2} \mu_{1} \left( {\mu_{1} - 3\mu_{2} } \right)} \right) \\ & \quad + 3yh_{2} \mu_{1} \mu_{2} + h_{1} \mu_{2} \left( {3y\mu_{1} + 2h_{1} \mu_{2} } \right)) + 2G_{2} (y - h_{1} \\ & \quad - h_{2} )h_{2}^{2} \mu_{1} \mu_{2} (2U\mu_{1} (h_{2}^{2} (\mu_{1} - 3\mu_{2} ) + 3yh_{1} \mu_{2} + (3y - 2h_{1} )h_{2} \mu_{2} ) \\ & \quad + G_{1} (2(y - h_{2} )^{2} h_{2}^{2} \mu_{1} - h_{1}^{3} (y - 2h_{2} )\mu_{2} + h_{1}^{2} (yh_{2} (2\mu_{1} - 5\mu_{2} ) - h_{2}^{2} (\mu_{1} \\ & \quad - 3\mu_{2} ) + 2y^{2} \mu_{2} ) + 2h_{1} (y - h_{2} )h_{2} (y\mu_{1} + (y - h_{2} )\mu_{2} ))) + \mu_{2} (4UG_{1} (y \\ & \quad - h_{1} - h_{2} )\mu_{1} \mu_{2} (2(y - h_{2} )^{2} h_{2}^{2} \mu_{1} - yh_{1}^{3} \mu_{2} + h_{1}^{2} (2y - 3h_{2} )(h_{2} (\mu_{1} - \mu_{2} ) \\ & \quad + y\mu_{2} ) + 2h_{1} (y - h_{2} )h_{2} (y\mu_{1} (y - h_{2} )\mu_{2} )) - 12\mu_{1} (h_{1} ( - 2kh_{1}^{2} T_{l} \\ & \quad - U^{2} y^{2} \mu_{1} + yh_{1} (2kT_{l} - 2kT_{u} + U^{2} \mu_{1} ))\mu_{2}^{2} + h_{2}^{3} \mu_{1} ( - 2KT_{l} \mu_{1} + U^{2} (\mu_{1} \\ & \quad - \mu_{2} )\mu_{2} ) - h_{2} \mu_{2} (U^{2} y^{2} \mu_{1} \mu_{2} + 2kh_{1}^{2} T_{l} (2\mu_{1} + \mu_{2} ) - 2yh_{1} \mu_{1} (2kT_{l} \\ & \quad - 2KT_{u} + U^{2} \mu_{2} )) + h_{2}^{2} \mu_{1} ( - 2kyT_{u} \mu_{1} + U^{2} \mu_{2} (( - y + h_{1} )\mu_{1} + (2y \\ & \quad - h_{1} )\mu_{2} ) + 2kT_{l} ((y - h_{1} )\mu_{1} - 2h_{1} \mu_{2} ))) + G_{1}^{2} (y - h_{1} \\ & \quad - h_{2} )(2(y - h_{2} )^{3} h_{2}^{3} \mu_{1}^{2} + 2h_{1} (y - h_{2} )^{2} h_{2}^{2} \mu_{1} (y\mu_{1} + 2(y \\ & \quad - h_{2} )\mu_{2} ) + h_{1}^{4} \mu_{2} (3h_{2}^{2} (\mu_{1} - \mu_{2} ) - 2y^{2} \mu_{2} + 5yh_{2} \mu_{2} + 2h_{1}^{2} (y \\ & \quad - h_{2} )h_{2} (y^{2} \mu_{2} (2\mu_{1} + \mu_{2} ) + h_{2}^{2} \mu_{2} (2\mu_{1} + \mu_{2} ) + yh_{2} (\mu_{1}^{2} - 4\mu_{1} \mu_{2} - 2\mu_{2}^{2} )) \\ & \quad + 2h_{1}^{3} (y^{3} \mu_{2}^{2} - 4y^{2} h_{2} \mu_{2}^{2} - 2h_{2}^{3} \mu_{2}^{2} + yh_{2}^{2} (\mu_{1}^{2} + 5\mu_{2}^{2} ))))) \\ \end{aligned} $$
$$ \begin{aligned} T_{2} \left( y \right) & = \frac{1}{{24k(h_{1} + h_{2} )\mu_{1} \mu_{2} (h_{2} \mu_{1} + h_{1} \mu_{2} )^{2} }}(yG_{2}^{2} \mu_{1} ( - h_{2}^{3} ( - 2y^{3} + 4y^{2} h_{2} - 3yh_{2}^{2} \\ & \quad + h_{2}^{3} )\mu_{1}^{2} + 2h_{1}^{3} (y^{3} - 4y^{2} h_{2} + 6yh_{2}^{2} - 4h_{2}^{3} )\mu_{2}^{2} + h_{1} h_{2}^{2} \mu_{1} ((2y^{3} - 4y^{2} h_{2} \\ & \quad + 3yh_{2}^{2} - 2h_{2}^{3} )\mu_{1} + 4(y - h_{2} )^{3} \mu_{2} ) + h_{1}^{2} h_{2} \mu_{2} ((4y^{3} - 12y^{2} h_{2} + 12yh_{2}^{2} \\ & \quad - 7h_{2}^{3} )\mu_{1} + 2(y^{3} - 4y^{2} h_{2} + 6yh_{2}^{2} - 3h_{3}^{2} )\mu_{2} )) \\ & \quad - 2yG_{2} \mu_{1} \mu_{2} ( - 2U\mu_{1} (h_{2}^{2} (2y^{2} - 3yh_{2} + h_{2}^{2} )\mu_{1} + h_{1}^{2} (2y^{2} - 6yh_{2} \\ & \quad + 3h_{2}^{2} )\mu_{2} + h_{1} h_{2} (y(2y - 3h_{2} )\mu_{1} + 2(y^{2} - 3yh_{2} + 2h_{2}^{2} )\mu_{2} )) \\ & \quad + G_{1} h_{1}^{2} (h_{2}^{2} (2y^{2} - 3yh_{2} + h_{2}^{2} )\mu_{1} + h_{1}^{2} (2y^{2} - 6yh_{2} + 5h_{2}^{2} )\mu_{2} \\ & \quad + h_{1} h_{2} ((2y^{2} - 3yh_{2} + 2h_{2}^{2} )\mu_{1} + 2(y^{2} - 3yh_{2} + 2h_{2}^{2} )\mu_{2} ))) \\ & \quad + \mu_{2} ( - 4UyG_{1} h_{{^{1} }}^{2} \mu_{1} \mu_{2} (h_{1} (3y - 4h_{2} )\mu_{1} + 3(y - h_{2} )h_{2} \mu_{1} - h_{1}^{2} \mu_{2} ) \\ & \quad + yG_{1}^{2} h_{1}^{4} (3(y - 2h_{1} )h_{2} \mu_{1} \mu_{2} - h_{2}^{2} \mu_{1} (2\mu_{1} + 3\mu_{2} ) + h_{1} \mu_{2} (3y\mu_{1} - h_{1} \mu_{2} )) \\ & \quad - 12\mu_{1} ( - 2kh_{3}^{2} T_{l} \mu_{1}^{2} + h_{1} \mu_{2} ( - U^{2} y^{2} \mu_{1}^{2} - 2kh_{1} (( - y + h_{1} ))T_{l} + yT_{u} )\mu_{2} \\ & \quad + U^{2} yh_{1} \mu_{1} \mu_{2} ) - h_{2} \mu_{2} (U^{2} y^{2} \mu_{1}^{2} - 2yh_{1} \mu_{1} (2kT_{l} - 2kT_{u} + U^{2} \mu_{1} ) \\ & \quad + 2kh_{1}^{2} T_{l} (2\mu_{1} + \mu_{2} )) + h_{2}^{2} \mu_{1} (y\mu_{1} ( - 2kT_{u} + U^{2} \mu_{2} ) + 2kT_{l} ((y - h_{1} )\mu_{1} \\ & \quad - 2h_{1} \mu_{2} ))))) \\ \end{aligned} $$

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Mulchrone, K.F., Mukherjee, S. Shear Senses and Viscous Dissipation of Layered Ductile Simple Shear Zones. Pure Appl. Geophys. 172, 2635–2642 (2015). https://doi.org/10.1007/s00024-015-1035-8

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