Channel flow extrusion model to constrain dynamic viscosity and Prandtl number of the Higher Himalayan Shear Zone

Abstract

Constraining magnitudes of mechanical and thermo-mechanical parameters of rocks and shear zones are the important goals in structural geology and tectonics (Talbot in J Struct Geol 21:949–957, 1999). Such parameters aid dynamic scaling of analogue tectonic models (Ramberg in Gravity, deformation and the Earth’s crust in theory, experiments and geological applications, 2nd edn. Academic Press, London, 1981), which are useful to unravel tectonics in further details (Schultz-Ela and Walsh in J Struct Geol 24:247–275, 2002). The channel flow extrusion of the Higher Himalayan Shear Zone (HHSZ, = Higher Himalaya) can be explained by a top-to-S/SW simple shear (i.e. the D₂ deformation) in combination with a pressure gradient induced flow against gravity. Presuming its Newtonian incompressible rheology with parallel inclined boundaries, the viscosity (μ) of this shear zone along a part of the Himalayan chain through India, Nepal and Bhutan is estimated to vary widely between ~10¹⁶ and 10²³ Pa s, and its Prandtl number (P _r ) within ~10²¹–10²⁸. The estimates utilized ranges of known thickness (6–58 km) of the HHSZ, that of its top subzone of ductile shear of normal shear sense (STDS_U: 0.35–9.4 km), total rate of slip of its two boundaries (0.7–131 mm year⁻¹), pressure gradient (0.02–6 kb km⁻¹), density (2.2–3.1 g cm⁻³) and thermal diffusivity (0.5 × 10⁻⁶–2.1 × 10⁻⁶ m s⁻²) along the orogenic trend. Considering most of the parameters specifically for the Sutlej section (India), the calculated viscosity (μ) and the Prandtl number (P _r ) of the HHSZ are deduced to be μ: ~10¹⁷–10²³ Pa s and P _r  ~ 10²²–10²⁸. The upper limits of the estimated viscosity ranges are broadly in conformity with a strong Tibetan mid-crust from where a part of the HHSZ rocks extruded. On the other hand, their complete ranges match with those for its constituent main rock types and partly with those for the superstructure and the infrastructure. The estimated mechanical and thermo-mechanical parameters of the HHSZ will help to build dynamically scaled analogue models for the Himalayan deformation of the D₂–phase.

Appendices

Appendix 1

Questions raised about the channel flow extrusion mechanism of the HHSZ

(i) The ‘hairpin’ P–T-t path of the Higher Himalaya could also be generated by burial followed by extrusion by any other mechanism (review by Harris 2007). More specifically, the thermal, metamorphic and the chronologic evolution of the Langtang section of the HHSZ in central Nepal fit with the critical taper model. In contrast, the P–T evolution of the Barun Gneiss inside the HHSZ, and the temperature profile from the Karnali section (Nepal) of the HHSZ fit with a channel flow model (Groppo et al. 2012; Yakymchuk and Godin 2012). (ii) The vertical extents of the major Himalayan thrusts at depth are speculative for two reasons. First, the depths that the two faults bounding the HHSZ, viz. the Main Central Thrust (MCT) and the South Tibetan Detachment System-Upper (STDS_U), reached and confined the channel flow has remained unconstrained (Sharma 2009). Secondly, whereas in the western Himalaya, the Main Himalayan Thrust (the base of the subhorizontal channel) is well established, its presence in the eastern Himalaya has been questioned by Kayal (2008) based on studies on micro-earthquakes. Finally, at the Kathmandu klippen, the MCT and the leucogranites indicative of channel flow are not observed (Jhonson and Harley 2012); (iii) The geographic extents of partially molten rocks within the Asian as well as the Eurasian plates have remained indeterminate. This is because (iiia) Sudha et al. (2011) deciphered low density materials within fractures near the MCT in the Alaknanda river section (India), and Seshunarayana et al. (2011) from the Bhagirathi section (India); (iiib) based on geochemical studies, Guo and Wilson (2011) recently proposed that the Higher Himalayan leucogranites are derived from a melt of >80 % from the HHSZ itself and <20 % from the Lesser Himalayan rocks by metasomatic replacements. (iiic) a lower crustal zone of low viscosity (of the order of 10¹⁶ to 10¹⁷ Pa s) has been envisaged in Mongolia (Vergnolle et al. 2003) but that is ~2,000 km northeast to the HHSZ. (iv) Seismic studies of southern Tibet revealed that the low velocity zones in mid-crust, a possible indicator of rocks in a partially molten state (Zhao et al. 2004), occur as discontinuous pods and cannot support channel flow throughout the Himalaya (Hetényi et al. 2011; Zhang et al. 2012). A ‘soft Tibet’ model required for channel flow is not supported by geophysical evidences (Tapponnier 2012). (v) A low Poisson’s ratio of 0.24 below NE Tibet possibly indicates a felsic rheology devoid of any flow (Pan and Niu 2011). (vi) Crustal thickness of the Tibet can be explained by Cenozoic shortening and channel flow is not a requisite (Lease et al. 2012 and references therein). Harrison (2006) argued that (viia) the crustal low velocity zone could be due to aqueous phases rather than partially molten rocks (also, reviews by Unsworth 2010, Yang et al. 2012); and (viib) no zircon from the Gangdese batholith, characteristic of the partially molten rocks, has been documented from the HHSZ. However, Jamieson et al. (2006) negated the second argument by using the ‘material tracking method’ of modelling to show that ~30 Ma of channel flow could not expose the Asian materials on the surface. Though it is still unclear whether the low velocity zone below the southern Tibet is a genuine indication of partially molten rocks (Bai et al. 2010’s review), a mixture of aqueous phases and partially molten rocks can also sustain a flow (Unsworth 2010).

(viii) Finite element modelling of the Tibetan tectonics reveals a high viscosity of ~5 × 10²³ Pa s at mid-crustal depth, much higher than a previous estimate of 10¹⁹–10²¹ Pa s by Hilley et al. (2005), which has been considered to be unsuitable for channel flow (Copley et al. 2011). (ix) Considering the rate of uplift and the elevation of the Tibetan plateau, Rey et al. (2010) deduced a Moho temperature of 500–600 °C before the plateau thickened. This temperature was considered by them unsuitable for a long-distance channel flow. (x) Based on U-Th–Pb monazite dates, it has been shown that the HHSZ in the eastern Himalaya extruded by channel flow and critical taper mechanisms in a flipping mode (Beaumont and Jamieson 2010; Chambers et al. 2011). This could hold true for the Himalayan orogen where protracted erosion between ~16–18 Ma and at ~3 Ma led to channel flow, and waning of erosion in the intervening phases allowed a critical taper situation (Beaumont and Jamieson 2010; Clift 2010). Even if one considers that channel flow operated within certain time intervals, whether it is persisting at present is unknown (Kohn 2008; Imayama et al. 2010). (xi) Lithological correlation between the Indian basement and the Himalaya led Yin et al. (2007) to conclude that unlike channel flow, the eastern Himalaya underwent thick skinned tectonics. (xii) Integrating structural, metamorphic and tectonic data, Herman et al. (2010) favoured the ‘duplex model’ over the channel flow model. (xiii) Carosi et al. (2010) argued that merely 2- to 4-km-thick Higher Himalayan Crystalline in Lower Dolpo (western Nepal) cannot support channel flow. Secondly, activation of the Toijem Shear Zone inside the Higher Himalaya at ~26 Ma much before than that of the MCTZ and the STDS_U at ~23–17 Ma does not fit with a simple channel flow of the Higher Himalaya there. (xiv) The channel flow model assumes that flow takes place inside rigid boundaries. However, Mandal et al. (2009) argued that phyllites of the Daling Group as the lower boundary of the channel in the Sikkim–Darjeeling Himalaya is deformed ductilely and did not act as a rigid block; (xv) A zone of flexure slip fold inside the HHSZ in the Dhauliganga section (India) indicated that mere channel flow probably did not operate in that section (Mukherjee 2010c). (xvi) Whether the channel flow can model explain the genesis of sigmoidal, parallelogram and lenticular shear fabrics inside the HHSZ has been questioned (Mukherjee 2009; Mukherjee and Koyi 2010a, b) but not investigated. (xvii) In their fig. 11c, White et al. (2012) demonstrated that slab rollback can well explain Himalayan gneiss domes without any need to introduce channel flow.

Appendix 2

The ‘Poisson equation’ of rectilinear flow of an incompressible Newtonian viscous fluid in the z-direction through a very long parallel rigid boundary inclined shear zone is given by (Eq. 6.190 of Papanastasiou et al. 2000):

$$ (\partial^{2} U_{z} /\partial x^{2} ) + (\partial^{2} U_{z} /\partial y^{2} ) = \mu^{ - 1} [\partial P/\partial z - dg\;{\text{Sin}}\theta ] $$

(1)

‘x’ and ‘y’ are perpendicular directions that lie on the cross-section of the shear zone; U _z —fluid along z-direction; ‘μ’—fluid viscosity; (∂P/∂x)—pressure gradient leading t o extrusion; ‘d’: fluid density; ‘g’: gravitational acceleration; and ‘θ’: shear zone dip.

Considering only the YZ section, (∂ ² U _z /∂x ²) = 0. Therefore:

$$ (\partial^{2} U_{z} /\partial y^{2} ) = \mu^{ - 1} [\partial P/\partial z - d \, g{\text{ Sin}}\theta ] $$

(2)

Integrating twice, considering the shear zone to be of 2y ₀ units thick, and at y = y ₀, U _z  = −U ₁ and at y = −y ₀, U _z  = U ₂ gives the profile:

$$ U_{z} = 0.5\mu^{ - 1} (\partial P/\partial z - d \, g{\text{ Sin}}\theta )\left( {y^{2} -y_{0}^{2} } \right) + 0.5\left\{ {\left( {U_{2} -U_{1} } \right) - y \, y_{0}^{ - 1} \left( {U_{1} + U_{2} } \right)} \right\} $$

(3)

Being a quadratic equation, it represents a parabola, whose vertex has the following coordinate:

x-ordinate: 0.5 (U ₁ − U₂) + 0.125 μ y ⁻²₀ (U ₁ + U ₂)² (∂P/∂z − d g Sinθ)⁻¹ − 0.5 y ²₀ μ⁻¹ (∂P/∂z − d g Sinθ)

y-ordinate: 0.5 μ y ⁻¹₀ (U ₁ + U ₂) (∂P/∂z − d g Sinθ)⁻¹

The vertex lies inside the shear zone if

$$ y_{0} > 0.5\mu y_{0}^{ - 1} \left( {U_{1} + U_{2} } \right)(\partial P/\partial z - d \, g{\text{ Sin}}\theta )^{ - 1} $$

(4)

It lies on one of the boundaries if

$$ y_{0} = 0.5\mu y_{0}^{ - 1} \left( {U_{1} + U_{2} } \right)(\partial P/\partial z - d \, g{\text{ Sin}}\theta )^{ - 1} $$

(5)

And it lies outside the shear zone if

$$ y_{0} < 0.5\mu y_{0}^{ - 1} \left( {U_{1} + U_{2} } \right)(\partial P/\partial z - d \, g{\text{ Sin}}\theta )^{ - 1} $$

(6)

In case of Eq. (4), the thickness of the STDS_U is

$$ T = [y_{0} --0.5\mu y_{0}^{ - 1} \left( {U_{1} + U_{2} } \right)(\partial P/\partial z - d \, g{\text{ Sin}}\theta )^{ - 1} ]. $$

(7)