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Localized rheological weakening by grain-size reduction during lithospheric extension
Tectonophysics 386 (2004) 117 – 145 www.elsevier.com/locate/tecto
Localized rheological weakening by grain-size reduction during lithospheric extension Tadashi Yamasaki* National Institute of Polar Research, Kaga 1-9-10, Itabashi-ku, Tokyo 173-8515, Japan Received 19 November 2002; accepted 26 May 2004 Available online 9 August 2004
Abstract Grain-size reduction may be a possible mechanism for the origin of localized deformation in the ductile regime. I investigated the effects of grain-size reduction due to dynamic recrystallization, cataclasis, and syntectonic metamorphic reaction on the stress envelope in the lithospheric mantle during extension by using a simple one-dimensional model. In this model, the lithosphere extends uniformly with a constant strain rate, and a fall in rock strength appears as a decrease in stress. Because grain-size distribution at the onset of extension is unknown, I regarded the steady state grain-size due to dynamic recrystallization as the initial size. Then, I evaluated the maximum effects of grain-size reduction by dynamic recrystallization during extension, and consequently examined the effects of grain-size reduction by cataclasis and metamorphic reaction under conditions when dynamic recrystallization occurs significantly. I find that it is difficult to bring about localized rheological weakening by grainsize reduction owing to dynamic recrystallization. In contrast, grain-size reduction by cataclasis can cause localized weakening during extension. There is a wide-ranging rate of grain-size reduction by means of cataclasis that causes localized weakening just below the Moho. I specified the reaction from spinel-lherzolite to plagioclase-lherzolite that plays a role in grain-size reduction by syntectonic metamorphism. The reaction occurs at depths less than 35 km, which is independent of the initial thermal state of the lithosphere. Localized rheological weakening can occur if the following conditions are satisfied: (1) grain-size before the reaction is greater than 0.7 mm under dry conditions and greater than 0.5 mm under wet conditions, and it decreases down to those values by the reaction; (2) grain-size decreases down to less than initial grain-size, when the dominant deformation mechanism is GSS creep at the onset of extension. It is also noted that dry conditions are more favourable for localized weakening. D 2004 Elsevier B.V. All rights reserved. Keywords: Rheological weakening; Grain-size; Dynamic recrystallization; Cataclasis; Spinel lherzolite–plagioclase lherzolite reaction; Lithospheric extension
1. Introduction * Present address: Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland. E-mail address: [email protected]. 0040-1951/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2004.05.006
Localized deformation in the lithosphere is often observed, and it is important for understanding various geological and geophysical phenomena.
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McKenzie (1977), Cloetingh et al. (1989), and Toth and Gurnis (1998) indicated that localized ductile deformation is important for the initiation of plate subduction. Hobbs et al. (1986) and Sleep (1997) emphasized that localized shear deformation could result in the occurrence of earthquakes. In addition, localized deformation may play an important role in determining the style of extension of the lithosphere: pure shear (McKenzie, 1978) or simple shear (Wernicke, 1985). It seems that the most important difference between pure and simple shear models relates to the presence of the lithosphere-cutting shear zone (e.g., Wernicke, 1985). In this study, I examined the origin of strain localization during lithospheric extension. Geological and geophysical observations have demonstrated that localized deformation occurs beneath continental extensional areas. The xenoliths of deformed and undeformed mantle rock directly provide evidence in support of strain localization (e.g., Pike and Schwartzmann, 1977; Cabanes and Mercier, 1988; Downes, 1990). Tectonites and mylonites exposed at the surface also constitute important evidence for the presence of shear zones in the lithosphere. Peridotite mylonites from the Alps (Vissers et al., 1991) and from the North Pyrenean Zone (Vissers et al., 1997; Newman et al., 1999) show that localized deformation occurs in the upper mantle during extension. Seismic reflectivity of shear zones has been investigated by several authors (e.g., Fountain et al., 1984; Wilshire and Kirby, 1989; Ji et al., 1997). Fountain et al. (1984) indicated that mineralogic and petrofabric characteristics of mylonite zones affect seismic reflectivity. Based on studies of mantle xenoliths, Wilshire and Kirby (1989) suggested that velocity contrasts between fractured and unfractured peridotite could be detected by seismic refraction and/or reflection techniques. Ji et al. (1997) showed that the layering of lithologic units with different acoustic impedance could result in strong seismic reflectivity, like that observed in the Morin Shear Zone in the Grenville Tectonic Province. Therefore, seismic reflection profiling may be one of the most attractive techniques for exploring the presence of shear zones within the lithosphere (e.g., Smythe et al., 1982; Brewer et al., 1983; Beach, 1986; Gibbs, 1987; Klemperer and Matthews, 1987; Reston, 1993;
Brun et al., 1992; MONA LISA Working Group, 1997). Strain localization has been successfully modelled in the brittle regime (e.g., Lavier et al., 2000). Brittle deformation mostly occurs in a localized manner. Therefore, it has been well described (e.g., Paterson, 1978). On the other hand, however, deformation in the ductile regime usually occurs homogeneously, and hence mechanisms for strain localization in the ductile regime have been poorly understood (e.g., Poirier, 1980). Localized ductile deformation may be accompanied by localized rheological weakening in the lithosphere, and grain-size refinement may play an important role in reducing the strength of rocks (e.g., Kirby, 1985). In ductile deformation, several creep mechanisms can control deformation, and the mechanism that produces the highest strain rate is the rate controlling one for given temperatures and applied stress conditions. Under relatively low temperatures, high stress and relatively large grain-size, deformation is mainly controlled by grain-size insensitive (GSI) dislocation creep. In contrast, the dominant deformation mechanism is grain-size sensitive (GSS) diffusion creep under moderate to high temperatures, low stress and small grain-size. Whereas Kirby (1983) and Carter and Tsenn (1987) indicated that the GSI creep seems to be dominant in typical lithosphere conditions, many authors recently have been paying attention to GSS creep in light of the many geological and geophysical problems occurring in the lithosphere (e.g., Rutter and Brodie, 1988; Hopper and Buck, 1993; Kameyama et al., 1997; Braun et al., 1999; Monte´si and Hirth, 2003; Hall and Parmentier, 2003). Rutter and Brodie (1988) indicated that GSS creep could play an important role in strain localization. Hopper and Buck (1993) discussed the role of GSS creep in the uppermost mantle for the initiation of rifting. Kameyama et al. (1997) suggested the importance of GSS rheology for the development of the shear zones. Braun et al. (1999) investigated the conditions for the transition from GSI creep to GSS creep due to dynamic recrystallization during progressive shear deformation in olivine-rich rocks, and concluded that localized deformation could occur under very specific conditions. Most recently, from the simulations of laboratory experiments, Monte´si and Hirth (2003) showed that the grain-size evolution can make the dominant
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creep mechanism changed, and they indicated that the ductile shear zone including the transient strainweakening or -hardening may be important for the postseismic deformation. In addition, Hall and Parmentier (2003) investigated the importance of GSS rheology for the formation of convective instability. GSS creep is extremely sensitive to the size of mineral grain, and to keep the strain rate constant, the required stress significantly decreases with a decrease in grain-size. In the deformational regime where GSI creep is dominant, if the grain-size is reduced significantly by some means, the rate-controlling creep mechanism may shift from GSI creep to GSS creep, followed by a significant reduction of rock strength (see Fig. 1). Some observations in strongly deformed shear zones show evidence for weakening due to this type of grain-size reduction (e.g., Vissers et al., 1995; Jin et al., 1998; Newman et al., 1999). In addition, recent experiments on olivine showed that a reduction in grain-size results in significant attenuation of seismic shear waves (e.g., Jackson et al., 2002; Tan et al., 2001). Therefore, localized ductile shear zones in the lithosphere caused by grain-size reduction could be observed by seismic study.
Grain Size Fig. 1. Schematic diagram showing the effects of grain-size reduction on rheological weakening (after Rutter and Brodie, 1988). The thick solid line represents the boundary between grain-size insensitive (GSI) dislocation creep and grain-size sensitive (GSS) diffusion creep. Thin solid lines show constant strain rates (e1˙ Ne˙2Ne3˙ Ne4˙ ). When grain-size becomes smaller than the transition grain-size by some means, significant rheological weakening is occurred. The dashed arrows show the weakening effects of grain-size reduction via a shift in the dominant deformation mechanism from GSI creep to GSS creep. If the stress is set to be constant (A to B), the weakening due to grain-size reduction occurs by increase in strain rate. On the other hand, if the strain rate is constant (A to C), the weakening occurs by reduction in stress.
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Previous studies on strain localization in the ductile regime during lithospheric extension were based on first-order rheological strain softening (Govers and Wortel, 1993, 1995; Frederiksen and Braun, 2001). Govers and Wortel (1995) considered grain-size reduction by dynamic recrystallization as a mechanism for reducing rock strength. In their modelling study, it was assumed that a shift of the dominant creep mechanism in the mantle from GSI creep to GSS creep starts after 10% strain, and its transition is completed when the amount of strain achieves 50%. Although they concluded that the shear zone could develop due to strain softening, rock strength controlled by the grain-size evolution as a function of stress and temperature was not considered. More recently, Frederiksen and Braun (2001) performed a similar calculation using a simple strain–viscosity relationship to investigate the effects of strain required to initiate strain softening, the magnitude of strength reduction, and the rate of strain softening. Whereas they successfully demonstrated the occurrence of lithosphere-scale simple shear deformation during extension, the concrete mechanisms for the strain softening were not considered. Therefore, they did not reveal what controls these parameters. In this study, I focus on the localized rheological weakening in the ductile regime due to grain-size reduction in the lithospheric mantle during extension. I specifically investigate the effects of grain-size reduction brought about by dynamic recrystallization (e.g., Karato et al., 1980; Kirby, 1985; Rutter and Brodie, 1988; van der Wal et al., 1993; De Bresser et al., 1998; Rutter, 1999), cataclasis (e.g., Hippler and Knipe, 1990; Goodwin and Wenk, 1995; Bos and Spiers, 2001), and syntectonic metamorphic reaction (e.g., Brodie and Rutter, 1987; Handy, 1989; Drury et al., 1991; Vissers et al., 1997; Newman et al., 1999). I examine how lithospheric stress envelopes may be modified by grain-size reduction with respect to these three mechanisms, and explore the possibility as to whether localized rheological weakening, favourable for strain localization, can occur or not. In addition, it may be important to note that I evaluate the effect of grain-size change on the profiles of strength in the lithosphere during thermal cooling. Based on the model results in this study, I will provide several implications for the rifting process.
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2. Model descriptions
Table 1 Model parameter values used in this study
The schematic figure of a simple one-dimensional model adopted in this study is shown in Fig. 2. In this model, the whole lithosphere is thinned by a pure shear manner in response to an applied constant strain rate. Because localized rheological weak zones appear as smaller stress regions under the constant strain rate condition, I investigate the temporal evolution of lithospheric stress envelope during the extension. Ductile stress is assumed to be dependent not only on strain rate and temperature but also on mineral grain-size. Grain-size evolutions owing to dynamic recrystallization, cataclasis and syntectonic metamorphic reaction are included in the model. Grain-size and stresses are evaluated at the moving material point where the thermal cooling occurs during the extension. Because I focused on ductile weakening by grain-size reduction in the mantle, ductile deformation of the crust is assumed to occur only by the GSI creep mechanism. I considered a stratified lithosphere with three layers (see Fig. 2): a quartzite upper crust (Koch et al., 1989), an anorthite lower crust (Shelton and Tullis, 1981) and an olivine mantle (Karato et al., 1986). Although more recent results of flow law parameters on olivine than Karato et al. (1986) are
Parameter
Meaning
Value
H j q uc q lc qm a Cp R t uc t lc D p
Crustal heat production Thermal diffusivity Density of upper crust Density of lower crust Density of mantle coefficient of thermal expansion Specific heat Universal gas constant Initial thickness of upper crust Initial thickness of lower crust Constant of d l Stress exponent of d l
6.47107 W m3 106 m2 s2 2800 kg m3 2900 kg m3 3300 kg m3 3.0105 8C1 1300 J kg1 K1 8.314 J mol1 K1 15 km 15 km 15 mm MPa1.33a 1.33a
a
van der Wal et al. (1993).
available (Mei and Kohlstedt, 2000a,b; Hirth and Kohlstedt, 1995), the differences among them, however, seem to be minor. Therefore, the model results in this study are mostly independent on the used flow law parameters. 2.1. Thermal calculation Temperature distribution in the model must be calculated at any stage of deformation because both GSI and GSS creeps are functions of temperature. The temperature distribution during extension is controlled by the following one-dimensional heat transport equation: BT B2 T BT H ¼j 2 m þ Bt Bz Bz qCp
Fig. 2. Simple one-dimensional model of lithospheric extension used in this study. Pure shear deformation is assumed, and so the vertical velocity of lithospheric material is a linear function of depth. The applied extensional strain rate is assumed constant in the model. Thermal boundary conditions are such that the temperature at surface and bottom of the lithosphere is maintained to 0 and 1350 8C, respectively. Three-layered lithosphere is considered; a quartzite upper crust, an anorthite lower crust and an olivine mantle. Both GSI and GSS creeps are adopted in the mantle, but only the GSI creep is adopted in the crust.
ð1Þ
where T is the temperature, t is the time, j is the thermal diffusivity, z is the depth, H is the amount of internal heating due to radioactive elements, q is the density, and C p is the specific heat. m is the vertical velocity of material and is related to the strain rate e˙ by m=e˙ z for pure shear deformation. Stretching factor b is defined by b=exp(e˙ t). I assume a uniform distribution of radiogenic elements in the entire crust and ignore the presence of heat sources in the mantle. Parameter values used in this study are shown in Table 1. The initial temperature distribution is obtained by the steady state solution of Eq. (1) with a given mantle heat flux q m and a fixed surface temperature of 0 8C. The initial depth of the bottom of the lithosphere is
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defined by the depth of 1350 8C isotherms, and the lithosphere thickness is dependent on mantle heat flow. In the time-dependent calculation, the upper and lower boundaries of the model, i.e., the lithosphere, are maintained at constant temperatures of 0 and 1350 8C, respectively. I solved Eq. (1) by using an explicit 4-point upwind finite difference method (Fletcher, 1991). I assumed that an initial crustal thickness is 30 km, and the thickness of the upper crust is 15 km. In most calculations, I adopted the initial lithospheric thickness to be 125 km. However, in the calculations for investigating the effects of spinel- to plagioclaselherzolite reaction, several different values of the initial lithospheric thickness have been adopted, because the lithospheric thickness is very important through the geometrical relationship between the geotherm and the Clapeyron curve. 2.2. Stress calculation The deformation mechanism of rocks is, amongst others, sensitive to its composition, temperature, pressure and strain rate. Whether deformation occurs by brittle or ductile manner is dependent on the stress level owing to the two deformation mechanisms; the mechanism that gives a smaller stress at an applied strain rate is the controlling one. Therefore, the stress at each depth in the lithosphere is given by the lesser of the brittle and ductile stresses. At low temperature and pressure conditions, applied stress makes the rocks deform in a brittle fashion. The brittle stress would be controlled by frictional sliding on existing faults rather than by fractures (e.g., Goetze and Evans, 1979; Carter and Tsenn, 1987). The stress-level required to exceed frictional resistance on a fault is insensitive to rock type and temperature, and it is given by the Byerlee’s (1978) law. Following Brace and Kohlstedt (1980), I simply expressed the Byerlee’s law under tensional stress states as an approximated form of rb ¼ Bð1 kÞz
ð2Þ
where r b is the brittle stress, B is a constant with a value of 24 MPa/km and z is the depth in km. k is the density ratio of pore water to rock matrix of which value used in this study is 0.38 under wet conditions and zero under dry conditions (e.g., Ranalli, 1995).
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On the other hand, at high temperatures and pressures, rocks respond to the applied stress by ductile flow. I assumed that ductile flow is controlled by both GSI and GSS creep, and these two creep mechanisms are governed by the same extensional stress. Therefore, the total strain rate e˙ is given by the sum of the strain rate due to GSI creep e˙gsi and the strain rate due to GSS creep e˙gss and as, e˙ ¼ e˙ gsi þ e˙ gss
ð3Þ
The strain rate due to GSI creep is a function of temperature and stress, and it has a nonlinear dependence on stress: Qgsi e˙ gsi ¼ Agsi rnd exp ð4Þ RT where r d is the ductile stress, A is a constant, Q is the activation energy, R is the universal gas constant, T is the absolute temperature, and n is the power of stress. The strain rate due to GSS creep is dependent not only on temperature and stress, but also on grainsize, and it is expressed by the following rheological law: Qgss e˙ gss ¼ Agss rd d m exp ð5Þ RT where d is the average mineral grain-size and m is the power of grain-size. The ductile stress is calculated using Eqs. (3)–(5) for given strain rate, grain-size, and temperature. The rheological parameters used in this study are given in Table 2. In Fig. 3, I depict ductile stress as a function of temperature for different grain-size. Rheological parameters are both for dry and wet olivines (Karato et al., 1986). Applied strain rate is 51015 (1/s). Here, I define the ratio of the strain rate due to the GSI creep mechanism to total strain rate as rue˙gsi/ (e˙gsi+e˙gss). If the GSI creep ratio r is more than 0.5, the deformation occurs mainly by GSI creep. In contrast, the deformation is entirely due almost to GSS creep when the GSI creep ratio is less than 0.5. Smaller ductile stress is predicted for higher temperature in both GSI creep and GSS creep regimes. If grain-size and temperature are fixed to be constant, the ductile stress is larger for dry condition. For a given temperature, lower stress is predicted for smaller grain-
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Table 2 Parameters of GSI creep and GSS creep used in this study Meaning
Quartza
Anorthiteb
Olivinec
Wet GSI creep flow parameters A gsi (Pan s1 mmm) n m Q gsi (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
1.11021 2.61 0 145
5.61023 3.2 0 238
1.91015 3 0 420
Dry GSI creep flow parameters A gsi (Pan s1 mmm) n m Q gsi (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
5.581024 2.72 0 134
5.61023 3.2 0 238
2.41016 3.5 0 540
Wet GSS creep flow parameters A gss (Pan s1 mmm) n m Q gss (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
– – – –
– – – –
1.5109 1 3 250
Dry GSS creep flow parameters A gss (Pan s1 mmm) n m Q gss (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
– – – –
– – – –
7.7108 1 2 290
a b c
Koch et al. (1989). Shelton and Tullis (1981). Karato et al. (1986).
size in the domain where the deformation occurs mainly by GSS creep. However, the stress is almost independent on grain-size in the domain where GSI creep is a dominant deformation mechanism. 2.3. Grain-size change as a result of dynamic recrystallization It is widely believed that dynamic recrystallization can cause significant grain-size reduction. Dynamic recrystallization occurs due to grain boundary migration and subgrain rotation (e.g., Guillope´ and Poirier, 1979; Poirier, 1980; Hirth and Tullis, 1992). Kirby (1985) and Rutter and Brodie (1988) indicated that transition from GSI creep to GSS creep brought about by dynamic recrystallization could play an important role in the localization of ductile deformation. Kameyama et al. (1997) and Braun et al. (1999) investigated the effects of grain-size change by dynamic recrystallization on the evolution of the shear deformation. In their models, it is assumed that
the stress is uniform all over the considered volume. Therefore, the distribution of grain-size in the volume would be getting uniform as a progression of the deformation, and the deformation also becomes uniform. This is because the governing equation for the recrystallized grain-size is a function of only the stress level. In their studies, the occurrence of localized weakening depends on the initially given grain-size distribution, and thus they investigated whether the localized weak zone could lead the strain localization or not. On the other hand, in the present study, the stress and its dependent grain-size cannot become uniform within the lithosphere, because the applied strain rate is assumed to be uniform in the entire lithosphere and the temperature increases with depth. Then, I investigate whether the localized weak zone can be formed by the grain-size reduction during the uniform extension of the lithosphere or not. In order to calculate the temporal evolution of grain-size, I used an equation in which the grain-size d tends toward d l with increasing strain, and the rate
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Fig. 3. Stress contour as a function of temperature and grain-size. Rheological parameters are for olivine (Karato et al., 1986): (a) dry condition and (b) wet condition. Applied strain rate is 51015 (1/s). The thick dashed line indicates that rue˙gsi/(e˙gsi+e˙gss) is 0.5. The bGSI regimeQ indicates that the deformation occurs mainly by GSI creep (i.e. rN0.5), while the bGSS regimeQ indicates that the deformation occurs mainly by GSS creep (i.e. rb0.5). The thin solid curves are lines of constant stress. The numbers attached to each contour indicate the value of stress in the unit of MPa.
of grain-size change is controlled by (e.g., Kameyama et al., 1997; Braun et al., 1999), e˙ d˙ ¼ ðd dl Þ eT
ð6Þ
where e˙ is the strain rate and e T and is the amount of strain required for grain-size change. Karato et al. (1980) and van der Wal et al. (1993) experimentally proposed quantitative relationships between applied stress and recrystallized grain-size for olivine, and showed that grain-sizes from 0.5 to less than 0.01 mm can be expected for upper mantle conditions. In their studies, recrystallized grain-size is deduced from piezometric measurements in a form dependent only upon stress-level, as dl ¼ Drp
creep. Based on this hypothesis, d l is determined by the following equation (e.g., De Bresser et al., 2001; Gueydan et al., 2001):
ð7Þ
where D and p are constants (see Table 1). Smaller grain-size for higher stress is predicted by Eq. (7). Recently, De Bresser et al. (2001) suggested the hypothesis that dynamic recrystallization tends toward a balance between grain-size reduction and grain growth processes, and the average recrystallized grain-size will be stable at the condition that the strain rate due to GSI creep is equal to that due to GSS
dl ¼
Agss Agsi
m1
ln
r m exp
Qgsi Qgss mRT
ð8Þ
It is noted that whereas Eq. (7) is fully empirical one, Eq. (8) is modification of Eq. (7) with respect to the microstructural development. The grain-size evolution predicted by Eq. (8) is as follows; in the regime where GSI creep is dominant, present grain-size is larger than recrystallized grain-size and so reduction of grain-size would occur. In a GSS creep dominant regime, conversely, grain-size increases because present grain-size is smaller than recrystallized grain-size. This hypothesis is supported by experiments on a Mg alloy (De Bresser et al., 1998), and De Bresser et al. (2001) demonstrated that independently obtained relations between stress and recrystallized grain-size for olivine (Post, 1977; Karato et al., 1980; Ross et al., 1980) and calcite (Schmid et al., 1980; Friedman and Higgs, 1981; Rutter, 1995) correspond to the GSI– GSS transition region. Hence, it seems reasonable to adopt Eq. (8) for dynamically recrystallized grain-
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size. In most calculations in this study, I used Eq. (8) to examine the effects of grain-size change by dynamic recrystallization on the stress envelope in the lithosphere. For the purpose of comparison, however, I also showed results for cases where recrystallized grain-size is governed by Eq. (7). Experimental studies of Karato et al. (1980) and De Bresser et al. (1998) suggested that steady-state flow was achieved after about 40% strain. On the other hand, van der Wal et al. (1993) reported that for shortening experiments the steady state recrystallized grain-size for a given applied stress is obtained after about 3% strain. However, values for parameter e T are not well constrained. In order to evaluate the effects of e T the two different values, 0.03 and 0.4, are adopted in this study. The timescale for the grain-size change by the dynamic recrystallization is dependent on its value. Therefore, it is important to vary e T for evaluating the competition between the effects of dynamic recrystallization and thermal cooling. Twiss and Sellars (1978) discussed a lower limit of stress for dynamic recrystallization. The occurrence of dynamic recrystallization requires a condition where the decrease in free energy due to the removal of dislocations must exceed the increase in free energy caused by creation of a new grain boundary. The results of theoretical calculations indicated that if an applied stress level is such that recrystallized grainsize will be larger than the order of 10 mm (typical value for usual observed mantle xenoliths) dynamic recrystallization might not occur (Twiss and Sellars, 1978). Therefore, I set the upper limit of grain-size to be 10 mm. If recrystallized grain-size d l is larger than its maximum value, the rate of change in grainsize d is adopted to be zero in this study. Because there is no direct way to obtain the initial size of mineral grains in the lithosphere, I regarded the steady state grain-size due to dynamic recrystallization as the initial size. Recrystallized grain-size is dependent on stress-level, and stress-level changes with grain-size if the applied strain rate and temperature is fixed to be constant. Therefore, steady state grain-size for a given strain rate and temperature settles down in the intersection between the constitutive equation (Eq. (3)) and the equation governing recrystallized grain-size (Eq. (7) or Eq. (8)). Whereas Karato (1989a) suggested that cyclic softening–hardening could occur because of grain-size reduction and
subsequent grain growth, such cyclic events cannot occur if stress changes with grain-size. Thus, I can obtain steady state grain-size and ductile stress at any depth in the lithosphere using the initial steady state geotherm and the applied extensional strain rate, even when initial grain-size is unknown. In fact, however, Ter Heege et al. (2002) showed that the strengthening and subsequent weakening can be brought about by a change in dominant creep mechanism from GSI creep to GSS creep as the number of small grains in a volume increases with strain. It should be, therefore, emphasized that the calculations in this study are done under the condition that an average grain-size, not grain-size distribution, is considered at any depth. In Fig. 4, I show the behaviour of Eqs. (3), (7) and (8) as a function of grain-size and stress for dry and wet conditions. Applied strain rate is 51015 (1/s). When the temperature is 800 8C, the steady state grain-size governed by Eq. (8) and the corresponding ductile stress are 3.85 mm and 63 MPa for dry condition, respectively. For wet condition, the grain-size and the stress are 1.43 mm and 7.17 MPa, respectively. At a lower temperature of 600 8C, the steady state grainsize becomes smaller both for dry and wet conditions. If the initial steady state ductile stress is larger than the brittle stress, I adopt the actual stress to be brittle stress and grain-size at the depths to be 10 mm. During the progression of extension by pure shear manner, the lithosphere thins out and any material within it cools. Because pressure decreases with the upward movement of the materials and temperature decreases by thermal diffusion, materials, initially situated in the brittle regime, would always be deformed in a brittle fashion during extension. Therefore, the decision for initial grain-size in the brittle regime is not problematic as far as the grainsize changes by dynamic recrystallization. 2.4. Grain-size change as a result of cataclasis De Bresser et al. (2001) emphasized that cataclasis is a possible mechanism for the reduction of grain-size leading to rheological weakening. On the basis of an experimental study, Bos and Spiers (2001) suggested that weakening of fault rocks through a switch in the deformation mechanism from rate-independent behaviour to GSS creep could be caused by grain-size reduction due to cataclasis. Based on field observations
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Fig. 4. Behaviour of Eqs. (3), (7) and (8) as a function of grain-size and stress: (a) and (c) dry condition; and (b) and (d) wet condition. Applied strain rate is 51015 (1/s). Temperature is (a) and (b) 800 and (c) and (d) 600 8C. Because ductile stress changes with grain-size if the applied strain rate and the temperature are held to be constant, steady state grain-size is obtained at the intersection between Eqs. (3) and (7) or Eq. (8).
and laboratory experiments, Goodwin and Wenk (1995) indicated that grain-size reduction as a result of cataclasis would play an important role in the formation of the Santa Rosa mylonite zone. Whereas the cataclastic nature of grain-size reduction may not be apparent in natural rocks because diffusion processes can hide the microstructural evidence of cataclasis, such as angular grains and microcracks (Bos and Spiers, 2001), it seems sure that grain-size reduction can occur by cataclasis. In order to understand the process of grain-size reduction by brittle deformation, many studies have tried to analyse grain-size distribution of natural and experimental fault rocks (Sammis et al., 1986; Biegel et al., 1989; Marone and Scholz, 1989; Hippler and Knipe, 1990; Blenkinsop, 1991; Ray, 1999). However, it seems that the relationship between
grain-size reduction and the magnitude of displacement on a fault plane is poorly constrained. Because it is unknown how the grain-size is reduced with increase in strain, it is hard to incorporate the effects of grain-size reduction by cataclasis into the model. So then, I investigated its effects on the stress envelope in a following way. In the initial distribution of steady state grain-size due to dynamic recrystallization, I adopted several different values of d c for the grain-size in the brittle regime. In time-dependent calculations, I drew the stress envelope at any stage of deformation without grain-size change, and tried to find the critical stretching factor b c required for keeping the localized weak zone to be present. Then, it may be reasonable to state that localized rheological weakening can occur if the
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grain-size is reduced to d c before the stretching factor reaches b c. As described above, brittle deformation is typically due to the slip of pre-existing fault, and therefore it intrinsically occurs in localized manner. However, the deformation in the brittle regime occurs homogeneously over the scale of the entire lithosphere (e.g., Blenkinsop and Rutter, 1986). In the present study, I evaluated whether the localized reduction in ductile stress can be occurred by the grain-size reduction due to cataclasis during uniform deformation in the brittle regime. 2.5. Grain-size change as a result of metamorphic reaction It has been recognized that syntectonic metamorphic reactions could cause reduction of grain-size (e.g., Brodie and Rutter, 1987; Handy, 1989; Drury et al., 1991). The most well-known reactions in the uppermost mantle are garnet–spinel–plagioclase lherzolite reactions. Specifically, I focused on grain-size reduction associated with the reaction from spinel-lherzolite to plagioclase-lherzolite. In fact, the Turon de Te´coue`re peridotite in the North Pyrenean Zone has a very small grain-size (b10 Am), and Newman et al. (1999) concluded that such fine grain would be caused by the spinel- to plagioclase-lherzolite reaction. Similarly, Furusho and Kanagawa (1999) reported that the formation of lherzolite mylonites derived from the Hidaka metamorphic belt of central Hokkaido, Japan, is associated with the phase transformational reaction from spinel- to plagioclase-lherzolite. The spinel- to plagioclase-lherzolite reaction occurs with decreasing pressures, and therefore the continuous reaction described for these rocks may be important when upper mantle rocks move upwards to shallower levels, such as during lithospheric extension. The reaction boundary is determined by the intersection between the geotherm and Clapeyron curve. Many experimental and theoretical studies have attempted to determine the Clapeyron curve for spinel- to plagioclase-lherzolite reaction (e.g., Obata, 1976; Wood and Yuen, 1983; Gasparik, 1987). Wood and Yuen (1983) calculated a nonlinear Clapeyron curve for an idealized CaO–MgO–Al 2 O 3 –SiO 2 (CMAS) system. In this simplified system, the spinel–plagioclase boundary is univariant, and there-
fore the equilibrium reaction boundary is sharp. For more realistic mantle composition in CMAS+Na+Fe system (CMASNF), on the other hand, the univariant curve is broadened into a divariant P–T band (see Fig. 5) because of differential partitioning of additional components between the spinel and plagioclase phases. Thus, the reaction does not occur suddenly as in the CMAS system, but it is smeared out over an interval of depth. In this study, I used the result of theoretical calculation for the CMASNF system (Wood and Yuen, 1983). Once the spinel-lherzolite moves into the plagioclase stability field, the reaction could occur and consequently, grain-size reduction could happen. Therefore, I assume that grain-size reduction occurs when the material passes the boundary at which the geotherm intersects the higher-pressure limit of the plagioclase stability field. For metamorphic reactions, the grain-size of the product is controlled by a balance between the rates of nucleation and grain growth. If the rate of nucleation is higher than that of grain growth, smaller grain-size is obtained. Nucleation and growth rates will be strongly dependent on the cooling rate if deformation is absent. Because a high cooling rate will result in larger overstepping of the reaction, resultant nucleation and grain growth rates become higher and slower, respectively. If reaction occurs during deformation, the strain may cause a higher nucleation to grain growth ratio, and a very fine grain-size could be obtained. This is because the increase in dislocation densities and new
Fig. 5. Calculated spinel–plagioclase divariant Clapeyron curves in the CMAS+Na+Fe system by Wood and Yuen (1983).
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grain boundaries can promote nucleation even if activation energy is low (e.g., Brodie, 1980). A fast strain rate may be favourable for a fine grain-size, because fast transport of reaction products away from the reactants would prevent further reaction (e.g., Newman et al., 1999). In addition, the polyphase nature of the rocks may help to maintain a very fine grain-size by pinning of grain boundaries (e.g., Olgaard and Evans, 1988; Olgaard, 1990). In spite of these qualitative discussions (for more detailed discussions see Newman et al., 1999), it is actually unknown how much grain-size reduction occurs with the spinel- to plagioclase-lherzolite reaction. Whereas Riedel and Karato (1996) incorporated the growth rate Y and the nucleation rate I into the model and investigated the grain-size evolution associated with olivine to spinel phase transformation, very complex system for the change in grain-size by spinel- to plagioclase-lherzolite reaction prevents us to know the exact value of Y and I. So then, in this study, I assume that the olivine grain-size decreases down to 0.01 mm as a result of the reaction, as confirmed by Newman et al. (1999) on the North Pyrenean Zone mylonite and by Furusho and Kanagawa (1999) on the mylonite in Uenzaru peridotite complex. I will discuss the condition required for localized rheological weakening later. Although the garnet- to spinel-lherzolite reaction is also an important phase transformation in the upper mantle (e.g., Green and Ringwood, 1967), it does not seem common as a mechanism for grain-size-related softening. Additionally, Pe´rez-Gussinye´ and Reston (2001) emphasized the serpentinization of peridotites as a possible origin of strain localization. However, the fall in strength associated with serpentinization is related to its coefficient of friction rather than grainsize reduction.
3. Model results 3.1. Effects of grain-size reduction as a result of dynamic recrystallization 3.1.1. Initial state of grain-size and stress in the lithosphere In Fig. 6, I show initial steady state grain-size due to dynamic recrystallization and the corresponding
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Fig. 6. Initial steady state grain-size and stress in the lithosphere for an applied strain rate of 51015 (1/s). Grain-size distributions are shown in (a) and (c), and stress envelopes are in (b) and (d). Two different equations governing the steady state grain-size by dynamic recrystallization are used; (a) and (b) are for the case of Eq. (8), and (c) and (d) are for the case of Eq. (7). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Because ductile deformation of the crustal material occurs only from GSI creep, grain-size in the crust is not depicted.
stress envelope in the lithosphere. Black and grey lines represent the results for dry and wet conditions, respectively. Because I assume that ductile deformation of the crust occurs only by means of GSI creep, grain-size in the crust is not shown. Fig. 6(a) shows initial grain-size predicted by Eq. (8). For dry condition, grain-size gradually increases from 1 mm at depth of 55 km to 10 mm at depth of 85 km. Because grain-size becomes more than 10 mm at depths greater than 85 km, I presume grain-size to be 10 mm at the depths according to the assumption in
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Fig. 7. Temporal evolution of temperature and stress envelope in the lithosphere: (a) temperature, (b) stress envelope when the recrystallized grain-size is controlled by Eq. (8), and (c) stress envelope when the recrystallized grain-size is controlled by Eq. (7). Applied strain rate is 51015 (1/s). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho.
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this study. For wet condition, grain-size gradually increases from 0.5 mm at depth of 45 km to 5.5 mm at the bottom of the lithosphere. As can be seen in the stress envelopes shown in Fig. 6(b), the localized weak zone between higher stress zones does not appear in the lithospheric mantle, and ductile stress gradually decreases with depth. If a change in grain-size due to dynamic recrystallization is governed by Eq. (7), very small grain-size is predicted around depths of 30 km both for dry and wet conditions (see Fig. 6(c)). This is because the temperature is low at such depths (see also Fig. 4). The grain-size gradually increases with depth, and becomes more than 10 mm at depths greater than 95 km for dry condition and at depths greater than 85 km
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for wet condition. Therefore, I presume grain-size to be 10 mm at such depths according to the assumption in this study. The stress envelopes shown in Fig. 6(d) indicate that localized weakening does not occur in the lithospheric mantle. 3.1.2. Temporal evolutions of grain-size and stress in the lithosphere Because I regard steady state grain-size owing to dynamic recrystallization as the initial size in the present study, I evaluated only the effect of change in stress on grain-size by thermal cooling in the timedevelopment calculation. Temporal evolutions of temperature and stress envelope within the lithosphere are shown in Fig. 7. Applied strain rate is 51015
Fig. 8. Temporal evolution of the GSI creep ratio ((a) and (c)) and grain-size ((b) and (d)) distributions. Applied strain rate is 51015 (1/s). Recrystallized grain-size due to dynamic recrystallization is governed by Eq. (8). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Two different values are adopted for the strain e T: e T=0.4 ((a) and (b)); e T=0.03 ((c) and (d)).
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Fig. 8 (continued).
(1/s). Adopted governing equations for the equilibrium grain-size are Eqs. (7) and (8) for Fig. 7(b) and (c), respectively. Because I focused on the change in grain-size as a result of dynamic recrystallization, the change in grain-size in the brittle regime is ignored. It is assumed that grain-size of the materials gone into the model from the below is the same size at the bottom of lithosphere at the onset of extension. As can be seen in the figures, strength of the lithosphere decreases with time, which is associated with increase in geothermal gradient as a progression of extension. However, the localized weak zone between higher stress zones in the lithospheric mantle does not appear at any depth ranges, which is independent on adopted equations governing the equilibrium grain-size. It seems that the stress changes mostly are not brought about by grain-size change. Strength reduction is mainly caused by the upward movement of each
material point because of lower brittle strength at shallower depth as indicated by Eq. (2). From the temporal evolution of stress envelope (Fig. 7(b) and (c)), however, it may be difficult to understand the effect of thermal diffusion during the extension on the stress change through the grain-size change. So then, I show the temporal evolution of the distribution of the GSI creep ratio (r) and grain-size in the lithosphere (Fig. 8). The change in grain-size due to dynamic recrystallization is governed by Eq. (8). At time=0 my, the GSI creep ratio is constant to be 0.5 with depth for wet condition. This is because the initial grain-size should be given according to the nature of Eq. (8), in which the strain rate due to GSI creep is equal to that due to GSS creep. However, for dry condition, the GSI creep ratio at the onset of extension decreases at depths between 85 and 110 km, and ductile deformation at
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depths greater than 110 km occurs mostly by GSS creep. This is because grain-size does not exceed 10 mm in this study. When e T is 0.4, the ratio of GSI creep becomes more than that of GSS creep at depths between 40 and 75 km after 5 my. This is because stress increases by thermal cooling during extension. If grain-size is held to be constant, the increase in stress results in a higher ratio of GSI creep. Although grain-size will be controlled by Eq. (8) in such a way that the strain rate due to GSI creep is equal to that due to GSS creep, the rate of the change in stress exceeds that of the change in grain-size because finite strain (e T=0.4) is required for dynamic recrystallization. Grain-sizes are almost unchanged by
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the effect of thermal diffusion (see Fig. 8(b)). As seen in Fig. 8(c), on the other hand, the magnitude of increase in the GSI creep ratio for e T = 0.03 is smaller than that for e T = 0.4. For e T = 0.03, grain-size is rapidly recovered to its recrystallized value in response to the change in stress (see Fig. 8(d)). Generally, temporal evolution of the deformation mechanism is such that the ratio of GSI creep increases with time, and the switch from GSI creep to GSS creep does not occur at any depth. In Fig. 9, I show the same figures as Fig. 8 to illustrate the case that the recrystallized grain-size is governed by Eq. (7). At the onset of extension, the mechanism for ductile deformation is mostly GSS creep at any depth for dry condition. However, the
Fig. 9. Temporal evolution of the GSI creep ratio ((a) and (c)) and grain-size ((b) and (d)) distributions. Applied strain rate is 51015 (1/s). Recrystallized grain-size due to dynamic recrystallization is governed by Eq. (7). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Two different values are adopted for the strain e T: e T=0.4 ((a) and (b)); e T=0.03 ((c) and (d)).
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Fig. 9 (continued).
ratio of GSI creep increases at depths between 85 and 95 km along with the increase in grain-size, but at greater depths, it decreases in accordance with the assumption that the maximum grain-size is 10 mm (see Fig. 6(c)). For wet condition, ductile deformation at depths shallower than 55 km occurs mostly by GSS creep at the onset of extension. The ratio of GSI creep gradually increases with depth along with the increase in grain-size, and ductile deformation at depths greater than 75 km occurs mostly by GSI creep. For dry condition, a region where the GSI creep ratio is not zero at time=0 my develops a higher GSI creep ratio with time when e T is 0.4 (see Fig. 9(a)). For wet condition, a region where the ductile deformation occurs mostly by GSS creep finally diminishes after 14 my. Such an increase in the GSI creep ratio is attributed to the increase in stress as a result of thermal cooling. However, the ratio of each
ductile deformation mechanism is mostly constant with time for e T = 0.03 (see Fig. 9(c)), because the change in grain-size associated with the change in stress by thermal cooling occurs more rapidly. Grainsize change for e T = 0.03 occurs much more than for e T = 0.4 (see Fig. 9(b) and (d)). In general, the GSI creep ratio increases with time, and I am unable to find a switch from GSI to GSS creep at any stage of extension under the conditions applied in the numerical experiments. 3.2. Effects of grain-size reduction as a result of cataclasis In Fig. 10, I show the temporal evolution of grainsize, GSI creep ratio and stress envelope in the lithosphere for an applied strain rate of 51015 (1/s). I assume that the initial grain-size is given by the
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Fig. 10. Temporal evolution of (a) grain-size, (b) GSI creep ratio and (c) stress envelope in the lithosphere. An applied strain rate is 51015 (1/s). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Initial grain-size in the brittle regime is adopted to be 0.01 mm, and the temporal change in grain-size is ignored. The initial grain-size is given by the steady state example due to dynamic recrystallization governed by Eq. (8).
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steady state grain-size due to dynamic recrystallization governed by Eq. (8), and hence I examine the effects of grain-size reduction by cataclasis under such condition that dynamic recrystallization significantly occurs. The initial grain-size in the brittle regime, d c, is adopted to be 0.01 mm (see Fig. 10(a)), and the temporal change in grain-size is ignored. At time=0 my, the localized weak zones in the lithospheric mantle are formed at depths between 38 and 55 km under dry condition and at depths between 30 and 45 km for wet condition, where ductile deformation occurs mostly by GSS creep (see Fig. 10(b)). From the figure at time=0 my, it can be said that if grain-size in the brittle regime is instantaneously reduced to 0.01 mm by cataclasis, such localized weakening can be promoted for any size of mineral grain in the brittle regime at the initiation of extension. Similarly, if grain-size is reduced to 0.01 mm during the first 5 my, localized weakening in the lithospheric mantle can occur at depths between 18 and 25 km under dry condition and at depths between 13 and 20 km for wet condition. Furthermore, if grain-size is reduced to 0.01 mm during first 10 my, localized weakening in the lithospheric mantle can occur at depths around 10 km both for wet and dry conditions. Comparing with the results in Section 3.1, it is clear that the grain-size reduction by the cataclasis is very important for the localized rheological weakening in the lithospheric mantle. However, if it takes 14 my to reduce the grainsize to 0.01 mm, localized weakening in the lithospheric mantle cannot occur at any depth. This is because the temperature at shallow depths is too low to cause ductile stress to be less than brittle one even if the grain-size is very small (see also Fig. 10(a)). I show the critical time t c as a function of grain-size d c in Fig. 11, where localized rheological weakening in the lithospheric mantle can occur if grain-size in the brittle regime is reduced to d c before the extensional time reaches t c. The corresponding critical stretching factor b c for strain rate of 51015 (1/s) is also depicted in the figure. The shaded area represents possible conditions for localized weakening in the lithospheric mantle during extension. For example, if grain-size in the brittle regime is reduced to about 0.6 mm under dry conditions and to about 0.3 mm under wet conditions before the extensional time reaches 3 million years (the corresponding stretching factor is 1.6), it is possible for grain-size reduction by
Fig. 11. Relationship between the critical extensional time t c and grain-size d c, in which localized ductile weakening can occur if grain-size in the brittle regime is reduced to d c before the extensional time reaches t c. Shaded area illustrates the conditions for localized weakening in the lithosphere during extension. Corresponding critical stretching factor b c for strain rate of 51015 (1/s) is shown in the figure.
cataclasis to cause the localized weakening. However, if grain-size is reduced at the most to 0.9 mm under dry conditions and to 0.5 mm under wet conditions, the localized weakening cannot occur. In addition, a critical time t c more than 12 million years (the corresponding critical stretching factor b c is about seven) prevents the occurrence of localized weakening even if the grain-size is reduced to 0.01 mm. 3.3. Effects of grain-size reduction as a result of spinel- to plagioclase-lherzolite reaction Temporal evolution of a geometrical relationship between the geotherm and the Clapeyron curve for the spinel–plagioclase reaction is illustrated in Fig. 12. Initial lithospheric thickness is taken as 125 km, and the applied strain rate is 51015 (1/s). At time=0 my, the geotherm does not intersect with the Clapeyron curve. In this state, the whole lithospheric mantle consists of only spinel-lherzolite. After about 6 my, the geotherm intersects with the higher-pressure boundary of the divariant P–T band. As can be seen in the figure, the geotherm and the higher-pressure boundary of the divariant P–T band are crossing each
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Fig. 12. Temporal evolution of the geometrical relationship between the geotherm and spinel–plagioclase Clapeyron curve for an applied strain rate of 51015 (1/s). Divariant P–T band for CMASNF system is depicted in each figure.
other by two points, and the stability domain for plagioclase is between those for spinel. The geometrical relationship between the geotherm and the spinel–plagioclase Clapeyron curve is dependent on the initial thickness of the lithosphere. I show the time-dependent depth of the upper and lower reaction boundaries for the different initial lithospheric thickness of 80, 100, 125, and 150 km in Fig. 13(a). Temperatures at the reaction boundaries are also illustrated in Fig. 13(b). The stretching factor for e˙ =51015 (1/s) is depicted in each figure. As shown
in the figures, the time at the initiation of the reaction is sensitive to the initial thickness of the lithosphere; the reaction starts earlier for thinner lithosphere and vice versa. This is because the initial geothermal gradient is larger for the thinner lithosphere. For the initial thickness of 125 km, the depth of the lower reaction boundary rapidly increases to 33 km, and then decreases slowly to a depth less than 30 km. The depth of the upper reaction boundary, on the other hand, decreases gradually down to about 5 km. Temperature at the lower reaction boundary increases
Fig. 13. Temporal evolution of (a) depth of reaction boundary and (b) temperature at reaction boundary for the different initial lithospheric thickness. Corresponding stretching factor for strain rate of 51015 (1/s) is shown in each figure.
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Fig. 14. Temporal evolution of (a) grain-size, (b) GSI creep ratio, and (c) stress in the lithosphere. Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. The initial thickness of lithosphere is adopted to be 125 km, and the applied strain rate is 51015 (1/s). Initial grain-size is given by the steady state grain-size due to dynamic recrystallization governed by Eq. (8).
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towards an asymptotic value with time, and temperature at the upper reaction boundary decreases down to about 400 8C. This behaviour is mostly insensitive to the initial thickness of the lithosphere. Therefore, it seems that the effect of the grain-size reduction by the reaction is mostly independent of the initial thickness of the lithosphere, but the initial thickness does control the time when the grain-size reduction initiates. In Fig. 14, I show the time-dependent grain-size, GSI creep ratio and stress in the lithosphere. The initial thickness of lithosphere is 125 km, and the applied strain rate is 51015 (1/s). I assume that grain-size reduction occurs when the material passes the boundary where the geotherm intersects the higher-pressure limit of the plagioclase stability field. Initial olivine grain-size is given by the steady state grain-size due to dynamic recrystallization governed by Eq. (8). Therefore, I evaluated the effects of grainsize reduction by the spinel- to plagioclase-lherzolite reaction during which time the dynamic recrystallization occurs significantly. In addition, it should be noted that the reaction products are olivine with a smaller grain-size but are not the development of plagioclase out of spinel. After 5.6 my, grain-size reduction as a result of the reaction occurs at depths between 25 and 30 km (see Fig. 14(a)), and the deformation mechanism at that depth range is mostly GSS creep (see Fig. 14(b)). Localized rheological weakening in the lithospheric mantle appears clearly at those depths both for dry and wet conditions (see Fig. 14(c)), but the magnitude of strength reduction is larger under dry condition. After 6.2 my, localized rheological weakening in the lithospheric mantle can be seen at depths between 18 and 33 km under dry condition. However, the localized weak zone under wet condition disappears on the scale of stress with a few MPa, whereas grain-size is very small and ductile deformation occurs mostly by GSS creep at depths between 18 and 33 km. After 10 my, although the dominant deformation mechanism is GSS creep at depths between 14 and 33 km, the localized weak zone disappears both for dry and wet conditions on that scale. Such disappearance of the localized weak zone on that scale is attributed to the fact that ductile stress is very low for high temperatures at depths greater than 33 km, and the lower boundary for the localized weak zone cannot be seen.
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4. Discussion 4.1. Possibility of localized rheological weakening by grain-size reduction during lithospheric extension In order to explore the possibility if grain-size reduction can cause localized rheological weakening favourable for strain localization, I investigated the effects of grain-size evolution as a result of dynamic recrystallization, cataclasis, and spinel- to plagioclaselherzolite reaction on the stress envelope in the lithospheric mantle. In the simple one-dimensional model adopted in this study, the applied strain rate is constant with time, and therefore the reduction of strength appears as a drop in stress. From the model results, I found that grain-size reduction by dynamic recrystallization cannot cause localized weakening, even when a significant change in grain-size occurs. The change in grain-size by dynamic recrystallization is sensitive to the stress and temperature that continuously change with depth in the lithosphere. It is, therefore, intrinsically difficult to cause rheological weakening at a particular depth range. Because the temperature within the lithosphere gradually increases with depth and it is hard to create the region where the temperature is particularly low or high in the lithosphere, the equilibrium grain-size also gradually changes with depth, if the uniform strain rate is applied over the entire lithosphere. It is, therefore, impossible to cause the localized rheological weakening at a particular depth range. Even if the localized weak zone is initially given, it will become to be smeared out by the effect of dynamic recrystallization and will be finally diminished. That is, the localized rheological weakening cannot occur in the steady state flow. Whereas the transient weakening could occur, this is dependent on the initial distribution of grainsize. The grain-size should be decreased locally at the initial state. In order to show this, it is enough to demonstrate the equilibrium grain-size distribution inferred from the initial temperature distribution and applied strain rate and the time-dependent effect of thermal diffusion during the extension. The effect of thermal cooling on grain-size is not great, whereas the thermal cooling during extension can make ductile stress increased to maintain the strain rate constant. Increase in the GSI creep ratio is attributed to the fact that the rate of change in stress by
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thermal cooling exceeds that of change in grain-size because finite strain is required for the completion of dynamic recrystallization. In fact, the increase in stress by thermal cooling is suppressed more significantly by the effect of dynamic recrystallization for the model with smaller e T (see Figs. 8 and 9). Although I show that grain-size reduction by dynamic recrystallization cannot cause localized weakening, the results in this study do not mean that it cannot cause rheological weakening. Because recrystallized grain-size governed by Eq. (8) is such that the strain rate due to GSI creep is equal to that due to GSS creep, substantial rheological weakening through a switch in the deformation mechanism from GSI creep to GSS creep cannot occur. Weakening can occur only in the GSI creep dominant regime. However, possible rheological weakening is minor, as pointed out by several previous studies (e.g., De Bresser et al., 2001; Ter Heege et al., 2002). The spinel- to plagioclase-lherzolite reaction has potential to cause localized weakening during extension. This reaction occurs at depths shallower than 35 km, and temperatures at the upper and lower reaction boundaries are between about 700 and 400 8C and between 700 and 1300 8C, respectively (see Fig. 13). The time for the initiation of grain-size reduction by the reaction is dependent on initial thickness of the lithosphere; the greater time is required for the thicker lithosphere. Time-dependent temperature at the reaction boundaries, depicted in Fig. 13, would be sensitive to the applied strain rate through the thermal diffusion during the extension. However, it may not be necessary to consider thermal cooling during extension for most of the existing rifted-basins (e.g., Jarvis and McKenzie, 1980). It is well known that the rate of reaction is strongly sensitive to temperature. If the reaction occurs under low temperatures, the rate of reaction is low and hence significant grain-size reduction might not occur in the geologically short time. In this study, I show that the spinel- to plagioclase-lherzolite transformation at the upper reaction boundary occurs at temperatures between about 700 and 400 8C (see Fig. 13(b)). However, Ahrens and Schubert (1975) discussed the temperature-dependent reaction rate of the gabbroeclogite transformation, and concluded that the reaction takes place in geologically short time only when the temperature is above 600 8C. Therefore,
significant grain-size reduction might not be attainable at temperatures less than 600 8C because of low reaction rate at these temperatures. In this study, I presume the grain-size to decrease down to 0.01 mm by the reaction. Although experimental studies on the amount of grain-size reduction by the reaction are not available, it is very important for localized rheological weakening. Therefore, I evaluated the required conditions to cause significant weakening by the reaction. Significant grain-size reduction by the spinel- to plagioclase-lherzolite reaction may occur at temperatures between 600 and 1300 8C as discussed above. At temperatures higher than 600 8C, the deformation occurs mainly by GSS creep, when grain-size is less than 0.7 mm under dry conditions and less than 0.5 mm under wet conditions, as can be seen in Fig. 3. Therefore, in order to obtain significant reduction of stress, grain-size should decrease down to those values by the reaction if the initial grain-size is more than those values. On the other hand, if the dominant deformation mechanism is GSS creep at the onset of extension, grain-size should decrease down to less than initial grain-size to produce the localized weak zone. As shown in Fig. 14, a created localized weak zone with a strength contrast of more than a few MPa is not maintained for a long time, and it disappears within a few million years. Significant strength contrast between in weaker and stronger zones would be required for the localization of deformation. In fact, Frederiksen and Braun (2001) indicated that the degree of viscosity reduction with strain is important factor for determining how the deformation is localized. However, it might be unknown how much strength reduction is required for the localization of deformation at this moment. If the localization of deformation requires that the strength difference between in weaker and in stronger zones is more than a few MPa, it might be hard to obtain a significant fall in rock strength at temperatures higher than 950 8C under dry conditions and higher than 800 8C under wet conditions, respectively (see Fig. 3), in spite of substantial grain-size reduction by the reaction. So then, I could state that the most suitable condition for localized deformation may be at the moment of the initial state of the reaction, if the localized weak zone is truly important for the localization of deformation.
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As discussed above, dry conditions have a wider range of grain-size and temperature for significant reduction of ductile stress by the reaction than the wet conditions do. Model results in this study show that larger strength reduction is predicted and the created weak zone could be maintained for longer period under dry conditions. Therefore, dry conditions are more favourable for localized rheological weakening, and this is consistent with the observation that reaction-enhanced softening described for the Turon de Te´coue`re peridotite is not attributed to a hydration reaction and may have occurred in the absence of aqueous fluids (Vissers et al., 1997; Newman et al., 1999). 4.2. Comparison with the results of previous dynamical modelling In previous numerical models (Govers and Wortel, 1993, 1995; Frederiksen and Braun, 2001), the firstorder strain softening was adopted at any depth in the lithospheric mantle. With this assumption, weakening occurs where more strain intrinsically accumulates in the two-dimensional model. Therefore, it is obvious that the rheological weakening is likely to take place even though the effect of strain softening is not incorporated into the model. From the results in this study, if grain-size reduction occurs by cataclasis, a localized weak zone would be formed at depths where deformation occurs by brittle failure, that is, just below the Moho depth. If grain-size reduction is brought about by the spinel–plagioclase lherzolite reaction, rheological weakening would occur only at depths shallower than 35 km. Furthermore, it would be difficult for grain-size reduction by dynamic recrystallization to cause a significant rheological weakening, except when grain growth is inhibited. Therefore, it seems that significant rheological weakening by grainsize reduction could only occur at a restricted depth range. Although Frederiksen and Braun (2001) demonstrated the occurrence of simple shear deformation using a simple strain-softening model, it may be difficult to form a shear zone cutting-off the entire lithospheric mantle. Results in this study seem to be more consistent with the studies by Govers and Wortel (1993, 1995), in which strain softening can cause localized deformation, but deformation on the lithosphere scale is mostly by pure shear.
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4.3. Other weakening mechanisms than grain-size reduction In this paper, I examined only the effects of grainsize reduction on the stress envelope in the lithospheric mantle. Several other softening mechanisms have been proposed (e.g., Poirier, 1980; Brodie and Rutter, 1985), including geometrical softening, thermal softening, and reaction softening. I discuss here, whether these other mechanisms can cause localized rheological weakening or not. Geometrical softening is due to the formation of the lattice-preferred orientation of minerals (e.g., Poirier, 1980). When deformation occurs by GSS creep, lattice-preferred orientation cannot develop and the resultant structure of rocks is mostly isotropic (Karato, 1988). However, if deformation occurs by GSI creep, a strong lattice preferred orientation could be formed. Lattice-preferred orientation will result in seismic anisotropy (Karato, 1987, 1989b; Nicolas and Christensen, 1987), and Karato (1992) proposed the hypothesis that the Lehmann discontinuity is attributed to the preferred orientation of olivine as a result of a change in the deformation mechanism. Rocks in shear zones often show preferred crystallographic orientation (e.g., Pieri et al., 2001). However, Wenk et al. (1991) concluded that geometrical softening has a minor effect on the reduction of strength, and laboratory experiments (Zhang et al., 2000) demonstrated that the development of the lattice-preferred orientation in olivine might result in hardening rather than weakening. Thus, geometrical softening may not be important for localized weakening. Strain localization in the ductile regime has been successfully modelled using thermal softening, in which a drop in strength is related to the increase in temperature by shear heating through a temperaturedependent viscosity (e.g., Yuen et al., 1978; Fleitout and Froidevaux, 1980; Kameyama et al., 1997). However, in these studies, the effects of shear heating were evaluated under the condition that the shear zone was initially assumed. Shear heating might not be important in the absence of localized deformation (e.g., Brun and Cobbold, 1980; Govers and Wortel, 1995; Monte´si and Zuber, 2002). Therefore, shear heating is important for maintaining the shear zones, but not for the origin of localized weak zones, whereas Leloup et al. (1999) indicated the importance
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of shear heating for the reaction-enhanced ductility developed along the strike-slip shear zones. Aside from grain-size reduction, it has been recognized that syntectonic metamorphic reactions in polymineralic rocks can also cause a strong reduction in rock strength either from that reaction products are intrinsically weaker than the reactants or from that nostrained new grains can deform more easily by intracrystalline glide than the original ones (Brodie and Rutter, 1985; Handy, 1989; Vissers et al., 1997). These effects were not considered in this study. Although it is unknown how much these softening mechanisms could bring about the reduction of rock strength, it is sure that weakening could occur at the same time and at the same position as those caused by grain-size reduction in this study, as far as the corresponding reaction is the spinel- to plagioclaselherzolite reaction. 4.4. Implications for the style of lithospheric extension Many studies have attempted to classify the development of specific basins in terms of either pure shear or simple shear. However, many rift-related basins have not been successfully classified in the style of extension. For example, Beach (1986) concluded that the structure in the North Sea Basin is consistent with a simple shear model from seismic reflection data. Barton and Wood (1984) explained, on the other hand, the subsidence data in the North Sea Basin in terms of the pure shear model. Latin and White (1990) quantitatively investigated adiabatic decompression melting in both the simple shear and the pure shear model, and concluded that it seems very difficult to reconcile the observed amount and composition of magma in the North Sea with a simple shear mechanism, based upon experimental results (McKenzie and Bickle, 1988; Furlong and Fountain, 1986). Therefore, it is not easy to determine the style of extension based on various geological and geophysical observations. Such difficulty may be attributed to the lack of understanding of the mechanism for the occurrence of simple shear deformation during extension. Although several kinematic models have been proposed, including pure shear (McKenzie, 1978), simple shear (Wernicke, 1985), and combinations of these two models (e.g., Lister et al., 1986; Kusznir and Egan,
1989), such models cannot provide dynamical aspects of the extensional process. Even the models with dynamical aspects (e.g., Braun and Beaumont, 1987; Bassi et al., 1993; Bassi, 1995) intrinsically predict symmetric extension. Whereas Frederiksen and Braun (2001) showed that simple strain softening could cause strain localization in the lithosphere, the origin of asymmetric extension is still not well known. Braun and Beaumont (1989) indicated that preexisting weakness in the crust and mantle is important for asymmetric deformation of the lithosphere. However, there is no evidence for the presence of such weakness at the initiation of the extension, as pointed out by Govers and Wortel (1993). Frederiksen and Braun (2001) proposed that simple shear deformation occurs when conditions favourable for strain localization are satisfied. Contrarily, the lithosphere is extended by a pure shear manner under conditions that are not favourable for strain localization. In this hypothesis, the deformed lithosphere has both components of pure shear and simple shear. In fact, many of continental rifts and passive margins have been interpreted in terms of both pure shear and simple shear deformation models (e.g., Wernicke and Burchfiel, 1982; Barton and Wood, 1984; Beach, 1986; Lister et al., 1986; Mutter et al., 1989; White, 1990; Torres et al., 1993; Collier et al., 1994). Therefore, it may be reasonable to say that most of the extensional basins are developed by both pure shear and simple shear deformations, and these have a particular amount of each deformational component. Based on a dynamical modelling study, Frederiksen and Braun (2001) concluded that the amount of strain that is accommodated by pure shear and simple shear is controlled by the strain required for the initiation of strain softening. If the strain softening occurs after a larger amount of strain, a higher amount of pure shear deformation is predicted. In the present study, I show that the stretching factor for the initiation of rheological softening is dependent on the initial thermal condition of the lithosphere for grain-size reduction by the spinel–plagioclase lherzolite reaction; a larger stretching factor is required for colder lithosphere and vice versa. Therefore, it is implied that the lithosphere may be mostly deformed uniformly for cold lithosphere, and hot lithosphere is more favourable for simple shear deformation, if localized rheological weakening is really important for a switch from pure
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shear to simple shear deformation. Various ratios of simple shear deformation can be predicted when localized weakening occurs by grain-size reduction due to cataclasis, which is dependent on an unknown rate of grain-size reduction. In this study, I cannot examine how a localized weak zone is connected laterally in the extended lithosphere. Furthermore, a common characteristic of a kinematic model of lithospheric extension is such that deformation is imposed by prescribing a velocity field (e.g., Gueydan et al., 2001), and the constitutive equations are not related to the deformation. Therefore, for the simple model presented in this study, I cannot examine how the created weak zone influences subsequent dynamic deformation of the lithosphere. So then, I should develop a two-dimensional dynamic model of lithospheric extension for further investigation into a transition from pure shear to simple shear deformation in a future study. If the detailed geological phenomenon associated with the transition from pure shear to simple shear deformation is revealed, the confusing observational evidence may be systematically interpreted.
5. Conclusions I investigated the effects of grain-size reduction as a result of dynamic recrystallization, cataclasis and spinel- to plagioclase-lherzolite reaction on the rheological behaviour of the lithosphere during a uniform extensional process. The present study may provide important constraints on the style of extension if localized rheological weakening can lead to the onset of simple shear deformation. The conclusions in this paper are as follows: (i)
(ii)
Grain-size reduction by dynamic recrystallization cannot cause localized rheological weakening that may be favourable for strain localization, even if significant grain-size change could occur. Grain-size reduction by cataclasis can cause localized rheological weakening just below the Moho depth. Because the rate of grain-size reduction by cataclasis is poorly constrained, it is difficult, at this moment, to specify the time when localized weakening occurs.
(iii)
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Grain-size reduction by the spinel- to plagioclase-lherzolite reaction can produce a weak zone at a particular depth range during extension. This reaction occurs at depths shallower than 35 km, which is independent of the initial thermal state in the lithosphere. Localized rheological weakening can occur under the following conditions. If grain-size before the reaction is greater than 0.7 mm under dry conditions and greater than 0.5 mm under wet conditions, it should decrease down to those values by the reaction. On the other hand, if the dominant deformation mechanism is GSS creep at the onset of extension, it is enough that grainsize decreases down to less than initial grainsize by the reaction. It is also important to note that dry conditions are more favourable to form an unmistakably localized weak zone.
Acknowledgements I would like to thank Tetsuzo Seno, Yoshitaka Takeda, and Takeshi Ikeda for carefully reading the manuscript and stimulating discussions. I also thank Masanori C. Kameyama and Hans De Bresser for their constructive reviews for improving the manuscript.
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Rutter, E.H., Brodie, K.H., 1988. The role of tectonic grain size reduction in the rheological stratification of the lithosphere. Geol. Rundsch. 77, 295 – 308. Sammis, C.G., Osborne, R.H., Anderson, J.L., Banerdt, M., White, P., 1986. Self-similar cataclasis in the formation of fault-gouge. Pure Appl. Geophys. 124, 53 – 78. Schmid, S.M., Paterson, M.S., Boland, J.M., 1980. High temperature flow and dynamic recrystallization in Carrara marble. Tectonophysics 65, 245 – 280. Shelton, G., Tullis, J., 1981. Experimental flow laws for crustal rocks. EOS 62, 396. Sleep, N., 1997. Application of a unified rate and state friction theory to the mechanics of fault zones with strain localization. J. Geophys. Res. 102, 2875 – 2895. Smythe, D.K., Dobinson, A., McQuillin, R., Brewer, J.A., Matthews, D.H., Blundell, D.J., Kelk, B., 1982. Deep structure of Scottish Caledonides revealed by the MOIST reflection profile. Nature 299, 338 – 340. Tan, B.H., Jackson, I., Fitz Gerald, J.D., 2001. High-temperature viscoelasticity of fine-grained polycrystalline olivine. Phys. Chem. Miner. 28, 641 – 664. Ter Heege, J.H., De Bresser, J.H.P., Spiers, C.J., 2002. The influence of dynamic recrystallization on the grain size distribution and rheological behaviour of Carrara marble deformed in axial compression. In: De Meer, S., Drury, M.R., De Bresser, J.H.P., Pennock, G.M. (Eds.), Spec. Publ.-Geol. Soc., vol. 200, pp. 331 – 353. Torres, J., Bois, S., Burrus, J., 1993. Initiation and evolution of the Valencia Trough (western Mediterranean): constraints from deep seismic profiling and subsidence analysis. Tectonophysics 228, 57 – 80. Toth, J., Gurnis, M., 1998. Dynamics of subduction initiation at preexisting fault zones. J. Geophys. Res. 103, 18053 – 18067. Twiss, R.J., Sellars, C.M., 1978. Limits of applicability of the recrystallized grain size geopiezometer. Geophys. Res. Lett. 5, 337 – 340. van der Wal, D., Chopra, P., Drury, M., Fitz Gerald, J., 1993. Relationships between dynamically recrystallized grain size and deformation conditions in experimentally deformed olivine rocks. Geophys. Res. Lett. 20, 1479 – 1482. Vissers, R.L.M., Drury, M.R., Hoogerduijn Starting, E.H., van der Wal, D., 1991. Shear zones in the upper mantle: a case study in an Alpine lherzolite massif. Geology 19, 990 – 993. Vissers, R.L.M., Drury, M.R., Hoogerduijn Starting, E.H., Spiers, C.J., van der Wal, D., 1995. Mantle shear zones and their effect on lithosphere strength during continental breakup. Tectonophysics 249, 155 – 171. Vissers, R.L.M., Drury, M.R., Newman, J., Fliervoet, T.F., 1997. Mylonitic deformation in upper mantle peridotites of the North Pyrenean Zone (France): implications for strength and strain localization in the lithosphere. Tectonophysics 279, 303 – 325. Wenk, H.-R., Bennett, K., Canova, G.R., Molinari, A., 1991. Modelling plastic deformation of peridotite with the selfconsistent theory. J. Geophys. Res. 96, 8337 – 8349. Wernicke, B., 1985. Uniform-sense normal simple shear of the continental lithosphere. Can. J. Earth Sci. 22, 108 – 125.
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Localized rheological weakening by grain-size reduction during lithospheric extension Tadashi Yamasaki* National Institute of Polar Research, Kaga 1-9-10, Itabashi-ku, Tokyo 173-8515, Japan Received 19 November 2002; accepted 26 May 2004 Available online 9 August 2004
Abstract Grain-size reduction may be a possible mechanism for the origin of localized deformation in the ductile regime. I investigated the effects of grain-size reduction due to dynamic recrystallization, cataclasis, and syntectonic metamorphic reaction on the stress envelope in the lithospheric mantle during extension by using a simple one-dimensional model. In this model, the lithosphere extends uniformly with a constant strain rate, and a fall in rock strength appears as a decrease in stress. Because grain-size distribution at the onset of extension is unknown, I regarded the steady state grain-size due to dynamic recrystallization as the initial size. Then, I evaluated the maximum effects of grain-size reduction by dynamic recrystallization during extension, and consequently examined the effects of grain-size reduction by cataclasis and metamorphic reaction under conditions when dynamic recrystallization occurs significantly. I find that it is difficult to bring about localized rheological weakening by grainsize reduction owing to dynamic recrystallization. In contrast, grain-size reduction by cataclasis can cause localized weakening during extension. There is a wide-ranging rate of grain-size reduction by means of cataclasis that causes localized weakening just below the Moho. I specified the reaction from spinel-lherzolite to plagioclase-lherzolite that plays a role in grain-size reduction by syntectonic metamorphism. The reaction occurs at depths less than 35 km, which is independent of the initial thermal state of the lithosphere. Localized rheological weakening can occur if the following conditions are satisfied: (1) grain-size before the reaction is greater than 0.7 mm under dry conditions and greater than 0.5 mm under wet conditions, and it decreases down to those values by the reaction; (2) grain-size decreases down to less than initial grain-size, when the dominant deformation mechanism is GSS creep at the onset of extension. It is also noted that dry conditions are more favourable for localized weakening. D 2004 Elsevier B.V. All rights reserved. Keywords: Rheological weakening; Grain-size; Dynamic recrystallization; Cataclasis; Spinel lherzolite–plagioclase lherzolite reaction; Lithospheric extension
1. Introduction * Present address: Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland. E-mail address: [email protected]. 0040-1951/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.tecto.2004.05.006
Localized deformation in the lithosphere is often observed, and it is important for understanding various geological and geophysical phenomena.
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McKenzie (1977), Cloetingh et al. (1989), and Toth and Gurnis (1998) indicated that localized ductile deformation is important for the initiation of plate subduction. Hobbs et al. (1986) and Sleep (1997) emphasized that localized shear deformation could result in the occurrence of earthquakes. In addition, localized deformation may play an important role in determining the style of extension of the lithosphere: pure shear (McKenzie, 1978) or simple shear (Wernicke, 1985). It seems that the most important difference between pure and simple shear models relates to the presence of the lithosphere-cutting shear zone (e.g., Wernicke, 1985). In this study, I examined the origin of strain localization during lithospheric extension. Geological and geophysical observations have demonstrated that localized deformation occurs beneath continental extensional areas. The xenoliths of deformed and undeformed mantle rock directly provide evidence in support of strain localization (e.g., Pike and Schwartzmann, 1977; Cabanes and Mercier, 1988; Downes, 1990). Tectonites and mylonites exposed at the surface also constitute important evidence for the presence of shear zones in the lithosphere. Peridotite mylonites from the Alps (Vissers et al., 1991) and from the North Pyrenean Zone (Vissers et al., 1997; Newman et al., 1999) show that localized deformation occurs in the upper mantle during extension. Seismic reflectivity of shear zones has been investigated by several authors (e.g., Fountain et al., 1984; Wilshire and Kirby, 1989; Ji et al., 1997). Fountain et al. (1984) indicated that mineralogic and petrofabric characteristics of mylonite zones affect seismic reflectivity. Based on studies of mantle xenoliths, Wilshire and Kirby (1989) suggested that velocity contrasts between fractured and unfractured peridotite could be detected by seismic refraction and/or reflection techniques. Ji et al. (1997) showed that the layering of lithologic units with different acoustic impedance could result in strong seismic reflectivity, like that observed in the Morin Shear Zone in the Grenville Tectonic Province. Therefore, seismic reflection profiling may be one of the most attractive techniques for exploring the presence of shear zones within the lithosphere (e.g., Smythe et al., 1982; Brewer et al., 1983; Beach, 1986; Gibbs, 1987; Klemperer and Matthews, 1987; Reston, 1993;
Brun et al., 1992; MONA LISA Working Group, 1997). Strain localization has been successfully modelled in the brittle regime (e.g., Lavier et al., 2000). Brittle deformation mostly occurs in a localized manner. Therefore, it has been well described (e.g., Paterson, 1978). On the other hand, however, deformation in the ductile regime usually occurs homogeneously, and hence mechanisms for strain localization in the ductile regime have been poorly understood (e.g., Poirier, 1980). Localized ductile deformation may be accompanied by localized rheological weakening in the lithosphere, and grain-size refinement may play an important role in reducing the strength of rocks (e.g., Kirby, 1985). In ductile deformation, several creep mechanisms can control deformation, and the mechanism that produces the highest strain rate is the rate controlling one for given temperatures and applied stress conditions. Under relatively low temperatures, high stress and relatively large grain-size, deformation is mainly controlled by grain-size insensitive (GSI) dislocation creep. In contrast, the dominant deformation mechanism is grain-size sensitive (GSS) diffusion creep under moderate to high temperatures, low stress and small grain-size. Whereas Kirby (1983) and Carter and Tsenn (1987) indicated that the GSI creep seems to be dominant in typical lithosphere conditions, many authors recently have been paying attention to GSS creep in light of the many geological and geophysical problems occurring in the lithosphere (e.g., Rutter and Brodie, 1988; Hopper and Buck, 1993; Kameyama et al., 1997; Braun et al., 1999; Monte´si and Hirth, 2003; Hall and Parmentier, 2003). Rutter and Brodie (1988) indicated that GSS creep could play an important role in strain localization. Hopper and Buck (1993) discussed the role of GSS creep in the uppermost mantle for the initiation of rifting. Kameyama et al. (1997) suggested the importance of GSS rheology for the development of the shear zones. Braun et al. (1999) investigated the conditions for the transition from GSI creep to GSS creep due to dynamic recrystallization during progressive shear deformation in olivine-rich rocks, and concluded that localized deformation could occur under very specific conditions. Most recently, from the simulations of laboratory experiments, Monte´si and Hirth (2003) showed that the grain-size evolution can make the dominant
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creep mechanism changed, and they indicated that the ductile shear zone including the transient strainweakening or -hardening may be important for the postseismic deformation. In addition, Hall and Parmentier (2003) investigated the importance of GSS rheology for the formation of convective instability. GSS creep is extremely sensitive to the size of mineral grain, and to keep the strain rate constant, the required stress significantly decreases with a decrease in grain-size. In the deformational regime where GSI creep is dominant, if the grain-size is reduced significantly by some means, the rate-controlling creep mechanism may shift from GSI creep to GSS creep, followed by a significant reduction of rock strength (see Fig. 1). Some observations in strongly deformed shear zones show evidence for weakening due to this type of grain-size reduction (e.g., Vissers et al., 1995; Jin et al., 1998; Newman et al., 1999). In addition, recent experiments on olivine showed that a reduction in grain-size results in significant attenuation of seismic shear waves (e.g., Jackson et al., 2002; Tan et al., 2001). Therefore, localized ductile shear zones in the lithosphere caused by grain-size reduction could be observed by seismic study.
Grain Size Fig. 1. Schematic diagram showing the effects of grain-size reduction on rheological weakening (after Rutter and Brodie, 1988). The thick solid line represents the boundary between grain-size insensitive (GSI) dislocation creep and grain-size sensitive (GSS) diffusion creep. Thin solid lines show constant strain rates (e1˙ Ne˙2Ne3˙ Ne4˙ ). When grain-size becomes smaller than the transition grain-size by some means, significant rheological weakening is occurred. The dashed arrows show the weakening effects of grain-size reduction via a shift in the dominant deformation mechanism from GSI creep to GSS creep. If the stress is set to be constant (A to B), the weakening due to grain-size reduction occurs by increase in strain rate. On the other hand, if the strain rate is constant (A to C), the weakening occurs by reduction in stress.
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Previous studies on strain localization in the ductile regime during lithospheric extension were based on first-order rheological strain softening (Govers and Wortel, 1993, 1995; Frederiksen and Braun, 2001). Govers and Wortel (1995) considered grain-size reduction by dynamic recrystallization as a mechanism for reducing rock strength. In their modelling study, it was assumed that a shift of the dominant creep mechanism in the mantle from GSI creep to GSS creep starts after 10% strain, and its transition is completed when the amount of strain achieves 50%. Although they concluded that the shear zone could develop due to strain softening, rock strength controlled by the grain-size evolution as a function of stress and temperature was not considered. More recently, Frederiksen and Braun (2001) performed a similar calculation using a simple strain–viscosity relationship to investigate the effects of strain required to initiate strain softening, the magnitude of strength reduction, and the rate of strain softening. Whereas they successfully demonstrated the occurrence of lithosphere-scale simple shear deformation during extension, the concrete mechanisms for the strain softening were not considered. Therefore, they did not reveal what controls these parameters. In this study, I focus on the localized rheological weakening in the ductile regime due to grain-size reduction in the lithospheric mantle during extension. I specifically investigate the effects of grain-size reduction brought about by dynamic recrystallization (e.g., Karato et al., 1980; Kirby, 1985; Rutter and Brodie, 1988; van der Wal et al., 1993; De Bresser et al., 1998; Rutter, 1999), cataclasis (e.g., Hippler and Knipe, 1990; Goodwin and Wenk, 1995; Bos and Spiers, 2001), and syntectonic metamorphic reaction (e.g., Brodie and Rutter, 1987; Handy, 1989; Drury et al., 1991; Vissers et al., 1997; Newman et al., 1999). I examine how lithospheric stress envelopes may be modified by grain-size reduction with respect to these three mechanisms, and explore the possibility as to whether localized rheological weakening, favourable for strain localization, can occur or not. In addition, it may be important to note that I evaluate the effect of grain-size change on the profiles of strength in the lithosphere during thermal cooling. Based on the model results in this study, I will provide several implications for the rifting process.
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2. Model descriptions
Table 1 Model parameter values used in this study
The schematic figure of a simple one-dimensional model adopted in this study is shown in Fig. 2. In this model, the whole lithosphere is thinned by a pure shear manner in response to an applied constant strain rate. Because localized rheological weak zones appear as smaller stress regions under the constant strain rate condition, I investigate the temporal evolution of lithospheric stress envelope during the extension. Ductile stress is assumed to be dependent not only on strain rate and temperature but also on mineral grain-size. Grain-size evolutions owing to dynamic recrystallization, cataclasis and syntectonic metamorphic reaction are included in the model. Grain-size and stresses are evaluated at the moving material point where the thermal cooling occurs during the extension. Because I focused on ductile weakening by grain-size reduction in the mantle, ductile deformation of the crust is assumed to occur only by the GSI creep mechanism. I considered a stratified lithosphere with three layers (see Fig. 2): a quartzite upper crust (Koch et al., 1989), an anorthite lower crust (Shelton and Tullis, 1981) and an olivine mantle (Karato et al., 1986). Although more recent results of flow law parameters on olivine than Karato et al. (1986) are
Parameter
Meaning
Value
H j q uc q lc qm a Cp R t uc t lc D p
Crustal heat production Thermal diffusivity Density of upper crust Density of lower crust Density of mantle coefficient of thermal expansion Specific heat Universal gas constant Initial thickness of upper crust Initial thickness of lower crust Constant of d l Stress exponent of d l
6.47107 W m3 106 m2 s2 2800 kg m3 2900 kg m3 3300 kg m3 3.0105 8C1 1300 J kg1 K1 8.314 J mol1 K1 15 km 15 km 15 mm MPa1.33a 1.33a
a
van der Wal et al. (1993).
available (Mei and Kohlstedt, 2000a,b; Hirth and Kohlstedt, 1995), the differences among them, however, seem to be minor. Therefore, the model results in this study are mostly independent on the used flow law parameters. 2.1. Thermal calculation Temperature distribution in the model must be calculated at any stage of deformation because both GSI and GSS creeps are functions of temperature. The temperature distribution during extension is controlled by the following one-dimensional heat transport equation: BT B2 T BT H ¼j 2 m þ Bt Bz Bz qCp
Fig. 2. Simple one-dimensional model of lithospheric extension used in this study. Pure shear deformation is assumed, and so the vertical velocity of lithospheric material is a linear function of depth. The applied extensional strain rate is assumed constant in the model. Thermal boundary conditions are such that the temperature at surface and bottom of the lithosphere is maintained to 0 and 1350 8C, respectively. Three-layered lithosphere is considered; a quartzite upper crust, an anorthite lower crust and an olivine mantle. Both GSI and GSS creeps are adopted in the mantle, but only the GSI creep is adopted in the crust.
ð1Þ
where T is the temperature, t is the time, j is the thermal diffusivity, z is the depth, H is the amount of internal heating due to radioactive elements, q is the density, and C p is the specific heat. m is the vertical velocity of material and is related to the strain rate e˙ by m=e˙ z for pure shear deformation. Stretching factor b is defined by b=exp(e˙ t). I assume a uniform distribution of radiogenic elements in the entire crust and ignore the presence of heat sources in the mantle. Parameter values used in this study are shown in Table 1. The initial temperature distribution is obtained by the steady state solution of Eq. (1) with a given mantle heat flux q m and a fixed surface temperature of 0 8C. The initial depth of the bottom of the lithosphere is
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defined by the depth of 1350 8C isotherms, and the lithosphere thickness is dependent on mantle heat flow. In the time-dependent calculation, the upper and lower boundaries of the model, i.e., the lithosphere, are maintained at constant temperatures of 0 and 1350 8C, respectively. I solved Eq. (1) by using an explicit 4-point upwind finite difference method (Fletcher, 1991). I assumed that an initial crustal thickness is 30 km, and the thickness of the upper crust is 15 km. In most calculations, I adopted the initial lithospheric thickness to be 125 km. However, in the calculations for investigating the effects of spinel- to plagioclaselherzolite reaction, several different values of the initial lithospheric thickness have been adopted, because the lithospheric thickness is very important through the geometrical relationship between the geotherm and the Clapeyron curve. 2.2. Stress calculation The deformation mechanism of rocks is, amongst others, sensitive to its composition, temperature, pressure and strain rate. Whether deformation occurs by brittle or ductile manner is dependent on the stress level owing to the two deformation mechanisms; the mechanism that gives a smaller stress at an applied strain rate is the controlling one. Therefore, the stress at each depth in the lithosphere is given by the lesser of the brittle and ductile stresses. At low temperature and pressure conditions, applied stress makes the rocks deform in a brittle fashion. The brittle stress would be controlled by frictional sliding on existing faults rather than by fractures (e.g., Goetze and Evans, 1979; Carter and Tsenn, 1987). The stress-level required to exceed frictional resistance on a fault is insensitive to rock type and temperature, and it is given by the Byerlee’s (1978) law. Following Brace and Kohlstedt (1980), I simply expressed the Byerlee’s law under tensional stress states as an approximated form of rb ¼ Bð1 kÞz
ð2Þ
where r b is the brittle stress, B is a constant with a value of 24 MPa/km and z is the depth in km. k is the density ratio of pore water to rock matrix of which value used in this study is 0.38 under wet conditions and zero under dry conditions (e.g., Ranalli, 1995).
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On the other hand, at high temperatures and pressures, rocks respond to the applied stress by ductile flow. I assumed that ductile flow is controlled by both GSI and GSS creep, and these two creep mechanisms are governed by the same extensional stress. Therefore, the total strain rate e˙ is given by the sum of the strain rate due to GSI creep e˙gsi and the strain rate due to GSS creep e˙gss and as, e˙ ¼ e˙ gsi þ e˙ gss
ð3Þ
The strain rate due to GSI creep is a function of temperature and stress, and it has a nonlinear dependence on stress: Qgsi e˙ gsi ¼ Agsi rnd exp ð4Þ RT where r d is the ductile stress, A is a constant, Q is the activation energy, R is the universal gas constant, T is the absolute temperature, and n is the power of stress. The strain rate due to GSS creep is dependent not only on temperature and stress, but also on grainsize, and it is expressed by the following rheological law: Qgss e˙ gss ¼ Agss rd d m exp ð5Þ RT where d is the average mineral grain-size and m is the power of grain-size. The ductile stress is calculated using Eqs. (3)–(5) for given strain rate, grain-size, and temperature. The rheological parameters used in this study are given in Table 2. In Fig. 3, I depict ductile stress as a function of temperature for different grain-size. Rheological parameters are both for dry and wet olivines (Karato et al., 1986). Applied strain rate is 51015 (1/s). Here, I define the ratio of the strain rate due to the GSI creep mechanism to total strain rate as rue˙gsi/ (e˙gsi+e˙gss). If the GSI creep ratio r is more than 0.5, the deformation occurs mainly by GSI creep. In contrast, the deformation is entirely due almost to GSS creep when the GSI creep ratio is less than 0.5. Smaller ductile stress is predicted for higher temperature in both GSI creep and GSS creep regimes. If grain-size and temperature are fixed to be constant, the ductile stress is larger for dry condition. For a given temperature, lower stress is predicted for smaller grain-
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Table 2 Parameters of GSI creep and GSS creep used in this study Meaning
Quartza
Anorthiteb
Olivinec
Wet GSI creep flow parameters A gsi (Pan s1 mmm) n m Q gsi (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
1.11021 2.61 0 145
5.61023 3.2 0 238
1.91015 3 0 420
Dry GSI creep flow parameters A gsi (Pan s1 mmm) n m Q gsi (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
5.581024 2.72 0 134
5.61023 3.2 0 238
2.41016 3.5 0 540
Wet GSS creep flow parameters A gss (Pan s1 mmm) n m Q gss (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
– – – –
– – – –
1.5109 1 3 250
Dry GSS creep flow parameters A gss (Pan s1 mmm) n m Q gss (kJ mol1)
Preexponential constant Stress exponent Grain-size exponent Activation energy
– – – –
– – – –
7.7108 1 2 290
a b c
Koch et al. (1989). Shelton and Tullis (1981). Karato et al. (1986).
size in the domain where the deformation occurs mainly by GSS creep. However, the stress is almost independent on grain-size in the domain where GSI creep is a dominant deformation mechanism. 2.3. Grain-size change as a result of dynamic recrystallization It is widely believed that dynamic recrystallization can cause significant grain-size reduction. Dynamic recrystallization occurs due to grain boundary migration and subgrain rotation (e.g., Guillope´ and Poirier, 1979; Poirier, 1980; Hirth and Tullis, 1992). Kirby (1985) and Rutter and Brodie (1988) indicated that transition from GSI creep to GSS creep brought about by dynamic recrystallization could play an important role in the localization of ductile deformation. Kameyama et al. (1997) and Braun et al. (1999) investigated the effects of grain-size change by dynamic recrystallization on the evolution of the shear deformation. In their models, it is assumed that
the stress is uniform all over the considered volume. Therefore, the distribution of grain-size in the volume would be getting uniform as a progression of the deformation, and the deformation also becomes uniform. This is because the governing equation for the recrystallized grain-size is a function of only the stress level. In their studies, the occurrence of localized weakening depends on the initially given grain-size distribution, and thus they investigated whether the localized weak zone could lead the strain localization or not. On the other hand, in the present study, the stress and its dependent grain-size cannot become uniform within the lithosphere, because the applied strain rate is assumed to be uniform in the entire lithosphere and the temperature increases with depth. Then, I investigate whether the localized weak zone can be formed by the grain-size reduction during the uniform extension of the lithosphere or not. In order to calculate the temporal evolution of grain-size, I used an equation in which the grain-size d tends toward d l with increasing strain, and the rate
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Fig. 3. Stress contour as a function of temperature and grain-size. Rheological parameters are for olivine (Karato et al., 1986): (a) dry condition and (b) wet condition. Applied strain rate is 51015 (1/s). The thick dashed line indicates that rue˙gsi/(e˙gsi+e˙gss) is 0.5. The bGSI regimeQ indicates that the deformation occurs mainly by GSI creep (i.e. rN0.5), while the bGSS regimeQ indicates that the deformation occurs mainly by GSS creep (i.e. rb0.5). The thin solid curves are lines of constant stress. The numbers attached to each contour indicate the value of stress in the unit of MPa.
of grain-size change is controlled by (e.g., Kameyama et al., 1997; Braun et al., 1999), e˙ d˙ ¼ ðd dl Þ eT
ð6Þ
where e˙ is the strain rate and e T and is the amount of strain required for grain-size change. Karato et al. (1980) and van der Wal et al. (1993) experimentally proposed quantitative relationships between applied stress and recrystallized grain-size for olivine, and showed that grain-sizes from 0.5 to less than 0.01 mm can be expected for upper mantle conditions. In their studies, recrystallized grain-size is deduced from piezometric measurements in a form dependent only upon stress-level, as dl ¼ Drp
creep. Based on this hypothesis, d l is determined by the following equation (e.g., De Bresser et al., 2001; Gueydan et al., 2001):
ð7Þ
where D and p are constants (see Table 1). Smaller grain-size for higher stress is predicted by Eq. (7). Recently, De Bresser et al. (2001) suggested the hypothesis that dynamic recrystallization tends toward a balance between grain-size reduction and grain growth processes, and the average recrystallized grain-size will be stable at the condition that the strain rate due to GSI creep is equal to that due to GSS
dl ¼
Agss Agsi
m1
ln
r m exp
Qgsi Qgss mRT
ð8Þ
It is noted that whereas Eq. (7) is fully empirical one, Eq. (8) is modification of Eq. (7) with respect to the microstructural development. The grain-size evolution predicted by Eq. (8) is as follows; in the regime where GSI creep is dominant, present grain-size is larger than recrystallized grain-size and so reduction of grain-size would occur. In a GSS creep dominant regime, conversely, grain-size increases because present grain-size is smaller than recrystallized grain-size. This hypothesis is supported by experiments on a Mg alloy (De Bresser et al., 1998), and De Bresser et al. (2001) demonstrated that independently obtained relations between stress and recrystallized grain-size for olivine (Post, 1977; Karato et al., 1980; Ross et al., 1980) and calcite (Schmid et al., 1980; Friedman and Higgs, 1981; Rutter, 1995) correspond to the GSI– GSS transition region. Hence, it seems reasonable to adopt Eq. (8) for dynamically recrystallized grain-
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size. In most calculations in this study, I used Eq. (8) to examine the effects of grain-size change by dynamic recrystallization on the stress envelope in the lithosphere. For the purpose of comparison, however, I also showed results for cases where recrystallized grain-size is governed by Eq. (7). Experimental studies of Karato et al. (1980) and De Bresser et al. (1998) suggested that steady-state flow was achieved after about 40% strain. On the other hand, van der Wal et al. (1993) reported that for shortening experiments the steady state recrystallized grain-size for a given applied stress is obtained after about 3% strain. However, values for parameter e T are not well constrained. In order to evaluate the effects of e T the two different values, 0.03 and 0.4, are adopted in this study. The timescale for the grain-size change by the dynamic recrystallization is dependent on its value. Therefore, it is important to vary e T for evaluating the competition between the effects of dynamic recrystallization and thermal cooling. Twiss and Sellars (1978) discussed a lower limit of stress for dynamic recrystallization. The occurrence of dynamic recrystallization requires a condition where the decrease in free energy due to the removal of dislocations must exceed the increase in free energy caused by creation of a new grain boundary. The results of theoretical calculations indicated that if an applied stress level is such that recrystallized grainsize will be larger than the order of 10 mm (typical value for usual observed mantle xenoliths) dynamic recrystallization might not occur (Twiss and Sellars, 1978). Therefore, I set the upper limit of grain-size to be 10 mm. If recrystallized grain-size d l is larger than its maximum value, the rate of change in grainsize d is adopted to be zero in this study. Because there is no direct way to obtain the initial size of mineral grains in the lithosphere, I regarded the steady state grain-size due to dynamic recrystallization as the initial size. Recrystallized grain-size is dependent on stress-level, and stress-level changes with grain-size if the applied strain rate and temperature is fixed to be constant. Therefore, steady state grain-size for a given strain rate and temperature settles down in the intersection between the constitutive equation (Eq. (3)) and the equation governing recrystallized grain-size (Eq. (7) or Eq. (8)). Whereas Karato (1989a) suggested that cyclic softening–hardening could occur because of grain-size reduction and
subsequent grain growth, such cyclic events cannot occur if stress changes with grain-size. Thus, I can obtain steady state grain-size and ductile stress at any depth in the lithosphere using the initial steady state geotherm and the applied extensional strain rate, even when initial grain-size is unknown. In fact, however, Ter Heege et al. (2002) showed that the strengthening and subsequent weakening can be brought about by a change in dominant creep mechanism from GSI creep to GSS creep as the number of small grains in a volume increases with strain. It should be, therefore, emphasized that the calculations in this study are done under the condition that an average grain-size, not grain-size distribution, is considered at any depth. In Fig. 4, I show the behaviour of Eqs. (3), (7) and (8) as a function of grain-size and stress for dry and wet conditions. Applied strain rate is 51015 (1/s). When the temperature is 800 8C, the steady state grain-size governed by Eq. (8) and the corresponding ductile stress are 3.85 mm and 63 MPa for dry condition, respectively. For wet condition, the grain-size and the stress are 1.43 mm and 7.17 MPa, respectively. At a lower temperature of 600 8C, the steady state grainsize becomes smaller both for dry and wet conditions. If the initial steady state ductile stress is larger than the brittle stress, I adopt the actual stress to be brittle stress and grain-size at the depths to be 10 mm. During the progression of extension by pure shear manner, the lithosphere thins out and any material within it cools. Because pressure decreases with the upward movement of the materials and temperature decreases by thermal diffusion, materials, initially situated in the brittle regime, would always be deformed in a brittle fashion during extension. Therefore, the decision for initial grain-size in the brittle regime is not problematic as far as the grainsize changes by dynamic recrystallization. 2.4. Grain-size change as a result of cataclasis De Bresser et al. (2001) emphasized that cataclasis is a possible mechanism for the reduction of grain-size leading to rheological weakening. On the basis of an experimental study, Bos and Spiers (2001) suggested that weakening of fault rocks through a switch in the deformation mechanism from rate-independent behaviour to GSS creep could be caused by grain-size reduction due to cataclasis. Based on field observations
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Fig. 4. Behaviour of Eqs. (3), (7) and (8) as a function of grain-size and stress: (a) and (c) dry condition; and (b) and (d) wet condition. Applied strain rate is 51015 (1/s). Temperature is (a) and (b) 800 and (c) and (d) 600 8C. Because ductile stress changes with grain-size if the applied strain rate and the temperature are held to be constant, steady state grain-size is obtained at the intersection between Eqs. (3) and (7) or Eq. (8).
and laboratory experiments, Goodwin and Wenk (1995) indicated that grain-size reduction as a result of cataclasis would play an important role in the formation of the Santa Rosa mylonite zone. Whereas the cataclastic nature of grain-size reduction may not be apparent in natural rocks because diffusion processes can hide the microstructural evidence of cataclasis, such as angular grains and microcracks (Bos and Spiers, 2001), it seems sure that grain-size reduction can occur by cataclasis. In order to understand the process of grain-size reduction by brittle deformation, many studies have tried to analyse grain-size distribution of natural and experimental fault rocks (Sammis et al., 1986; Biegel et al., 1989; Marone and Scholz, 1989; Hippler and Knipe, 1990; Blenkinsop, 1991; Ray, 1999). However, it seems that the relationship between
grain-size reduction and the magnitude of displacement on a fault plane is poorly constrained. Because it is unknown how the grain-size is reduced with increase in strain, it is hard to incorporate the effects of grain-size reduction by cataclasis into the model. So then, I investigated its effects on the stress envelope in a following way. In the initial distribution of steady state grain-size due to dynamic recrystallization, I adopted several different values of d c for the grain-size in the brittle regime. In time-dependent calculations, I drew the stress envelope at any stage of deformation without grain-size change, and tried to find the critical stretching factor b c required for keeping the localized weak zone to be present. Then, it may be reasonable to state that localized rheological weakening can occur if the
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grain-size is reduced to d c before the stretching factor reaches b c. As described above, brittle deformation is typically due to the slip of pre-existing fault, and therefore it intrinsically occurs in localized manner. However, the deformation in the brittle regime occurs homogeneously over the scale of the entire lithosphere (e.g., Blenkinsop and Rutter, 1986). In the present study, I evaluated whether the localized reduction in ductile stress can be occurred by the grain-size reduction due to cataclasis during uniform deformation in the brittle regime. 2.5. Grain-size change as a result of metamorphic reaction It has been recognized that syntectonic metamorphic reactions could cause reduction of grain-size (e.g., Brodie and Rutter, 1987; Handy, 1989; Drury et al., 1991). The most well-known reactions in the uppermost mantle are garnet–spinel–plagioclase lherzolite reactions. Specifically, I focused on grain-size reduction associated with the reaction from spinel-lherzolite to plagioclase-lherzolite. In fact, the Turon de Te´coue`re peridotite in the North Pyrenean Zone has a very small grain-size (b10 Am), and Newman et al. (1999) concluded that such fine grain would be caused by the spinel- to plagioclase-lherzolite reaction. Similarly, Furusho and Kanagawa (1999) reported that the formation of lherzolite mylonites derived from the Hidaka metamorphic belt of central Hokkaido, Japan, is associated with the phase transformational reaction from spinel- to plagioclase-lherzolite. The spinel- to plagioclase-lherzolite reaction occurs with decreasing pressures, and therefore the continuous reaction described for these rocks may be important when upper mantle rocks move upwards to shallower levels, such as during lithospheric extension. The reaction boundary is determined by the intersection between the geotherm and Clapeyron curve. Many experimental and theoretical studies have attempted to determine the Clapeyron curve for spinel- to plagioclase-lherzolite reaction (e.g., Obata, 1976; Wood and Yuen, 1983; Gasparik, 1987). Wood and Yuen (1983) calculated a nonlinear Clapeyron curve for an idealized CaO–MgO–Al 2 O 3 –SiO 2 (CMAS) system. In this simplified system, the spinel–plagioclase boundary is univariant, and there-
fore the equilibrium reaction boundary is sharp. For more realistic mantle composition in CMAS+Na+Fe system (CMASNF), on the other hand, the univariant curve is broadened into a divariant P–T band (see Fig. 5) because of differential partitioning of additional components between the spinel and plagioclase phases. Thus, the reaction does not occur suddenly as in the CMAS system, but it is smeared out over an interval of depth. In this study, I used the result of theoretical calculation for the CMASNF system (Wood and Yuen, 1983). Once the spinel-lherzolite moves into the plagioclase stability field, the reaction could occur and consequently, grain-size reduction could happen. Therefore, I assume that grain-size reduction occurs when the material passes the boundary at which the geotherm intersects the higher-pressure limit of the plagioclase stability field. For metamorphic reactions, the grain-size of the product is controlled by a balance between the rates of nucleation and grain growth. If the rate of nucleation is higher than that of grain growth, smaller grain-size is obtained. Nucleation and growth rates will be strongly dependent on the cooling rate if deformation is absent. Because a high cooling rate will result in larger overstepping of the reaction, resultant nucleation and grain growth rates become higher and slower, respectively. If reaction occurs during deformation, the strain may cause a higher nucleation to grain growth ratio, and a very fine grain-size could be obtained. This is because the increase in dislocation densities and new
Fig. 5. Calculated spinel–plagioclase divariant Clapeyron curves in the CMAS+Na+Fe system by Wood and Yuen (1983).
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grain boundaries can promote nucleation even if activation energy is low (e.g., Brodie, 1980). A fast strain rate may be favourable for a fine grain-size, because fast transport of reaction products away from the reactants would prevent further reaction (e.g., Newman et al., 1999). In addition, the polyphase nature of the rocks may help to maintain a very fine grain-size by pinning of grain boundaries (e.g., Olgaard and Evans, 1988; Olgaard, 1990). In spite of these qualitative discussions (for more detailed discussions see Newman et al., 1999), it is actually unknown how much grain-size reduction occurs with the spinel- to plagioclase-lherzolite reaction. Whereas Riedel and Karato (1996) incorporated the growth rate Y and the nucleation rate I into the model and investigated the grain-size evolution associated with olivine to spinel phase transformation, very complex system for the change in grain-size by spinel- to plagioclase-lherzolite reaction prevents us to know the exact value of Y and I. So then, in this study, I assume that the olivine grain-size decreases down to 0.01 mm as a result of the reaction, as confirmed by Newman et al. (1999) on the North Pyrenean Zone mylonite and by Furusho and Kanagawa (1999) on the mylonite in Uenzaru peridotite complex. I will discuss the condition required for localized rheological weakening later. Although the garnet- to spinel-lherzolite reaction is also an important phase transformation in the upper mantle (e.g., Green and Ringwood, 1967), it does not seem common as a mechanism for grain-size-related softening. Additionally, Pe´rez-Gussinye´ and Reston (2001) emphasized the serpentinization of peridotites as a possible origin of strain localization. However, the fall in strength associated with serpentinization is related to its coefficient of friction rather than grainsize reduction.
3. Model results 3.1. Effects of grain-size reduction as a result of dynamic recrystallization 3.1.1. Initial state of grain-size and stress in the lithosphere In Fig. 6, I show initial steady state grain-size due to dynamic recrystallization and the corresponding
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Fig. 6. Initial steady state grain-size and stress in the lithosphere for an applied strain rate of 51015 (1/s). Grain-size distributions are shown in (a) and (c), and stress envelopes are in (b) and (d). Two different equations governing the steady state grain-size by dynamic recrystallization are used; (a) and (b) are for the case of Eq. (8), and (c) and (d) are for the case of Eq. (7). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Because ductile deformation of the crustal material occurs only from GSI creep, grain-size in the crust is not depicted.
stress envelope in the lithosphere. Black and grey lines represent the results for dry and wet conditions, respectively. Because I assume that ductile deformation of the crust occurs only by means of GSI creep, grain-size in the crust is not shown. Fig. 6(a) shows initial grain-size predicted by Eq. (8). For dry condition, grain-size gradually increases from 1 mm at depth of 55 km to 10 mm at depth of 85 km. Because grain-size becomes more than 10 mm at depths greater than 85 km, I presume grain-size to be 10 mm at the depths according to the assumption in
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Fig. 7. Temporal evolution of temperature and stress envelope in the lithosphere: (a) temperature, (b) stress envelope when the recrystallized grain-size is controlled by Eq. (8), and (c) stress envelope when the recrystallized grain-size is controlled by Eq. (7). Applied strain rate is 51015 (1/s). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho.
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this study. For wet condition, grain-size gradually increases from 0.5 mm at depth of 45 km to 5.5 mm at the bottom of the lithosphere. As can be seen in the stress envelopes shown in Fig. 6(b), the localized weak zone between higher stress zones does not appear in the lithospheric mantle, and ductile stress gradually decreases with depth. If a change in grain-size due to dynamic recrystallization is governed by Eq. (7), very small grain-size is predicted around depths of 30 km both for dry and wet conditions (see Fig. 6(c)). This is because the temperature is low at such depths (see also Fig. 4). The grain-size gradually increases with depth, and becomes more than 10 mm at depths greater than 95 km for dry condition and at depths greater than 85 km
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for wet condition. Therefore, I presume grain-size to be 10 mm at such depths according to the assumption in this study. The stress envelopes shown in Fig. 6(d) indicate that localized weakening does not occur in the lithospheric mantle. 3.1.2. Temporal evolutions of grain-size and stress in the lithosphere Because I regard steady state grain-size owing to dynamic recrystallization as the initial size in the present study, I evaluated only the effect of change in stress on grain-size by thermal cooling in the timedevelopment calculation. Temporal evolutions of temperature and stress envelope within the lithosphere are shown in Fig. 7. Applied strain rate is 51015
Fig. 8. Temporal evolution of the GSI creep ratio ((a) and (c)) and grain-size ((b) and (d)) distributions. Applied strain rate is 51015 (1/s). Recrystallized grain-size due to dynamic recrystallization is governed by Eq. (8). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Two different values are adopted for the strain e T: e T=0.4 ((a) and (b)); e T=0.03 ((c) and (d)).
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Fig. 8 (continued).
(1/s). Adopted governing equations for the equilibrium grain-size are Eqs. (7) and (8) for Fig. 7(b) and (c), respectively. Because I focused on the change in grain-size as a result of dynamic recrystallization, the change in grain-size in the brittle regime is ignored. It is assumed that grain-size of the materials gone into the model from the below is the same size at the bottom of lithosphere at the onset of extension. As can be seen in the figures, strength of the lithosphere decreases with time, which is associated with increase in geothermal gradient as a progression of extension. However, the localized weak zone between higher stress zones in the lithospheric mantle does not appear at any depth ranges, which is independent on adopted equations governing the equilibrium grain-size. It seems that the stress changes mostly are not brought about by grain-size change. Strength reduction is mainly caused by the upward movement of each
material point because of lower brittle strength at shallower depth as indicated by Eq. (2). From the temporal evolution of stress envelope (Fig. 7(b) and (c)), however, it may be difficult to understand the effect of thermal diffusion during the extension on the stress change through the grain-size change. So then, I show the temporal evolution of the distribution of the GSI creep ratio (r) and grain-size in the lithosphere (Fig. 8). The change in grain-size due to dynamic recrystallization is governed by Eq. (8). At time=0 my, the GSI creep ratio is constant to be 0.5 with depth for wet condition. This is because the initial grain-size should be given according to the nature of Eq. (8), in which the strain rate due to GSI creep is equal to that due to GSS creep. However, for dry condition, the GSI creep ratio at the onset of extension decreases at depths between 85 and 110 km, and ductile deformation at
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depths greater than 110 km occurs mostly by GSS creep. This is because grain-size does not exceed 10 mm in this study. When e T is 0.4, the ratio of GSI creep becomes more than that of GSS creep at depths between 40 and 75 km after 5 my. This is because stress increases by thermal cooling during extension. If grain-size is held to be constant, the increase in stress results in a higher ratio of GSI creep. Although grain-size will be controlled by Eq. (8) in such a way that the strain rate due to GSI creep is equal to that due to GSS creep, the rate of the change in stress exceeds that of the change in grain-size because finite strain (e T=0.4) is required for dynamic recrystallization. Grain-sizes are almost unchanged by
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the effect of thermal diffusion (see Fig. 8(b)). As seen in Fig. 8(c), on the other hand, the magnitude of increase in the GSI creep ratio for e T = 0.03 is smaller than that for e T = 0.4. For e T = 0.03, grain-size is rapidly recovered to its recrystallized value in response to the change in stress (see Fig. 8(d)). Generally, temporal evolution of the deformation mechanism is such that the ratio of GSI creep increases with time, and the switch from GSI creep to GSS creep does not occur at any depth. In Fig. 9, I show the same figures as Fig. 8 to illustrate the case that the recrystallized grain-size is governed by Eq. (7). At the onset of extension, the mechanism for ductile deformation is mostly GSS creep at any depth for dry condition. However, the
Fig. 9. Temporal evolution of the GSI creep ratio ((a) and (c)) and grain-size ((b) and (d)) distributions. Applied strain rate is 51015 (1/s). Recrystallized grain-size due to dynamic recrystallization is governed by Eq. (7). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Two different values are adopted for the strain e T: e T=0.4 ((a) and (b)); e T=0.03 ((c) and (d)).
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Fig. 9 (continued).
ratio of GSI creep increases at depths between 85 and 95 km along with the increase in grain-size, but at greater depths, it decreases in accordance with the assumption that the maximum grain-size is 10 mm (see Fig. 6(c)). For wet condition, ductile deformation at depths shallower than 55 km occurs mostly by GSS creep at the onset of extension. The ratio of GSI creep gradually increases with depth along with the increase in grain-size, and ductile deformation at depths greater than 75 km occurs mostly by GSI creep. For dry condition, a region where the GSI creep ratio is not zero at time=0 my develops a higher GSI creep ratio with time when e T is 0.4 (see Fig. 9(a)). For wet condition, a region where the ductile deformation occurs mostly by GSS creep finally diminishes after 14 my. Such an increase in the GSI creep ratio is attributed to the increase in stress as a result of thermal cooling. However, the ratio of each
ductile deformation mechanism is mostly constant with time for e T = 0.03 (see Fig. 9(c)), because the change in grain-size associated with the change in stress by thermal cooling occurs more rapidly. Grainsize change for e T = 0.03 occurs much more than for e T = 0.4 (see Fig. 9(b) and (d)). In general, the GSI creep ratio increases with time, and I am unable to find a switch from GSI to GSS creep at any stage of extension under the conditions applied in the numerical experiments. 3.2. Effects of grain-size reduction as a result of cataclasis In Fig. 10, I show the temporal evolution of grainsize, GSI creep ratio and stress envelope in the lithosphere for an applied strain rate of 51015 (1/s). I assume that the initial grain-size is given by the
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Fig. 10. Temporal evolution of (a) grain-size, (b) GSI creep ratio and (c) stress envelope in the lithosphere. An applied strain rate is 51015 (1/s). Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. Initial grain-size in the brittle regime is adopted to be 0.01 mm, and the temporal change in grain-size is ignored. The initial grain-size is given by the steady state example due to dynamic recrystallization governed by Eq. (8).
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steady state grain-size due to dynamic recrystallization governed by Eq. (8), and hence I examine the effects of grain-size reduction by cataclasis under such condition that dynamic recrystallization significantly occurs. The initial grain-size in the brittle regime, d c, is adopted to be 0.01 mm (see Fig. 10(a)), and the temporal change in grain-size is ignored. At time=0 my, the localized weak zones in the lithospheric mantle are formed at depths between 38 and 55 km under dry condition and at depths between 30 and 45 km for wet condition, where ductile deformation occurs mostly by GSS creep (see Fig. 10(b)). From the figure at time=0 my, it can be said that if grain-size in the brittle regime is instantaneously reduced to 0.01 mm by cataclasis, such localized weakening can be promoted for any size of mineral grain in the brittle regime at the initiation of extension. Similarly, if grain-size is reduced to 0.01 mm during the first 5 my, localized weakening in the lithospheric mantle can occur at depths between 18 and 25 km under dry condition and at depths between 13 and 20 km for wet condition. Furthermore, if grain-size is reduced to 0.01 mm during first 10 my, localized weakening in the lithospheric mantle can occur at depths around 10 km both for wet and dry conditions. Comparing with the results in Section 3.1, it is clear that the grain-size reduction by the cataclasis is very important for the localized rheological weakening in the lithospheric mantle. However, if it takes 14 my to reduce the grainsize to 0.01 mm, localized weakening in the lithospheric mantle cannot occur at any depth. This is because the temperature at shallow depths is too low to cause ductile stress to be less than brittle one even if the grain-size is very small (see also Fig. 10(a)). I show the critical time t c as a function of grain-size d c in Fig. 11, where localized rheological weakening in the lithospheric mantle can occur if grain-size in the brittle regime is reduced to d c before the extensional time reaches t c. The corresponding critical stretching factor b c for strain rate of 51015 (1/s) is also depicted in the figure. The shaded area represents possible conditions for localized weakening in the lithospheric mantle during extension. For example, if grain-size in the brittle regime is reduced to about 0.6 mm under dry conditions and to about 0.3 mm under wet conditions before the extensional time reaches 3 million years (the corresponding stretching factor is 1.6), it is possible for grain-size reduction by
Fig. 11. Relationship between the critical extensional time t c and grain-size d c, in which localized ductile weakening can occur if grain-size in the brittle regime is reduced to d c before the extensional time reaches t c. Shaded area illustrates the conditions for localized weakening in the lithosphere during extension. Corresponding critical stretching factor b c for strain rate of 51015 (1/s) is shown in the figure.
cataclasis to cause the localized weakening. However, if grain-size is reduced at the most to 0.9 mm under dry conditions and to 0.5 mm under wet conditions, the localized weakening cannot occur. In addition, a critical time t c more than 12 million years (the corresponding critical stretching factor b c is about seven) prevents the occurrence of localized weakening even if the grain-size is reduced to 0.01 mm. 3.3. Effects of grain-size reduction as a result of spinel- to plagioclase-lherzolite reaction Temporal evolution of a geometrical relationship between the geotherm and the Clapeyron curve for the spinel–plagioclase reaction is illustrated in Fig. 12. Initial lithospheric thickness is taken as 125 km, and the applied strain rate is 51015 (1/s). At time=0 my, the geotherm does not intersect with the Clapeyron curve. In this state, the whole lithospheric mantle consists of only spinel-lherzolite. After about 6 my, the geotherm intersects with the higher-pressure boundary of the divariant P–T band. As can be seen in the figure, the geotherm and the higher-pressure boundary of the divariant P–T band are crossing each
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Fig. 12. Temporal evolution of the geometrical relationship between the geotherm and spinel–plagioclase Clapeyron curve for an applied strain rate of 51015 (1/s). Divariant P–T band for CMASNF system is depicted in each figure.
other by two points, and the stability domain for plagioclase is between those for spinel. The geometrical relationship between the geotherm and the spinel–plagioclase Clapeyron curve is dependent on the initial thickness of the lithosphere. I show the time-dependent depth of the upper and lower reaction boundaries for the different initial lithospheric thickness of 80, 100, 125, and 150 km in Fig. 13(a). Temperatures at the reaction boundaries are also illustrated in Fig. 13(b). The stretching factor for e˙ =51015 (1/s) is depicted in each figure. As shown
in the figures, the time at the initiation of the reaction is sensitive to the initial thickness of the lithosphere; the reaction starts earlier for thinner lithosphere and vice versa. This is because the initial geothermal gradient is larger for the thinner lithosphere. For the initial thickness of 125 km, the depth of the lower reaction boundary rapidly increases to 33 km, and then decreases slowly to a depth less than 30 km. The depth of the upper reaction boundary, on the other hand, decreases gradually down to about 5 km. Temperature at the lower reaction boundary increases
Fig. 13. Temporal evolution of (a) depth of reaction boundary and (b) temperature at reaction boundary for the different initial lithospheric thickness. Corresponding stretching factor for strain rate of 51015 (1/s) is shown in each figure.
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Fig. 14. Temporal evolution of (a) grain-size, (b) GSI creep ratio, and (c) stress in the lithosphere. Black and grey curves represent the results for dry and wet conditions, respectively. The horizontal dotted line indicates the depth of the Moho. The initial thickness of lithosphere is adopted to be 125 km, and the applied strain rate is 51015 (1/s). Initial grain-size is given by the steady state grain-size due to dynamic recrystallization governed by Eq. (8).
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towards an asymptotic value with time, and temperature at the upper reaction boundary decreases down to about 400 8C. This behaviour is mostly insensitive to the initial thickness of the lithosphere. Therefore, it seems that the effect of the grain-size reduction by the reaction is mostly independent of the initial thickness of the lithosphere, but the initial thickness does control the time when the grain-size reduction initiates. In Fig. 14, I show the time-dependent grain-size, GSI creep ratio and stress in the lithosphere. The initial thickness of lithosphere is 125 km, and the applied strain rate is 51015 (1/s). I assume that grain-size reduction occurs when the material passes the boundary where the geotherm intersects the higher-pressure limit of the plagioclase stability field. Initial olivine grain-size is given by the steady state grain-size due to dynamic recrystallization governed by Eq. (8). Therefore, I evaluated the effects of grainsize reduction by the spinel- to plagioclase-lherzolite reaction during which time the dynamic recrystallization occurs significantly. In addition, it should be noted that the reaction products are olivine with a smaller grain-size but are not the development of plagioclase out of spinel. After 5.6 my, grain-size reduction as a result of the reaction occurs at depths between 25 and 30 km (see Fig. 14(a)), and the deformation mechanism at that depth range is mostly GSS creep (see Fig. 14(b)). Localized rheological weakening in the lithospheric mantle appears clearly at those depths both for dry and wet conditions (see Fig. 14(c)), but the magnitude of strength reduction is larger under dry condition. After 6.2 my, localized rheological weakening in the lithospheric mantle can be seen at depths between 18 and 33 km under dry condition. However, the localized weak zone under wet condition disappears on the scale of stress with a few MPa, whereas grain-size is very small and ductile deformation occurs mostly by GSS creep at depths between 18 and 33 km. After 10 my, although the dominant deformation mechanism is GSS creep at depths between 14 and 33 km, the localized weak zone disappears both for dry and wet conditions on that scale. Such disappearance of the localized weak zone on that scale is attributed to the fact that ductile stress is very low for high temperatures at depths greater than 33 km, and the lower boundary for the localized weak zone cannot be seen.
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4. Discussion 4.1. Possibility of localized rheological weakening by grain-size reduction during lithospheric extension In order to explore the possibility if grain-size reduction can cause localized rheological weakening favourable for strain localization, I investigated the effects of grain-size evolution as a result of dynamic recrystallization, cataclasis, and spinel- to plagioclaselherzolite reaction on the stress envelope in the lithospheric mantle. In the simple one-dimensional model adopted in this study, the applied strain rate is constant with time, and therefore the reduction of strength appears as a drop in stress. From the model results, I found that grain-size reduction by dynamic recrystallization cannot cause localized weakening, even when a significant change in grain-size occurs. The change in grain-size by dynamic recrystallization is sensitive to the stress and temperature that continuously change with depth in the lithosphere. It is, therefore, intrinsically difficult to cause rheological weakening at a particular depth range. Because the temperature within the lithosphere gradually increases with depth and it is hard to create the region where the temperature is particularly low or high in the lithosphere, the equilibrium grain-size also gradually changes with depth, if the uniform strain rate is applied over the entire lithosphere. It is, therefore, impossible to cause the localized rheological weakening at a particular depth range. Even if the localized weak zone is initially given, it will become to be smeared out by the effect of dynamic recrystallization and will be finally diminished. That is, the localized rheological weakening cannot occur in the steady state flow. Whereas the transient weakening could occur, this is dependent on the initial distribution of grainsize. The grain-size should be decreased locally at the initial state. In order to show this, it is enough to demonstrate the equilibrium grain-size distribution inferred from the initial temperature distribution and applied strain rate and the time-dependent effect of thermal diffusion during the extension. The effect of thermal cooling on grain-size is not great, whereas the thermal cooling during extension can make ductile stress increased to maintain the strain rate constant. Increase in the GSI creep ratio is attributed to the fact that the rate of change in stress by
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thermal cooling exceeds that of change in grain-size because finite strain is required for the completion of dynamic recrystallization. In fact, the increase in stress by thermal cooling is suppressed more significantly by the effect of dynamic recrystallization for the model with smaller e T (see Figs. 8 and 9). Although I show that grain-size reduction by dynamic recrystallization cannot cause localized weakening, the results in this study do not mean that it cannot cause rheological weakening. Because recrystallized grain-size governed by Eq. (8) is such that the strain rate due to GSI creep is equal to that due to GSS creep, substantial rheological weakening through a switch in the deformation mechanism from GSI creep to GSS creep cannot occur. Weakening can occur only in the GSI creep dominant regime. However, possible rheological weakening is minor, as pointed out by several previous studies (e.g., De Bresser et al., 2001; Ter Heege et al., 2002). The spinel- to plagioclase-lherzolite reaction has potential to cause localized weakening during extension. This reaction occurs at depths shallower than 35 km, and temperatures at the upper and lower reaction boundaries are between about 700 and 400 8C and between 700 and 1300 8C, respectively (see Fig. 13). The time for the initiation of grain-size reduction by the reaction is dependent on initial thickness of the lithosphere; the greater time is required for the thicker lithosphere. Time-dependent temperature at the reaction boundaries, depicted in Fig. 13, would be sensitive to the applied strain rate through the thermal diffusion during the extension. However, it may not be necessary to consider thermal cooling during extension for most of the existing rifted-basins (e.g., Jarvis and McKenzie, 1980). It is well known that the rate of reaction is strongly sensitive to temperature. If the reaction occurs under low temperatures, the rate of reaction is low and hence significant grain-size reduction might not occur in the geologically short time. In this study, I show that the spinel- to plagioclase-lherzolite transformation at the upper reaction boundary occurs at temperatures between about 700 and 400 8C (see Fig. 13(b)). However, Ahrens and Schubert (1975) discussed the temperature-dependent reaction rate of the gabbroeclogite transformation, and concluded that the reaction takes place in geologically short time only when the temperature is above 600 8C. Therefore,
significant grain-size reduction might not be attainable at temperatures less than 600 8C because of low reaction rate at these temperatures. In this study, I presume the grain-size to decrease down to 0.01 mm by the reaction. Although experimental studies on the amount of grain-size reduction by the reaction are not available, it is very important for localized rheological weakening. Therefore, I evaluated the required conditions to cause significant weakening by the reaction. Significant grain-size reduction by the spinel- to plagioclase-lherzolite reaction may occur at temperatures between 600 and 1300 8C as discussed above. At temperatures higher than 600 8C, the deformation occurs mainly by GSS creep, when grain-size is less than 0.7 mm under dry conditions and less than 0.5 mm under wet conditions, as can be seen in Fig. 3. Therefore, in order to obtain significant reduction of stress, grain-size should decrease down to those values by the reaction if the initial grain-size is more than those values. On the other hand, if the dominant deformation mechanism is GSS creep at the onset of extension, grain-size should decrease down to less than initial grain-size to produce the localized weak zone. As shown in Fig. 14, a created localized weak zone with a strength contrast of more than a few MPa is not maintained for a long time, and it disappears within a few million years. Significant strength contrast between in weaker and stronger zones would be required for the localization of deformation. In fact, Frederiksen and Braun (2001) indicated that the degree of viscosity reduction with strain is important factor for determining how the deformation is localized. However, it might be unknown how much strength reduction is required for the localization of deformation at this moment. If the localization of deformation requires that the strength difference between in weaker and in stronger zones is more than a few MPa, it might be hard to obtain a significant fall in rock strength at temperatures higher than 950 8C under dry conditions and higher than 800 8C under wet conditions, respectively (see Fig. 3), in spite of substantial grain-size reduction by the reaction. So then, I could state that the most suitable condition for localized deformation may be at the moment of the initial state of the reaction, if the localized weak zone is truly important for the localization of deformation.
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As discussed above, dry conditions have a wider range of grain-size and temperature for significant reduction of ductile stress by the reaction than the wet conditions do. Model results in this study show that larger strength reduction is predicted and the created weak zone could be maintained for longer period under dry conditions. Therefore, dry conditions are more favourable for localized rheological weakening, and this is consistent with the observation that reaction-enhanced softening described for the Turon de Te´coue`re peridotite is not attributed to a hydration reaction and may have occurred in the absence of aqueous fluids (Vissers et al., 1997; Newman et al., 1999). 4.2. Comparison with the results of previous dynamical modelling In previous numerical models (Govers and Wortel, 1993, 1995; Frederiksen and Braun, 2001), the firstorder strain softening was adopted at any depth in the lithospheric mantle. With this assumption, weakening occurs where more strain intrinsically accumulates in the two-dimensional model. Therefore, it is obvious that the rheological weakening is likely to take place even though the effect of strain softening is not incorporated into the model. From the results in this study, if grain-size reduction occurs by cataclasis, a localized weak zone would be formed at depths where deformation occurs by brittle failure, that is, just below the Moho depth. If grain-size reduction is brought about by the spinel–plagioclase lherzolite reaction, rheological weakening would occur only at depths shallower than 35 km. Furthermore, it would be difficult for grain-size reduction by dynamic recrystallization to cause a significant rheological weakening, except when grain growth is inhibited. Therefore, it seems that significant rheological weakening by grainsize reduction could only occur at a restricted depth range. Although Frederiksen and Braun (2001) demonstrated the occurrence of simple shear deformation using a simple strain-softening model, it may be difficult to form a shear zone cutting-off the entire lithospheric mantle. Results in this study seem to be more consistent with the studies by Govers and Wortel (1993, 1995), in which strain softening can cause localized deformation, but deformation on the lithosphere scale is mostly by pure shear.
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4.3. Other weakening mechanisms than grain-size reduction In this paper, I examined only the effects of grainsize reduction on the stress envelope in the lithospheric mantle. Several other softening mechanisms have been proposed (e.g., Poirier, 1980; Brodie and Rutter, 1985), including geometrical softening, thermal softening, and reaction softening. I discuss here, whether these other mechanisms can cause localized rheological weakening or not. Geometrical softening is due to the formation of the lattice-preferred orientation of minerals (e.g., Poirier, 1980). When deformation occurs by GSS creep, lattice-preferred orientation cannot develop and the resultant structure of rocks is mostly isotropic (Karato, 1988). However, if deformation occurs by GSI creep, a strong lattice preferred orientation could be formed. Lattice-preferred orientation will result in seismic anisotropy (Karato, 1987, 1989b; Nicolas and Christensen, 1987), and Karato (1992) proposed the hypothesis that the Lehmann discontinuity is attributed to the preferred orientation of olivine as a result of a change in the deformation mechanism. Rocks in shear zones often show preferred crystallographic orientation (e.g., Pieri et al., 2001). However, Wenk et al. (1991) concluded that geometrical softening has a minor effect on the reduction of strength, and laboratory experiments (Zhang et al., 2000) demonstrated that the development of the lattice-preferred orientation in olivine might result in hardening rather than weakening. Thus, geometrical softening may not be important for localized weakening. Strain localization in the ductile regime has been successfully modelled using thermal softening, in which a drop in strength is related to the increase in temperature by shear heating through a temperaturedependent viscosity (e.g., Yuen et al., 1978; Fleitout and Froidevaux, 1980; Kameyama et al., 1997). However, in these studies, the effects of shear heating were evaluated under the condition that the shear zone was initially assumed. Shear heating might not be important in the absence of localized deformation (e.g., Brun and Cobbold, 1980; Govers and Wortel, 1995; Monte´si and Zuber, 2002). Therefore, shear heating is important for maintaining the shear zones, but not for the origin of localized weak zones, whereas Leloup et al. (1999) indicated the importance
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of shear heating for the reaction-enhanced ductility developed along the strike-slip shear zones. Aside from grain-size reduction, it has been recognized that syntectonic metamorphic reactions in polymineralic rocks can also cause a strong reduction in rock strength either from that reaction products are intrinsically weaker than the reactants or from that nostrained new grains can deform more easily by intracrystalline glide than the original ones (Brodie and Rutter, 1985; Handy, 1989; Vissers et al., 1997). These effects were not considered in this study. Although it is unknown how much these softening mechanisms could bring about the reduction of rock strength, it is sure that weakening could occur at the same time and at the same position as those caused by grain-size reduction in this study, as far as the corresponding reaction is the spinel- to plagioclaselherzolite reaction. 4.4. Implications for the style of lithospheric extension Many studies have attempted to classify the development of specific basins in terms of either pure shear or simple shear. However, many rift-related basins have not been successfully classified in the style of extension. For example, Beach (1986) concluded that the structure in the North Sea Basin is consistent with a simple shear model from seismic reflection data. Barton and Wood (1984) explained, on the other hand, the subsidence data in the North Sea Basin in terms of the pure shear model. Latin and White (1990) quantitatively investigated adiabatic decompression melting in both the simple shear and the pure shear model, and concluded that it seems very difficult to reconcile the observed amount and composition of magma in the North Sea with a simple shear mechanism, based upon experimental results (McKenzie and Bickle, 1988; Furlong and Fountain, 1986). Therefore, it is not easy to determine the style of extension based on various geological and geophysical observations. Such difficulty may be attributed to the lack of understanding of the mechanism for the occurrence of simple shear deformation during extension. Although several kinematic models have been proposed, including pure shear (McKenzie, 1978), simple shear (Wernicke, 1985), and combinations of these two models (e.g., Lister et al., 1986; Kusznir and Egan,
1989), such models cannot provide dynamical aspects of the extensional process. Even the models with dynamical aspects (e.g., Braun and Beaumont, 1987; Bassi et al., 1993; Bassi, 1995) intrinsically predict symmetric extension. Whereas Frederiksen and Braun (2001) showed that simple strain softening could cause strain localization in the lithosphere, the origin of asymmetric extension is still not well known. Braun and Beaumont (1989) indicated that preexisting weakness in the crust and mantle is important for asymmetric deformation of the lithosphere. However, there is no evidence for the presence of such weakness at the initiation of the extension, as pointed out by Govers and Wortel (1993). Frederiksen and Braun (2001) proposed that simple shear deformation occurs when conditions favourable for strain localization are satisfied. Contrarily, the lithosphere is extended by a pure shear manner under conditions that are not favourable for strain localization. In this hypothesis, the deformed lithosphere has both components of pure shear and simple shear. In fact, many of continental rifts and passive margins have been interpreted in terms of both pure shear and simple shear deformation models (e.g., Wernicke and Burchfiel, 1982; Barton and Wood, 1984; Beach, 1986; Lister et al., 1986; Mutter et al., 1989; White, 1990; Torres et al., 1993; Collier et al., 1994). Therefore, it may be reasonable to say that most of the extensional basins are developed by both pure shear and simple shear deformations, and these have a particular amount of each deformational component. Based on a dynamical modelling study, Frederiksen and Braun (2001) concluded that the amount of strain that is accommodated by pure shear and simple shear is controlled by the strain required for the initiation of strain softening. If the strain softening occurs after a larger amount of strain, a higher amount of pure shear deformation is predicted. In the present study, I show that the stretching factor for the initiation of rheological softening is dependent on the initial thermal condition of the lithosphere for grain-size reduction by the spinel–plagioclase lherzolite reaction; a larger stretching factor is required for colder lithosphere and vice versa. Therefore, it is implied that the lithosphere may be mostly deformed uniformly for cold lithosphere, and hot lithosphere is more favourable for simple shear deformation, if localized rheological weakening is really important for a switch from pure
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shear to simple shear deformation. Various ratios of simple shear deformation can be predicted when localized weakening occurs by grain-size reduction due to cataclasis, which is dependent on an unknown rate of grain-size reduction. In this study, I cannot examine how a localized weak zone is connected laterally in the extended lithosphere. Furthermore, a common characteristic of a kinematic model of lithospheric extension is such that deformation is imposed by prescribing a velocity field (e.g., Gueydan et al., 2001), and the constitutive equations are not related to the deformation. Therefore, for the simple model presented in this study, I cannot examine how the created weak zone influences subsequent dynamic deformation of the lithosphere. So then, I should develop a two-dimensional dynamic model of lithospheric extension for further investigation into a transition from pure shear to simple shear deformation in a future study. If the detailed geological phenomenon associated with the transition from pure shear to simple shear deformation is revealed, the confusing observational evidence may be systematically interpreted.
5. Conclusions I investigated the effects of grain-size reduction as a result of dynamic recrystallization, cataclasis and spinel- to plagioclase-lherzolite reaction on the rheological behaviour of the lithosphere during a uniform extensional process. The present study may provide important constraints on the style of extension if localized rheological weakening can lead to the onset of simple shear deformation. The conclusions in this paper are as follows: (i)
(ii)
Grain-size reduction by dynamic recrystallization cannot cause localized rheological weakening that may be favourable for strain localization, even if significant grain-size change could occur. Grain-size reduction by cataclasis can cause localized rheological weakening just below the Moho depth. Because the rate of grain-size reduction by cataclasis is poorly constrained, it is difficult, at this moment, to specify the time when localized weakening occurs.
(iii)
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Grain-size reduction by the spinel- to plagioclase-lherzolite reaction can produce a weak zone at a particular depth range during extension. This reaction occurs at depths shallower than 35 km, which is independent of the initial thermal state in the lithosphere. Localized rheological weakening can occur under the following conditions. If grain-size before the reaction is greater than 0.7 mm under dry conditions and greater than 0.5 mm under wet conditions, it should decrease down to those values by the reaction. On the other hand, if the dominant deformation mechanism is GSS creep at the onset of extension, it is enough that grainsize decreases down to less than initial grainsize by the reaction. It is also important to note that dry conditions are more favourable to form an unmistakably localized weak zone.
Acknowledgements I would like to thank Tetsuzo Seno, Yoshitaka Takeda, and Takeshi Ikeda for carefully reading the manuscript and stimulating discussions. I also thank Masanori C. Kameyama and Hans De Bresser for their constructive reviews for improving the manuscript.
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