Towards scaling laws for subduction initiation on terrestrial planets: constraints from two-dimensional steady-state convection simulations
- Teresa Wong^{1}Email author and
- Viatcheslav S Solomatov^{1}
DOI: 10.1186/s40645-015-0041-x
© Wong and Solomatov. 2015
Received: 24 October 2014
Accepted: 7 April 2015
Published: 2 July 2015
Abstract
The strongly temperature-dependent viscosity of rocks leads to the formation of nearly rigid lithospheric plates. Previous studies showed that a very low yield stress might be necessary to weaken and mobilize the plates, for example, due to water. However, the magnitude of the yield stress remains poorly understood. While the convective stresses below the lithosphere are relatively small, sublithospheric convection can induce large stresses in the lithosphere indirectly, through thermal thinning of the lithosphere. The magnitude of the thermal thinning, the stresses associated with it, and the critical yield stress to initiate subduction depend on several factors including the viscosity law, the Rayleigh number, and the aspect ratio of the convective cells. We conduct a systematic numerical analysis of lithospheric stresses and other convective parameters for single steady-state convection cells. Such cells can be considered as part of a multi-cell, time-dependent convective system. This allows us a better control of convective solutions and a relatively simple scaling analysis. We find that subduction initiation depends much stronger on the aspect ratio than in previous studies and speculate that plate tectonics initiation may not necessarily require significant weakening and can, at least in principle, start if a sufficiently long cell develops during planetary evolution.
Keywords
Convection Stress Lithosphere Subduction Plate tectonicsBackground
Plate tectonics is central to many aspects of the geology and evolution of terrestrial planets. While Earth is the only planet where plate tectonics is observed, its driving mechanism and timing of initiation are still poorly understood. Subduction is thought to be the fundamental process for plate tectonics initiation because the slab pull of subducting slab contributes most to the forces that drive plate movements. On the Earth, initiation of subduction is greatly facilitated by tectonic forces associated with plate motions already occurring elsewhere (Mueller and Phillips 1991; Hall et al. 2003). Various models for subduction initiation has been proposed (e.g., McKenzie 1977; Turcotte 1977; Ogawa 1990; Mueller and Phillips 1991; Kemp and Stevenson 1996; Toth and Gurnis 1998; Stern 2004; Solomatov 2004a; Ueda et al. 2008; Nikolaeva et al. 2010), many of which involve existing plate boundaries or weak zones. Incipient subduction zones are often found near transform faults or fracture zones because of their physical weakness (e.g., Mueller and Phillips 1991; Gurnis et al. 2004).
On one-plate planets such as Venus and Mars, the absence of plate tectonics is likely to be due to the difficulty of subduction initiation in the absence of forces due to plate motions. In other words, the problem of plate tectonics initiation can be viewed as the problem of the very first occurrence of subduction. Due to the high sensitivity of viscosity to temperature, the lithosphere acts as the cold rigid thermal boundary layer that has a very high strength. On these planets, mantle convection is likely to be in the stagnant lid regime (e.g., Morris and Canright 1984; Fowler 1985; Solomatov 1995). One possible mechanism for the very first episode of subduction is due to the lithospheric stresses generated by mantle convection (Ogawa 1990; Fowler and O’Brien 2003; Solomatov 2004a). The magnitude of these stresses is relatively small compared to the lithospheric strength suggested by laboratory and field observations (e.g., Kohlstedt et al. 1995; Gurnis et al. 2004), and thus, it is usually believed that to initiate subduction, some weakening mechanisms must be present in the lithosphere.
Much effort has been devoted to understand the weakening mechanisms of the lithosphere. Several studies showed that the frictional shear stress resisting subduction at transform faults and fracture zones have to be less than 10 MPa for subduction to occur (Toth and Gurnis 1998; Hall et al. 2003; Gurnis et al. 2004). Stress drop estimates from earthquakes also indicate that fault strength may be approximately 10 MPa (Kanamori 1994; Kanamori and Brodsky 2004). Models are able to describe global reduction in the lithospheric strength, as well as localized weak zones such that plate-like features can be generated from mantle convection in a self-consistent manner (e.g., Trompert and Hansen 1998; Moresi and Solomatov 1998; Tackley 2000a; Bercovici 2001; Branlund et al. 2001; Regenauer-Lieb et al. 2001; Regenauer-Lieb and Kohl 2003; Regenauer-Lieb et al. 2006; Korenaga 2007; Landuyt et al. 2008). Various approaches have been used to deal with the creation of weak zones (Bercovici 2001; Bercovici and Ricard 2005; Landuyt et al. 2008; Branlund et al. 2001; Regenauer-Lieb and Kohl 2003). The two-phase damage theory with a grain-size dependent rheology was developed to explain the formation of weak plate boundaries and track the evolution of deformation (e.g., Bercovici and Ricard 2005; Landuyt et al. 2008; Bercovici 2012). Some studies suggested that water might play an important role in the localization of deformation (Regenauer-Lieb et al. 2001; Regenauer-Lieb and Kohl 2003; Regenauer-Lieb et al. 2006). Water also weakens the lithosphere by lowering the activation energy (Regenauer-Lieb et al. 2001; Regenauer-Lieb and Yuen 2004; Regenauer-Lieb et al. 2006) and increasing the pore fluid pressure (Kohlstedt et al. 1995).
One approach to quantify the weakening of lithosphere is to set a yield value to the rheology of the lithosphere to simulate brittle behavior (Fowler 1993; Trompert and Hansen 1998; Moresi and Solomatov 1998; Richards et al. 2001; Tackley 2000a,b; Fowler and O’Brien 2003; Solomatov 2004a; Stein et al. 2004; O’Neill et al. 2007; Stein and Hansen 2008). The yield stress can be regarded as a simplification of mechanisms that describe the strength of the lithosphere. Convection with yield stress is usually categorized into three regimes: mobile lid regime, transitional regime with some episodic failure, and stagnant lid regime (Moresi and Solomatov 1998; Tackley 2000b; Stein et al. 2004). Stein and Hansen (2008) further subdivided the transitional regime into episodically mobile and stable plate mobilization regimes. To access the conditions of a planet to have plate tectonics, some researchers presented regime diagrams in terms of Rayleigh number, viscosity contrast, and yield stress (e.g., Stein et al. 2004; O’Neill and Lenardic 2007).
A number of studies attempted to derive scaling relations for convective stresses and yield stress to extrapolate to planetary conditions (e.g., Moresi and Solomatov 1998; Fowler and O’Brien 2003; Solomatov 2004a,b; O’Neill et al. 2007; Valencia and O’Connell 2009; Korenaga 2010b; Van Heck and Tackley 2011; Stamenkovic and Breuer 2014). Yet the accurate description of these convection-induced stresses inside the lithosphere and thus the yield stress is lacking.
This study seeks to understand the stress distribution of the steady-state convecting cell with respect to various convective parameters using the pseudoplastic rheology as a first step. The goal of this study is to find a scaling law for the lithospheric stress (hereafter referred as lid stress) and the critical yield stress, which is the highest yield value at which the stagnant lid could be mobilized. Note that an alternative and, perhaps, more intuitive approach would be to assume that the yield stress is known and to try to figure out under what dynamic conditions it can be reached. However, (a) the ‘normal’ yield stress is so high that it is nearly impossible to reach and (b) given the uncertainties in the weakening mechanisms and thus the actual magnitude of the yield stress, it should be treated as an unknown.
In this study, we first examine the stress structure in steady-state stagnant lid convection and explore scaling relationships between convective parameters especially in relation to aspect ratio to develop a scaling theory for lid stress and critical yield stress. We then compare the theoretical scaling laws with numerical results. In addition, we investigate the accuracy of the Frank-Kamenetskii approximation for modeling the initiation of plate tectonics.
Rheology
Frank-Kamenetskii approximation
Many numerical studies used use a relatively low viscosity contrast to observe plate behavior, which has limited applications to realistic planetary situations. Moresi and Solomatov (1998) investigated the convective regimes with viscosity contrast ranging from 3×10^{4} to 3×10^{7}, and in Tackley (2000a), the viscosity contrast was limited to 10^{4}, whereas Richards et al. (2001) and Stein and Hansen (2008) used viscosity contrast on the order of 10^{5}. The viscosity contrast across the terrestrial lithosphere is many orders of magnitude higher.
The low-viscosity contrast is used because high-viscosity contrasts are difficult to treat in numerical calculations (Moresi and Solomatov 1995). Thus, the Arrhenius function is often approximated by the Frank-Kamenetskii function, which reduces the viscosity contrast by many orders of magnitude compared to Arrhenius viscosity function. This makes the problem of convection with strongly temperature-dependent viscosity more computationally tractable.
This method of expanding the terms in the exponent preserves the interior viscosity and the change of viscosity with temperature close to T _{ i }, where convection actively takes place. Some studies expanded the terms inside the exponents differently (e.g., King 2009). However, it is important to use Equations 4 and 5 to ensure the asymptotic accuracy of Frank-Kamenetskii approximation (Morris 1982; Morris and Canright 1984; Fowler 1985; Frank-Kamenetskii 1969).
Frank-Kamenetskii approximation was shown to be sufficiently accurate for the interior of the convective layer with large viscosity contrast (Solomatov and Moresi 1996; Ratcliff et al. 1997; Reese et al. 1999). Recent studies have examined convection with Arrhenius rheology and suggested slightly different scaling laws compared to convection with Frank-Kamenetskii viscosity (Korenaga 2009; Stein and Hansen 2013). Here, we assess the accuracy of the Frank-Kamenetskii approximation in predicting the values of critical yield stress.
Pseudoplastic rheology and plastic yielding
where \(\dot {e}\) is the second invariant of the strain rate tensor. The yield stress defines a change on deformation mechanism based on the second invariant of the deviatoric stress tensor, which corresponds to the Von Mises yield criterion. In this study, we consider two types of yield stress: a constant yield stress τ _{ y } or a depth-dependent yield stress with a constant gradient τ y′.
Methods
Formulation of the problem
Equations of thermal convection
where i and j, are indices of the coordinate axes.
The boundary conditions are as follows. For a cell with only base heating, the top and bottom surfaces are isothermal. The temperature of the top surface as T _{0} and that of the bottom surface as T _{1}. The temperature difference is Δ T=T _{1}−T _{0}. The vertical boundaries are thermally insulated. All surfaces are free-slip. The velocity normal to a cell boundary is zero.
Non-dimensionalization
where e _{ i } is a unit vector in the direction of gravity.
Matching Arrhenius viscosity and exponential viscosity in non-dimensional form
T _{ i } differs from the bottom temperature T _{1} by a rheological temperature difference Δ T _{rh}, which is on the order of θ ^{−1}.
Results
Steady-state convection
We use the finite element code CITCOM (Moresi and Solomatov 1995) to simulate convection in a 64a×64 box, where a is the aspect ratio. Several high viscosity cases were ran with 128a×128 resolution for more accurate results. All cases were run until they reached a steady state. We consider the range of parameters in which convection is in the stagnant lid regime (Solomatov 1995).
To compare the stresses in exponential and Arrhenius viscosity, we choose a range of T _{0} and calculate their corresponding E that gives the same θ according to Equation 29 with T _{ i }≈1.
Aspect ratio
Rayleigh number
Viscosity contrast
Lid stress scaling theory
Fowler (1985) obtained a polynomial expression for the stress in the lid below the surface stress boundary layer in large lid slope approximation, which allows a comparatively simple scaling relation for stress. In order to solve the equations of convection, Fowler took Ra and θ to be asymptotically large and assumed the magnitude of lid slope to be either on the order of lid thickness or rheological sublayer thickness. For the Earth and some smaller terrestrial planets as well as most numerical simulations, Ra may not be as high as the asymptotic theory require, and thus, some other theory may be needed for lid stress scaling. Moreover, as we find in this study, the lid slope does not follow either of these two end-member cases and thus needs to be scaled based on numerical simulations.
where τ _{ i } is the stress at the interior temperature T _{ i }.
Thus, the lid stress is determined by Ra, d T/d y, λ, and y, which will be defined in the following discussion. This scaling is similar to that obtained from the analytical solutions of Fowler (1985).
Lid base temperature
Numerical results of power law coefficients in scalings of different parameters with Ra, aspect ration a , and Frank-Kamenetskii parameter θ
Parameter | c in 10 ^{ c } | Ra | a | θ |
---|---|---|---|---|
Nu | 0.23 ± 0.07 | 0.24 ± 0.01 | -0.17 ± 0.03 | -1.17 ± 0.05 |
Δ T _{ rh }/Δ T | 0.25 ± 0.17 | 0.010 ± 0.02 | 0.39 ± 0.08 | -0.84 ± 0.12 |
δ _{0}/d | -0.26 ± 0.10 | -0.22 ± 0.01 | 0.27 ± 0.04 | 1.12 ± 0.06 |
δ _{lid}/d | -0.74 ± 0.06 | -0.23 ± 0.01 | 0.12 ± 0.03 | 1.44 ± 0.04 |
δ _{lid,max}/d | -0.62 ± 0.07 | -0.19 ± 0.01 | 0.33 ± 0.03 | 1.26 ± 0.05 |
δ _{rh}/d | 0.02 ± 0.33 | -0.20 ± 0.04 | 0.60 ± 0.15 | 0.34 ± 0.21 |
Lid slope | -0.98 ± 0.13 | -0.07 ± 0.02 | 0.14 ± 0.06 | 0.63 ± 0.09 |
τ _{lid} | 0.36 ± 0.21 | 0.68 ± 0.03 | 0.92 ± 0.09 | -0.15 ± 0.13 |
τ _{ y,c r } | 0.40 ± 0.22 | 1.09 ± 0.03 | 1.67 ± 0.10 | -1.49 ± 0.14 |
τ y,c r′ | 2.34 ± 0.32 | 1.34 ±0.04 | 1.45 ± 0.14 | -3.47 ± 0.21 |
Lid slope and lid thickness
Thermal gradient
The temperature is approximately a linear function of depth and the thermal gradient in the lid is about constant with depth. At mid-width (x=0.5a), it is approximately equal to the Nusselt number, which is the non-dimensional horizontally averaged surface temperature gradient. Since we are looking at temperature changes from the interior to the bottom of the lid which includes the rheological sublayer, we also check the thermal gradient in the rheological sublayer Δ T _{rh}/δ _{rh} to note any difference in the scaling relations. As before, we choose the values Δ T _{rh}/δ _{rh} at the mid-width to exclude boundary effects for scaling purposes.
In previous theories, Δ T _{rh}/Δ T∼θ ^{−1} and δ _{rh}/δ _{lid}∼θ ^{−1}. The determination of C follows the description in the previous section on lid base temperature, and it is found to be dependent on aspect ratio and θ. Therefore, Δ T _{rh}/Δ T and δ _{rh}/δ _{lid} will also have a dependence on a and θ, and their scaling relations are summarized in Table 1.
Stress scaling
As shown in Figure 10, there is a slight difference in the thermal gradient in the lid and in the rheological sublayer; therefore, Equations 38 and 39 may result in slightly different scaling exponents. We check both scaling relations to see whether the stresses at the lid base and those in the lid can be scaled similarly.
Power law coefficients in scaling laws for stresses: numerical results vs theory
Parameter | Ra | a | θ | Method |
---|---|---|---|---|
τ _{lid} | 0.68 ± 0.03 | 0.92 ± 0.09 | -0.15 ± 0.13 | Numerical |
0.73 ± 0.09 | 1.14 ± 0.29 | 0.146 ± 0.42 | Theory (in terms of Δ T _{ rh }/δ _{ rh }) | |
0.77 ± 0.12 | 1.18 ± 0.39 | 0.14 ± 0.56 | Theory (in terms of Nu) | |
τ _{ y,c r } | 1.09 ± 0.03 | 1.67 ± 0.10 | -1.49 ± 0.14 | Numerical |
0.94 ± 0.10 | 1.81 ± 0.32 | -1.13 ± 0.47 | Theory (in terms of Δ T _{ rh }/δ _{ rh }) | |
0.97 ± 0.13 | 1.85 ± 0.42 | -1.16 ± 0.61 | Theory (in terms of Nu) | |
τ y,c r′ | 1.34 ±0.04 | 1.45 ± 0.14 | -3.47 ±0.21 | Numerical |
1.13 ± 0.11 | 1.48 ±0.35 | -2.39 ± 0.51 | Theory (in terms of Nu) | |
1.16 ± 0.14 | 1.52 ±0.45 | -2.38 ± 0.65 | Theory (in terms of Δ T _{ rh }/δ _{ rh }) |
Convection with yield stress
We use the steady-state solutions as the starting point before imposing a yield stress to simulate plastic yielding. For the yield stress gradient, a small cohesion term (surface yield stress) was introduced to stabilize the solution.
Regimes of convection with constant yield stress or constant yield stress gradient
In this transitional regime, the yield stress slightly changes the interior dynamics as can be seen in various convective parameters of the interior region of the cell. The yield stress increases the lid slope whereas the lid thickness remains approximately the same. When the yield stress is slightly above the critical value, these changes caused by yield stress are negligible and convection remains in the stagnant lid regime. Therefore in deriving the critical yield stress scalings, we refer to the steady-state structure that has a yield stress just above the critical value, so that the scaling relations for various convective parameters (Table 1) from steady-state stagnant lid convection can still be used.
Time evolution of lid weakening and failure
A possible contribution to the uncertainty in determining the critical yield stress is that at the vicinity of the critical value, the behavior may be difficult to interpret. In some cases that as the yield stress gets close to a critical value, the surface velocity increases slowly and it may take more than 10^{5} timesteps to reach a point of overturn, whereas typically, it takes less than 10^{4} timesteps to a drastic increase in surface velocity (Figure 18 right). This may be due to the behavior of dynamic system near a critical point.
Critical depth of plastic failure zone
where δ _{0}=δ _{lid}+δ _{rh}. The model of Fowler and O’Brien (2003) predicts that δ _{ pl } is defined by the temperature that gives the interior viscosity. For Newtonian rheology, this means that the plastic zone has to extend through the base of the lid. Solomatov (1995,2004a) suggested that δ _{ pl } only has to penetrate to the isotherm at which the viscosity contrast with the interior viscosity is e ^{4(n+1)}, where n is the stress exponent for non-Newtonian viscosity.
Since the critical viscosity contrast is neither a constant or a function of θ, we look at the depth of the plastic zone to derive scaling relations for the yield stress. We investigate the plastic depth δ _{pl} as a fraction of the lid thickness. For scaling purposes, the lid thickness was previously defined at the middle of the convecting cell. Here, since δ _{pl} is defined at the downwelling edge, we have to determine a lid thickness at the edge δ _{lid,max}. This is done by extrapolating the mid-width lid slope to the downwelling edge (Figure 10). As shown is Figure 24, the plastic depth is approximately 0.3 to 0.5 of the lid thickness. We take approximate values for our scaling relations rather than scaling these properties with convective parameters because the trends observed in Figure 24 maybe due to insufficient viscosity contrasts which place convection on the boundary of transitional regime, especially for θ=13 in which the viscosity is reduced to approximately 10^{4} by the critical yield stress.
To find the lid stress at δ _{pl} using Equation 34, we also need to determine the distance of the base of the plastic zone from the convective interior y _{pl}. The lid base is at δ _{rh} from the interior, so we express y _{pl} in terms of δ _{rh} to give a sense of distance in relation to rheological sublayer thickness. While there is a general trend of increasing of y _{pl}/δ _{rh} with θ, it is difficult to discern a correlation as y _{pl}/δ _{rh} fluctuates; thus, we take y _{pl}≈3δ _{rh} for our scaling relations.
The dependence of δ _{pl}, y _{ pl }, and Δ η _{pl} on Ra and aspect ratio are very weak and therefore assumed negligible.
Scaling for critical yield stress and critical yield stress gradient
Critical yield stress scaling theory
where C _{1}, C _{2}, C _{3} ranges from 4 to 16, 8 to 53, and 16 to 178, respectively.
From the previous sections, since Δ T _{rh}/Δ T, δ _{rh}/d, δ _{lid}/d, and λ are all scaled in terms of Ra, a, and θ with scaling exponents summarized in Table 1. τ _{lid}, τ _{ y,c r }, and τ y,c r′ can be scaled in terms of Ra, a, and θ. The results are listed in Table 2.
Numerical results for critical yield stress and yield stress gradient: Arrhenius vs. exponential viscosity
For the cases tested at resolution that is doubled, we find that the values for R _{ τ } and \(R_{\tau ^{\prime }}\) are within 5% difference. Therefore, 64×64a resolution is sufficient for our single-cell steady-state convection analysis.
Discussion and conclusions
Comparison with other studies of stress scaling laws
The studies dealing with convective stress scaling often aim to provide an expression for stress in terms of radius and mass of a planet to predict the likelihood of plate tectonics. They reached various conclusions (e.g., O’Neill et al. 2007; O’Neill and Lenardic 2007; Valencia and O’Connell 2007;2009; Korenaga 2010a; Van Heck and Tackley 2011; Stamenkovic and Breuer 2014). One of the main difficulties in deriving convincing scaling laws for plate tectonics initiation was a poor understanding of lid stresses and how they are related to lid failure. In the present study, we have addressed these issues using two-dimensional steady-state convective cell simulations. This is the simplest system to analyze, and yet even for this system, the derivation of scaling laws proved to be complicated and not well described by the existing asymptotic theories. Below, we discuss some differences between our study and previous studies and summarize our scaling laws in a dimensional form.
In some studies (e.g., Moresi and Solomatov 1998; Trompert and Hansen 1998; Tackley 2000b; Fowler and O’Brien 2003), the authors assumed that subduction occurs when the stresses in the convective interior exceed the yield stress. This means that subduction begins when not only the lid but also the interior of the convective cell fails. However, subduction initiation may not necessarily require the failure of interiors but instead may only require failure of just a small portion of the lid. The stresses in the lid are several orders of magnitude higher than the stresses in the interior, and also, they scale differently. Thus, the assumption regarding what part of the convective cell must fail in order for subduction to begin is critically important. In this study, we have investigated this assumption quantitatively, based on a detailed analysis of stresses and other parameters in the convective cell, and then formulated the critical conditions for subduction initiation.
In agreement with Fowler (1985), we have shown that the lid slope is a key factor in determining the stresses in the lid. However, our model has several important differences from Fowler (1985). The theoretical solution in Fowler (1985) is a similarity solution and does not take into the finite horizontal extent of the lid. Our model has vertical boundaries, and thus, the structure of the lid in our model is more complex. Also, the solution in Fowler (1985) is an asymptotic solution requiring very high values of parameters, such as Ra and θ, and a satisfaction of certain asymptotic conditions, which are not reached in our simulations and may not necessarily be reached on planets. Thus, our scaling laws are not asymptotic in this sense. Also, solutions in Fowler (1985) are obtained for two end-member cases, the large lid slope case, and the small lid slope case. We find that the lid slope behaves in a more complex way and is between these two end-member cases. We have determined a scaling law for the lid slope numerically and used it to derive the scaling law for the stresses in the lid.
Our analysis suggests that the stresses in the lid increase approximately as a square of the distance from the bottom of the lid (Equation 34 and Figure 14). This agrees with the asymptotic analysis of Fowler (1985) but is different from the stress distribution in Solomatov (2004a). In Solomatov (2004a), the stress distribution was more complex because the convective cell was heated from within rather than from the bottom and the internal heating affected the temperature-induced density distribution in the lid. At Rayleigh numbers higher than those reached in Solomatov (2004a), the lid is expected to become sufficiently thin so that the heat production inside the lid would be negligible compared to the heat flux at the base of the lid. Thus, we expect that for convection with internal heating, the stress distribution in the lid should approach the quadratic distribution that we observe for convection with bottom heating.
We find that subduction initiation requires that only a part of the lid undergoes plastic failure, roughly 0.3 to 0.5 of the total lid thickness. This generally agrees with the analysis in Solomatov (2004a) and confirms that the plastic failure does not have to extend all the way to the bottom of the lid as was assumed in Fowler and O’Brien (2003). However, unlike Solomatov (2004a), we determine the distance to the boundary of the plastic failure zone by measuring it from the base of the lid and scaling it in terms of the rheological boundary layer thickness. We find that such an approach is more appropriate because the mobility of the lid is largely controlled by the viscosity contrast between the zone of failure and the convective interior of the cell, which in turn is scaled with the rheological boundary layer thickness.
Estimates for the Earth
To compare our results with those obtained in Solomatov (2004a,b), we convert the critical yield stress τ _{ y,c r } and critical yield stress gradient τ y,c r′ into their dimensional forms (Equations 20 and 22) and estimate the critical yield strength and the critical coefficient of friction μ for subduction initiation on the Earth.
The interior viscosity cannot be reliably estimated from the viscosity law alone and is usually determined from better constrained properties such as lithospheric thickness. Therefore, following Solomatov (2004a), we use the scaling law for δ _{0} (Table 1) and present the results in terms of the thickness of the thermal boundary layer δ _{0}∼100 km instead of the mantle viscosity η _{1}.
The scaling law for the critical yield stress depends strongly on aspect ratio a. Previous studies have scaled the aspect ratio from half-space cooling of lithosphere (Korenaga 2010b; Stamenkovic and Breuer 2014) or estimated from numerical simulations (Solomatov 2004a,b), whereas it was assumed to be on the order of 1 in Valencia and O’Connell (2009). We use the horizontal width of the convective cells as l _{hor}=a d∼100 km as a very rough value to compare our estimates with those in Solomatov (2004a,b). This value was inferred from observational constraints on the present-day horizontal scale of sublithospheric convective structures (Solomatov 2004a).
Parameters used to estimate τ _{ y,cr } and τy,cr′ for Earth as in Solomatov ( 2004a ,b)
α | 3×10^{−5} |
κ | 10^{−6} m^{2} s ^{−1} |
δ _{lid} | 100 km |
k | 3 W m ^{−1} K ^{−1} |
E | 430 kJ mol ^{−}1 |
d | approximately 500 km |
g | 10 m s ^{−2} |
ρ | 3,300 kg m ^{−3} |
T _{0} | 300 K |
T _{ i } | 1,700 K |
l _{hor} | 100 km |
from Equation 50.
If we take into account the fact that Frank-Kamenetskii approximation that we used to derive the scaling laws overestimate the critical yield stress and the critical friction coefficient (Figure 31), then both ours and the estimates in Solomatov (2004a,b) should be further reduced by a factor of 2 (Figure 31), depending on the values of the viscosity parameters and the Rayleigh number.
One major difference between our scaling laws and the scaling laws obtained in Solomatov (2004a) is a much stronger dependence of the critical yield stress and critical friction coefficient on the width of the convecting layer l _{hor} - they scale roughly as approximately \(l_{\text {hor}}^{2}\) as opposed to the previous scaling approximately l _{hor}. This means that the critical values of the yield stress and friction coefficient would increase by 2 to 4 orders of magnitude if the width of the convective cells increased by 1 to 2 orders of magnitude (for example, in the past history of the Earth),and thus, at least in principle, could reach the experimentally observed values that are on the order of 1,000 MPa for τ _{ y,c r } and μ∼0.6 to 0.85 (e.g., Byerlee 1978; Goetze and Evans 1979; Kohlstedt et al. 1995; Mei et al. 2010) and values constrained by loading models with in situ stress measurements of Hawaiian Islands (Zhong and Watts 2013) which are 0.25 to 0.7 for μ and 100 to 200 MPa for lithospheric stress. This implies that the chances of plate tectonics might be higher than we thought before. Time-dependent calculations and a more realistic formulation of the problem are required to better understand the implications of these results for plate tectonics initiation.
Uncertainties in stress scaling
The scaling laws derived here are applicable to Newtonian rheology; therefore, the activation energy for diffusion creep is used in our calculations. However, it should be noted that dislocation creep is probably the dominant mechanism in the lithosphere (Karato and Wu 1993). For the Earth, wet dislocation creep may be preferable (Solomatov and Moresi 2000) while for other terrestrial planets such as Venus might have dry lithosphere. To apply on a wider range of planets including icy bodies, scaling laws based on non-Newtonian rheology will be required.
In previous scaling theories, the lid slope is often considered to be small because the lid thickness is assumed to be relatively small. Even in the large lid slope end member case in Fowler’s theory, δ _{lid} is assumed to be small relative to the thickness of the convecting layer.
However, our simulation indicates that the slope may be significant, so the derivations may need to be modified to take this into account.
Free-slip boundary conditions are often used in solving equations for thermal convection, but this restricts the vertical motion of the surface. Recent studies have used the free-surface boundary conditions, which is closer to natural surface condition as both normal and shear stress on the surface is reduced to zero (Zhong et al. 1996; Schmeling et al. 2008; Kaus et al. 2010; Crameri et al. 2012; Kramer et al. 2012). It maybe computationally expensive to implement this for the time being, but it could be worthwhile to explore its effect on scaling relations for stresses in the future.
Our numerical results show that τ _{ y,c r } of Arrhenius viscosity approaches that of exponential viscosity as the Frank-Kamenetskii parameter θ increases. This enables us to use exponential viscosity law to extrapolate to high Arrhenius viscosity contrast conditions. Besides the Frank-Kamenestskii approximation, the viscosity contrasts can be reduced in other ways, one of which is to set a cutoff viscosity. The stress structure resulting from the cutoff viscosity will have to be examined. We can then compare the accuracy of these approximations and apply them to extrapolate the results to planetary parameters.
Our results generally support previous conclusions that in order for the convective regime on the terrestrial planets in the inner Solar System to change from stagnant lid convection to plate tectonics, the yield stress of the lithosphere should be much smaller (several MPa) than that predicted by laboratory experiments on rock deformation (hundreds of MPa as predicted by Byerlee’s law). However, our results suggest a much stronger dependence of the critical yield stress on the horizontal width of the convective cells. This opens a possibility of subduction initiation even for the large, experimentally measured, lithospheric strength provided that a sufficiently long convective cell forms in a time-dependent mantle convection. In the future, it would be important to investigate the role of initiation conditions and statistical fluctuations of convective cells for the initiation of subduction in time-dependent convection.
Declarations
Acknowledgements
The authors thank NSF for partial support and the two anonymous reviewers for their constructive comments to improve the manuscript.
Authors’ Affiliations
References
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