https://orcid.org/0009-0002-1464-1995
SUMMARY
Effective viscosity of the upper mantle is a critical parameter for comprehending the dynamics of the lithosphere and plate tectonics. In recent years, magnetotelluric (MT) surveys have gained attention as a potential tool for determining viscosity. However, a direct physical basis for the relationship between effective viscosity and electrical resistivity still needs to be improved. To address this issue, we have established the basis that connects electrical resistivity and effective viscosity under different thermochemical conditions and the principle of electrical neutrality. The creep and electrical conductions of rocks in the lithospheric mantle are all the thermally activated atomic-scale random motion of particles in solids or melts, controlled by the slowest and fastest particles, respectively. Due to the sizeable electrostatic interaction energy, the concentrations of the two particles must always satisfy the charge-neutral condition. Hence, we derived two ideal models for anhydrous and hydrous olivine. Then, to demonstrate their applicability, we converted the resistivity cross-section constructed from long-period MT data to the effective viscosity distribution in the Tarim–Tianshan–Junggar area. We found that the results matched those obtained by the previous method well. Overall, this study provides valuable insights into the potential of MT surveys for determining the effective viscosity of the upper mantle.
1. INTRODUCTION
The effective viscosity is a crucial parameter for multiscale geodynamic modelling, fine-scale lithospheric deformation and large-scale geophysical data interpretation (Xu et al. 2018). The effective viscosity of the upper mantle depends mainly on components, stress, temperature, pressure, water content and oxygen fugacity (Karato 2010). Previous models of mantle viscosity were primarily based on observations of post-glacial rebound, post-seismic deformation, or extrapolation of laboratory data (Hirth & Kohlstedt 2015; Hu et al. 2016). These models had limited resolution and generally failed in the case of lamellar decoupling within a lithospheric interior. Therefore, it is particularly attractive for estimating effective viscosity through high-resolution geophysical imaging. This will provide fundamental insights into Earth's dynamics and contribute to a better understanding of geophysical processes.
If the strain rate is assumed to be constant, the effective viscosity is mainly determined by water, temperature and pressure. Additionally, changes in the physical and thermochemical state that alter viscosity affect electrical conductivity. High-temperature and pressure experiments on major minerals (e.g. olivine, pyroxene and garnet) have suggested that the general form of effective viscosity (Karato & Wu 1993; Xu et al. 2018) is
where |$\sigma $| is stress, |$\dot \varepsilon $| is strain rate, d is grain size, |${C_w}$| is water content; E and V are activation energy and activation volume, respectively; P and T are pressure and temperature; R is the ideal gas constant; A, m, q, r and n are all laboratory-derived parameters.
The electrical resistivity (|$\rho $|) takes a similar form (Xu et al. 2018):
where the parameters are taking the physical meanings as those at the same positions in eq. (1).
Some studies have applied electrical resistivity to estimate the effective viscosity in the upper mantle with considerable success (Unsworth et al. 2005; Bai et al. 2010; Liu & Hasterok 2016; Xiao et al. 2018; Xu et al. 2018). These results are all derived from the empirical presumption that the electrical resistivity in the lithospheric mantle is positively correlated with effective viscosity. For example, the following relationship between the two normalized parameters was suggested (Liu & Hasterok 2016; Xu et al. 2018):
where |${\eta _0}$| is reference viscosity; |$\rho $| is the electrical resistivity inverted from magbetotelluric (MT) data; |${\rho _0}$| is the reference electrical resistivity; and |${C_0}$| and |${C_1}$| are two coefficients that should be constrained from other data. In the scheme of Liu & Hasterok (2016), |${\eta _0}$| and |${\rho _0}$| are closely related to the regional average, and |${C_0}$| and |${C_1}$| are determined from geodynamic modelling. For the scheme of Xu et al. (2018), |${\eta _0}$| and |${\rho _0}$| are the reference values associated with the regionally minimal water content, and |${C_0}$| and |${C_1}$| are calibrated from laboratory-measured data, which eliminated the need for geodynamic modelling, where some parameters are difficult to estimate from various observation and/or tests. However, volatiles other than water, such as carbon dioxide and fluorine, and iron-bearing minerals and sulfides (Özaydın & Selway 2020; Xu et al. 2020), all seriously affect the rock electrical resistivity under the temperature and pressure conditions of the lithospheric mantle. Additionally, Selway et al. (2020) utilized a petrologically constrained genetic algorithm approach to interpret MT data. This approach enabled them to obtain information on the upper mantle temperature, hydrogen content and the presence of partial melt. Furthermore, they calculated effective viscosity using an experimental model, which essentially serves as an extrapolation of laboratory results. In short, the full expression of the physical relationships between the various mechanisms of electrical conduction and plastic deformation under the conditions of the upper mantle has not yet been established.
From a microscopic perspective, we here propose a theoretical basis for converting electrical resistivity into effective viscosity for nominal anhydrous minerals (NAMs). We then demonstrate its applicability by converting the electrical resistivity, derived from long-period MT data into an effective viscosity distribution in the Tarim–Tianshan–Junggar region of Northwest China.
2. METHODOLOGY AND THEORY
2.1 The component dependence of olivine creep rate and electrical conductivity
Under thermochemical conditions in the lithosphere, minerals will undergo plastic deformation (Karato 2008). As the component with the largest proportion in volume, the physical parameters of olivine control the rheological and electrical properties of the upper mantle (Karato et al. 1986; Hirth & Kohlstedt 2003). The creep and electrical conduction of olivine in the lithospheric mantle are both related to thermally activated atomic-scale random motion of particles in solids or melts (Chakraborty 2008; Zhang 2010).
The observation of crystallographic preferred orientation in mantle xenoliths (Jin et al. 1998), the evidence for strong seismic anisotropy (Gung et al. 2003), and the results of the thermo-mechanical numerical simulation (Ruh et al. 2022) have suggested that dislocation creep is the dominant deformation mechanism in the lithospheric mantle. While high-temperature experiments have indicated that the electrical properties are controlled by proton conduction under hydrous conditions and by small polaron conduction under anhydrous conditions.
To evaluate the relationship between electrical conduction and effective viscosity, we examine the processes of dislocation creep and proton/small polaron conduction. Hirth & Kohlstedt (2015) have recently presented a model based on dislocation climb to account for the measured values of the stress exponent and the sluggish rate of Si lattice diffusion in olivine under both anhydrous and hydrous conditions. Following the model of Hirth & Kohlstedt (2015), if the dislocation velocity is determined by the velocity of climb (|${v_\mathrm{c}}$|), then |$\dot \varepsilon \propto {v_\mathrm{c}} \propto D_{\textrm{pipe}}^\mathrm{Si}$|, and
where |${V_\mathrm{m}}$| is molar volume (|$44 \times {10^{ - 6}}{\rm{\,\,}}{\mathrm{ m}^3}/\mathrm{ mol}$|), R0 is the dislocation spacing (|${R_0} \approx {d^{ - \frac{1}{2}}}$|), |${r_0}$| is the dislocation core radius (|${r_0} \approx 2b$|), and the effect of pipe diffusion can be accounted for by substituting the effective diffusivity (|${{\rm{\mathit{ D}}}^{{\rm\mathrm{Si}}}} \approx d\pi r_0^2D_{\textrm{pipe}}^\mathrm{Si}$|, with the assumption that |${V_\mathrm{c}}$| is limited by pipe diffusion of |$\mathrm{ Si}$| atoms), and the relationship between dislocation density and stress is described by the equation |$d{b^2} = B{( {\frac{\sigma }{\mu }} )^x}$|(with B, q are constants, and |$\mu $| is shear modulus).
It is worth noting that the diffusivity of |$\mathrm{ Si}$| is equal to the product of the diffusivity of |$\mathrm{ Si}$| vacancies and the concentration of |$\mathrm{ Si}$| vacancies, that is |$D_{\textrm{pipe}}^\mathrm{Si} = D_{{V_\mathrm{Si}}}^\mathrm{Si}[ {{V_\mathrm{Si}}} ]$|(Tasaka et al. 2015), where |$D_{{V_\mathrm{Si}}}^\mathrm{Si}$| is the diffusivity of |$\mathrm{ Si}$| vacancies and |$[ {{V_\mathrm{Si}}} ]$| is the concentration of |$\mathrm{ Si}$| vacancies. Kröger & Vink (1956) notation was used to express the species and point defects. Thus,
It is obvious that, for climb-controlled dislocation creep, the strain rate is proportional to the concentration of silicon vacancies, that is |${\rm{\dot \varepsilon }} \propto [ {{V_\mathrm{Si}}} ]$|.
Also, we examine the processes of electrical conduction under anhydrous and hydrous lithospheric mantle. The electrical conductivity |${\sigma _\mathrm{e}}$| is proportional to the concentration of carriers N and their mobility |${\mu _\mathrm{e}}$|(Dai et al. 2020). For rocks in the upper mantle, electrical conduction is mainly attributed to ionic diffusion of Fe2+, Mg2+, or protons, and to hopping of electrons or electron holes (Yoshino 2010):
or
where q is the charge of the carriers, D is the diffusion coefficient of the charged species, and f is a non-dimensional constant representing the geometrical factor (|$f\sim 1$|). Thus, the conductivity of the lithospheric mantle is controlled by the concentration of carriers (N), that is |${\sigma _\mathrm{e}} \propto N$|.
Additionally, as long as we know the relationship between the concentration of carriers (N) and the Si vacancies (|$[ {{V_\mathrm{Si}}} ]$|), we can obtain the relationship between the effective viscosity and the resistivity at the macro level.
2.2 The relationship between Si vacancies and charged carriers: point-defect chemistry
In an equilibrium state, the concentrations of charged particles are controlled by thermochemical equilibrium (Nishihara et al. 2008). In other words, the concentrations of the different types of particles are determined by electroneutrality, which requires that the total positive charge equals the total negative charge in a mineral crystal (Schmalzried & Pelton 1981). As long as the principle of electrical neutrality holds, if the concentrations of |${\rm\mathrm{Si}}$| vacancies change, the concentrations of current carriers must change synchronously.
Under anhydrous conditions, the dominant positive and negative charged defects are ferric iron |${\rm{Fe}}_{\rm{M}}^ \cdot $| and M-site vacancy |${\rm{\mathit{ V}}}_{\rm{M}}^{{\rm{^{\prime\prime}}}}$|, respectively (Fig. 1), such that the charge-neutrality condition is given by |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {V_\mathrm{m}^{{\rm{^{\prime\prime}}}}} ]$|. Given the charge-neutrality condition for anhydrous upper mantle, the deformation is rate-limited by diffusion of |$\mathrm{ Si}$| vacancies (|$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$|) (Tasaka et al. 2015), and the electrical conduction is dominated by diffusion of small polarons (|$\mathrm{ Fe}_\mathrm{m}^ \cdot $|) (Dohmen & Chakraborty 2007; Costa & Chakraborty 2008; Fei et al. 2012).
A schematic diagram showing the concentrations of various defects as a function of water content in olivine (modified from Karato et al. 2006). As the water content change, the dominant charged defects ferric iron |${\bf{Fe}}_{\bf{M}}^ \cdot $| and M-site vacancy |${\bf{V}}_{\bf{M}}^{^{\prime\prime}}$| at low-water content become |${\bf{Fe}}_{\bf{M}}^ \cdot $| and |${\bf{H}}_{\bf{M}}^{\prime}$| at high-water content.
And under hydrous conditions, the dominant positive and negative charged defects become |${\rm{Fe}}_{\rm{M}}^ \cdot $| and |${\rm{H}}_{\rm{M}}^{\rm{^{\prime}}}$|, respectively (Fig. 1), such that charge-neutrality is given by |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\rm{^{\prime}}}} ]$|. Given the charge-neutrality condition for hydrous silicates, the deformation is rate-limited by diffusion of |$\mathrm{ Si}$| vacancies (|$( {3{\rm{H}}} )_{{\rm\mathrm{Si}}}^{\rm{^{\prime}}}$|) (Tasaka et al. 2015), and the electrical conduction may be dominated by diffusion of water-related defects (|${{\rm{H}}^ \cdot }$|) (Dai & Karato 2014).
Based on the charge-neutrality condition and point-defect chemistry, we can predict the relationship between carriers’ concentrations and Si vacancies concentrations. Under anhydrous conditions, the relationship between |$[ {V_\mathrm{Si}^{{\rm{^{\prime\prime\prime \prime}}}}} ]$| and |$[ {Fe_\mathrm{m}^ \cdot } ]$| is |$[ {V_\mathrm{Si}^{{\rm{^{\prime\prime\prime \prime}}}}} ] \propto {[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ]^2}$|; while under hydrous conditions, the relationship between |$[ {( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}} ] \propto {[ {{\mathrm{ H}^ \cdot }} ]^{\frac{5}{3}}}$| (see Appendix A for details).
2.3 Resistivity-viscosity conversion model
Denoting |$[ {{V_\mathrm{Si}}} ] = k \cdot {N^m}$| and following equations (6) and (7), we obtain the formula for a constant strain rate,
Then, we can simplify the above equation as follows:
where |${\rm{\rho }}$| is electrical resistivity, and C represents the constant pre-exponential factors (|$C = {\dot \varepsilon _0}^{\frac{{1 - n}}{n}}{( {\frac{{{B^2}2{\pi ^2}{V_\mathrm{m}}r_0^2D_{{V_\mathrm{Si}}}^\mathrm{Si}}}{{{\mu ^{2x}}{b^4}\textit{RTln}( {\frac{{{R_0}}}{{{r_0}}}} )}}} )^{ - \frac{1}{n}}} \cdot {( {k{{( {\frac{R}{{{\rm{fD}}{{\rm{q}}^2}}}} )}^m}} )^{ - \frac{1}{n}}}$|). The values of m and n have been shown in Table 1. Due to the unknown values of C, we review the experimental data of olivine electrical resistivity and viscosity under wet and dry conditions to fit the above equation, that is |${{\rm{\eta }}_0} = C \cdot {T^{\frac{{1 - m}}{n}}} \cdot \rho _0^{\frac{m}{n}}$|. Thus,
where |${{\rm{\eta }}_0}\,\,and\,\,{\rho _0}$| denote reference viscosity and electrical resistivity, respectively, obtained from laboratory results and varying with temperature, pressure, etc.
The relationships between effective viscosity and electrical resistivity under different conditions of charge-neutrality (|$\frac{{{{\bf{\eta }}_{{\bf{eff}}}}}}{{{\eta _0}}} = {( {\frac{\rho }{{{\rho _0}}}} )^{\frac{m}{n}}}$|). The |${\sigma _{\mathrm{ wet}0}}$| and |${\sigma _{\mathrm{ dry}0}}$| represent the reference electrical resistivities for the ‘wet’ and ‘dry’ conditions, respectively. The |${\eta _{\mathrm{ wet}0}}$| and |${\eta _{\mathrm{ dry}0}}$| represent the reference viscosities for the wet and dry conditions, respectively.
| Defect control dislocation creep | Carriers | Charge-neutrality condition | m | Laboratory-based relation |
|---|---|---|---|---|
| |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}$| | |${{\rm{H}}^ \cdot }$| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| | 5/3 | |${\eta _{\mathrm{ wet}0}}$|/|${\sigma _{\mathrm{ wet}0}}$| |
| |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$| | |${\rm{Fe}}_{\rm{M}}^ \cdot $| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {\mathit{ V}_\mathrm{m}^{^{\prime\prime}}} ]$| | 2 | |${\eta _{\mathrm{ dry}0}}$|/|${\sigma _{\mathrm{ dry}0}}$| |
| Defect control dislocation creep | Carriers | Charge-neutrality condition | m | Laboratory-based relation |
|---|---|---|---|---|
| |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}$| | |${{\rm{H}}^ \cdot }$| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| | 5/3 | |${\eta _{\mathrm{ wet}0}}$|/|${\sigma _{\mathrm{ wet}0}}$| |
| |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$| | |${\rm{Fe}}_{\rm{M}}^ \cdot $| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {\mathit{ V}_\mathrm{m}^{^{\prime\prime}}} ]$| | 2 | |${\eta _{\mathrm{ dry}0}}$|/|${\sigma _{\mathrm{ dry}0}}$| |
The relationships between effective viscosity and electrical resistivity under different conditions of charge-neutrality (|$\frac{{{{\bf{\eta }}_{{\bf{eff}}}}}}{{{\eta _0}}} = {( {\frac{\rho }{{{\rho _0}}}} )^{\frac{m}{n}}}$|). The |${\sigma _{\mathrm{ wet}0}}$| and |${\sigma _{\mathrm{ dry}0}}$| represent the reference electrical resistivities for the ‘wet’ and ‘dry’ conditions, respectively. The |${\eta _{\mathrm{ wet}0}}$| and |${\eta _{\mathrm{ dry}0}}$| represent the reference viscosities for the wet and dry conditions, respectively.
| Defect control dislocation creep | Carriers | Charge-neutrality condition | m | Laboratory-based relation |
|---|---|---|---|---|
| |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}$| | |${{\rm{H}}^ \cdot }$| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| | 5/3 | |${\eta _{\mathrm{ wet}0}}$|/|${\sigma _{\mathrm{ wet}0}}$| |
| |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$| | |${\rm{Fe}}_{\rm{M}}^ \cdot $| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {\mathit{ V}_\mathrm{m}^{^{\prime\prime}}} ]$| | 2 | |${\eta _{\mathrm{ dry}0}}$|/|${\sigma _{\mathrm{ dry}0}}$| |
| Defect control dislocation creep | Carriers | Charge-neutrality condition | m | Laboratory-based relation |
|---|---|---|---|---|
| |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}$| | |${{\rm{H}}^ \cdot }$| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| | 5/3 | |${\eta _{\mathrm{ wet}0}}$|/|${\sigma _{\mathrm{ wet}0}}$| |
| |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$| | |${\rm{Fe}}_{\rm{M}}^ \cdot $| | |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {\mathit{ V}_\mathrm{m}^{^{\prime\prime}}} ]$| | 2 | |${\eta _{\mathrm{ dry}0}}$|/|${\sigma _{\mathrm{ dry}0}}$| |
Overall, we sought a macroscopic resistivity–viscosity transformation model based on the concentrations of microscopic point defects. It should be noted that the concentrations of these point defects are temperature-dependent, implying that the equilibrium coefficient |${K_i}$| for each reaction is not static. Instead, they vary significantly with factors such as temperature and composition (Dohmen & Chakraborty 2007). From a defect chemistry perspective, the temperature dependency of k (where |$[ {{V_\mathrm{Si}}} ] = k \cdot {N^m}$|) suggests that the C in eq. (9) is also influenced by temperature. In contrast, eq. (10) lacks explicit temperature-dependent terms. Instead, the impact of temperature is primarily reflected in how it affects the reference viscosity (|${\eta _0}$|) and reference resistivity (|${\rho _0}$|). We will delve deeper into the influence of temperature on the effective viscosity transformation from electrical resistivity through specific examples.
3. EXAMPLE
3.1 The effective viscosity structure of lithospheric mantle in Northwest Xinjiang
In order to test the rationality of our model, we used a recent high-resolution cross-section of resistivity (Fig. 2c), which was extracted from the latest MT images (Fig. 2b) using a 3-D scheme with the nonlinear conjugate gradient algorithm in the ModEM computational framework (Egbert & Kelbert 2012; Kelbert et al. 2014). Surface heat flow (Fig. 2b) in the study area was interpolated from the China heat flow database (Jiang et al. 2019). The temperature distributions were extracted from the 3-D model of the lithospheric thermal structure of East Asia that used the latest thermal conductivity and radiogenic heat production measurements in mainland China (Sun et al. 2022). And the strain rate from the global positioning system (GPS) measurement is 10−15 s−1 (Zhu & Shi 2011; Wang & Shen 2020), which has been used previously in China continent (Deng et al. 2017) and Tianshan region (Pan et al. 2023). The reference electrical resistivities for the wet (|${\sigma _{\mathrm{ wet}0}})$| and dry (|${\sigma _{\mathrm{dry}0}})$|conditions were calculated based on Gardés et al. (2014). The reference viscosities for the wet (|${\eta _{\mathrm{wet}0}})$| and dry (|${\eta _{\mathrm{dry}0}})$|conditions were calculated based on Karato & Jung (2003), the parameters are shown in Table 2.
MT site distribution in Northwest Xinjiang and the representative cross-section of electrical resistivity. (a) Tectonic framework around and in mainland China (solid blue lines are active tectonic block boundaries, dashed blue lines are inferred regional boundaries, and orange solid lines are active tectonic boundaries). (b) Shaded topography, surface heat flow interpolated using spline method from the China heat flow database (Jiang et al. 2019), crustal thickness estimated by joint analysis of gravity and seismic data (Zhao et al. 2020). The triangular points represent MT stations (Liu et al. 2024). Active faults from Deng et al. (2003) are shown as black solid lines. (c) Cross-section of electrical resistivity along the red dashed line in Fig. 2(b) extracted from the preferred 3-D resistivity model from Liu et al. (2024). Black dashed lines represent the geotherms extracted from the latest 3-D model of the lithospheric thermal structure of East Asia (Sun et al. 2022).
Creep and conductivity law parameters of olivine aggregates
| Dislocation Creepa | |${\log _{10}}A$| | n | r | E (kJ mol−1) | V (cm3 mol−1) | Notes |
|---|---|---|---|---|---|---|
| Anhydrous | 6.1 ± 0.2 | 3.0 ± 0.1 | – | 510 ± 30 | 14 ± 2 | Karato et al. 2003 |
| Anhydrous | 5.04 | 3.5 | – | 530 | 15–20 | Kawazoe et al. 2009 |
| Hydrous | 0.56 ± 0.02 | 3.0 ± 0.1 | 1.2 ± 0.05 | 410 ± 40 | 11 ± 3 | Karato et al. 2003 |
| Hydrous | 2.7 | 3.0 ± 0.1 | 1.25 | 470 ± 40 | 38 | Mei & Kohlstedt 2000a |
| Electrical conductionb | |${\log _{10}}{A_\mathrm{e}}$| | |${r_\mathrm{e}}$| | |${\rm{\Delta }}H$| (kJ mol−1) | |$\alpha $| (kJ/mol/wt ppm1/3) | Notes | – |
| Anhydrous | 2.34 | – | 144 | – | Gardés et al. 2014 | – |
| Hydrous | −1.37 | 1 | 89 | 1.79 | Gardés et al. 2014 | – |
| Dislocation Creepa | |${\log _{10}}A$| | n | r | E (kJ mol−1) | V (cm3 mol−1) | Notes |
|---|---|---|---|---|---|---|
| Anhydrous | 6.1 ± 0.2 | 3.0 ± 0.1 | – | 510 ± 30 | 14 ± 2 | Karato et al. 2003 |
| Anhydrous | 5.04 | 3.5 | – | 530 | 15–20 | Kawazoe et al. 2009 |
| Hydrous | 0.56 ± 0.02 | 3.0 ± 0.1 | 1.2 ± 0.05 | 410 ± 40 | 11 ± 3 | Karato et al. 2003 |
| Hydrous | 2.7 | 3.0 ± 0.1 | 1.25 | 470 ± 40 | 38 | Mei & Kohlstedt 2000a |
| Electrical conductionb | |${\log _{10}}{A_\mathrm{e}}$| | |${r_\mathrm{e}}$| | |${\rm{\Delta }}H$| (kJ mol−1) | |$\alpha $| (kJ/mol/wt ppm1/3) | Notes | – |
| Anhydrous | 2.34 | – | 144 | – | Gardés et al. 2014 | – |
| Hydrous | −1.37 | 1 | 89 | 1.79 | Gardés et al. 2014 | – |
Unit of A is|$\,\,\mathrm{ M}\mathrm{ P}{\mathrm{ a}^{ - n - r}}{\mathrm{ s}^{ - 1}}$|, unit of |${A_\mathrm{e}}$| is S m−1.
Dislocation creep law parameters for olivine: |$\dot \varepsilon ( {T,P,{C_{\mathrm{ OH}}}} ) = AC_{\mathrm{ OH}}^r\exp \left( { - \frac{{E + PV}}{{RT}}} \right){\sigma ^n}$|.
Conductivity law parameters for olivine: |${\sigma _\mathrm{e}}( {T,P,{C_{\mathrm{ OH}}}} ) = {A_\mathrm{e}}C_{\mathrm{ OH}}^{re}{\rm{exp}}\left( { - \frac{{{\rm{\Delta }}H - \alpha C_{\mathrm{ OH}}^{\frac{1}{3}}}}{{RT}}} \right)$|.
Creep and conductivity law parameters of olivine aggregates
| Dislocation Creepa | |${\log _{10}}A$| | n | r | E (kJ mol−1) | V (cm3 mol−1) | Notes |
|---|---|---|---|---|---|---|
| Anhydrous | 6.1 ± 0.2 | 3.0 ± 0.1 | – | 510 ± 30 | 14 ± 2 | Karato et al. 2003 |
| Anhydrous | 5.04 | 3.5 | – | 530 | 15–20 | Kawazoe et al. 2009 |
| Hydrous | 0.56 ± 0.02 | 3.0 ± 0.1 | 1.2 ± 0.05 | 410 ± 40 | 11 ± 3 | Karato et al. 2003 |
| Hydrous | 2.7 | 3.0 ± 0.1 | 1.25 | 470 ± 40 | 38 | Mei & Kohlstedt 2000a |
| Electrical conductionb | |${\log _{10}}{A_\mathrm{e}}$| | |${r_\mathrm{e}}$| | |${\rm{\Delta }}H$| (kJ mol−1) | |$\alpha $| (kJ/mol/wt ppm1/3) | Notes | – |
| Anhydrous | 2.34 | – | 144 | – | Gardés et al. 2014 | – |
| Hydrous | −1.37 | 1 | 89 | 1.79 | Gardés et al. 2014 | – |
| Dislocation Creepa | |${\log _{10}}A$| | n | r | E (kJ mol−1) | V (cm3 mol−1) | Notes |
|---|---|---|---|---|---|---|
| Anhydrous | 6.1 ± 0.2 | 3.0 ± 0.1 | – | 510 ± 30 | 14 ± 2 | Karato et al. 2003 |
| Anhydrous | 5.04 | 3.5 | – | 530 | 15–20 | Kawazoe et al. 2009 |
| Hydrous | 0.56 ± 0.02 | 3.0 ± 0.1 | 1.2 ± 0.05 | 410 ± 40 | 11 ± 3 | Karato et al. 2003 |
| Hydrous | 2.7 | 3.0 ± 0.1 | 1.25 | 470 ± 40 | 38 | Mei & Kohlstedt 2000a |
| Electrical conductionb | |${\log _{10}}{A_\mathrm{e}}$| | |${r_\mathrm{e}}$| | |${\rm{\Delta }}H$| (kJ mol−1) | |$\alpha $| (kJ/mol/wt ppm1/3) | Notes | – |
| Anhydrous | 2.34 | – | 144 | – | Gardés et al. 2014 | – |
| Hydrous | −1.37 | 1 | 89 | 1.79 | Gardés et al. 2014 | – |
Unit of A is|$\,\,\mathrm{ M}\mathrm{ P}{\mathrm{ a}^{ - n - r}}{\mathrm{ s}^{ - 1}}$|, unit of |${A_\mathrm{e}}$| is S m−1.
Dislocation creep law parameters for olivine: |$\dot \varepsilon ( {T,P,{C_{\mathrm{ OH}}}} ) = AC_{\mathrm{ OH}}^r\exp \left( { - \frac{{E + PV}}{{RT}}} \right){\sigma ^n}$|.
Conductivity law parameters for olivine: |${\sigma _\mathrm{e}}( {T,P,{C_{\mathrm{ OH}}}} ) = {A_\mathrm{e}}C_{\mathrm{ OH}}^{re}{\rm{exp}}\left( { - \frac{{{\rm{\Delta }}H - \alpha C_{\mathrm{ OH}}^{\frac{1}{3}}}}{{RT}}} \right)$|.
For the charge-neutrality model where |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {V_\mathrm{m}^{^{\prime\prime}}} ]$| that exists under ‘dry’ condition, |${\rm{Fe}}_{\rm{M}}^ \cdot $| and |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}$| are the most important defects for rheology and electrical conduction (denoted by |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}/{\rm{Fe}}_{\rm{M}}^ \cdot $| model), respectively. Moreover, the reference viscosity and reference conductivity for olivine under anhydrous condition were derived from the relationships presented by Karato & Jung (2003) and Gardés et al. (2014), respectively. The effective viscosity was then estimated using eq. (10) and shown in Fig. 3(a).
Effective viscosities calculated from the resistivity cross-section under different electrical conduction models. (a)–(c) The effective viscosities transformed from eq. (10) for |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}/\mathrm{ Fe}_\mathrm{m}^ \cdot $| (anhydrous) and |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}$|/|${\mathrm{ H}^ \cdot }$| (hydrous) models and a combination of these two models at different depths. (d) and (e) are ‘soft’ and ‘hard’ lithosphere extracted from Deng et al. (2017). The five triangles in panels (a) and (d) mark the locations for comparison in Fig. 4.
For the charge-neutrality model where |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| that exists under ‘wet’ condition, |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$| and |${{\rm{H}}^ \cdot }$| play significance roles in rheology and electrical conduction (denoted by |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$|/|${{\rm{H}}^ \cdot }$| model), respectively. The reference viscosity and reference conductivity under hydrous condition were derived from the relationships presented by Karato & Jung (2003) and Gardés et al. (2014), respectively. Based on geochemical analyses of mantle inclusions in eastern China, the minimum water content is 10 ppm (Xia et al. 2019). Recent research has also pointed out that the water content of natural olivine samples from the northwestern Tarim, measured by secondary ion mass spectrometry (SIMS), is less than 168 ppm (Wang et al. 2022). In order to simplify, the water content of the uppermost lithospheric mantle was set to 200 ppm at a depth of 45 km and linearly decreased to 10 ppm at a depth of 200 km. The effective viscosity estimated by eq. (10) is shown in Fig. 3(b). The high-temperature and high-pressure experimental results revealed that the dominant conduction mechanism for olivine is proton conduction at relatively low temperature, switching to small polaron conduction around 1300 K (Yoshino et al. 2009). If we simply adopt this observation as a rule of thumb, as the temperature transitions from below 1300 K to above 1300 K, the viscosity of olivine transitions from being predominantly controlled by |$( {3{\rm{H}}} )_{{\rm\mathrm{Si}}}^{\rm{^{\prime}}}$| to |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}$|. The resulting estimated effective viscosity distribution is shown in Fig. 3(c).
Regardless of the differences among Figs 3(a)–(c), the effective viscosity at a large scale is positively correlated with electrical resistivity obtained from MT imaging. The effective viscosities of the topmost lithospheric mantle beneath the Tarim and Junggar regions are higher than those beneath the Tianshan region, while the effective viscosities at the base of the lithosphere in the Tianshan region are slightly higher than that beneath the Tarim and Junggar regions. Such variations in the upper mantle of the Chinese Tianshan region have already been noticed in previous studies (Liu et al. 2007). The effective viscosities under the |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}/{\rm{Fe}}_{\rm{M}}^ \cdot $| model range from 1020 to 1022.5 Pa s, while the effective viscosities under the |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$|/|${{\rm{H}}^ \cdot }$| model range from 1019.7 to 1022.8 Pa s. The result derived from the combination model (Fig. 3c) is very close to the results from |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}/{\rm{Fe}}_{\rm{M}}^ \cdot $| and |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$|/|${{\rm{H}}^ \cdot }$| models in corresponding zones, respectively. The two zones are separated at a depth that precisely coincides with 1300 K.
3.2 Comparisons with previous results
Extrapolating laboratory results is one of the widely used methods to obtain the effective viscosity of the lithospheric mantle. However, the results from extrapolation are only sensitive vertically due to the limited ability to resolve lateral temperature and pressure. The viscosity–depth profiles derived by laboratory results are shown in Fig. 4, and the temperature is the lateral mean average temperature along profile A–B. The difference of effective viscosities obtained by different laboratories is close to 1 logarithmic unit due to the differences in samples and experimental conditions. The range and decreasing trend of the effective viscosities in this study are close to the results obtained from the extrapolation of laboratory data. The lateral variations in effective viscosities within the shallow lithosphere are more pronounced, with differences in effective viscosities being approximately 1.5 logarithmic units, while the differences in effective viscosities in the deep lithosphere are small.
Comparison of the effective viscosities from this study (using combination model) with extrapolated viscosities from some laboratory results. The ‘dry (2003)’ and ‘wet (2003)’ correspond to the viscosity under dry and wet conditions from Karato & Jung (2003), respectively. The ‘wet (2000)’ corresponds to the viscosity under the wet condition from Mei & Kohlstedt (2000a), and the ‘dry (2009)’ corresponds to the viscosity under the dry condition from Kawazoe et al. (2009). The grey line represents the lateral mean average temperature along the lateral direction of profile A–B versus depth.
To further evaluate the reliability of the effective viscosities after conversion, the results were compared with ‘soft’ and ‘hard’ lithospheric strengths (Figs 3d and e) extracted from (Deng & Tesauro 2016). Due to the low-spatial resolution of the results, we compare two independent data sets at five sites along the MT transect for a more accurate comparison (Fig. 5). As illustrated in Figs 3 and 5, the effective viscosities of Deng & Tesauro (2016) continuously decrease at depths of 60–200 km. However, our |${\rm{V}}_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime \prime}}}}/{\rm{Fe}}_{\rm{M}}^ \cdot $| model shows a sharp decrease at the depths of 50–120 km followed by a gentle decrease at depths of 120–200 km. In contrast, the |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$|/|${{\rm{H}}^ \cdot }$| model shows a sharp decrease at depths of 50–120 km but followed by a gentle increase at depth of 120–200 km. Since olivine in the lower lithosphere is virtually dry, the |$( {3\mathrm{ H}} )_\mathrm{Si}^{\rm{^{\prime}}}$|/|${{\rm{H}}^ \cdot }$| model actually overestimates the effective viscosities of the lower lithosphere. The viscosity structures obtained by the two approaches are very similar, and their differences lie within the uncertainties, which confirms the physical validity of the resistivity-converted viscosity.
![Comparison of the effective viscosities from this study with Deng's model at five sites along the MT transect. (a) for $[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$ charge-neutrality model; (b) for $[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {V_\mathrm{m}^{^{\prime\prime}}} ]$ charge-neutrality model, and (c) for combination model. Shadowed powder blue and faded pink are used for the ranges of ‘hard’ and ‘soft’ lithospheric viscosity models (Deng & Tesauro 2016), respectively. The site positions along the MT transect are marked in colours.](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/gji/240/2/10.1093_gji_ggae438/2/m_ggae438fig5.jpeg?Expires=1747893244&Signature=U3OH19sg44VcK88UvzoHdOK0m-qJuPrRFiaqxJv~RreYmmWyNXWHjIpws7G9vEOXCFkOGWUNqdN~ACxstKWochtL97JGcV-2o~AjCzjvEtWjbjsf0gMqjzssLKkfwNqY53uKHAfBMy~JdAimWbQaY3YMAm27iUzf40iTFfVxnY1e~aJJNiJHrV95Bl1Igb8DPA1pQ57-fJtUAwzHWD075BGNv4gFp5ypS8alH7SeXXgq5j4-uT9XOAbtg3V2ZNQ4VDKnkq28uuHYcu6n5mXYE8hlv4bEtES~qOhcBHgiTF1Mu60MHYC8Uyr7k06ab8yMdTSOKRuEgWaH~Fw~9h6cKQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
Comparison of the effective viscosities from this study with Deng's model at five sites along the MT transect. (a) for |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| charge-neutrality model; (b) for |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {V_\mathrm{m}^{^{\prime\prime}}} ]$| charge-neutrality model, and (c) for combination model. Shadowed powder blue and faded pink are used for the ranges of ‘hard’ and ‘soft’ lithospheric viscosity models (Deng & Tesauro 2016), respectively. The site positions along the MT transect are marked in colours.
The differences may be due to the deviation of the laboratory relationship and/or the derived temperature field. The elimination of this error depends on further constraints on the water distribution within the lithosphere and advancements in high-temperature and high-pressure experiments. If the lithosphere–asthenosphere boundary (LAB) is at depths of 150–200 km in this region (An & Shi 2006; Steinberger & Becker 2018; Zhang et al. 2019), the results based on a pure dislocation creep model, as we have used, would be biased. This is due to the possibility of partial melting occurring at the LAB depth. If the lithosphere is not entirely mechanically coherent, the presumption of a constant strain rate is not applicable to the whole domain, regardless of the method used. Despite the uncertainties mentioned above, considering the density variations derived from P-wave traveltime tomography and joint inversion (Liu et al. 2007; Deng et al. 2017), we have good reason to justify the lateral variations of effective viscosities in the upper mantle under the Tarim–Tianshan–Junggar region.
3.3 The robustness of the result translated from electrical resistivity
3.3.1 The effects of water content on effective viscosity
Mineral physicists have recently discovered that most NAMs can absorb water, or rather hydroxyl, up to levels of dozens or even hundreds of ppm (Jones et al. 2012). Petrological evidence also suggests that olivine in the continental lithosphere contains about 60–100 ppm of water, while olivine in the lower lithosphere is virtually dry (Peslier et al. 2010). Previous studies have proposed that the electrical conductivity in hydrogen-rich olivine is mainly due to the diffusion of |${\mathrm{ H}^ \cdot }$| (Wang et al. 2006; Dai & Karato 2014), the mechanism predicts |${r_\mathrm{e}} = 0.75$|(|$\sigma \propto f_{{\mathrm{ H}_2}\mathrm{ O}}^{re}$|) for the defect chemical model (Karato 2008), while experimental observations indicated the exponent r = 0.6 ∼ 1. Thus, we must consider the uncertainties of the water content exponent on the reference electrical resistivity/effective viscosity under |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}/{\mathrm{ H}^ \cdot }$| model. By incorporating the experimental results into the resistivity-viscosity transformation model, we find that |$\Delta \eta \propto {( {\Delta {C_w}} )^{\frac{{m{r_\mathrm{e}} - r}}{n}}} = {( {\Delta {C_w}} )^{0.131429}}$|, |${r_\mathrm{e}} = 1$| from Gardés et al. (2014) and r = 1.25 from Karato & Jung (2003). Clearly, for the same electrical resistivity, |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}/{\mathrm{ H}^ \cdot }$| model with higher water content tends to increase the calculated effective viscosity, and the change in water content has only a weak effect on the result of |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}/{\mathrm{ H}^ \cdot }$| model.
3.3.2 The sensitivity of strain rate on effective viscosity
The effective viscosity of the lithosphere is related to the strain rate, which must be specified in the calculation. In past studies, a constant strain rate is often used for these calculations. For a small study area, the strain rate changes minimally and has little influence on the calculated results. However, the actual strain rate of the lithospheric mantle typically ranges from |${10^{ - 14}}$| to |${10^{ - 16}}\,\,{\mathrm{ s}^{ - 1}}$|, so the effect of the changing strain rate on the results should be considered in applications. Eqs (1) and (7) are derived under the assumption of a constant strain rate. Assuming this assumption is valid, the partial derivative of the strain rate can be computed by taking the logarithm of the expression for effective viscosity, thereby quantifying the impact of the strain rate on the calculated effective viscosity:
As reported by Karato & Jung (2003), the value of n is 3, we obtained |$\frac{{\partial ( {\lg {\eta _{\mathrm{ eff}}}} )}}{{\partial ( {lg\varepsilon _0^ \cdot } )}} \approx 0.67$|. This indicates that when the strain rate increases by one order of magnitude, the effective viscosity decreases by approximately 0.67 logarithmically, which corresponds to a decrease of approximately 0.2 times the original viscosity.
The strain rate of the Tianshan fold belt is approximately |${10^{ - 15}}\,\,{\mathrm{ s}^{ - 1}}$|, but the measured strain rate of the Tarim Basin and Junggar area is smaller than that of the Tianshan area, at about |$3.2 \times {10^{ - 16}}\,\,{\mathrm{ s}^{ - 1}}$| (Wang et al. 2020; Li et al. 2021). To measure the effect of the difference between the constant strain rate and the measured strain rate, the effective viscosities of the different blocks were replotted (Fig. 6). It is shown that the effective viscosity perturbation caused by the difference in strain rate is less than 0.5 logarithmic units.
Effective viscosity–depth diagram of Tarim Basin (a) and Junggar Basin (c) when the strain rate variation is considered. Solid lines represent the effective viscosities calculated at different locations along the profile using the constant strain rate (|${10^{ - 15}}\,\,{\mathrm{ s}^{ - 1}}$|) under combination model. Dash lines represent the effective viscosities at corresponding locations of Tarim Basin and Junggar Basin calculated using the measured strain rate (|$3.2 \times {10^{ - 16}}\,\,{\mathrm{ s}^{ - 1}}$|). The grey line is the lateral mean temperature versus depth profile from Sun et al. (2022).
3.3.3 The sensitivity of the temperature model on effective viscosity
The temperature results obtained by different methods or databases have great uncertainty in calculating the upper mantle thermal structure, with a maximum difference of several hundred Kelvin(K), which leads to larger uncertainties in the reference resistivity and reference viscosity. In order to determine the effect of the temperature model on the effective viscosity results, this paper utilizes the thermal structure model from other independent data sets (Deng & Tesauro 2016), which combines the thermal crustal model of Sun et al. (2013) with estimates of temperature in the upper mantle obtained by inverting seismic velocities from the shear model of Li et al. (2013) at depths between 100 and 300 km, to reconstruct the results presented in the previous section.
At the same depth, Deng's temperature model is about 200 K higher than Sun's, and differences of viscosity structures caused by temperature are less than 0.5 logarithmic units in both |$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}/\mathrm{ Fe}_{Mg}^ \cdot $| and |$( {3\mathrm{ H}} )_\mathrm{Si}^{\prime}/{\mathrm{ H}^ \cdot }$| electric neutral conditions, and the lateral variations of effective viscosity are consistent (Fig. 7). While the changes in the temperature model have significant impacts on the mixed model, that is because the mixed mode defines the hydrous and anhydrous models based on the temperature limits. Additionally, changes in the temperature models have significant impacts on the boundary between the hydrous and anhydrous models. The least-squares difference method is used to smooth the transition at the boundary between the two models, which results in the use of different neutral conditions at a certain depth due to temperature differences.
Comparison of the effective viscosity model derived from different temperature models. Panels (a) and (b) are the temperature models. Panels (c), (e) and (g) show the effective viscosities under different electric neutral assumptions with resistivity under Sun's temperature model. Panels (d), (f) and (h) show the effective viscosities under different electric neutral assumptions with resistivity under Deng's temperature model. Above, the temperature models are the terrain and surface heat flow, respectively.
4. DISCUSSION
According to previous research, creep is to be driven by diffusion of the slowest species, namely, Si in silicate minerals, and the flux of Si ions is dominantly along dislocations. Meanwhile, the electrical resistivity is dominated by the motion of the charged species with a fast diffusion rate. Based on the physical properties and the requirement of electrical neutrality, we have developed a theory that can convert an electrical resistivity model into an effective viscosity distribution for the lithospheric mantle under varying thermochemical conditions. Our method improves upon previous research (Liu & Hasterok 2016; Xu et al. 2018; Selway et al. 2020) by providing clearer physics and establishing different relationships under different thermochemical conditions that are consistent with the actual situation of the lithospheric mantle. The reliability of our method has been demonstrated by a real-world example.
It is noteworthy that our scheme cannot be directly applied to melt interconnection domains, where the melt volume fraction exceeds 0.5 per cent, which is the connectedness threshold for multiple conducting phases in melt-bearing olivine aggregate (Laumonier et al. 2017). Actually, some empirical models can be incorporated into our model to covert melt electrical resistivity into effective viscosity (Pommier et al. 2013). However, the melt fraction in the mantle is still difficult to determine at present. Therefore, in cases where the resistivity model exhibits a low-resistance anomaly due to partial melting, the effective viscosity derived from it will also exhibit a corresponding low-viscosity response. However, it is important to note that the range of these values serves as a mere reference, and an accurate estimation of the associated error is not currently feasible.
The fact is that the uncertainties of our method must be carefully evaluated. First, both the effective viscosity and electrical resistivity are strongly affected by temperature. However, direct measurements of lithospheric temperature at depth are not feasible, and the thermal structures of the lithosphere obtained by different methods vary significantly. Second, under the upper mantle thermochemical conditions, the deformation mechanism changes from diffusion creep at relatively low pressure/fine grain size to dislocation creep at high-temperature/coarse grain size. Similarly, the electrical conduction mechanism of the lithospheric mantle can be divided into proton conduction at low temperatures, small polaron conduction at relatively high temperatures, and ion conduction at high temperature above the solidus. However, the temperature/pressure domains for different conduction and deformation mechanisms are not precisely known. In other words, we know very little about the T–P phase boundaries and their characteristics that transition from one mechanism to another for electrical conduction and deformation.
Despite these uncertainties, it has been established that reconsidering the macro-level relationship between effective viscosity and resistivity from the perspective of microscale defect chemistry offers a practical approach. This revised perspective does not alter the assessment of the correlation between lateral lithospheric variations and structural features within acceptable error margins. It can be expected that the findings of this study will have substantial implications for dynamic numerical simulations and the investigation of extensive mantle structures.
5. CONCLUSIONS
We presented a model based on defect chemistry and electric neutrality to estimate the effective viscosity of the lithospheric mantle from electrical resistivity. This model was utilized to convert MT imaging of the Tianshan Mountains and its surrounding basins into the viscosity structure of the upper mantle. The reliability of the results is then verified through comparison with additional independent findings. Future research incorporating resistivity as a quantitative constraint will better reveal the viscosity of the lithospheric mantle, and help to understand the dynamic evolution of continents. With the continued development of mineral physics and the accumulation of geochemical and geophysical data, the proposed model will undoubtedly be further enhanced and refined, contributing to the ongoing progress in this crucial field of study.
ACKNOWLEDGMENTS
The authors wish to thank the National Natural Science Foundation of China for providing financial support by grants of 41974082 and 41830212. We appreciate Dr Graham Heinson and associate editor Dr Ute Weckmann for their constructive comments that have significantly improved the manuscript. We also appreciate Dr Y.F. Deng (Guangzhou Institute of Geochemistry, CAS) for sharing their released data for comparison.
AUTHOR CONTRIBUTIONS
Man Li (formal analysis, software, methodology, investigation, validation, visualization, writing—original draft), Yixian Xu (conceptualization, formal analysis, funding acquisition, methodology, project administration, writing—review and editing), Lian Liu (conceptualization, supervision, investigation, validation), Bo Yang (conceptualization, supervision, investigation, validation), Yi Zhang (conceptualization, supervision, investigation, validation) Yixin Zhu (conceptualization, supervision, investigation, validation), and Shuyu Liu (formal analysis, investigation, validation).
CONFLICT OF INTEREST
The authors declare no conflicts of interest relevant to this study.
DATA AVAILABILITY
The Liu's 3-D resistivity model is available at 10.17632/97cgtd28d3.1 (Liu et al. 2024). The profile resistivity model extracted from the 3-D resistivity model from Liu et al. (2024) is available at 10.5281/zenodo.8337293. The China Heat Flow data are available from China Heat Flow Database (China Heat Flow Database.). The temperature structures are available at 10.5281/zenodo.6459746 (Sun et al. 2022). The MATLAB scripts for effective viscosity formulation can be found at 10.5281/zenodo.8311167. Several Figures were created by Generic Mapping Tools (GMT) version 6 (Wessel et al. 2019), available at https://www.generic-mapping-tools.org/.
REFERENCES
APPENDIX A: POINT-DEFECT CHEMISTRY
Hydrogen in NAMs can be considered as the ‘point defect’ (Kohlstedt et al. 1996; Mei & Kohlstedt 2000b). Following requirement of charge balance, there are different models of hydrogen dissolution in silicates (Karato et al. 2006; Karato 2008). The first possibility is that two protons from one molecule go to M-site that is generally occupied by Mg or Fe.
By applying the law of mass action to equation (A1), we can get the equation:
where |${f_{{\mathrm{ H}_2}\mathrm{ O}}}$| is water fugacity, |$[ {( {2\mathrm{ H}} )_\mathrm{m}^ \times } ]$| is the concentration of |$( {2\mathrm{ H}} )_\mathrm{m}^ \times $| defect, |${a_{\mathrm{ MO}}}$| is the activity of MO, |${K_i}$| is the equilibrium constant for reaction (i).
Another possibility is that four protons enter the Si-site to occupy the quadrivalent silicon, the reaction is,
From the similar process, we obtain
In addition, as the temperature increase, the neutral defect |$( {2\mathrm{ H}} )_\mathrm{m}^ \times $| may become a charged defect through the ionization reaction,
A free proton |${\mathrm{ H}^ \cdot }$| reacts with a negatively charged defect such as |${\it{V}}_{\rm{M}}^{{\rm{^{\prime\prime}}}}$| to yield,
Thus,
Therefore, we can get the following equations by applying the law of mass action to eqs (A5)–(A7)
Returning to eq. (A4), the eqs (A11) and (A12) can be built:
Combining eqs (A11) and (A12),
Inserting eqs of (A2), (A4), (A8), (A9) into eq. (A13), we obtain
In thermal equilibrium, the charge-neutral condition must be satisfied, that is, the density of positive charge is equal to the density of negative charge everywhere, namely,
In many cases, the concentration of one type of positive-valence defects is often much higher than that of other positive-valence defects, the negative-valence defects are also the same, then the above equation can then be simplified. For anhydrous conditions, the charge-neutrality condition is |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = 2[ {V_\mathrm{m}^{^{\prime\prime}}} ]$|. Additionally, the deformation and electrical conduction under anhydrous upper mantle are dominated by diffusion of |$\mathrm{ Si}$| vacancies (|$V_\mathrm{Si}^{^{\prime\prime\prime \prime}}$|) (Tasaka et al. 2015) and small polaron (|$\mathrm{ Fe}_\mathrm{m}^ \cdot $|) (Dohmen & Chakraborty 2007; Costa & Chakraborty 2008; Fei et al. 2012), respectively. Given the charge-neutrality conditions, the dependence of |$[ {V_\mathrm{Si}^{^{\prime\prime\prime \prime}}} ]$| on |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ]$| can be determined from eq. (A14),
Up to now, we constructed the relationship between |$[ {V_\mathrm{Si}^{^{\prime\prime\prime \prime}}} ]$|, the defects associated with |$\mathrm{ Si}$|-site, and |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ]$|, the most important defects for electrical conduction of NAMs at dry upper mantle.
In contrast, when we consider a transition in deformation/conduction behaviour between wet and dry conditions, it is important to note that a process that works in one condition may not work alone so that the constitutive relations may not be additive. With the water fugacity (content) increase, the dominant charged defects change from ferric iron |${\rm{Fe}}_{\rm{M}}^ \cdot $| and M-site vacancy |${\rm{V}}_{\rm{M}}^{{\rm{^{\prime\prime}}}}$| to |${\rm{Fe}}_{\rm{M}}^ \cdot $| and |${\rm{H}}_{\rm{M}}^{\rm{^{\prime}}}$|, and the charge-neutrality condition is |$[ {\mathrm{ Fe}_\mathrm{m}^ \cdot } ] = [ {\mathrm{ H}_\mathrm{m}^{\prime}} ]$| (see Fig. 1). Given the charge-neutrality condition for anhydrous silicates, the deformation is rate limited by diffusion of |$\mathrm{ Si}$| vacancies (|$( {3{\rm{H}}} )_{{\rm\mathrm{Si}}}^{\rm{^{\prime}}}$|) (Tasaka et al. 2015), and the electrical conduction may be dominated by diffusion of water-related defect (|${{\rm{H}}^ \cdot }$|) (Dai & Karato 2014). The relationship between the concentration of the |$( {3{\rm{H}}} )_{{\rm\mathrm{Si}}}^{{\rm{^{\prime\prime\prime}}}}$| and the |${{\rm{H}}^ \cdot }$| is discussed as follow.
In |${( {{\rm{Fe}},{\rm{Mg}}} )_2}{\rm\mathrm{Si}}{{\rm{O}}_4}$| system, the oxygen ion with a crystal can be written as,
Applying the law of mass action, one obtains,
Inserting eq. (A2) into eq. (A11), we get
Obviously, the |$[ {( {3\mathrm{H}} )_\mathrm{Si}^{\prime}} ]$| depends on |$[ {{\rm{H}^ \cdot }} ]$| and water fugacity, while the |$[ {{\rm{H}^ \cdot }} ]$|also depends on water fugacity, the relationship between |$[ {( {3\mathrm{H}} )_\mathrm{Si}^{\prime}} ]$| and |$[ {{\rm{H}^ \cdot }} ]$| is thus nonlinear.
Assuming that,
and taking account of eq. (A3), we obtain,
Following the law of mass action, the above reaction formula become
When the charge-neutrality conduction is given by |$[ {{\rm{Fe}}_{\rm{M}}^ \cdot } ] = [ {{\rm{H}}_{\rm{M}}^{\rm{^{\prime}}}} ]$|, combining eqs (A8), (A9) and (A11), we have
From eq. (A19),
where |${k_2} = \frac{{K_{A1}^2K_{A3}^{\frac{2}{3}}K_{A5}^2a_{\mathrm{ MO}}^{\frac{4}{3}}a_{\mathrm{ Si}{\mathrm{ O}\mathrm{ }_2}}^{\frac{2}{3}}}}{{K_{A6}^{\frac{2}{3}}K_{A21}^{\frac{2}{3}}K_{A21}^{\frac{1}{3}}K_{A4}^{\frac{1}{3}}}}$|.
