Hanyong Liu

Kai Zhang

Jannick Ingrin

Xiaozhi Yang
email: xzyang@nju.edu.cn

Electrical conductivity of omphacite and garnet indicates limited deep water recycling by crust subduction

It is usually stated that oceanic crust recycling at subduction zones introduces large quantities of water into Earth's interior. However, it remains clouded how much water is recycled to the deep mantle. This is largely due to the challenge in directly sampling, and analyzing the water content of, a deep subducting crust. The subducting crust below ~30 km depth is dominated by omphacite and garnet in eclogite-facies rocks. Here we have determined the electrical conductivity of omphacite and garnet in representative subduction-related eclogites, each with varying contents of Fe and H2O that are key in electrical conduction. Considering the measured conductivity, the eclogite chemistry by geochemical investigations and the highly resistive property of subducting crusts by geophysical surveys, we demonstrate that, at 70-120 km depths in the subducting crust, the H2O contents of omphacite and garnet are strikingly small, with the maximum value being <400 and <80 ppm in the former and latter, respectively. The very small water contents indicate extremely water-poor conditions, or very low water activity, during the eclogite-facies metamorphism and in the system. This further implies the absence of appreciated amounts of hydrous phases such as amphibole and chlorite in the matrix, because of the strong ability of omphacite and garnet in hosting water as documented in natural samples. We suggest that the recycling of water to the deep mantle by oceanic crust subduction is limited. The results are important for modeling the conductivity of subducting slabs and understanding the deep water circulation.

Introduction

The transport of water from the hydrosphere to the mantle plays a critical role in affecting mantle dynamics, magma genesis and Earth's habitability. Water is recycled back to the mantle by slab subduction. At very shallow depths (e.g., <20 km), the input of water into the mantle is primarily controlled by pore water [START_REF] Peacock | Fluid processes in subduction zones[END_REF][START_REF] Cai | Water input into the Mariana subduction zone estimated from ocean-bottom seismic data[END_REF]. In a downgoing slab at greater depths, however, water chiefly occurs as OH groups bound in the structure of the constitutive minerals, both hydrous and nominally anhydrous, note that H2 dissolution in minerals is significant only at very high pressure and reducing conditions [START_REF] Yang | Molecular hydrogen in mantle minerals[END_REF]. As a slab descends, water is gradually lost by the breakdown of hydrous minerals such as amphibole and chlorite at depths mostly <100 km, due to their instabilities at high temperature [START_REF] Peacock | Fluid processes in subduction zones[END_REF][START_REF] Liu | Stability of hydrous phases in subducting oceanic crust[END_REF][START_REF] Frost | The stability of hydrous mantle phases[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF]. Driven by its low density, the released water rises, metasomatizes wallrocks and causes melting in the wedge. This causes the development and widespread distribution of subduction zone volcanism and ore deposits.

It is widely believed that the crust layer in the slab is important for water recycling, because most of the slab water is initially stored there [START_REF] Ito | Alteration of oceanic crust and geologic cycling of chlorine and water[END_REF][START_REF] Peacock | Fluid processes in subduction zones[END_REF][START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF][START_REF] Van Keken | High-resolution models of subduction zones: Implications for mineral dehydration reactions and the transport of water into the deep mantle[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF]. Beyond ~30 km depth, the slab crust is dominated by eclogites (via phase change), consisting mainly of nominally anhydrous omphacite and garnet that are the main water carriers in eclogites after hydrous minerals decompose [START_REF] Lu | Water solubility in pyrope to 100 kbar[END_REF][START_REF] Bromiley | An experimental investigation of hydroxyl solubility in jadeite and Na-rich clinopyroxenes[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF]. As such, the water contents of omphacite and garnet are key to understanding water recycling to the deep mantle. Omphacite and garnet in terrain eclogites, that were once subducted to mantle depths and then returned to surface, could hold up to several thousands of ppm H2O as structural OH, leading to the suggestion that the subducting crust is able to carry large quantities of water into the deep mantle beyond the stability of the hydrous phases [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF]. The high water contents of omphacite and garnet in the eclogites may, however, have suffered from complex secondary fluid-rock interactions postdating eclogite peak-metamorphism (e.g., [START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF][START_REF] Schmädicke | Water transport by subduction: clues from garnet of Erzgebirge UHP eclogite[END_REF]. It is thus obscure how much water is present in omphacite and garnet in the deep subducting crust.

The electrical conductivity of nominally anhydrous minerals (NAMs) is sensitive to water [START_REF] Karato | The role of hydrogen in the electrical conductivity of the upper mantle[END_REF][START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF][START_REF] Yoshino | The effect of water on the electrical conductivity of olivine aggregates and its implications for the electrical structure of the upper mantle[END_REF][START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF]Yang et al., , 2012;;Yang and McCammon, 2012;[START_REF] Zhang | Electrical conductivity of enstatite as a function of water content: implications for the electrical structure in the upper mantle[END_REF][START_REF] Zhang | Electrical conductivity of omphacite as a function of water content and implications for high conductivity anomalies in the Dabie-Sulu UHPM belts and Tibet[END_REF][START_REF] Zhao | Electrical conductivity of mantle clinopyroxene as a function of water content and its implication on electrical structure of uppermost mantle[END_REF][START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF]. Meanwhile, electromagnetic surveys offer a direct window into Earth's interior. In this context, constraints can be provided on the water contents of omphacite and garnet in the subducting crust (and water recycling), if the conductivity structure of subducting crusts is resolved and the electrical property of subduction-related materials is determined. In a recent work, [START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF] have quantified the effect of water on the conductivity of omphacite and garnet, and provided a preliminary constraint on the likely water content in the subducting crust. However, Liu et al. have not evaluated the effect of Fe, which along with H are crucial in electrical conduction, and the question arises how convincing their results might be applied to Earth's interior. In this report, we have by experimental work measured the conductivity of omphacite and garnet from representative eclogites, which plus the samples of [START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF] show a wide range of contents in Fe and H2O. The data are used to infer the water contents of the minerals in the subducting crust, by combining with geophysically-based electrical structure and slab petrology. The results demonstrate clearly that a small amount of water is present in omphacite and garnet in deep subducting crusts.

Experiments and Methods

Starting materials and sample characterization

Three starting eclogites were from Bixiling (omp1 + grt1) and Shuanghe (omp3 + grt3) in China and Weißenstein (omp2 + grt2) in Germany, and were all related to slab subduction and exhumation [START_REF] Franz | High-pressure phengite decomposition in the Weissenstein eclogite, Münchberger Gneiss Massif, Germany[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF]. The Weißenstein eclogite was studied in [START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF]. These samples were chosen by considering Fe contents of constituting omphacite and garnet and quality of grains from >30 eclogites (note: the electrical property of silicates is independent of the origin). Natural eclogites were not used for conductivity studies directly, because of the occurrence of accessory materials (e.g., secondary hydrous phases, fluid inclusions and/or other impurities) and zoned water distribution that affect conductivity analyses. We analyzed the conductivity of omphacite and garnet separately and then modeled the bulk conductivity. Omphacite and garnet are representative in composition, in particular in Fe that almost covers the content ranges in subduction-involved eclogites (omp1 and omp3: Appendix).

Optically clear omphacite/garnet grains, ~100-700 µm in size, were handpicked under a binocular microscope. Each sample is chemically homogeneous in major-/minorelements (Table 1), as measured by a JEOL JXA-8230 electron microprobe (15 kV accelerating voltage, 10 nA current and <5 µm beam size). Grains were ground to powder, mostly 30-80 µm, and H-annealed with minor distilled water in Ni capsules (ID 4.2 mm, OD 5.0 mm and length 10 mm) at 1-3.5 GPa and 650-700 °C (150-180 h duration) in a piston cylinder press. By this, water homogenization was achieved in each sample, and samples with different water contents were prepared. Dry samples were not prepared, because OH is common in natural omphacite and garnet. Recovered samples were cored into cylinders of 3 mm diameter and 1.7 mm length for subsequent conductivity runs, note that the conductivity of silicate minerals by lattice conduction is independent of grain size above ~5 µm that is typical for minerals in the deep Earth [START_REF] Yang | Grain size effect on the electrical conductivity of clinopyroxene[END_REF].

A precise analysis of water content is key for evaluating its effect on conductivity.

The general principle for water quantification in silicate minerals by Fourier-transform infrared (FTIR) spectroscopy is given by the modified Beer-Lambert law: CH = Abstotal/I, where CH is OH content, Abstotal is the total OH integrated absorbance normalized to 1 cm thickness, and I is the mineral-specific integral molar absorption coefficient. As such, CH is determined by Abstotal, once the coefficient I is externally calibrated. For optically anisotropic minerals, the intensity of OH bands is sensitive to the orientation of the IR active dipole relative to incident beam, and Abstotal is the sum of thicknessnormalized integrated absorbance along the three principal axes (a, b and c). This requires orienting samples and polarized FTIR analyses along the axes, which are difficult for small grains and are time-consuming for sample preparation. Recently, [START_REF] Shuai | Quantitative analysis of H-species in anisotropic minerals by polarized infrared spectroscopy along three orthogonal directions[END_REF] have shown that Abstotal can be well characterized by polarized FTIR analyses along any three mutually perpendicular directions of a given sample, without the necessity to orient the samples; in case very small grains which are unable to be prepared for polarized FTIR work, [START_REF] Qiu | Quantitative analysis of H-species in anisotropic minerals by unpolarized infrared spectroscopy: An experimental evaluation[END_REF] have demonstrated that Abstotal can be obtained by unpolarized FTIR analyses on randomly oriented grains.

Water contents were measured by a Bruker Vertex 70V FTIR spectrometer coupled with a Hyperion 2000 microscope. Analyses were performed on optically clear grains with an aperture of 60 × 60 µm and a resolution of 4 cm -1 (128 scans for each spectrum by a globar source, KBr-Ge beam splitter, MCT detector and wire-grid Ze-Se polarizer).

For anisotropic omphacite, polarized spectra were acquired for relatively large grains in annealed samples before conductivity runs [START_REF] Shuai | Quantitative analysis of H-species in anisotropic minerals by polarized infrared spectroscopy along three orthogonal directions[END_REF], and unpolarized spectra on 15-20 randomly oriented grains were obtained for fine-grained samples after conductivity runs [START_REF] Qiu | Quantitative analysis of H-species in anisotropic minerals by unpolarized infrared spectroscopy: An experimental evaluation[END_REF]. For cubic garnet, unpolarized spectra were recorded over 6-9 grains. Water contents were calibrated by the mineral-specific coefficients of [START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF] for omphacite and garnet in eclogites, and uncertainty is mostly <10%, as documented in [START_REF] Shuai | Quantitative analysis of H-species in anisotropic minerals by polarized infrared spectroscopy along three orthogonal directions[END_REF] and [START_REF] Qiu | Quantitative analysis of H-species in anisotropic minerals by unpolarized infrared spectroscopy: An experimental evaluation[END_REF]. For a comparison under the same framework, the same calibration was used to recalculate the H2O content of minerals in natural eclogites in available studies. This yields values different from some early work, e.g., the 1840 ppm OH of omphacite in [START_REF] Smyth | Incorporation of hydroxyl in upper-mantle clinopyroxenes[END_REF], estimated by the OH peak linear intensity, is actually ~310 ppm H2O; in contrast, the H2O contents of omphacite and garnet in recent reports [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF][START_REF] Schmädicke | Water transport by subduction: clues from garnet of Erzgebirge UHP eclogite[END_REF] are less affected. The general results of this work are not affected by adopting the calibration coefficients.

Conductivity experiments at elevated conditions

Slabs are rather oxidizing above ~200 km depth, with redox state similar to the Ni-NiO buffer [START_REF] Mccammon | The paradox of mantle redox[END_REF]. Therefore, conductivity runs were carried out with Ni-NiO at 1-2.5 GPa and 250-850 °C in an end-loaded piston cylinder press. The assembly design resembles that in our previous studies [START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF](Yang et al., , 2012;;[START_REF] Yang | Orientation-related electrical conductivity of hydrous olivine, clinopyroxene and plagioclase and implications for the structure of the lower continental crust and uppermost mantle[END_REF][START_REF] Yang | Grain size effect on the electrical conductivity of clinopyroxene[END_REF]Yang and McCammon, 2012;[START_REF] Li | Unusually high electrical conductivity of phlogopite: the possible role of fluorine and geophysical implications[END_REF][START_REF] Li | Fluorine follows water: Effect on electrical conductivity of silicate minerals by experimental constraints from phlogopite[END_REF][START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF], and only a brief description is offered. In each run, a BN-Ni double capsule was adopted to maintain sample geometry and yield a relatively sealed chamber. BN causes reducing conditions only when O2 is rich (e.g., in air) and run duration is long above ~1000 °C, by 4BN + 3O2 = 2B2O3 + 2N2, and BN itself in a sealed system does not influence the redox state. It has been demonstrated that, for the design at <1000 °C and short duration (typical of conductivity runs), the reaction of Pt and Fe in samples is negligible and the redox state in the chamber is well buffered, as tested with pyroxenes and olivine [START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF](Yang et al., , 2012;;[START_REF] Yang | Orientation-related electrical conductivity of hydrous olivine, clinopyroxene and plagioclase and implications for the structure of the lower continental crust and uppermost mantle[END_REF]Yang and McCammon, 2012).

Before assembly, Al2O3 parts were heated at 1000 °C to remove absorbed water, and during assembly, no cement/glue was used to immobilize the various parts to avoid volatile release at high temperature that affects the analyses. Completed assembly was heated at 136 °C overnight, and was heated again at 200 °C after loading into the press.

Impedance spectra were recorded in the heating-cooling cycles, with a Solartron 1260 Impedance/Gain Phase analyser by frequency sweeping at 10 6 -1 Hz and 0.5 V applied voltage. Temperature of the analyses was up to 850 °C, to minimize water diffusion loss of samples upon heating. Duration was typically ~1 min or less per analysis (depending on temperature). After each run, recovered capsule was polished for optical, FTIR and backscattered electron examinations. Ni-NiO pairs were present, and sample distortions and grain growths were negligible, as observed in similar work [START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF](Yang et al., , 2012;;[START_REF] Yang | Orientation-related electrical conductivity of hydrous olivine, clinopyroxene and plagioclase and implications for the structure of the lower continental crust and uppermost mantle[END_REF][START_REF] Yang | Grain size effect on the electrical conductivity of clinopyroxene[END_REF]Yang and McCammon, 2012;[START_REF] Li | Fluorine follows water: Effect on electrical conductivity of silicate minerals by experimental constraints from phlogopite[END_REF][START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF]. Resistance (R) was obtained from the impedance spectra (see below), and conductivity (σ) was calculated by σ = L/SR, where L and S are the effective sample length and cross-section area, respectively. Uncertainty is usually <5% for conductivity and <20 °C for temperature.

Results

Representative FTIR spectra of annealed samples are shown in Fig. 1. The spectra show absorption peaks at 3610-3625, 3500-3520 and 3440-3460 cm -1 in the omphacites and at 3610-3640 and 3530-3550 cm -1 in the garnets, which are typical of OH bands in the eclogite minerals [START_REF] Smyth | Incorporation of hydroxyl in upper-mantle clinopyroxenes[END_REF][START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF][START_REF] Schmädicke | Water transport by subduction: clues from garnet of Erzgebirge UHP eclogite[END_REF]. In each mineral, the peak frequency positions are similar, suggesting similar H incorporation mechanism, and the relative difference of peak intensity between different samples are due to different water contents and/or grain orientations. Sample H2O contents are basically the same prior to and after each conductivity run (Table 2). Fe and other major-/minor-elements are more sluggish than H, and their contents are unchanged in the runs. Representative complex spectra are presented in Fig. 2. The spectra usually show a high frequency arc and a low frequency tail, although the tail is sometimes not obvious, and the spectral shapes are similar to those reported for other minerals [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF][START_REF] Yoshino | The effect of water on the electrical conductivity of olivine aggregates and its implications for the electrical structure of the upper mantle[END_REF][START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF]Yang et al., , 2012;;[START_REF] Yang | Orientation-related electrical conductivity of hydrous olivine, clinopyroxene and plagioclase and implications for the structure of the lower continental crust and uppermost mantle[END_REF][START_REF] Zhang | Electrical conductivity of enstatite as a function of water content: implications for the electrical structure in the upper mantle[END_REF][START_REF] Zhang | Electrical conductivity of omphacite as a function of water content and implications for high conductivity anomalies in the Dabie-Sulu UHPM belts and Tibet[END_REF][START_REF] Zhao | Electrical conductivity of mantle clinopyroxene as a function of water content and its implication on electrical structure of uppermost mantle[END_REF][START_REF] Li | Unusually high electrical conductivity of phlogopite: the possible role of fluorine and geophysical implications[END_REF][START_REF] Li | Fluorine follows water: Effect on electrical conductivity of silicate minerals by experimental constraints from phlogopite[END_REF][START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF]. The arc and tail was by sample lattice conduction and electrode effects, respectively.

The conductivity of NAMs (and other Earth materials) is quantitatively described by the Arrhenius relation:

(1) where s0 is a constant, ΔH is the activation enthalpy, R is the ideal gas constant, and T is temperature. Conductivities in the first heating were sometimes affected by residual moisture in the chamber, yielding data not obeying Eq. (1) as observed in similar studies (Yang et al., 2012;[START_REF] Yang | Grain size effect on the electrical conductivity of clinopyroxene[END_REF][START_REF] Li | Fluorine follows water: Effect on electrical conductivity of silicate minerals by experimental constraints from phlogopite[END_REF][START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF], and the influenced data were excluded. Conductivity data and the fittings to Eq. ( 1) are shown in Fig. 3, and fit parameters are given in Table 2. Sample conductivity is much greater than the assembly background conductivity [START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF]. The data are consistent between different heating-cooling cycles in each run (see also Appendix), suggesting the absence of system hysteresis. The conductivity is slightly larger at 2.5 than at 1 GPa, indicated by the runs with an OH-bearing omphacite (omp3). This is consistent with the previous study on an OH-bearing pyrope that shows similar conductivity enhancement by pressure [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF]. In contrast, the conductivity of dry olivine or garnet decreases slightly with increasing pressure [START_REF] Xu | Pressure effect on electrical conductivity of mantle olivine[END_REF][START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF]. The difference is caused by the different pressure dependence between small polaron and H conduction in the minerals (see Section 4), but the effect of pressure (at a small variation) on the conductivity of silicate minerals is in general insignificant.

Fe-and H-dependent conductivity

In Fe-and H-bearing NAMs, the conductivity is related to the conduction of small

0 exp( / R ) H T s s = × -D
polarons (electron-hole hopping between Fe 2+ and Fe 3+ ) and protons (H). Our data show that, at otherwise identical conditions, the conductivity of both omphacite and garnet increases with Fe and H2O contents (Fig. 3), agreeing with studies on other NAMs [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF][START_REF] Yoshino | The effect of water on the electrical conductivity of olivine aggregates and its implications for the electrical structure of the upper mantle[END_REF][START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF]Yang et al., , 2012;;Yang and McCammon, 2012;[START_REF] Zhang | Electrical conductivity of enstatite as a function of water content: implications for the electrical structure in the upper mantle[END_REF][START_REF] Zhao | Electrical conductivity of mantle clinopyroxene as a function of water content and its implication on electrical structure of uppermost mantle[END_REF]. The data trends of our omphacites are consistent with those of synthetic omphacites at 3 GPa [START_REF] Zhang | Electrical conductivity of omphacite as a function of water content and implications for high conductivity anomalies in the Dabie-Sulu UHPM belts and Tibet[END_REF], and the conductivity of the omphacite (omp3) with 7.92 wt.% FeO and 70 ppm H2O is close to that of an augite with 8.31 wt.% FeO and 75 ppm H2O at 1 GPa (Yang and McCammon, 2012) (Fig. 3a). The conductivity of the omphacite (omp1) with 1.46 wt.% and 200 ppm H2O resembles that of the one (omp2) with 4.27 wt.% and 150-290 ppm H2O, implying an insignificant effect of Fe at that content range. At a similar 155-160 ppm H2O, the conductivity of the Fe-rich garnet (15.40 wt.% FeO) is less than that reported by [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF] for a Fe-poor pyrope (7.42 wt.% FeO) at 8 GPa (Fig. 3b). Probably, this is related to the different pressures (1 vs. 8 GPa) and H-species (note that different H-defects could have different mobility, contributing differently to the conduction), e.g.,

narrow OH bands at 3700-3500 cm -1 in this study vs. broad bands at 3800-3000 cm -1 in [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF]. The conductivity of our garnets is close at similar contents of wt.%) and H2O (100-110 ppm, with similar OH band shapes, Fig. 1).

The activation enthalpy is Fe-dependent in omphacite, ~87, 81-84 and 65 kJ/mol at 1.64, 4.27 and 7.92 wt.% FeO, respectively, but is broadly Fe-independent in garnet, 88-91 kJ/mol over a large range of Fe contents, 15.40-26.58 wt.% FeO (Table 1). Therefore, the effect of Fe on the conductivity of omphacite and garnet is mineral dependent.

For the same omphacite or garnet in eclogite samples under similar conditions, the conductivity is linearly proportional to water content below ~1000 °C and at moderate H2O contents [START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF][START_REF] Zhang | Electrical conductivity of omphacite as a function of water content and implications for high conductivity anomalies in the Dabie-Sulu UHPM belts and Tibet[END_REF]. This has also been reported for many other NAMs such as augite, diopside, enstatite, plagioclase, pyrope and olivine [START_REF] Yang | The effect of water on the electrical conductivity of lower crustal clinopyroxene[END_REF](Yang et al., , 2012;;Yang and McCammon, 2012;[START_REF] Zhang | Electrical conductivity of enstatite as a function of water content: implications for the electrical structure in the upper mantle[END_REF][START_REF] Zhao | Electrical conductivity of mantle clinopyroxene as a function of water content and its implication on electrical structure of uppermost mantle[END_REF]. For an OH-bearing mineral with a fixed content of Fe (and other major elements), the relative contribution of small polartron conduction is usually much smaller than that of proton conduction at relatively low temperature. Eq. ( 1) can be approximated by:

(2

)
where ΔHH is the activation enthalpy by H conduction and is numerically equal to ΔH in Eq. ( 1). As such, the conductivity of both omphacite and garnet can be modeled for a range of H2O contents at each of the starting FeO contents (Fig. 4). The essentially same conductivity of omphacite with 1.64 and 4.27 wt.% FeO at the same H2O content, as noted above, indicates a more significant role of H than Fe at such Fe content levels.

The N-type semiconductor relation is used to model the effect of both Fe and H on the conductivity of both minerals:

(3)

where XFe is the molar fraction of Fe (= Fe/(Fe + Mg)), ΔE is the activation energy, and A, n, a and b are constants. The insensitivity of omphacite conductivity to FeO content at 1.64-4.27 wt.% makes it hard to model the whole dataset with Eq. (3). In this context, the modeling is carried out for omphacite using the conductivity data of our samples starting with omp1 and omp3, that restrict the boundaries of Fe contents in subductionrelated omphacites, and the synthetic omphacite (5.81 wt.% FeO and 420 ppm H2O with similar OH band positions) of [START_REF] Zhang | Electrical conductivity of omphacite as a function of water content and implications for high conductivity anomalies in the Dabie-Sulu UHPM belts and Tibet[END_REF], and for garnet using the data of all our samples (the [START_REF] Dai | Electrical conductivity of pyrope-rich garnet at high temperature and high pressure[END_REF]

H H C H T s µ × -D 1/3 1/3 exp[ ( ) / R ] n Fe H Fe H A X C E X C T s a b = × × × -D - -
results by Eq. ( 2) and (3) are well consistent at the same H2O and boundary Fe contents, especially at >500 °C (Fig. 4).

The bulk conductivity of eclogite is calculated from omphacite and garnet with the [START_REF] Hashin | A variational approach to the theory of the effective magnetic permeability of multiphase materials[END_REF] bounds, which offer the narrowest restrictions of a twophase composite despite the geometrical configurations:

(4)

where sHS+ and sHSare the upper and lower bounds, respectively, given s1>s2 (x is the volume proportion). A key prerequisite for the modeling is the equilibrium distribution of major/minor elements and OH between coexisting minerals. This is a factor that must be considered when modeling the bulk conductivity of any matrix consisting of several phases. The calculation is carried out by Eq. ( 2) using the conductivity data of minerals at different H2O contents for each starting Fe content, in which case the effect of Fe is constrained at the same time (Fig. 4). Eq. ( 3) is not adopted, because of the difficulty in controlling chemical equilibrium between the two minerals. Each of the mineral pairs, omp1-grt1, omp2-grt2 and omp3-grt3, was separated from the same starting eclogite, and the major/minor elements are already in equilibrium. The water partition coefficient between omphacite and garnet in eclogite, ~0.2 from [START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF], is adopted to obtain the equilibrium H2O contents (assuming 100, 200 and 400 ppm in omphacite).

The bulk conductivity of eclogite is not affected by likely minor impurities, because if present, they are usually isolated (not connected) in the matrix and do not contribute to bulk conduction. The results are shown in Fig. 5a and b for omphacite-rich and garnetrich rocks, respectively. This produces the modeled bulk conductivity at variable Fe and H2O contents and high temperature. In particular, The FeO contents reflect the range of the two minerals in subducting crusts (as noted before). Given other conditions, sHS+

1 1 1 2 2 1 1 1 1 1 2 1 1 2 2 2 [( ) / (3 )] [( ) / (3 )] HS HS x x x x s s s s s s s s s s - - + - - - = + - + = + - +
and sHSof each sample show a small variation above 600 °C, and are up to ~0.1 S/m at 900 °C (for the modeled composition). Considering the positive pressure effect on the conductivity of OH-bearing omphacite and garnet as mentioned above, the bulk conductivity is only slightly greater at higher pressure.

Water contents in subducting omphacite and garnet

Electromagnetic induction is sensitive to conductive rather than resistive bodies in Earth's interior. This makes it hard to accurately resolve the conductivity of subducting crusts that are highly resistive. Usually, high conductivity is not detected in subducting slabs, and slab conductivity is much smaller than that of mantle wedge and surrounding mantle in the inversion results. Conductivity of the subducting crust, either fast or slow subduction in diverse settings, is estimated to be mostly 10 -4 to 10 -3 S/m at ~70-120 km depth [START_REF] Vanyan | A geoelectric model of the Cascadia subduction zone[END_REF][START_REF] Mcgary | Pathway from subducting slab to surface for melt and fluids beneath Mount Rainier[END_REF][START_REF] Ichiki | Electrical image of subduction zone beneath northeastern Japan[END_REF][START_REF] Vargas | Fluid distribution in the Central Andes subduction zone imaged with magnetotellurics[END_REF].

This is a reasonable estimate by imposing a 1000, 5000 or 10000 Ω slab resistivity in the inversion models [START_REF] Ichiki | Electrical image of subduction zone beneath northeastern Japan[END_REF]. A typical electrical structure of a subducting crust is given in Fig. 5c (2014) by applying different inversion methods, the crust portion at ~70-120 km depth is very resistive in most of their models. Moreover, Fig. 5c is for the Cascadia area, but broadly similar results on resistive slab vs. depth have been reported for other tectonic environments (e.g., [START_REF] Vanyan | A geoelectric model of the Cascadia subduction zone[END_REF][START_REF] Ichiki | Electrical image of subduction zone beneath northeastern Japan[END_REF][START_REF] Vargas | Fluid distribution in the Central Andes subduction zone imaged with magnetotellurics[END_REF].

It has been shown that, despite a wide range of ages, geometries and convergence rates, slabs actually share many common features with their thermal structures [START_REF] Peacock | Thermal and petrologic structure of subduction zones[END_REF][START_REF] Syracuse | The global range of subduction zone thermal models[END_REF][START_REF] Penniston-Dorland | The global range of subduction zone thermal structures from exhumed blueschists and eclogites: Rocks are hotter than models[END_REF]. At ~70-120 km depth, the temperature is mostly ~600-900 °C in the crust layer, for subduction zones over various tectonic settings. The FeO content is ~1-8 wt.% of omphacite and 14-27 wt.% of garnet in terrane eclogites that were once brought to ~200 km depth [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF]and Appendix), suggesting that deeply subducting eclogites are in fact not depleted in Fe. As such, to cause the 10 -4 -10 -3 S/m conductivity of subducting crusts at ~600-900 °C, the only possibility is that omphacite and garnet are water-poor. According to the modeling in Fig. 5a-b, the H2O contents are <400 ppm in omphacite and <80 ppm in garnet at 600 °C and <100 ppm in omphacite and <20 ppm in garnet at 900 °C for the bottom Fe boundaries (omp1-grt1), and <30 ppm for the top Fe boundaries (omp3-grt3). These H2O-content ranges are meaningful, in particular if considering the weaker effect of a minor variation in Fe than in H content (Section 4), and can be even smaller, if the effect of pressure on the conductivity of OHbearing omphacite and garnet is considered (Section 3). The maximum <400 ppm H2O content resembles that estimated in [START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF], which is based on omp2 in contrast to omp1 in this study (due to the more significant effect of H at the Fe content levels as noted above). The inferred H2O contents of omphacite and garnet in subducting crusts (~70-120 km depth) are much less than those up to 2000-3000 ppm in terrane eclogites [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF]. This suggests that OH-rich omphacite and garnet in the terrane samples were H-enriched by secondary processes during their exhumation to surface, in line with the arguments by analyzing mineral OH patterns and D/H ratios [START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF][START_REF] Schmädicke | Water transport by subduction: clues from garnet of Erzgebirge UHP eclogite[END_REF]. Interestingly, the H2O values resemble the mostly <360 ppm of omphacite and <65 ppm of garnet in volcano-hosted xenolith eclogites [START_REF] Smyth | Incorporation of hydroxyl in upper-mantle clinopyroxenes[END_REF][START_REF] Bell | Water in Earth's mantle: the role of nominally anhydrous minerals[END_REF][START_REF] Koch-Müller | Hydroxyl in omphacites and omphacitic clinopyroxenes of upper mantle to lower crustal origin beneath the Siberian platform[END_REF]. This makes more sense if xenolith eclogites represent the various dehydrated products of subducted crusts that were accidently entrained by volcanoes and transferred to the surface; however, things could be complicated because a general consensus has not been reached for the origin of xenolith eclogites, e.g., some may form by melt accumulation [START_REF] Jacob | Nature and origin of eclogite xenoliths from kimberlites[END_REF].

Consequently, the conductivity data and inferred water contents of omphacite and garnet are broadly consistent with the constraints from other approaches. The inversion of electromagnetic data at subduction zones has long been subjected to the difficulty in adopting a good reference resistor in the starting model. In most cases, the inversion is carried out by highlighting the conductivity contrast between the resistive slab and the conductive paths in the wedge (linked to melting and/or dehydrated fluids). In order to identify fine-scale variation of conductivity in the subducting slab, e.g., due to change of temperature and release of water with subduction, it is important to impose a reliable slab conductivity in the inversion model. The water contents of minerals in the slab, the electrical properties of eclogites and the conductivity laws, as provided here, allow to model the conductivity of oceanic crusts as a function of temperature and water content as the starting reference resistor for the inversion of electromagnetic data.

Implications for deep water recycling

The very low water contents of omphacite and garnet at ~70-120 km depth imply a much smaller water content in the subducting crust than the original value dominated by hydrous phases (e.g., amphibole and chlorite), e.g., ~1-2 wt.% H2O [START_REF] Peacock | Fluid processes in subduction zones[END_REF].

Water in the subducting crust is mostly released at shallow depths, and the dehydration of hydrous phases and omphacite and garnet in eclogites produces conductive channels in the wedge [START_REF] Mcgary | Pathway from subducting slab to surface for melt and fluids beneath Mount Rainier[END_REF]. Eclogite-facies metamorphism, which involves all materials in the system, occurs at >30 km depths in the subducting crust (e.g., [START_REF] Peacock | Fluid processes in subduction zones[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF]. The inferred small water contents of omphacite and garnet are indicative of extremely water-poor conditions, or very low water activity, during the metamorphism and in the system. Otherwise, the water contents of the two minerals would dramatically increase, due to their strong ability in accommodating OH [START_REF] Katayama | Hydroxyl in clinopyroxene from the deep subducted crust: evidence for H2O transport into mantle[END_REF][START_REF] Katayama | Water content in natural eclogite and implication for water transport into the deep upper mantle[END_REF][START_REF] Sheng | H2O contents and D/H ratios of nominally anhydrous minerals from ultrahigh-pressure eclogites of the Dabie orogen, eastern China[END_REF], and their conductivity would be greatly enhanced, leading to conductive zones in the subducting slab that is inconsistent with geophysical mappings [START_REF] Vanyan | A geoelectric model of the Cascadia subduction zone[END_REF][START_REF] Mcgary | Pathway from subducting slab to surface for melt and fluids beneath Mount Rainier[END_REF][START_REF] Ichiki | Electrical image of subduction zone beneath northeastern Japan[END_REF][START_REF] Vargas | Fluid distribution in the Central Andes subduction zone imaged with magnetotellurics[END_REF]. The water-poor conditions also imply very minor to negligible amounts of hydrous phases in the system, because of their extensive dehydration and water escape at shallow depths. This agrees with the arguments that subducting crusts may contain no hydrous minerals beyond ~3 GPa and 700 °C, as advocated by considering dehydration in natural rocks, numerical modeling and phase stabilities [START_REF] Liu | Stability of hydrous phases in subducting oceanic crust[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF].

In this context, the amount of water carried to the deep mantle by subducting crust is regulated by omphacite and garnet in eclogites, but not hydrous phases. This implies a minor amount of water that could be brought to the deep mantle, although it might be significant at very shallow depths [START_REF] Peacock | Fluid processes in subduction zones[END_REF][START_REF] Cai | Water input into the Mariana subduction zone estimated from ocean-bottom seismic data[END_REF]. Assuming the total length, thickness and density of subducting oceanic crusts of 40000 km [START_REF] Syracuse | The global range of subduction zone thermal models[END_REF], 6 km and 3 g/cm 3 , respectively, and the average rate of subduction of 5 cm/year, the mass of water transferred to the mantle exceeding ~70 km depth in the past 3 billion years would correspond to <3% of the oceans. A direct implication is that the exchange of water between Earth's exterior and interior reservoirs might not be efficient. This fits well with the high dehydration efficiency (>92%) of subducting crusts at shallow depths, estimated from the H2O/Ce ratios of mid-ocean ridge basalts [START_REF] Dixon | Recycled dehydrated lithosphere observed in plume-influenced mid-ocean-ridge basalt[END_REF], and the preservation of protosolar nebular water in the deep mantle over Earth's evolution, constrained by the D/H ratios of plume-related lavas [START_REF] Hallis | Evidence for primordial water in Earth's deep mantle[END_REF]. The limited water transfer by crust deep subduction does not mean a water-poor deep Earth, because surface water (and atmosphere) formed by degassing of the Earth [START_REF] Allègre | Rare gas systematics: formation of the atmosphere, evolution and structure of the Earth's mantle[END_REF] and its deep interior was already hydrous.

Consequently, the water flux into the deep mantle is much smaller than that simply estimated from the stabilities of hydrous minerals such as amphibole, chlorite and dense hydrous magnesium silicates, by assuming their continuous presence in appreciated amounts in the downgoing crust. The stability fields of hydrous phases are established by experiments at water saturated/super-saturated conditions, e.g., with up to >10 wt.% water in sealed capsules [START_REF] Liu | Stability of hydrous phases in subducting oceanic crust[END_REF][START_REF] Frost | The stability of hydrous mantle phases[END_REF][START_REF] Schmidt | Devolatilization During Subduction[END_REF]. This requirement is hard to always fulfil in the subducting crust that is not a closed system and contains an initial H2O content of ~1-2 wt.% prior to subduction [START_REF] Peacock | Fluid processes in subduction zones[END_REF], which is mostly released at shallow depths. Thus, there is a gap between the water-poor state in the deep slab and the water-rich state in laboratory. Finally, if slab mantle bears serpentine, e.g., by the serpentinization of peridotites providing there is enough water for serpentine formation prior to subduction, then its role in conveying water to Earth's interior should be evaluated. Key to this issue is to map the actual amount of serpentine in the slab mantle and the dynamics of the low density with subduction, e.g., 2.6 g/cm 3 of serpentine vs. 3.4 g/cm 3 of peridotite and 3.6 g/cm 3 of eclogite. This is beyond the scope of this work, and more studies are required.

Figure captions

  (after McGary et al. (2014) who focused on conductive channels in the wedge). We want to note that, although[START_REF] Evans | Electrical structure of the central Cascadia subduction zone: The EMSLAB Lincoln Line revisited[END_REF] have offered a variety of outputs (mainly in the wedge) with the same electromagnetic data of McGary et al.

Fig. 1

 1 Fig. 1 Representative FTIR spectra of annealed omphacite and garnet. (a) polarized

Fig. 2

 2 Fig. 2 Representative complex spectra of (a) omphacite (B260) and (b) garnet (B256).

Fig. 3

 3 Fig. 3 Measured conductivity of (a) omphacite and (b) garnet at 1-2.5 GPa and 250-

Fig. 4

 4 Fig. 4 Modeled conductivity of (a) omphacite and (b) garnet at various FeO and H2O

Fig. 5

 5 Fig. 5 Electrical conductivity of eclogites with different modal composition and Fe

  Fig.1

Table 1

 1 Composition of the starting omphacite and garnet (by wt.%) Assuming all Fe as FeO. Data are the average and standard deviation (italic) by electron microprobe analyses. omp, omphacite; grt, garnet.

		SiO2	TiO2 Al2O3	FeO	MnO MgO	CaO	Na2O	K2O	Total
	omp1	56.84 0.03	9.59	1.64	0.02	10.57 15.39 5.72	0.01	99.81
		0.42	0.02	0.07	0.03	0.01	0.30	0.09	0.03	0.01
	omp2* 55.64 0.07	11.38 4.27	0.02	8.44	12.98 7.31	<0.01 100.11
		0.16	0.03	0.35	0.28	0.02	0.17	0.40	0.24	<0.01
	omp3	55.37 0.05	7.87	7.92	0.04	8.39	13.54 6.64	<0.01 99.82
		0.26	0.01	0.08	0.15	0.04	0.03	0.08	0.16	<0.01
	grt1	39.48 0.05	22.22 15.40 0.29	9.33	12.24 <0.01 <0.01 99.01
		0.15	0.03	0.07	0.17	0.01	0.18	0.19	<0.01 <0.01
	grt2*	37.64 0.03	22.73 26.55 0.74	6.81	5.63	0.03	<0.01 100.16
		0.20	0.02	0.12	0.36	0.08	0.17	0.32	0.02	<0.01
	grt3	37.49 0.07	21.63 26.58 0.14	5.02	8.79	0.04	<0.01 99.76
		0.15	0.04	0.10	0.07	0.04	0.10	0.03	0.02	<0.01

*: data were reproduced from

[START_REF] Liu | Electrical conductivity of OH-bearing omphacite and garnet in eclogite: the quantitative dependence on water content[END_REF]

(see text).

Acknowledgements

We thank Lingmin Zhang for assistance with microprobe analyses and Charles Geiger for some text polishing on an early version. X.Y. is deeply indebted to Masahiro Ichiki and Philip Wannamaker for clarifying the electrical structure of subducting slab and the technical issues of electromagnetic work. X.Y. thanks Joseph Smyth and Monika Koch-Müller for discussions. Editorial handling by James Badro and comments by Fabrice Gaillard and one anonymous reviewer helped to improve the manuscript. This study was supported by National Science Foundation of China (41725008) and National Key