Keywords: mantle lithosphere, osmium isotopes, refertilization, harzburgites, lherzolites

Highlights

-Correlations between Os isotopes and fertility indices in non-cratonic peridotites -Such correlations inconsistent with recent refertilization of ancient harzburgites -Can be explained by ancient refertilization followed by radiogenic ingrowth -Suggests harzburgite depletion causally related to subsequent refertilization -Whole rock correlations and internal Os isotope heterogeneity are not inconsistent

Introduction

The origin and evolution of the non-cratonic subcontinental lithospheric mantle (SCLM) underlying most of the Earth's continental mass remains poorly understood despite decades of study. This mantle domain, which is sampled by xenoliths in alkali basalts and by orogenic peridotite massifs, is composed predominantly of peridotites with compositions varying from depleted harzburgites to fertile lherzolites. For many years, this range of compositions was generally interpreted to result from varying degrees of melt extraction from a fertile lherzolite precursor. Nevertheless, possible inconsistencies in this scenario were soon recognized, notably the often-observed enrichment of incompatible elements in harzburgites (e.g. [START_REF] Frey | Ultramafic inclusions from San Carlos, Arizona: petrologic and geochemical data bearing on their petrogenesis[END_REF], which is inconsistent with simple melt extraction models. The development of melt percolation models [START_REF] Navon | Geochemical consequences of melt percolation: The upper mantle as a chromatographic column[END_REF][START_REF] Vernières | A plate model for the simulation of trace element fractionation during partial melting and magma transport in the Earth's upper mantle[END_REF] demonstrated how the geochemical composition of a peridotite column could be profoundly modified by the percolation of magmas out of equilibrium with the original peridotite. In addition, field studies, such as that of the Lherz massif by Le [START_REF] Le Roux | The Lherz spinel lherzolite: Refertilized rather than pristine mantle[END_REF], have provided direct observational and petrological evidence for the transformation of harzburgites into lherzolites. Such studies have led to an evolution in current understanding of mantle peridotites. The wide range of compositions encountered in ultramafic rock suites is now often attributed to the refertilization of depleted harzburgites by percolating mafic melts, rather than to variable degrees of melt extraction from a putative precursor of primitive mantle composition. (The term "refertilization" as used in this contribution refers to the transformation of harzburgite into lherzolite through percolative magmatic processes.)

Although the importance of refertilization has been increasingly recognized, constraining the timing of such processes remains a critical issue. One possibility is that melt refertilization is linked temporally to the original magmatic processes that created the lithosphere. If so, this might imply that refertilization is an intrinsic part of the lithosphere formation mechanism.

Alternatively, melt refertilization could be related to tectonic episodes affecting the lithosphere hundreds of millions of years after its formation. In this case, refertilization would have no connection with the original harzburgite-forming melt depletion event. Thus our understanding of how the non-cratonic SCLM formed is deeply dependent on determining when refertilization occurred. Moreover, as depleted harzburgites are less dense than fertile lherzolites [START_REF] Poudjom Djomani | The density structure of subcontinental lithosphere through time[END_REF] the timing of refertilization will have significant implications for the stability and long-term survival of the SCLM relative to the convecting mantle.

Osmium isotopes offer the possibility of constraining the timing of refertilization of the SCLM.

Since osmium concentrations of peridotites are much higher than those of mantle melts and fluids, the Os isotope compositions of lithospheric peridotites are to some extent protected from modification by metasomatic agents that very frequently perturb the Sr, Nd and Pb isotope compositions of these rocks. Results from dozens of Re-Os studies of orogenic massifs and ultramafic xenoliths derived from the SCLM are now available. Modeling these results allows some broad conclusions to be drawn concerning the timing of refertilization of the non-cratonic lithosphere.

A brief overview of the Re-Os isotope system

The isotope 187 Os is formed by the β -decay of 187 Re, with a half-life of 41.6 Ga (λ=1.666x10 -11 a -1 ; [START_REF] Smoliar | Re-Os ages of Group IIA, IIIA, IVA, and IVB iron meteorites[END_REF]. The Re-Os system differs from all other isotope systems commonly used to study mantle geochemistry in that both the parent and the daughter elements have a strong affinity for sulfide. Osmium is an exceptionally chalcophile element, with recent results suggesting D sulf/sil (sulfide liquid/silicate liquid) values approaching 10 6 [START_REF] Mungall | Partitioning of platinum-group elements and Au between sulfide liquid and basalt and the origins of mantle-crust fractionation of the chalcophile elements[END_REF]. This means that during partial melting of peridotite, Os will be retained in the residue as long as the slightest trace of base metal sulfide remains. Furthermore, even after all sulfide is removed, Os remains in the residue in ultratrace platinum group minerals (PGM) such as laurites and alloys [START_REF] Luguet | Residual platinum-group minerals from highly depleted harzburgites of the Lherz massif (France) and their role in HSE fractionation of the mantle[END_REF]. In addition, while observational evidence suggests that olivine is not a significant host of Os [START_REF] Burton | The compatibility of rhenium and osmium in natural olivine and their behaviour during mantle melting and basalt genesis[END_REF][START_REF] Puchtel | Platinum group element geochemistry of komatiites from the Alexo and Pyke Hill areas, Ontario, Canada 11Associate[END_REF], experimental partitioning data for Ir and and Ru, two platinum group elements closely related to Os, indicate that they are mildly compatible in olivine and spinel (see compilation in [START_REF] Brenan | Experimental Results on Fractionation of the Highly Siderophile Elements (HSE) at Variable Pressures and Temperatures during Planetary and Magmatic Differentiation[END_REF]. This suggests that Os may also be compatible in these phases under some conditions. For all of these reasons, Os behaves as a highly compatible element during mantle melting.

The chalcophility of Re is strongly dependant on the fugacities of oxygen and sulfur [START_REF] Fonseca | How chalcophile is rhenium? An experimental study of the solubility of Re in sulphide mattes[END_REF], and varies systematically with the relative value of these two parameters [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF]. As discussed more fully below, [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF] argues that this relative value is constrained within a limited range under most upper mantle conditions, corresponding to D sulf/sil values for Re between about 380 and 820. Given the relatively low abundances of sulfides in mantle peridotites, and the incompatible nature of Re in other upper mantle phases [START_REF] Mallmann | The effect of oxygen fugacity on the partitioning of Re between crystals and silicate melt during mantle melting[END_REF]; see also compilation in [START_REF] Brenan | Experimental Results on Fractionation of the Highly Siderophile Elements (HSE) at Variable Pressures and Temperatures during Planetary and Magmatic Differentiation[END_REF], this means that unlike Os, Re behaves as a moderately incompatible element during mantle melting.

The contrasting behaviors of Re and Os during melting have several consequences. The high compatibility of Os insures that the Os concentrations of peridotites are elevated relative to those of most melts and fluids, offering some degree of protection to 187 Os/ 188 Os ratios from modification during metasomatism. Retention of Os during melting also explains the very low Os concentrations (~30 pg/g; Peucker-Ehrenbrink and Jahn, 2001), and thus high Re/Os ratios of the continental crust, which with time have produced very radiogenic crustal Os isotope compositions relative to mantle values. As the estimated Re concentrations of the continental crust (~200 pg/g; Peucker-Ehrenbrink and Jahn, 2001) are comparable to mantle values, and the volume of the crust is small compared to that of the mantle, mass balance considerations imply that extraction of the continental crust had little effect on the average Re/Os ratio of the mantle. This is true even if the continental crust is assumed to have been extracted from only the upper mantle, and the resulting upper mantle Re/Os ratio is considered. In consequence, the average 187 Os/ 188 Os ratio of the upper mantle, is expected to be similar to that inferred for the primitive upper mantle (PUM). This is another respect in which the Re/Os system differs from the Rb/Sr and Sm/Nd isotope systems, for which the average upper mantle isotope ratios differ markedly from the inferred PUM values [START_REF] Zindler | Chemical geodynamics[END_REF]. Of course, the inference that the average 187 Os/ 188 Os value of the upper mantle should have been only slightly modified by crustal extraction does not imply that the Os composition of the upper mantle is uniform. Substantial heterogeneity is expected within the convecting mantle, notably between depleted residues of melt extraction and recycled oceanic crust, and this is evident in the heterogeneity observed between and among abyssal peridotites and basalts [START_REF] Becker | Re-Pt-Os Isotopic and Highly Siderophile Element Behavior in Oceanic and Continental Mantle Tectonites[END_REF][START_REF] Gannoun | Highly Siderophile Element and Os Isotope Systematics of Volcanic Rocks at Divergent and Convergent Plate Boundaries and in Intraplate Settings[END_REF].

In the following discussion, the term primitive upper mantle (PUM) will be preferred to bulk silicate Earth (BSE), though the compositions of the two reservoirs are often taken to be equivalent. Unlike the BSE, the PUM is not an actual reservoir, but instead is a theoretical concept referring to the upper part of the mantle prior to its depletion by extraction of the crust.

Nevertheless the composition of this theoretical reservoir can be directly characterized by the study of upper mantle samples, while estimation of the BSE composition requires more assumptions about possible meteoritic building blocks. Therefore throughout the discussion the term PUM will be used. No implications are intended regarding possible differentiation of the primordial mantle or the extent to which mantle convection affects the whole rather than the upper part of the mantle.

Os isotope correlations with bulk rock fertility

It has long been recognized that in many non-cratonic peridotite suites whole rock Os isotope compositions are correlated with indices of fertility, and in fact such correlations were used to estimate the 187 Os/ 188 Os composition of the PUM (~0.1296 ±0.008; [START_REF] Meisel | Osmium isotopic compositions of mantle xenoliths: a global perspective[END_REF]. Such relationships are rarely seen for highly incompatible lithophile elements such as Sr, Nd and Pb.

Broadly speaking, these relationships result from the contrasting behavior of Re and Os during melt extraction, and are often preserved because of the relative resistance of osmium isotopes to metasomatic modification. In the following discussion, we will use Al 2 O 3 content as a proxy for bulk peridotite fertility, as aluminum is a major element and bulk rock Al 2 O 3 data are available for almost all sample suites, but other parameters, such as MgO and CaO contents, Yb and Lu concentrations, olivine Fo# and spinel Cr# have also been used as fertility proxies (e.g. [START_REF] Armytage | Osmium isotope evidence for Early to Middle Proterozoic mantle lithosphere stabilization and concomitant production of juvenile crust in Dish Hill, CA peridotite xenoliths[END_REF][START_REF] Byerly | Evidence from mantle xenoliths for lithosphere removal beneath the central Rio Grande Rift[END_REF][START_REF] Gao | Re-Os evidence for replacement of ancient mantle lithosphere beneath the North China craton[END_REF][START_REF] Handler | Proterozoic lithosphere in Marie Byrd Land, West Antarctica: Re-Os systematics of spinel peridotite xenoliths. Chemical Geology, Highly Siderophile elements in the Earth and Meteorties[END_REF][START_REF] Peslier | Re-Os constraints on harzburgite and lherzolite formation in the lithospheric mantle: A study of Northern Canadian Cordillera xenoliths[END_REF][START_REF] Wu | The chemical-temporal evolution of lithospheric mantle underlying the North China Craton[END_REF]. Most of the discussion will center on whole rock 187 Os/ 188 Os compositions, even though it has been demonstrated that large Os isotope variations exist among sulfides from the same rock sample [START_REF] Alard | New insights into the Re-Os systematics of subcontinental lithospheric mantle from in-situ analyses of sulfides[END_REF][START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF][START_REF] Luguet | Dating mantle peridotites using Re-Os isotopes: The complex message from whole rocks, base metal sulfides, and platinum group minerals[END_REF] and that the sulfide and Platinum Group Mineral (PGM) phases hosting Os and the other PGE in mantle peridotites display great complexity [START_REF] Lorand | Chalcophile and Siderophile Elements in Mantle Rocks: Trace Elements Controlled By Trace Minerals[END_REF].

Reconciliation of this extensive internal heterogeneity in Os host phases and isotope composition with broader whole rock relationships will be discussed below. Nevertheless, the existence of internal heterogeneity should not obscure the fact that systematic correlations between whole rock Os isotope compositions and major element fertility indices are nearly ubiquitous and must be explained.

Osmium isotope results are now available from studies of more than 50 off-cratonic peridotite suites, permitting some broad generalizations to be drawn. Most of the xenolith data summarized below were assembled for a recent review article [START_REF] Luguet | Highly Siderophile Element and 187Os Signatures in Non-cratonic Basalt-hosted Peridotite Xenoliths: Unravelling the Origin and Evolution of the Post-Archean Lithospheric Mantle[END_REF], while the orogenic peridotite data were taken from the references given by [START_REF] Becker | Re-Pt-Os Isotopic and Highly Siderophile Element Behavior in Oceanic and Continental Mantle Tectonites[END_REF] for settings described in that reference as convergent margin tectonites or continental/continent-ocean transitional tectonites. A complete list of references for both sample types is given in appendix A of the supplemental online material (SOM).

Figure 1.

Whole rock 187 Os/ 188 Os ratios as a function of Al 2 O 3 content for peridotites from the non-cratonic subcontinental lithosphere. Data for ultramafic xenoliths are mostly from the database of [START_REF] Luguet | Highly Siderophile Element and 187Os Signatures in Non-cratonic Basalt-hosted Peridotite Xenoliths: Unravelling the Origin and Evolution of the Post-Archean Lithospheric Mantle[END_REF], while those for orogenic peridotites are from settings described as convergent margin tectonites or continental/continent-ocean transitional tectonites by [START_REF] Becker | Re-Pt-Os Isotopic and Highly Siderophile Element Behavior in Oceanic and Continental Mantle Tectonites[END_REF]. The complete list of localities and the corresponding references are given in Appendix A of the supplemental online material (SOM).

All of the 187 Os/ 188 Os data are plotted as a function of Al 2 O 3 content in Fig. 1. Most of the data lie on a broad trend between highly depleted compositions with unradiogenic 187 Os/ 188 Os values, and fertile lherzolite with an Os isotope composition similar to the Primitive Upper Mantle (PUM) composition defined by [START_REF] Meisel | Osmium isotopic compositions of mantle xenoliths: a global perspective[END_REF]. Nevertheless, many samples plot substantially above this general trend, particularly among the xenolith data. In contrast, only a few plot significantly below the trend. These tendencies are confirmed by considering the various sample suites individually. Most localities display well defined trends, though the slopes vary between sites. The significance of these trends was evaluated by tracing a line between the proposed PUM composition and the least radiogenic sample in each suite (excluding suites with fewer than six samples and those with no samples with Al 2 O 3 contents less than 2%), and determining the deviation in 187 Os/ 188 Os of each sample point from the line.

Among xenolith suites, 51% of the samples were found to plot within ±1.5% of the trend, while 34% plot above the trend and only 13% plot below it. Among orogenic peridotite massifs, 57% plot within ±1.5% of the trend, 29% plot above the trend, and 13% plot below it. Examples of two xenolith suites are shown in Fig. 2, one from Vitim, Siberia [START_REF] Pearson | Re-Os isotope systematics and platinum group element fractionation during mantle melt extraction: a study of massif and xenolith peridotite suites[END_REF] which displays a very strong correlation, and another from Lianshan, eastern China [START_REF] Reisberg | Re-Os and S systematics of spinel peridotite xenoliths from east central China: Evidence for contrasting effects of melt percolation[END_REF]. The latter shows a more typical case of a well-defined correlation coupled with a substantial number of points plotting above the trend. Like many non-cratonic xenolith suites, the Lianshan peridotites have suffered extensive recent Os and Re loss during both pre-and post-eruptive processes. Because of its representative nature, the Lianshan suite will be used as an example in much of the following discussion, but the same conclusions could be drawn from any of the many suites displaying well-defined linear trends passing through the PUM composition. [START_REF] Meisel | Osmium isotopic compositions of mantle xenoliths: a global perspective[END_REF]. In both cases, the trend line is drawn between the least radiogenic sample and the PUM composition, and the gray field represents 187 Os/ 188 Os values within ±1.5% of this line. The samples from Vitim, Siberia studied by [START_REF] Pearson | Re-Os isotope systematics and platinum group element fractionation during mantle melt extraction: a study of massif and xenolith peridotite suites[END_REF] define an exceptionally good correlation. The samples from Lianshan, China [START_REF] Reisberg | Re-Os and S systematics of spinel peridotite xenoliths from east central China: Evidence for contrasting effects of melt percolation[END_REF] represent a more typical case, which includes a significant number of samples that plot well above the trend, although most points still plot close to the line. As the Lianshan case is more common, it is used as an example for the modeling, but the same conclusions can be drawn using any of the numerous observed correlations.

What creates correlations between Os isotopes and fertility indices?

Simple mixing between unradiogenic peridotites and radiogenic mafic magmas

Radiogenic ingrowth after partial melt extraction

Another possibility, proposed by [START_REF] Reisberg | Longevity of sub-continental mantle lithosphere from osmium isotope systematics in orogenic peridotite massifs[END_REF], is that the correlations are essentially isochron analogs, formed by radiogenic ingrowth after ancient partial melting events that left variably depleted peridotites. As discussed above, Re is expected to behave as a moderately incompatible element during partial melting, so Re concentrations in the residue should decrease as melting progresses. Osmium, on the other hand, is highly compatible, so Os contents of peridotites should increase slightly during melting simply due to the removal of Os given in figure caption and in Appendix A of SOM; model provided as spreadsheet "non-modal partial melting" in Appendix B of SOM). The model includes silicate phases and spinel as well as sulfide. Rhenium partition coefficients for silicate phases and spinel are based on the work of [START_REF] Mallmann | The effect of oxygen fugacity on the partitioning of Re between crystals and silicate melt during mantle melting[END_REF], assuming oxygen fugacity corresponding to the QFM buffer.

For osmium, very few results exist for silicate and spinel partitioning, so the values used are based on available ranges compiled by [START_REF] Brenan | Experimental Results on Fractionation of the Highly Siderophile Elements (HSE) at Variable Pressures and Temperatures during Planetary and Magmatic Differentiation[END_REF] for iridium, a closely related element whose abundances are usually well correlated with those of Os in peridotite suites.

Nevertheless, though silicate and spinel partitioning are considered in the model, sulfide has by far the largest control on Re and Os abundances of the residue during melting. Sulfide is progressively removed during melting, to the extent imposed by the SCSS (Sulfur Concentration at Sulfide Saturation) of the magma (Fig. 4a). Rhenium concentrations will decrease with progressive melting (Fig. 4b), as the decreasing sulfide content lowers the Re bulk rock partition coefficient. This figure shows curves corresponding to D sulf/sil (sulfide liquid/silicate liquid partition coefficient) values for Re of 820 and 380, which as noted above are the expected upper and lower limits of this parameter for most conditions of mantle melting [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF]. The curve corresponding to a D sulf/sil value of 380 shows a nearly linear decrease in Re concentration in the residue with increasing melt extraction. In constrast to Re, Os is extremely chalcophile, so it is retained in the residue as long as the slightest trace of sulfide remains (Fig. 4c), and likely even longer if platinum group minerals such as laurite and

Os-Ir alloys are exsolved. As magmaphile components are removed, the concentration of Os in the residue increases. As a result of this contrasting behavior, the Re/Os ratio of the residue is negatively correlated with the extent of melt extraction (Fig. 4d), giving rise with time (1.7 Ga in the example shown), to a positive correlation between 187 Os/ 188 Os and Al 2 O 3 (Fig. 4e) due to the radioactive decay of 187 Re. For simplicity, it is assumed here that Al 2 O 3 decreases linearly with extent of melt extraction, starting with an estimated primitive mantle value of 4.2% and decreasing to zero at 25% extraction. The model is continued to 20% melt extraction, which is equivalent to the point of exhaustion of base metal sulfides in the residue using the chosen parameters; no attempt is made to model greater extents of melting for which Os is dominantly hosted by rare, tiny Platinum Group Minerals [START_REF] Luguet | Residual platinum-group minerals from highly depleted harzburgites of the Lherz massif (France) and their role in HSE fractionation of the mantle[END_REF] or possibly olivine or spinel.

A D sulf/sil value of 380 for Re is used as this best reproduces the linear trends seen in numerous peridotite suites. This D sulf/sil value for Re also falls within the range (200 to 450) calculated from non-modal batch melting modeling of Re contents in Pyreneen peridotites [START_REF] Burnham | The petrogenesis of the eastern Pyrenean peridotites: an integrated study of their whole-rock geochemistry and Re-Os isotope composition[END_REF].

From Fig. 4e, it is apparent that this simple melting model, followed by a prolonged period of radiogenic ingrowth, can successfully reproduce the 187 Os/ 188 Os vs. Al 2 O 3 trend seen in the Lianshan peridotites. Thus if Os isotopes alone are considered, melt extraction to varying extents followed by ingrowth of 187 Os is a viable process for producing the variety of peridotites, ranging in composition from depleted harzburgite to fertile lherzolite, observed in most sample suites.

Transformation of harzburgite to lherzolite by refertilization

Though radiogenic ingrowth after ancient melt extraction can potentially explain the observed whole rock Os systematics, this explanation is unlikely to be applicable to all or even most cases. Abundant evidence now exists to suggest that many fertile lherzolites are not simple partial melting residues but are instead formed by the refertilization of harzburgites. This evidence, summarized by [START_REF] Bodinier | Orogenic, Ophiolitic, and Abyssal Peridotites[END_REF], includes structural observations of lherzolites replacing harzburgites, as in the Lherz massif [START_REF] Le Roux | The Lherz spinel lherzolite: Refertilized rather than pristine mantle[END_REF], mineral phase compositions that are inconsistent with trends expected from melt extraction, and textural evidence for crystallization of secondary phases from percolating melts, often in the form of intergranular pyroxene and spinel. Intergranular sulfides, notably pentlandite as well as chalcopyrite and pyrrhotite [START_REF] Lorand | Chalcophile and Siderophile Elements in Mantle Rocks: Trace Elements Controlled By Trace Minerals[END_REF][START_REF] Luguet | Dating mantle peridotites using Re-Os isotopes: The complex message from whole rocks, base metal sulfides, and platinum group minerals[END_REF], are often associated with the precipitated pyroxene and spinel. As sulfides are the main hosts for Re and Os in peridotites, a simple model was developed in order to explore whether the observed 187 Os/ 188 Os vs. Al 2 O 3 trends could be produced by the simultaneous precipitation of pyroxene, spinel and sulfide during melt percolation. Refertilization models of this type are meltconsuming [START_REF] Bodinier | Orogenic, Ophiolitic, and Abyssal Peridotites[END_REF] and will lead to an increase in peridotite volume due to the addition of these precipitated phases.

In the model, mafic magmas percolate upwards in a cooling lithosphere composed of depleted harzburgite. The partial melting event that formed the residual harzburgite is assumed to have been sufficiently profound to approximately homogenize the Os isotopic composition at the scale of the melting region, which is assumed to have followed the average upper mantle evolution path. Though sometimes not stated explicitly, this assumption of homogenization to an average mantle composition is inherent in all model age scenarios. A key feature of the model is that as spinel and pyroxene crystallize, the sulfur content of the magma will increase, leading to sulfur saturation when the SCSS is exceeded. The exsolved sulfide liquid will concentrate Os, and to a lesser extent Re, from the silicate liquid. A simplified plate model, composed of 100 stacked cells, is assumed. For each added melt increment, after pyroxene and spinel precipitation provoke exsolution of a sulfide liquid, Re and Os are allowed to equilibrate between all of these newly-formed phases and the remaining silicate liquid. (A variant of the model, in which the Re and Os in the liquid equilibrate with all of the phases present in the rock, is considered below.) This silicate liquid then migrates upward while the sulfide liquid remains behind because of its greater density. To start the next step, new silicate magma is added from below. No attempt is made to thermodynamically model changes in the major element composition of the bulk rock or the individual phases, as this would be far beyond the scope of this contribution. Instead, Al 2 O 3 is used as an index of the extent of refertilization, that is, alumina content is assumed to increase linearly with addition of precipitating phases from the percolating magma. This is achieved by adjusting the modal proportions of the crystallizing phases to assure that the Al 2 O 3 content of the magma remains constant. As shown in Appendix A (SOM), the modal compositions of fertile lherzolites produced by the model are not unreasonable, suggesting that this approximation is adequate for the purpose of modeling the behavior of the Re-Os system during melt percolation.

The parameters and assumptions of the model are explained in more detail in Appendix A. A spreadsheet is provided in Appendix B (SOM) that can be used to investigate the effects of modifying the inputs. This simple percolation model is used to test two cases: that of recent refertilization in which the unradiogenic harzburgite interacts with a melt of radiogenic Os composition, and that of ancient refertilization in which Re is added and the range of 187 Os/ 188 Os ratios is acquired through radiogenic ingrowth.

Can recent melt percolation transform ancient harzburgites into lherzolites?

The first case concerns refertilization episodes that are geologically recent, or at least much younger than and unrelated to the original profound melting events that depleted the harzburgites. In this case, the correlations between Os isotope compositions and whole rock fertility would have to reflect addition of sulfides with radiogenic Os derived from the melt together with addition of spinel and pyroxene. However, because of the extreme chalcophility of Os (D sulf/sil ~10 6 ; [START_REF] Mungall | Partitioning of platinum-group elements and Au between sulfide liquid and basalt and the origins of mantle-crust fractionation of the chalcophile elements[END_REF], essentially all of the Os would be incorporated into sulfides in the bottom-most increment of the percolation column (Fig. 5a). In the overlying increment, almost no Os is added, and the bulk rock Os concentration is decreased by the addition of Os-free spinel and pyroxene. As the amount of material added decreases with height in the column, this dilution effect becomes progressively less important moving upwards.

Since, with the exception of the lowest step, no radiogenic Os is added as the melts migrate up the column, nearly horizontal arrays are produced on 187 Os/ 188 Os vs. Al 2 O 3 diagrams (Fig. 3). This is true even for parameter choices most likely to favor the imprint of a radiogenic signature on the peridotites (percolating magma with exceptionally high Os concentration and radiogenic composition). These arrays deviate even more strongly from the data than the trends for the case of simple mixing also shown in Fig. 3, because exsolution of a sulfide liquid will rapidly strip essentially all of the Os from the percolating silicate melt. Though the partition coefficient is less well constrained, spinel also likely incorporates Os [START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF], reinforcing the compatible behavior of this element. Thus not only are the percolating magmas likely to start with low Os concentrations relative to those of peridotites, but also any Os contained in these melts will be trapped by sulfides and possibly spinel at the very bottom of the column. The nearly horizontal trends that recent melt percolation would predict (Fig. 3) are seldom, if ever, seen in peridotite suites (Fig. 1). If radiogenic Os were added long after the original melt depletion event that created the harzburgites, but nevertheless several hundred million years before the present, moderately sloped trends might be expected to result from these initially horizontal trends because of the effect of radiogenic ingrowth of 187 Os. However such trends would not pass through the PUM value, and would be notably less steep than the observed trends.

Another argument against addition of radiogenic sulfides concomitantly with precipitation of spinel and pyroxene is that under this scenario the observed 187 Os/ 188 Os vs. Al 2 O 3 correlations would require Os concentrations to increase systematically with Al 2 O 3 , which is not observed. refertilization of ancient depleted harzburgites to form lherzolites during much younger melt percolation is a process that almost never occurs.

Refertilization soon after melt extraction followed by radiogenic ingrowth

The second case to be tested is whether melt percolation occurring soon after melt extraction is 8a-c, for SCSS values varying from 1200 to 800 ppm, which corresponds roughly to depths of 0 to 120 km [START_REF] Li | Empirical equations to predict the sulfur content of mafic magmas at sulfide saturation and applications to magmatic sulfide deposits[END_REF] since this parameter is sensitive to pressure. Optimal results are found for a SCSS value of ~1000 ppm, but the variation of the curve throughout this range is limited. The second group consists of parameters that control the extent of refertilization, but SCSS is allowed to vary between 800 and 1200 ppm, which corresponds to depths of roughly 120 to 0 km [START_REF] Li | Empirical equations to predict the sulfur content of mafic magmas at sulfide saturation and applications to magmatic sulfide deposits[END_REF]. Though an optimal fit to the Lianshan data is found for SCSS ~1000, variation within the tested range has only a minor effect on the results. D-F) Example of a parameter (initial melt fraction) that has very little effect on the slope or curvature of the calculated refertilization trend, but a large effect on its extent. Sensitivity tests for other model parameters are presented in Appendix A.

The third type of behavior is displayed by D Re sulf/sil . As shown in Fig. 9, changing this parameter within the limits for this value proposed in the literature has a dramatic effect on the host basaltic glasses, a very shallow, slightly convex upwards trend is observed that falls far below the data points (Fig. 9a). In contrast, the very high D Re sulf/sil values of ~40,000 suggested by the experimental work of [START_REF] Sattari | Experimental Constraints on the Sulfide-and Chromite-Silicate Melt Partitioning Behavior of Rhenium and Platinum-Group Elements[END_REF] yield extremely concave upward curves that also deviate greatly from the data (Fig. 9d). As discussed by [START_REF] Fonseca | How chalcophile is rhenium? An experimental study of the solubility of Re in sulphide mattes[END_REF], the great range in D Re sulf/sil obtained from experimental studies reflects the sensitivity of this parameter to fS 2 and fO 2 . [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF] noted that a systematic relationship exists between D Re sulf/sil and the relative value of these two parameters, expressed as 1/2 logfS 2 -1/2 log fO 2 . He emphasized that fS 2 and fO 2 are linked in mantle melts by a heterogeneous equilibrium reaction relating FeO in the silicate liquid to FeS in the sulfide liquid. As FeO contents in primary mafic magmas are confined to a limited range, the value of 1/2 logfS 2 -1/2 log fO 2 is also confined to a restricted range. The limits of this range correspond to D Re sulf/sil values of about 380 to 820 for primary magmas with FeO contents of 11% and 8%, respectively [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF]. Figs. 9b and9c, respectively, compare model curves constructed with these D Re sulf/sil values to the test data from Lianshan. It is apparent that D Re sulf/sil values toward the lower end of this range, which should be associated with primary magmas with high FeO contents, produce curves that closely fit the data. As the FeO contents of primary magmas increase with pressure [START_REF] Jaques | Anhydrous melting of peridotite at 0-15 Kb pressure and the genesis of tholeiitic basalts[END_REF], it is plausible that melts circulating in the lower part of the lithosphere would be iron rich, and would thus have the relatively low D Re sulf/sil values required to fit the data. sulf/sil =380, lower limit of range suggested by [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF], corresponding to high FeO primary melts. This provides the best fit to the data and is the default value used in the text. C) D Re sulf/sil =820, upper limit of range suggested by [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF], corresponding to low FeO basalts. d) D Re sulf/sil =40000, suggested by experimental work of [START_REF] Sattari | Experimental Constraints on the Sulfide-and Chromite-Silicate Melt Partitioning Behavior of Rhenium and Platinum-Group Elements[END_REF].

It thus appears that refertilization following soon after the original melting event can indeed successfully explain the existence of correlations between Os isotope composition and fertility that are nearly ubiquitous in the whole rock record from the non-cratonic subcontinental lithospheric mantle. This is perhaps not surprising as the massive partial melting episodes needed for large scale development of depleted harzburgites are unlikely to stop abruptly.

Instead a cooling period would be expected, and during this time melts from the still hot underlying regions could migrate up through the cooling lithosphere, transforming harzburgites into lherzolites. Some support for ancient refertilization comes from the recent study by Le Roux et al. ( 2016) of websterite layers in the Lherz massif, which are thought to have been formed by the same refertilization process that produced the lherzolites. Using a combination of methods, these authors proposed a refertilization age of 1.5 to 1.8 Ga, much older than the late Variscan age originally postulated for this process [START_REF] Le Roux | The Lherz spinel lherzolite: Refertilized rather than pristine mantle[END_REF]. This new age overlaps with the Os T RD model age (1.75-2.07 Ga) that can be calculated from the 187 Os/ 188 Os ratio (0.1151) of the most refractory harzburgite studied from the massif [START_REF] Reisberg | Longevity of sub-continental mantle lithosphere from osmium isotope systematics in orogenic peridotite massifs[END_REF]. The age suggested by Le Roux et al. ( 2007) also agrees with the Os model age of 1.9 ±0.3 Ga calculated by [START_REF] Burnham | The petrogenesis of the eastern Pyrenean peridotites: an integrated study of their whole-rock geochemistry and Re-Os isotope composition[END_REF] for the eastern Pyrenean massifs. This agreement suggests that melt extraction, followed soon after by refertilization, is the process that produced the lherzolites in the type locality of this lithology.

Correlations with heavy Rare Earth Elements

In addition to Al 2 O 3 , the heavy Rare Earth Elements (HREE) Lu and Yb are sometimes used as fertility indices in ultramafic rocks, and their contents often correlate with 187 Os/ 188 Os ratios in samples from the non-cratonic mantle lithosphere (e.g. [START_REF] Armytage | Osmium isotope evidence for Early to Middle Proterozoic mantle lithosphere stabilization and concomitant production of juvenile crust in Dish Hill, CA peridotite xenoliths[END_REF][START_REF] Peslier | Re-Os constraints on harzburgite and lherzolite formation in the lithospheric mantle: A study of Northern Canadian Cordillera xenoliths[END_REF]. The most important factor controlling HREE behavior in upper mantle magmatic processes is the presence and abundance of garnet, as the HREE are compatible in this phase.

Though the depth of the spinel-garnet transition will depend on temperature and peridotite composition [START_REF] Ionov | Garnet Peridotite Xenoliths from the Vitim Volcanic Field, Baikal Region: the Nature of the Garnet--Spinel Peridotite Transition Zone in the Continental Mantle[END_REF][START_REF] Wood | Garnet in the Earth's Mantle[END_REF], garnet is usually stable at depths greater than ~70 km. Thus the model was adapted to test whether melt percolation in the garnet stability field could produce the observed correlations of Lu with 187 Os/ 188 Os.

Results of the modeling are shown in Fig. 7, and compared with the data from Lianshan. In this and garnet (Fig. 7d). If only spinel is present the calculated curve is strongly concave up.

Conversely if only garnet is present, the curve is strongly concave down. Neither case fits the data. In contrast, similar modal abundances of garnet and spinel produce an approximately linear correlation that passes through or close to the aligned data. It should be noted that garnet is not observed in the thin sections of the Lianshan xenoliths [START_REF] Reisberg | Re-Os and S systematics of spinel peridotite xenoliths from east central China: Evidence for contrasting effects of melt percolation[END_REF]. This might suggest that the source of these xenoliths experienced melt percolation near the spinelgarnet transition, and was subsequently uplifted into the spinel peridotite facies.

The inferred presence of both spinel and garnet suggests melt percolation in the mid to lower non-cratonic lithosphere. This is consistent with the conclusion drawn above from the optimal Re partition coefficient (~380), which suggests interaction with melts with high FeO contents, as might be expected at mid to lower lithospheric depths. It is also consistent with the fact that the data are best fit in the 187 Os/ 188 Os vs. Al 2 O 3 diagram (Fig. 8) when a SCSS value of ~1000 ppm is assumed, which would correspond to a depth of ~ 60 km [START_REF] Li | Empirical equations to predict the sulfur content of mafic magmas at sulfide saturation and applications to magmatic sulfide deposits[END_REF]. While none of these arguments is definitive, they are all suggestive of melt percolation in the mid to lower part of the non-cratonic lithosphere.

Equilibrium melt percolation model

In the model discussed above, only local equilibrium was assumed, that is, the Re and Os in the crystallized phases and exsolved sulfide liquid were taken to equilibrate only with the increment of silicate liquid from which these phases formed. This was done for simplicity, as it allowed each increment of liquid to be considered in isolation from the rest of the rock, and the total effect to be obtained simply by multiplying by the number of increments. However in reality, Re and Os in the newly formed phases may fully or partially equilibrate with the phases left by the preceding melt increments, as well as with the phases inherited from the original peridotite. This is more complex to model, as equilibrium concentrations must be calculated using modal proportions that change with each melt increment. Furthermore, the Re and Os concentrations of the liquid passed from any given cell to the overlying cell will change with time, as the modal composition of the peridotite with which the melt equilibrates changes. As a result, a separate spreadsheet is needed for each cell in the percolation column to allow the effects of each increment of melt addition to be calculated individually. For this reason, a full equilibrium model was tested only for the bottom 25 cells of the column, and the number of melt increments was fixed at 320.

As can be seen in Fig. 10, the results of the full equilibrium model are almost identical to those of the orignal model based on local equilibration between the melt and the newly formed phases. This is true even if the model parameters are changed, as long as these remain within geologically reasonable limits. The explanation for the good agreement lies in the strong affinity of both Re and Os for sulfides, implying that equilibration with the other phases present in the rock has little effect. If the D sulf/sil value for Re is decreased to an irrealistically low value (20 in the example shown in Fig. 10c and10d) not only do both curves plot far from the data, but they diverge substantially from each other. The equilibrium model is provided in Appendix C (SOM). Values are adjustable on the "Summary" sheet and results are directly compared with the original, local equilibrium model. sulf/sil =20, the two curves diverge sharply, and neither fits the data. D) Divergence of the curves reflects a contrast in the Re profiles with height between the two models when the D Re sulf/sil value is low.

Reconciling systematic whole rock Os isotope relationships with internal Os isotope heterogeneity

The preceding discussion suggests that the frequently observed whole rock correlations of Os isotope compositions with fertility indices in mantle peridotites can by explained by early 2002) it has been recognized that sulfides from individual peridotite samples often display a wide range of 187 Os/ 188 Os and Re/Os compositions. Subsequent studies (e.g. [START_REF] Alard | In situ Os isotopes in abyssal peridotites bridge the isotopic gap between MORBs and their source mantle[END_REF][START_REF] Gonzalez-Jimenez | The architecture of the European-Mediterranean lithosphere: A synthesis of the Re-Os evidence[END_REF][START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF][START_REF] Harvey | Unravelling the effects of melt depletion and secondary infiltration on mantle Re-Os isotopes beneath the French Massif Central[END_REF] have shown that such internal heterogeneity is the rule rather than the exception in peridotites from both oceanic and continental settings. As pointed out by [START_REF] Alard | New insights into the Re-Os systematics of subcontinental lithospheric mantle from in-situ analyses of sulfides[END_REF] and later workers, two main types of sulfides can be defined, Luguet and Pearson (2019) provide a recent review of studies relating the Re-Os systematics of sulfides to their petrographic contexts. Enclosed sulfides are generally taken to be residual phases left by ancient partial melting events. In contrast, interstitial sulfides are often assumed to have been added during later metasomatic events. This perspective has led some authors to suggest that whole rock Os compositions may in fact represent the superposition of several generations of sulfides and thus may have no temporal significance [START_REF] Alard | New insights into the Re-Os systematics of subcontinental lithospheric mantle from in-situ analyses of sulfides[END_REF][START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF][START_REF] Luguet | Dating mantle peridotites using Re-Os isotopes: The complex message from whole rocks, base metal sulfides, and platinum group minerals[END_REF].

In some cases, direct addition of sulfides bearing radiogenic Os is indeed likely to be a plausible explanation of the observed internal heterogeneity. Notably, the ~35% of whole rock samples that plot above 187 Os/ 188 Os vs. Al 2 O 3 trends defined in each suite may have experienced addition of radiogenic Os. Many of these samples are from localities situated close to destructive plate margins, where it has been shown that radiogenic Os from the subducting slab can be mobilized and incorporated in peridotites of the mantle wedge [START_REF] Brandon | Osmium recycling in subduction zones[END_REF][START_REF] Widom | The nature of metasomatism in the sub-arc mantle wedge: evidence from Re-Os isotopes in Kamchatka peridotite xenoliths[END_REF]. Such mobility is presumably due to the existence of oxidizing and/or chlorine rich conditions in these environments, which both favor Os mobility [START_REF] Wood | Thermodynamic calculations of the volatility of the platinum group elements (PGE): The PGE content of fluids at magmatic temperatures[END_REF][START_REF] Xiong | Experimental quantification of hydrothermal solubility of platinum-group elements with special reference to porphyry copper environments[END_REF]. Even in contexts not obviously related to subduction, such as the Montferrier locality of southern France [START_REF] Alard | Volatile-rich Metasomatism in Montferrier Xenoliths (Southern France): Implications for the Abundances of Chalcophile and Highly Siderophile Elements in the Subcontinental Mantle[END_REF], volatile rich and/or carbonatite metasomatism may add highly radiogenic sulfides. It is perhaps not surprising that most samples plotting above the trends are non-cratonic xenoliths (Fig. 1), since as has been long

recognized (e.g. [START_REF] Pearson | Re-Os isotope systematics and platinum group element fractionation during mantle melt extraction: a study of massif and xenolith peridotite suites[END_REF], the Os contents of peridotites from these tectonic settings are significantly lower than those of tectonic peridotites (2.3 ±1.6 ppb and 4.1 ±1.6 ppb respectively, based on compilation in [START_REF] Luguet | Highly Siderophile Element and 187Os Signatures in Non-cratonic Basalt-hosted Peridotite Xenoliths: Unravelling the Origin and Evolution of the Post-Archean Lithospheric Mantle[END_REF]. The low Os contents of the non-cratonic xenoliths, which result from both post-eruptive supergene weathering [START_REF] Handler | Behaviour of platinum-group elements in the subcontinental mantle of eastern Australia during variable metasomatism and melt depletion[END_REF] and pre-eruptive circulation of sulfur-undersaturated melts [START_REF] Ackerman | Highly siderophile element geochemistry of peridotites and pyroxenites from Horní Bory, Bohemian Massif: Implications for HSE behaviour in subduction-related upper mantle[END_REF]see Lorand and Luguet (2016) for discussion of this issue), may render the Os isotope systematics of these rocks particularly sensitive to modification.

These considerations suggest that the addition of radiogenic sulfides during a variety of metasomatic processes can explain both the internal Os isotope heterogeneity and the unexpectedly radiogenic bulk rock compositions of samples plotting above the trends in Fig. 1 and 2. It is therefore interesting to observe that samples plotting on the trends also show extensive internal Os isotope heterogeneity. This is apparent from the detailed study of [START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF], who performed Re/Os analyses on individual sulfides, silicate phases, spinels and their corresponding whole rocks from Kilbourne Hole in New Mexico (USA). Most of the whole rocks define a broad but undeniable correlation between 187 Os/ 188 Os and Al 2 O 3 (their Figure 2). The sulfides contain nearly the entire Os budget of the rocks, with the interstitial sulfides mostly having more radiogenic and more variable Os isotope compositions as well as much lower Os concentrations than the enclosed sulfides. The authors suggest that the radiogenic interstitial sulfides were added in multiple metasomatic events, and that their contribution therefore only obscures the meaning of the whole rock Os signature.

It is understandable that metasomatic addition of sulfides could destroy preexisting correlations between Os isotope ratios and Al 2 O 3 passing through the PUM composition. However it is difficult to imagine that this random, multifaceted process could systematically produce such trends in peridotite suites from throughout the world. Instead, where such correlations exist, the more radiogenic compositions of interstitial sulfides with high Re/Os, relative to enclosed sulfides, may simply reflect radiogenic ingrowth since the time of ancient refertilization, coupled with incomplete isotopic reequilibration with the surrounding rock. The melt percolation process described above would inevitably produce interstitial sufides almost completely bereft of Os, and thus with very high Re/Os ratios, while any enclosed residual sulfides would have high Os contents and low Re/Os ratios. Osmium isotope evolution with time in an individual sulfide would then depend not only on its Re/Os ratio, but also on the extent to which it has experienced inward and outward Os diffusion relative to the surrounding rock. This in turn depends on the temperature of the lithosphere relative to closure temperatures for Re and Os in the various silicate and sulfide phases. If the lithosphere remained well below the closure temperature since the time of the original melt percolation event, internal isochrons would be expected, with the interstitial sulfides reaching extremely radiogenic compositions.

For example, assuming that the mantle lithosphere beneath Kilbourne Hole had an age of 1.86

Ga, the T RD model age (assuming a mantle evolution curve passing throught the present day PUM value of 0.1296) of the least radiogenic whole rock sample from [START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF], the interstitial sulfides should currently have 187 Os/ 188 Os ratios ranging from 0.28 to 2168 (Fig. 11).

However this range contrasts strikingly with the range in fact observed (0.126 to 0.373).

Similarly, though not divided into enclosed and interstitial populations, sulfides from Massif Central (France) xenoliths yielded 187 Os/ 188 Os ratios ranging from 0.115 to 0.171 [START_REF] Harvey | Unravelling the effects of melt depletion and secondary infiltration on mantle Re-Os isotopes beneath the French Massif Central[END_REF], values much lower and less varied than those expected (0.12 to 25.9) from their Re/Os ratios for 1.7 Ga of radiogenic ingrowth (based on the oldest whole rock T RD age) in the underlying lithosphere. Thus it is clear that lithospheric temperatures were high enough to allow very substantial Os diffusion and isotopic reequilibration to occur. There is currently little direct information about Os diffusion rates in silicate rocks, but based on theoretical considerations [START_REF] Kogiso | Length scales of mantle heterogeneities and their relationship to ocean island basalt geochemistry[END_REF] suggest Os diffusion coefficients in olivine of log D = -16.5 to -16.4, similar to those of Sr or Pb, while the experimental study of [START_REF] Hayden | A diffusion mechanism for core-mantle interaction[END_REF] suggests that grain boundary diffusivity of Os may be several orders of magnitude higher. This supports the possibility that the extreme Os isotope compositions expected to develop in intergranular sufides with high Re/Os ratios (Fig. 11) might be mostly erased by diffusion. On the other hand, if lithospheric temperatures were well above the Re/Os closure temperature complete internal isotopic equilibrium at the hand sample scale would be expected at the time of eruption. As discussed above, this is also clearly not the case. This suggests that Re-Os diffusion rates at lithospheric temperatures were fast enough to erase most, but not all, of the expected heterogeneity between intergranular sulfides, and that enclosed sulfides were largely protected from equilibration by their Os poor silicate hosts. These conclusions are very similar to those reached by [START_REF] Burton | Osmium isotope disequilibrium between mantle minerals in a spinel-lherzolite[END_REF] from their detailed analysis of spinel, silicate phases, and interstitial and included sulfides from a spinel lherzolite from Kilbourne hole. Though the nearly complete Os isotopic homogenisation they observed between the silicates and the intergranular sulfides was not confirmed by the more recent work of [START_REF] Harvey | Osmium mass balance in peridotite and the effects of mantlederived sulphides on basalt petrogenesis[END_REF], the inference from this early study that Os diffusion is rapid enough to remove much, but not all, of the internal heterogeneity within lithospheric peridotites remains solid.

Thus Os isotopic heterogeneity among sulfides from an individual sample is not necessarily evidence for addition of radiogenic sulfides during metasomatic events unrelated to the original melt extraction episode. Instead, the somewhat chaotic but generally more radiogenic compositions of sulfides in interstitial positions are exactly what might be expected from hundreds of millions of years of radiogenic ingrowth in Re-rich sulfides coupled with extensive, but incomplete, isotopic reequilibration. There is therefore no contradiction between production of whole rock correlations between 187 Os/ 188 Os and fertility indices by radiogenic ingrowth after magma percolation immediately subsequent to melt extraction and the existence of internal Os isotope heterogeneity. This also suggests that Os model ages obtained from individual intergranular sulfide grains in non-cratonic lithospheric peridotites may have no temporal significance.

Summary and broader implications

Osmium isotopes provide a unique perspective on the transformation of harzburgite to lherzolite through melt percolation, and in particular place strong constraints on the timing of this process. Because of its extreme compatibility in the sulfide phase, essentially all of the Os carried by percolating sulfide-saturated melts will be trapped at the very bottom of the peridotite column. Therefore lherzolites formed by percolation of young, radiogenic melts through ancient harzburgites would be expected to retain the unradiogenic Os compositions of the harzburgites. The near absence of fertile lherzolites with unradiogenic Os isotope compositions thus indicates that transformation of ancient harzburgites into lherzolites by refertilization involving much younger magmas is a process that almost never occurs. Instead, near linear relationships, trending towards the PUM composition, between whole rock Os isotope compositions and indices of fertility are observed in most peridotite suites from the non-cratonic mantle. These correlations can be explained by ancient addition of Re-rich sulfides together with pyroxene and spinel during refertilization soon after, and likely related to, the profound melt extraction episodes that depleted the harzburgites. Radiogenic ingrowth of 187 Os over time would then produce the observed correlations. Ancient refertilization would result in a lithosphere composed largely of lherzolite, which would be denser than a lithosphere dominated by harzburgite [START_REF] Poudjom Djomani | The density structure of subcontinental lithosphere through time[END_REF]. The recognition from Os isotopes that the lherzolitic nature of the lithosphere must have been acquired early thus adds a complicating constraint that must be considered when trying to explain lithospheric stability over time.

In the modeled melt percolation process, crystallization of pyroxene and spinel triggers sulfide saturation in each step, producing correlations between Re/Os ratios and fertility indices that over time create the observed trends. The curvature of the trends produced by this model is strongly dependent on the value of D Re sulf/sil . To produce the observed linear relationships, the required D Re sulf/sil values are in the range of about 300 to 500, consistent with the values for sulf/sil predicted for high FeO melts under the combined fO 2 and fS 2 conditions expected for D Re non-arc mantle melting [START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF]. On the other hand, most parameters other than D Re sulf/sil have quite limited effect on the form of the 187 Os/ 188 Os vs. Al 2 O 3 relationship. This lack of sensitivity to variations in most parameters, including the extent of equilibration with the preexisting harzburgite, may explain why these linear trends are so widespread. The development of such trends is not inconsistent with the extensive internal isotopic heterogeneity very commonly observed between sulfides from the same rock [START_REF] Luguet | Dating mantle peridotites using Re-Os isotopes: The complex message from whole rocks, base metal sulfides, and platinum group minerals[END_REF]. In fact, such heterogeneity is the expected result of hundreds of millions of years of radiogenic ingrowth in sulfides with different Re/Os ratios coupled with extensive, but incomplete, diffusive reequilibration between phases in different petrographic contexts. The 187 Os/ 188 Os ratios of individual sulfides thus are unlikely to have time significance and probably cannot be associated with specific tectonic events.

Finally, it is interesting to speculate on the geodynamic context likely to produce the observed relationships between 187 Os/ 188 Os and fertility indices. Most authors think that the continental crust was formed in arc environments (see [START_REF] Arndt | Formation and Evolution of the Continental Crust[END_REF] for a review), but peridotite xenoliths from these settings often have highly radiogenic and unsystematic Os isotope compositions [START_REF] Brandon | Osmium recycling in subduction zones[END_REF][START_REF] Widom | The nature of metasomatism in the sub-arc mantle wedge: evidence from Re-Os isotopes in Kamchatka peridotite xenoliths[END_REF]. Furthermore, percolation modeling using the low sulf/sil values appropriate for oxidizing suprasubduction conditions produces highly curved D Re trends plotting far from the data. Hence it seems likely that the subcontinental non-cratonic lithosphere was not formed in arc settings. Instead, profound harzburgite-forming partial melting episodes followed shortly afterwards by percolation of the remaining melts through the cooling lithosphere are likely to occur above large mantle plumes. In this context, it is interesting that the Os isotope data suggest that nearly all of the non-cratonic lithosphere is of Paleo to mid Proterozoic age. [START_REF] Arndt | Episodic Earth evolution. Tectonophysics, Moho: 100 years after[END_REF] suggest that the distinctive peaks in the U-Pb zircon age record from that time period can be explained by the periodic generation of massive mantle plumes. Such plumes would not only produce intense melting events in the upper mantle, but would also accelerate plate tectonics leading to pulses of increased granite production. Thus it is possible that Proterozoic mantle lithosphere formed above mantle plumes and was then tectonically juxtaposed with nearby continental crust of similar age to produce the non-cratonic lithosphere observed today. In any case, the nearly ubiquitous linear relationships between Os isotopes and fertility indices place strong constraints on models of formation of the non-cratonic lithosphere underlying most of the Earth's continental area.

Figure 2 .

 2 Figure 2. Two examples of peridotite suites showing correlations between 187 Os/ 188 Os and Al 2 O 3 . The large blue square indicates the Primitive Upper Mantle (PUM) composition of[START_REF] Meisel | Osmium isotopic compositions of mantle xenoliths: a global perspective[END_REF]. In both cases, the trend line is drawn between the least radiogenic sample and the PUM composition, and the gray field represents 187 Os/ 188 Os values within ±1.5% of this line. The samples from Vitim, Siberia studied by[START_REF] Pearson | Re-Os isotope systematics and platinum group element fractionation during mantle melt extraction: a study of massif and xenolith peridotite suites[END_REF] define an exceptionally good correlation. The samples from Lianshan, China[START_REF] Reisberg | Re-Os and S systematics of spinel peridotite xenoliths from east central China: Evidence for contrasting effects of melt percolation[END_REF] represent a more typical case, which includes a significant number of samples that plot well above the trend, although most points still plot close to the line. As the Lianshan case is more common, it is used as an example for the modeling, but the same conclusions can be drawn using any of the numerous observed correlations.

  poor magmas. As a result, Re/Os ratios should decrease during melting concomitantly with fertile melt components, such as Al 2 O 3 . This leads to correlations of Re/Os with Al 2 O 3 , which with time produce correlations between 187 Os/ 188 Os and Al 2 O 3 .The melting process is explored numerically in Fig.4, which shows variations in S, Re and Os concentrations of the residue during non-modal batch melting of a spinel peridotite (parameters

Figure 4 .

 4 Figure 4. Effects of ancient partial melting of fertile peridotite followed by radiogenic ingrowth producing corelation between 187 Os/ 188 Os and Al 2 O 3 . A) Decrease in sulfide content with increasing fraction of melt extraction (note that melt fraction increases towards the left in this diagram). b) Decrease in [Re] of residue with increasing melt extraction, for D sulf /D sil (Re) values of 380 and 820, lower and upper boundaries of range proposed by Brenan (2008). C) Increase in [Os] of peridotite with increasing melt extraction, assuming D sulf /D sil (Os) = 10 6 . D) Decrease in Re/Os with progressive melt extraction for case of D sulf /D sil =380 for Re. E) 187 Os/ 188 Os vs. Al 2 O 3 correlation after 1.7 Ga of radiogenic ingrowth compared with Lianshan samples, assuming initial 187 Os/ 188 Os of 0.1171, the PUM value at 1.7 Ga. A non-modal batch partial melting model is used and Al 2 O 3 is taken to decrease linearly with extent of melting from fertile peridotite with 4.2% Al 2 O 3 to 0% at 25% melting. Other parameters are given in Appendix A (SOM).

Figure 5 .

 5 Figure 5. Modeled variation in whole rock [Os], [Re], Re/Os, Al 2 O 3 and [Lu] produced by melts percolating upwards in a harzburgite column. Assumed parameters are given in Appendix A (SOM).

Fig. 6 Height

 6 Fig.6shows the increase in[Os] in the peridotites that would result if the observed 187 Os/ 188 Os vs. Al 2 O 3 trends were formed by addition of radiogenic sulfides of three different assumed Os isotope compositions. To trace these curves, the value of 187 Os/ 188 Os corresponding to each radiogenic Os that would have to be added to reach each 187 Os/ 188 Os value was calculated. The steepness and curvature of the required increase of [Os] with Al 2 O 3 depends on the 187 Os/ 188 Os composition of the added sulfides, as less osmium is required if its composition is more radiogenic. This figure implies that lherzolites should have systematically higher Os contents than harzburgites, which is seldom, if ever, the case in peridotite suites (see examples of Os concentration variations with Al 2 O 3 in Fig.4of[START_REF] Luguet | Highly Siderophile Element and 187Os Signatures in Non-cratonic Basalt-hosted Peridotite Xenoliths: Unravelling the Origin and Evolution of the Post-Archean Lithospheric Mantle[END_REF]. The near absence of examples of suites in which [Os] increases with Al 2 O 3 , in addition to the absence of horizontal relationships between 187 Os/ 188 Os and Al 2 O 3 , leads to the conclusion that large scale

  likely to produce covariations between Re/Os and Al 2 O 3 , which would over time produce correlations between187 Os/ 188 Os and Al 2 O 3 by radiogenic ingrowth. Fig.5bshows how Re contents vary with height after melt percolation in the column. A D sulf/sil value of 380, which yielded the best results in the simple partial melting model is assumed in the case shown. Re concentrations are highest in the lowest step and decrease gradually with height towards the original harzburgite value. Thus the Re profile with depth contrasts markedly with that of Os, as a result of the difference in degree of chalcophility of the two elements. Fig.5also shows the Re/Os ratio as a function of height in the column, as well as the variation in Al 2 O 3 content that results from the precipitation of pyroxene and spinel. It is apparent from this figure that with the exception of the lowest, most fertile Os-rich step, Re/Os and Al 2 O 3 covary in a coherent manner. This leads to a nearly linear correlation between these two parameters, which with time will develop into a correlation between 187 Os/ 188 Os and Al 2 O 3 (Fig.7a and 7b). In Fig.7b, an age of 1.7 Ga was used to calculate the 187 Os/ 188 Os ratios, based on the Re depletion model age (T RD ;[START_REF] Walker | Nd, and Pb Isotope Systematics of Southern African Peridotite Xenoliths -Implications for the Chemical Evolution of Subcontinental Mantle[END_REF] of the most depleted harzburgite. T RD ages are obtained by comparing the measured 187 Os/ 188 Os values directly with the Os mantle evolution curve, and so intrinsically assume a Re content of zero for each sample before its emplacement at the surface. Therefore, T RD ages are always minimum values. Nevertheless, T RD ages should closely approximate melt extraction ages in the case of highly depleted harzburgites, as these rocks are expected to have lost nearly all of their Re during partial melting.

  little effect on the slope of the 187 Os/ 188 Os vs. Al 2 O 3 correlation. These parameters include initial melt fraction (porosity) of the bottom cell, fraction of silicate/spinel crystallization in each increment, and the number of times new increments of melt are added to the bottom of the column. As shown for the example of initial melt fraction (Fig.8d-f), these parameters have a very strong effect on the range of the calculated trend, but have only a minor effect on its slope.

Figure 8 .

 8 Figure 8. Parameter tests of the percolation model, considering ancient Re addition followed by radiogenic ingrowth of 187 Os. A-C) Example of a parameter (SCSS, Sulfur Concentration at Sulfide Saturation) that has only limited effect on either the slope or the extent of the calculated trend. SCSS is allowed to vary between 800 and 1200 ppm, which corresponds to depths of roughly 120 to 0 km[START_REF] Li | Empirical equations to predict the sulfur content of mafic magmas at sulfide saturation and applications to magmatic sulfide deposits[END_REF]. Though an optimal fit to the Lianshan data is found for SCSS ~1000, variation within the tested range has only a minor effect on the results. D-F) Example of a parameter (initial melt fraction) that has very little effect on the slope or curvature of the calculated refertilization trend, but a large effect on its extent. Sensitivity tests for other model parameters are presented in Appendix A.

  form of the curve relating 187 Os/ 188 Os to Al 2 O 3 . At very low D Re sulf/sil , such as the value of 40 suggested by Roy Barman et al. (1998) on the basis of Re contents in sulfide globules and their

Figure 9 .

 9 Figure 9. Dependence of modeled refertilization trend (after radiogenic ingrowth) on D Re sulf/sil . A) D Re sulf/sil =40, value suggested by analyses of sulfide globules and their host basaltic glasses (Roy-Barman et al., 1998). B) D Resulf/sil =380, lower limit of range suggested by[START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF], corresponding to high FeO primary melts. This provides the best fit to the data and is the default value used in the text. C) D Re sulf/sil =820, upper limit of range suggested by[START_REF] Brenan | Re-Os fractionation by sulfide melt-silicate melt partitioning: A new spin[END_REF], corresponding to low FeO basalts. d) D Re sulf/sil =40000, suggested by experimental work of[START_REF] Sattari | Experimental Constraints on the Sulfide-and Chromite-Silicate Melt Partitioning Behavior of Rhenium and Platinum-Group Elements[END_REF].

  dataset, 187 Os/ 188 Os correlates less well with Lu than with Al 2 O 3 , reflecting the considerable scatter in the relationship between Lu and Al 2 O 3 . Nevertheless, the approximately half of the data that are well aligned in Fig. 7d are fairly well fit by the model using the chosen parameters. The model partition coefficients for Lu in silicate phases and spinel were taken from McKenzie and O'nions (1991) as these are internally consistent, but the exact choice of K d (Lu) values has only a limited effect on the model results. On the other hand, the model results are highly sensitive to the relative proportions of the main Al 2 O 3 bearing phases, spinel

Figure 10 .

 10 Figure 10. Comparison of local equilibrium (original) model with full equilibrium model. A) If D Resulf/sil =380, the two models yield nearly indistinguishable results that fit the data well. B) Agreement between the two models reflects their nearly identical Re profiles with height in the column. C) If the assumed D Re sulf/sil =20, the two curves diverge sharply, and neither fits the data. D) Divergence of the curves reflects a contrast in the Re profiles with height between the two models when the D Re sulf/sil value is low.

  radiogenic ingrowth. Modeling results suggest that the development of these trends is fairly insensitive to the extent of equilibration with the host peridotite or to the values of most parameters, with the exception of the D sulf/sil value for Re. Nevertheless this relatively simple story must be reconciled with the much more chaotic Re-Os systematics observed on a mineral scale. Since the pioneering work ofAlard et al. (

  enclosed and interstitial, based on their petrographic setting. Enclosed sulfides, typically found as rounded inclusions in olivine, are composed of the exsolution products of monosulfide solid solution and mostly have high Os contents and low 187 Os/ 188 Os and Re/Os ratios. Interstitial sulfides are typically angular and often Cu and Ni-rich and are found in intergranular settings, often associated with clinopyroxene and spinel. They usually have low to very low Os concentrations, associated with high Re/Os ratios and radiogenic Os isotope compositions.

Figure 11 .

 11 Figure 11. A) Comparison of measured 187 Os/ 188 Os ratios (filled symbols) in enclosed and interstitial sulfides from peridotite xenoliths from Kilbourne Hole with values expected after 1.86 Ga of radiogenic ingrowth (open symbols.) Data from Harvey et al. (2011). B) Expanded view of (A).

  Ga for the Lianshan samples, based on the 187 Os/188 Os and corresponding T RD of the least radiogenic harzburgite) this results in a correlation between 187 Os/ 188 Os and Al 2 O 3 that passes through most of the Lianshan samples, as well as the PUM value. C) Expected correlation between 187 Re/ 188 Os and[Lu], assuming that the added Al 2 O 3 bearing phases are equally divided between spinel and garnet. D) Resulting correlation between 187 Os/ 188 Os and Lu compared with Lianshan data. Purple curve with diamonds represents precipitation of equal amounts of spinel and garnet. Red dashed curved shows case where the added alumina-bearing phase is composed only of spinel; gray dashed curve shows case where this phase is composed only of garnet. Model parameters are given in Appendix A.The sensitivity of the model to the choice of parameters is explored extensively in Appendix A of the SOM. Broadly speaking three types of behavior are observed, if parameter choices are kept within limits appropriate for the upper mantle excluding highly oxidized arc environments. The first consists of parameters that have a nearly negligible (D Os sulf/sil , K d Os cpx , K d Os sp , K d Re sp , [Os] liq , [Os] perid ) or minor (K d Re cpx , K d Re gt , [Re] liq , [Os] perid , Al 2 O 3liq , SCSS) effect on the slope, curvature or extent of the calculated curves. An example of this group (SCSS) is shown in Fig.

						Figure 6. Increase of bulk rock [Os] with Al 2 O 3 content
						required to produce linear correlations between
						187 Os/ 188 Os and Al 2 O 3 , if it is assumed that such
						correlations can be produced by simple addition of
						radiogenic sulfides together with spinel and pyroxene to
						depleted, unradiogenic harzburgite. Spreadsheet
						showing calculation is included in Appendix B (sheet
						"Rad sulfide addition"). Numbers on curves indicate
						assumed 187 Os/ 188 Os ratios of added sulfides. In the
						linear correlation on which these curves are based, the
						harzburgic endmember is assumed to have 187 Os/ 188 Os =
						0.117, [Os] = 3.5 ppb, and Al 2 O 3 = 0.7 wt.%, and the
						PUM is assumed to have 187 Os/ 188 Os = 0.1296 and Al 2 O 3
	0	1	2	3	4	= 4.2 wt.%. Positive co-variations between [Os] and
						Al 2 O 3 are rarely (if ever) observed in peridotite suites.

Figure 7. A) Expected correlation between 187 Re/ 188 Os and Al 2 O 3 , resulting from their coherent evolution with depth in the percolation column shown in Fig. 5. B) With time (1.7

Acknowledgements

I thank the editor, Catherine Chauvel, for inviting this review paper. Lyderic France, Juliette Pin, and all the members of the Magma discussion group at CRPG are thanked for their insightful questions and comments. The constructive suggestions of the reviewers, James Brenan and Igor Puchtel, are greatly appreciated.


Synthesis of N,N-diethylbenzylamine (CAS: 772-54-3) from N-benzyl-N,N,N-triethylammonium chloride via

Hofmann Elimination

Synthesis of N,N-diethylbenzylamine (CAS: 772-54-3) from N-benzyl-N,N,N-triethylammonium chloride via Hofmann Elimination

KOH can be readily dissolved in isopropanol to form a 1.07 M solution, addition of BnNEt 3 Cl (1.67 M in isopropanol) into this solution led to immediate participation of KCl and formation of BnNEt 3 OH solution. Warming this mixture at 45 ºC under reduced pressure led to Hofmann elimination and formation of N,Ndiethylbenzylamine. This constitute a convenient and effective way for the preparation of this fundamental reagent.

Experimental Procedure

A 250 mL flask equipped with a magnetic stir bar was charged with KOH (3.00 g, 0.053 mmol). i-PrOH (50 mL) was added and the mixture was stirred until all the solid dissolves to form a clear colorless solution. A solution of 1 (11.4 g in 30 mL i-PrOH) was added and a white participate was immediately formed. The mixture was stirred for 5 min and then filtered and the solid washed with i-PrOH (10 mL×2 ) and then combined to afford a clear solution. Most of the solvent was removed by a diaphragm vacuum pump at 35 ºC water bath. The water bath was then adjusted to 45 ºC, ethylene gas gradually generated. The mixture was kept at this temperature for 3 h, the reaction was completed. PE (50 mL) and H 2 O (50 mL) were added into the flask, the mixture was stirred and then the organic phase was separated. The aqueous phase was further extracted with PE (20 mL). The organic phase was combined, dried by Na 2 SO 4 and filtered and evaporated under reduced pressure to afford product 3 (6.93 g, 85% yield) as a colorless oil. The NMR spectral data of 3 was identical with the literature data 1 .

Acknowledgement

The author would like to thank Prof. Biao Yu at SIOC for his helpful guidance.

Financial support from the Youth Innovation Promotion Association of CAS (2021251) is acknowledged.

Author Information

11.4 g, 0.050 mol 6.93 g, 85% yield