MICHEL RABINOWICZ* AND MICHAEL J. TOPLIS

Transcription

1 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 PAGES171^116 9 doi:1.193/petrology/egp33 Melt Segregation in the Lower Part of the Partially Molten Mantle Zone beneath an Oceanic Spreading Centre: Numerical Modelling of the Combined Effects of Shear Segregation and Compaction MICHEL RABINOWICZ* AND MICHAEL J. TOPLIS LABORATOIRE DYNAMIQUE TERRESTRE ET PLANE TAIRE, CNRS^UMR 556, OBSERVATOIRE MIDI-PYRE NE ES, 14, AVENUE EDOUARD BELIN, 314 TOULOUSE, FRANCE RECEIVED JUNE, 7; ACCEPTED MAY 8, 9 The physical processes capable of concentrating melt within the lower part of the partially molten mantle horizon beneath an oceanic spreading centre have been considered using one-dimensional numerical models that take into account the effects of shear segregation during ductile deformation, compaction and liquid^solid surface tension. These models show that when the viscosity of the solid fraction of the mantle is constant, shear segregation is too slow to be a geologically relevant process. However, if, as suggested experimentally, the ductility of the solid fraction of the mantle increases in the presence of interstitial melt, shear segregation is found to develop on time-scales compatible with mantle upwelling.the critical melt fraction at which shear segregation occurs is predicted to be positively correlated with the melt-free viscosity of the mantle, a value of 5% being typical for simulations run with a melt-free viscosity of 1 18 Pa s. No characteristic length scale is found for this segregation, suggesting that melt will tend to concentrate at the length scale at which it is produced (i.e. that of mantle olivine grains). At face value this would imply that shear is not an efficient mechanism to concentrate liquid. However, segregation will occur parallel to the direction of compression, such that towards the bottom of the zone of partial melting beneath an oceanic spreading centre, horizontal veins are expected. Because of the local extraction of the interstitial melt, these horizontal veins will drastically reduce vertical permeability, a fact that will trigger compaction in that direction. In this case, rapid formation of melt-rich horizons approximately one compaction length in height is predicted. High permeability and high melt-free viscosity both act to efficiently drain liquid from throughout the lower region of the melting column, leading to the formation of thick widely spaced compaction waves. On the other hand, low values of these parameters lead to formation of thin closely spaced horizons derived from locally produced liquids. Liquid contained in these melt-rich horizons will be transported through the uppermost mantle, either in horizontal structures if the viscosity of the mantle is high (41 19 Pa s), or rapidly drained towards the surface if the direction of compression rotates vertically, a situation expected if the viscosity of the melt-free mantle is 1 18 Pa s. This complex interplay of shear segregation, compaction and regional stress fields may potentially help to explain certain geochemical and petrological features of rocks from mid-ocean ridges and ophiolite complexes, such as the apparent lack of interaction of mid-ocean ridge basaltlike liquids with the uppermost mantle and the planar geometry of dissolution channels observed in ophiolite complexes. KEY WORDS: melt; segregation; migration; shear; compaction; mantle; ophiolite; lherzolite; mid-ocean ridge; modelling INTRODUCTION Based upon a wealth of geochemical, geophysical and structural data from active and fossilized mid-ocean ridges, it is generally accepted that the formation of *Corresponding author. Michel.Rabinowicz@cnes.fr ß The Author 9. Published by Oxford University Press. All rights reserved. For Permissions, please journals.permissions@ oxfordjournals.org Downloaded from on May 18

2 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 JUNE 9 basaltic crust in this context results from the migration and accumulation of partial melts produced by adiabatic decompression of upwelling mantle at depths ranging from about 8 km to the base of the lithosphere [e.g. review by Langmuir et al. (199) and references therein]. Transport of melt in the mantle is possible because basaltic liquid extensively wets grains of olivine, forming an interconnected intergranular network even at small melt fractions (e.g. VonBargen & Waff, 1986; Wark & Watson, 1998; Laporte & Provost, ). In light of this fact, it has been proposed that partial melts produced during mantle upwelling are expelled towards the surface by a combination of buoyancy and compaction, resulting in an interstitial melt fraction that increases continuously from the bottom to the top of the partial molten zone (e.g. McKenzie, 1984). In the simplest scenario a steady state will be reached, for which the local concentration of liquid, the Darcy velocity and interstitial velocity of the melt can be calculated as a function of depth (Spiegelman & Kenyon, 199). If the permeability of the mantle is sufficiently high, the local concentration of liquid may remain low throughout the melting column, consistent with observations in abyssal and ophiolitic peridotites, which reveal that the proportion of interstitial melt in the mantle just below the Moho rarely exceeds % (e.g. Dick, 1989; Singh et al., 1998; Seyler et al., 1). Furthermore, for high-permeability systems, interstitial melt velocities of the order of metres per year are calculated, of comparable magnitude to velocities inferred from radioactive disequilibria in mid-ocean ridge basalt (MORB) and the timing of volcanic activity below the ice sheet in Iceland (Jull & MacKenzie, 1996; Kelemen et al., 1997). However, one weakness of the diffuse porous flow scenario is that predicted melt velocities are not fast enough to account for the apparent lack of equilibrium of MORB with a large part of the uppermost mantle (e.g. O Hara, 1965; Stolper, 198; Quick, 1981), nor for the significant role of fractional melting implied by the geochemistry of MORB and abyssal peridotites (e.g. Johnson et al., 199; Kelemen et al., 1997). Nevertheless, these geochemical characteristics can be reconciled with a steady-state scenario if liquids are chemically isolated from the surrounding mantle; for example, by transport through olivine-rich channels such as those exposed in ophiolite environments (e.g. Kelemen et al., 1995a, 1995b). Based upon this work a paradigm has emerged that mid-ocean ridges are more or less in a steady state, with liquid being transported through the uppermost mantle in high-porosity channels produced by dissolution-infiltration (Aharanov et al., 1995, 1997; Spiegelman et al., 1). However, the surface expression of mid-ocean ridge magmatism is clearly not continuous in time, suggesting that at some point during MORB petrogenesis the steadystate scenario is not valid. Furthermore, the variation of seismic shear-wave velocity in the oceanic mantle as a function of depth (e.g. Webb & Forsyth, 1998; Gu et al., 5) is not readily consistent with the idea that melt fraction increases towards the surface. For example, shearwave velocities generally decrease from the Moho to a depth typical of the dry solidus (Gu et al., 5), suggesting, if anything, that melt fraction may increase rather than decrease with depth. Although part of the variation in shear-wave velocity may be related to grain size, temperature and textural considerations (e.g. Faul & Jackson, 5), in the immediate vicinity of active ridges the magnitude of the decrease in shear-wave velocity is generally accepted to require the presence of liquid at significant depth. The amount of liquid is not well constrained, but interpretation of seismic data from the Pacific Ocean implies liquid fractions from a minimum of 1^% to a maximum of 5%, the exact value depending on the geometry of intergranular melt distribution (Forsyth et al., 1998; Webb & Forsyth, 1998; Gu et al., 5). Consideration of all the available geochemical and geophysical data therefore suggests that not all is known concerning the details of where melts are generated, how and when they accumulate, and how they are delivered to the ridge axis. Complete understanding of the physical and chemical processes associated with MORB petrogenesis is severely hampered by the fact that many of the processes involved occur at depths where no sampling of rocks is possible, and/or on time-scales not accessible by current methods of direct study. This state of affairs has led to the development of numerical models of the physical evolution of partially molten systems. Since the pioneering work of McKenzie (1984) and Scott & Stevenson (1984) much progress has been made in this field, although modelling of the movement of liquid in the mantle is still complicated by the fact that melt migration is affected by a variety of nonlinearly coupled processes that have widely variable length scales. Of the processes capable of concentrating melt at depth and delivering it to the surface, shear segregation, compaction, reaction-infiltration, and hydrofracture have been the focus of attention. Based upon this body of work it has been proposed that: (1) shear segregation can concentrate melt in veins orthogonal to the direction of extension at short length scales, potentially producing conduits for melt transport to the surface (Sleep, 1988; Stevenson, 1989; Richardson, 1998; Katz et al., 6); () a drop in the proportion of interstitial melt (as a result of partial crystallization, for example) may obstruct upward migration of the liquid, generating a series of discrete melt-rich compaction waves (e.g. Scott & Stevenson, 1989; Spiegelman, 1993; Khodakovskii et al., 1995; Connolly & Podladchikov, 1998; Rabinowicz & Ceuleneer, 5); (3) melt contained in such horizons may be transported in dykes resulting from hydro-fracturing of the partially molten mantle (Nicolas, 1986; Rubin, 1998), 17 Downloaded from on May 18

3 RABINOWICZ & TOPLIS MELT SEGREGATION IN MANTLE although this interpretation has been contested by several groups (e.g. Sleep, 1988; Kelemen et al., 1997; Rabinowicz & Ceuleneer, 5) who have argued that the excess pressure as a result of the buoyancy of a sub-ridge mush does not reach the hydro-fracturing threshold; (4) infiltration resulting from the reaction of a MORB-like melt with the uppermost mantle induces finger-like channels (Kelemen et al., 1995a, 1995b; Spiegelman et al., 1) potentially capable of transporting melt from significant depths to the surface, even when compaction plays a role (Spiegelman et al., 1; Spiegelman & Kelemen, 3). Although the studies mentioned above provide constraints on the potential mechanisms of melt concentration and extraction in the mid-ocean ridge environment, the relative importance and roles of different processes in the melting column, and the possible interactions between them, have not been fully explored. This is particularly true in the lower part of the melting column, where the development of instabilities would be of particular importance for the subsequent evolution of the mid-ocean ridge system. It is with this in mind that we have undertaken a new look at this problem, concentrating on the time-scales and relative roles of shear segregation and compaction. We begin by presenting the equations of Bercovici & Ricard (3), used as the starting point in this work. Equations of state and parameter values relevant to melt migration in the mantle are then discussed, allowing simplification to a tractable one-dimensional set of equations that may be used to model vertical melt segregation. Characteristic time and length scales of liquid concentration are calculated as a function of variations in solid viscosity, surface tension at liquid^solid interfaces, and bulk melt concentration, including the effects of continuous addition of liquid as a result of decompression melting. In particular, the combined effects of shear segregation and compaction are explored, as the former process is found to be capable of drastically reducing vertical permeability, leading to the generation of compaction waves. The development of melt reservoirs below 3 km depth, their transport to the surface, and the petrological consequences of our results, are then discussed. THE EQUATIONS Conservation equations Modelling the physical behaviour of systems experiencing partial melting is complicated by the volume increase associated with the transformation of solid into melt. To be rigorous, the compressibility of the solid and liquid fractions should be taken into account when calculating the volume increase due to melting (e.g. Connolly & Podladchikov, 1998), although it is more general practice to assume that both the solid and liquid are incompressible (e.g. Spiegelman et al., 1; Bercovici & Ricard, 3; Hier-Majumber et al., 6; Srameck et al., 7). In the latter case, and assuming a ductile mantle, the increase of volume during melting simply leads to an uplift of the lithosphere^ductile mantle interface. Even in the more complicated case considered by Connolly & Podladchikov (1998), melting generates an excess elastic pressure, which is relaxed on time-scales that are extremely short in comparison with those characterizing melt segregation and compaction. In this case too, melting simply results in a displacement of the ridge^sea interface. Given that explicit consideration of the volume changes associated with melting generates serious mathematical problems, which are particularly difficult to handle in the case of the development of shear segregation instabilities (Rabinowicz & Vigneresse, 4), we have therefore chosen to make the assumption that the transformation of solid to liquid has no associated volume increase, such that the equations describing the conservation of volume and of mass may be simply expressed. As described above, this assumption will not affect the physical evolution of the systems modelled here (in the deep part of the mantle melting column), as the principal effect of the volume change is simply a more or less instantaneous displacement of the ridge^sea interface. The conservation equations in this case may be þ ~r:ðfv f fþ þ ~r:ðð1 fþv m Þ¼ where is the volumetric melting rate during decompression in the mantle by unit of time t, v f and v m are the melt and solid velocities, respectively. The velocities may be decomposed into mean velocity and separation fields C and S defined as C ¼ fv f þð1 fþv m S ¼ fðv f v m Þ: ðaþ ðbþ Use of C and S has the advantage of separating calculation of the motion of the mush from that of the melt. In addition, in the case of one-dimensional vertical models, such as those illustrated below, C is a constant and equal to zero when the vertical reference frame moves with the mean flow. Combining equations (a) and (b), it may be shown that ~r:c þ C ~rf þ ~r:ðð1 fþsþ ¼ : ð3bþ Given that the rate of melt transport by the mean flow field C satisfies df þ C ~rf ð3cþ 173 Downloaded from on May 18

4 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 JUNE 9 equation (3b) may be written df dt þ ~r:ðð1 fþsþ ¼ : ð3dþ The equation of melt percolation Many attempts have been made in the literature to model melt flow in a deformable matrix (e.g. McKenzie, 1984; Stevenson, 1989; Richardson, 1998; Hall & Parmentier, ). The most recent assessment of the relevant governing equations (Bercovici et al., 1a and 1b; Ricard et al., 1; Bercovici & Ricard, 3) is particularly complete and incorporates both surface energy terms and explicit calculation of the excess pressure between liquid and solid. These equations will be used as the starting point of the work presented here. As discussed by Rabinowicz & Vigneresse (4), based upon the work of Bercovici & Ricard (3) an equation may be derived which balances the stresses caused by deformation of the solid fraction of the mush with those generated by friction of the melt during its flow inside the porous network: m kðfþ S ~r:ð1 fþzð ~rs þð~rsþ t Þþ ~r 3 ð1 fþz ~r:s þð1 fþ ~rp 1 o ðp ~rf þ ~rðsaþþ ¼ ð1 fþdrg f ~r:ð1 fþð ~rc þð~rcþ t Þ: In this equation: (1) m and Z represent the melt and solid viscosities, respectively; () g, dr and k(f) are the gravity vector, the density contrast between solid and melt, and the mush permeability, respectively; (3) p is the pressure contrast and o is the surface energy partitioning coefficient between melt and solid (Bercovici & Ricard, 3); (4) sa represents the intergranular surface tension, where s is the mean interface energy and a is the interfacial area per unit volume. In the following, we designate p as excess pressure. For two-phase flows, the excess pressure p is not a free variable, but is determined by an equation of state that relates it to the rate of variation of melt concentration df/dt (see below). The first two terms on the lefthand side of equation (4) represent the volumetric stresses as a result of viscous friction during intergranular-melt flow and compaction of the mush. The last three terms on the left-hand side of equation (4) represent the excess pressure gradient ~rp, the work of the effective pressure field and surface tension. The equation of states of melt migration in a sub-ridge mantle mush Modelling the movement of liquid in the mantle ideally requires consideration of a variety of non-linearly coupled processes that have widely variable length scales, from that of mantle flow (i.e. 1 km; Ghods & Arkani- Hamed, ; Katz et al. 6), through that of the ð4þ compaction process (which typically develops at a kilometric scale, Spiegelman, 1993; Khodakovskii et al., 1995) and dissolution channelling (proposed to occur at scales of 1^1 m; Spiegelman et al., 1; Braun & Kelemen, ) to that of shear segregation, proposed to occur at length scales less than 1 m (Stevenson, 1989; Richardson, 1998; Hall & Parmentier, ; Katz et al., 6). In light of this situation it is currently impossible to incorporate the action and counter-action of these different processes in a single two- or three-dimensional model of the mantle. On the other hand, the results of large-scale models of mantle flow constrain the direction and magnitude of stresses associated with ductile flow, and this information may be used as bounding constraints for models at smaller length scales, such as those developed in this work. A brief summary of the pertinent results of such large-scale models is presented below. Orientation and values of stresses generated by plastic flow in a sub-ridge mantle mush For the purposes of the present study, the principal parameters which require quantification are the orientation of the compression tensor (~s 1 ), and the value of the stress in the mantle below the ridge axis (defined as ST). These two parameters are in turn intimately related to the flow pattern of the mantle. At one extreme, the analytical solution of wedge flow in the mid-ocean ridge context calculated assuming a constant viscosity of 1 1 Pa s throughout the whole upper mantle (e.g. Spiegelman & McKenzie, 1987), leads to the conclusion that the strain velocity field (g) and the amplitude of ST have values close to zero in a vertical plane beneath the ridge axis. However, the low ratio of gravity anomaly to topography measured over hotspots shows that the viscosity of the uppermost km of the mantle is approximately a factor 3 lower than 1 1 Pa s (Ceuleneer et al.,1988). Moreover, the presence of interstitial melt may amplify the viscosity decrease in regions of partial melting and, in light of these considerations, a significant number of numerical models have been developed to constrain flow patterns as a function of viscosity variations in the upper mantle (e.g. Rabinowicz et al., 1984; Buck & Su, 1989; Scott & Stevenson, 1989; Sparks & Parmentier, 1991; Ceuleneer & Rabinowicz, 199; Scott, 199; Turcotte & Phipps Morgan, 199; Barnouin-Jha et al., 1997). Such models implicitly or explicitly assume that the viscosity of the mantle at depths greater than km is 41 1 Pa s, but lower at shallower depth. All of these models show that above the depth of the viscosity transition, mantle flow converges towards the vertical plane beneath the ridge axis (i.e. in the vicinity of the plane beneath the ridge axis, the flow narrows as the mantle ascends, Fig. 1). This flow pattern is the consequence of the increasing velocity of the mantle as it ascends, leading 174 Downloaded from on May 18

5 RABINOWICZ & TOPLIS MELT SEGREGATION IN MANTLE to values of g and ~s 1 that are4, a conclusion consistent with the reduction in grain size (as a result of deformation) and the onset of anisotropy at a depth of km, inferred from seismic data (Gaherty et al.,1996; Faul & Jackson, 5). As long as the mantle is accelerating upwards, the direction of maximal compression ~s 1 directly beneath the ridge axis will be horizontal (Ceuleneer & Rabinowicz, 199; Rabinowicz et al.,1993; Katzet al., 6). Below an active ridge, ~s 1 is therefore expected to be horizontally oriented at depth, although the effect of mantle buoyancy may lead to flow geometries more complicated than those predicted on the basis of plate spreading alone. In the simplest case, when the viscosity (Z )in the upper section of the mantle has values Pa s, the vertical velocity of the flow driven by plate spreading is always greater than that related to buoyancy-driven convection. In this case, the mantle directly beneath the ridge axis is in horizontal compression from km to the vicinity of the Moho (Scott & Stevenson, 1989; Scott, 199). Conversely, if the viscosity of the uppermost mantle is low (i.e. Z 1 18 Pa s), then buoyancy-related forces may dominate, leading to streamlines of mantle flow that define a rolling-mill shape (e.g. Buck & Su, 1989; Scott & Stevenson, 1989; Ceuleneer & Rabinowicz, 199; Ceuleneer et al., 1993; Rabinowicz et al., 1993). In this case there is a transition from a horizontally oriented ~s 1 to a vertically oriented ~s 1, a transition that is predicted to occur at a depth approximately halfway to that of the dry solidus (Fig. 1). The importance of this potential rotation of ~s 1 will be considered in detail below, but for the moment we stress the general result that in the lowermost part of the partially molten zone directly beneath an active spreading centre, ~s 1 will be horizontally oriented. Concerning the magnitude of the stress ST, twositua- tions can be envisaged. In the first, when the viscosity of the melt-free mantle is elevated (Z Pa s), the upwelling velocity as a result of buoyancy is very low (v 1cm/yr; Scott & Stevenson, 1989; Ito et al., 1999). In this case v is smaller than the half-spreading rate of the oceanic plate v sp (here taken equal to.5 cm/yr), implying that wedge flow driven by plate drift is the principal source of stress. To a first approximation ST wedge may be described by the expression Z v sp /h, where h is the height of the partially molten zone (h ¼ 6 km). When Z Pa s, values in excess of 1MPa are calculated for ST wedge. On the other hand, values of ST wedge are significantly lower if lower values of Z are assumed. For example, for Z 1 18 Pa s, ST wedge is only MPa. However, for the low-viscosity case, buoyancy caused by the presence of liquid can result in values of v greater than v sp, leading to the generation of additional stresses. The buoyancy-related stress has a value equal to Z v /r (McKenzie et al., 1974), where r is the radius of the upwelling diapir. In this respect, it is of note that for buoyancy-driven flows, v and r vary as Z ^/3 and Z 1/3, respectively (McKenzie et al., 1974), implying that ST(¼Z v /r) is independent of Z. The invariance of the amplitude of the stress field has been confirmed for the case of mantle plumes, even when more realistic (powerlaw, or temperature- and pressure-dependent) viscosity laws are used (Rabinowicz et al., 198; Monnereau et al., 1993). For the case of mid-ocean ridges of interest to us here, parameter values (Z ¼1 18 Pa s, v ¼5 cm/yr, and r ¼ 4 km) taken from Rabinowicz & Briais () may be used to calculate a value for ST of 75 MPa. Viscosity of the solid fraction of the mush and the dependence on melt fraction The viscosity of the solid fraction of the mantle, Z, isan essential parameter for modelling the excess pressure of the interstitial liquid. It is of note that experimentally determined deformation rates measured in response to an applied stress/strain can be equated to the viscosity of the solid fraction of the mush, even in a partially molten system, as long as all olivine grain faces remain in contact. On the other hand, the presence of liquid along grain edges may, nevertheless, have an influence on the deformation rate of olivine grains. For example, Cooper & Kohlstedt (1986) suggested that in the presence of a few per cent of melt (55%), dissolution^precipitation triggers along grain diffusion creep reducing the effective viscosity of the solid fraction by a factor of ^5. They proposed that viscosity varies in response to the increasing surface area of wetted grains, and this idea led to the suggestion that viscosity varies as a function of ^ˇf (Stevenson, 1989). In a subsequent experimental study by Hirth & Kohlstedt (1995), it was found that the viscosity of mixtures of olivine and basaltic melt showed a decrease in viscosity of approximately an order of magnitude when the proportion of liquid was 5%. This result was linked to the onset of the introduction of melt along certain twophase boundaries, caused by the anisotropy of olivine crystals. However, more recent experiments and models on pyroxene-bearing systems (e.g. Kelemen et al., 1997; Zimmerman & Kohlstedt, 4; Scott & Kohlstedt, 6) suggested that the viscosity drop at 5% is not as dramatic as that observed by Hirth & Kohlstedt (1995). In this case, an exponential decrease of viscosity with melt concentration f has been proposed: ZðfÞ ¼Z Z ðfþ ¼Z expð EfÞ ð5aþ where E is a constant ranging between and 45 (Fig. ). The value of E is controlled by the texture, grain size, surface tension and the topology of the melt^mineral interfaces. According to Zimmerman & Kohlstedt (4), softening is less marked for pyroxene-rich compositions (E ¼ ) compared with olivine-rich systems (E ¼ 45). 175 Downloaded from on May 18

6 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 JUNE 9 In general terms, the role of pyroxene on the rheology of peridotite or lherzolite is explained by how pyroxene influences the distribution of melt among two-, three- and four-phase boundaries, pyroxene-bearing three- and fourphase boundaries tending to be drier than equivalent olivine-bearing ones (Toramaru & Fujii, 1986; Zimmerman & Kohlstedt, 4). When equation (5a) is applied to natural systems, two competing effects should be taken into account. On the one hand, mantle xenoliths contain pyroxene grains much greater in size than those of olivine (Mercier & Nicolas, 1975). In this case, pyroxene^olivine interfaces will not be numerous, such that the overall rheology may be closer to that of pure olivine. On the other hand, at the base of the melting column, the mantle is more pyroxene-rich and laboratory data for pyroxene-bearing systems may indeed be the best analogue. An additional important consideration is the range of porosity over which equation (5a) is valid. In this respect it is of note that the combined experimental results of Zimmerman & Kohlstedt (4) and Scott & Kohlstedt (6) would seem to imply that solid viscosity decreases exponentially up to melt fractions approaching that of the deconsolidation threshold, 5% (Renner et al., ). However, it should be appreciated that above 15% liquid, local melt segregation may rapidly occur, even at laboratory time and length scales. If this is the case, the measured viscosity of partially molten samples does not represent the viscosity of the solid fraction, but an average of the solid viscosity and that of local melt-rich deconsolidated areas. Indeed, between 15 and 5% liquid, one or more discontinuous drops in viscosity may occur in response to the existence of melt-rich regions that allow more consolidated regions to slide past each other (Lejeune & Richet, 1995). As the deconsolidation threshold is approached the consolidated regions decrease in size, down to that of single grains, in which case the mush behaves as a crystal suspension. Beyond the deconsolidation threshold, the dynamics of the mush cannot be described by a two-phase formalism, although it is of note that compaction leads to more or less instantaneous separation of melt and solid in this case (Rabinowicz et al., 1). Despite these reservations, in most of the calculations presented in this study equation (5a) is adopted to describe solid viscosity up to the deconsolidation threshold. However, it should be appreciated that the equations describing melt percolation [i.e. equation (4)], depend not only on viscosity, but also its first-order derivative as a function of melt concentration, d(z(f))/df. With this in mind we believe that it is important to consider at least one alternative form for the variation of viscosity as a function of melt fraction, to explore the consequences of using a viscosity equation that has different (but nevertheless potentially realistic) values of d(z(f))/df relative to equation (5a). The equation chosen is inspired by the work of Hirth & Kohlstedt (1995) and has the form ZðfÞ ¼Z Z ðfþ p atan q 1 f f 1 ¼ p atanð qþ ð1 1=f Þþ1=fA ð5bþ where q and f are two constants 1, f is the melt concentration at which viscosity drops, and Z is the melt-free solid viscosity. Several examples of the variation of dimensionless viscosity Z (f) are illustrated in Fig., for values of q in the range 5^15, and f ¼ 1. Equation (5b) differs from (5a) in two important ways. First, it predicts that viscosity is more or less constant for small values of f. This possibility should be considered in our opinion, because the experimental data used to infer an exponential decrease in viscosity are derived from the study of partially molten materials for which the average grain size is significantly smaller than that of the Earth s mantle (1 mm vs 1mm). If grain boundary dissolution^precipitation plays an important role in deformation-assisted rheology, then the onset of rheological softening may occur at much lower melt fractions in laboratory studies than in the mantle. Second, equation (5b) predicts a constant solid viscosity at high melt fraction. In this respect we reiterate that the observed decrease in bulk viscosity demonstrated experimentally (Scott & Kohlstedt, 6) is in no way inconsistent with the idea that the viscosity of the solid fraction of the mush is insensitive to f above a certain melt fraction. Permeability The permeability of a melt network k(f) containing an interconnected fluid is generally expressed by a power law relationship as a function of melt concentration f (Bear, 197): kðfþ ¼ a f n ð6aþ C where C and n are dimensionless constants and a is the characteristic size of the solid grains. However, application of equation (6a) to partially molten mantle rocks has been the subject of considerable controversy. For example, despite the fact that basaltic melt in an olivine matrix forms a dihedral angle less than 68, it has been suggested that the effects of crystal anisotropy are so important that below a threshold value of f (in the range ^3%) permeability remains extremely low, before dramatically increasing (Faul, 1997, 1). On the other hand, based upon re-examination and reinterpretation of experiments performed on partially molten dunite and lherzolite, the microstructural evidence used to support the idea of 176 Downloaded from on May 18

7 RABINOWICZ & TOPLIS MELT SEGREGATION IN MANTLE a threshold jump in permeability has been refuted (Wark et al., 3). Accepting that equation (6a) may indeed be used to rationalize the variation of permeability, it is of note that theoretical considerations for a system of interconnected tubes predict that n ¼ (Turcotte & Phipps Morgan, 199), whereas experimental data at melt fractions greater than a couple of per cent systematically suggest that n ¼ 3 for liquids of dihedral angle 568 (e.g. Wark & Watson, 1998). Assuming that n ¼ 3, laboratory experiments on quartzite^fluid mixtures suggest a corresponding value for C of 7 (Liang et al., 1; Wark et al., 3). This value is of the same order of magnitude as that expressed by Carman s law (C ¼18), which describes the permeability of an aggregate with close-packed constant size spherical grains (Dullien, 1979), and that which may be calculated for a system of cubic crystals of constant size (C ¼ 33), although in detail experimentally determined permeabilities (Wark & Watson, 1998) are somewhat lower than those derived from purely theoretical considerations for low dihedral-angle systems (e.g. VonBargen & Waff, 1986; Zhu & Hirth, 3). This deviation from theoretical models is interpreted in terms of the contribution of crystal anisotropy and grain-size variations (e.g. Liang et al., 1; Wark et al., 3). In light of these considerations, a generalized form of equation (6a) has been retained for the modelling presented here: f 3 kðfþ ¼k 5 : ð6bþ 5 This equation has the advantage that the separate effects of grain size (a), and C are attenuated through combination into a single parameter, k 5, the reference permeability at 5% porosity. Nevertheless, the appropriate value of k 5 still requires some idea of the values for a and C relevant to the sub-ridge mantle. Concerning grain size, the study of natural lherzolites and peridotites clearly demonstrates the competing effects of textural ripening and deformation (e.g. Mercier & Nicolas, 1975; Harte, 1977; Ceuleneer & Cannat, 1997). For example, in lherzolites described by Mercier & Nicolas (1975), protogranular textures developed under static conditions have grain sizes typically exceeding mm with a minor proportion of grains having sizes up to 1cm, whereas highly deformed rocks of granoblastic texture have grains less than 1mm in size. Peridotites collected at the ridge axis have an intermediate porphyroclastic texture with 7% of grains being 1mm in size, accompanied by a population of grains 5mm in size (Ceuleneer & Cannat, 1997). These larger crystals are olivines, and are inferred to result from the coalescence and recrystallization of smaller grains. Lherzolites in the lower part of the mantle mush have resided in the mantle for significant times and as such may be expected to have large grain sizes. However, these rocks are also subject to significant deformation during upwelling of the mantle above the viscosity transition at km depth. Indeed, seismic evidence suggests a drastic decrease in grain size, from centimetres to millimetres, across a boundary at km (Faul & Jackson, 5) accompanied by the onset of anisotropy (Gaherty et al., 1996). In light of the available petrological and seismic data, we consider grain size in the range 1^3 mm to be typical in the lower part of the melting column. Turning now to the value of C, we recall that the value of 7 proposed by Wark et al. (3), based upon data of Wark & Watson (1998), is considered by many to be the most appropriate for the upper mantle. However, despite being difficult and elegant experiments, it should be appreciated that the work of Wark & Watson (1998) concerns monomineralic rocks (quartz) containing a low dihedralangle fluid, on samples that had been quenched from high pressure under static conditions. The mantle, on the other hand, contains a mixture of ferromagnesian phases that have variable wetting behaviour, and that are undergoing deformation at high pressure. The potential effects of these characteristics should therefore be taken into account when estimating the value of C relevant to the lower part of the melting column beneath an oceanic ridge. First, recent experimental data obtained in situ at high pressure on quartz^fluid systems (Giger et al., 7) imply that rock permeability is typically at least two orders of magnitude lower than that measured by Wark & Watson (1998). Although it is true that the permeabilities measured by Wark & Watson (1998) are in good agreement with theoretical estimates for texturally equilibrated systems, Giger et al. (7) made the suggestion that the samples of Wark & Watson (1998) experienced a degree of fracture associated with their quench from high pressure. Second, the effect of crystal anisotropy may be of importance. In this respect it should be recalled that the variation of viscosity as a function of melt fraction in olivine- and pyroxenebearing systems is rationalized in terms of a distribution of melt that deviates from simple models involving isotropic crystals (e.g. Hirth & Kohlstedt, 1995; Zimmerman & Kohlstedt, 4). However, the effect of this anisotropy on permeability is uncertain. On the one hand, the presence of melt along two phase boundaries will remove liquid from the interconnected part of the melt network, leading to a decrease in permeability (e.g. Faul, 1997, 1). On the other hand, the presence of dry three- or four-phase boundaries (e.g. Toramaru & Fujii, 1986; Daines & Kohlstedt, 1993) may result in an increase in the effective grain size, increasing permeability [equation (6a)]. More important for the situation modelled here is the large fraction of total pyroxene (5%) at the base of the mantle melting column. This is potentially important because basaltic liquid is known to wet pyroxene grain boundaries 177 Downloaded from on May 18

8 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 JUNE 9 less efficiently than equivalent olivine grain boundaries, the dihedral angle at opx^opx^opx triple junctions being of the order of 758 (Toramaru & Fujii,1986). Detailed modelling of the bulk permeability of pyroxene-bearing systems shows that between 4 and 6% pyroxene a dramatic decrease in permeability occurs at low porosity (Zhu & Hirth, 3), suggesting that the base of the melting column may be affected by this effect. Finally, deformation may also act to reduce permeability, either through promotion of dissolution^precipitation along grain boundaries or through a reduction in grain size [equation (6a)]. On the basis of these considerations we conclude that the upper bound to k 5 is 4 1^1 m, a value derived assuming a grain size of 3 mm and the value of C proposed by Wark et al. (3). On the other hand, smaller grain size and/or a greater value of C may lead to k 5 being up to several orders of magnitude lower than this upper bound for actively deforming mantle just above the dry solidus. We have therefore chosen a reference value for k 5 of 6 1^13 m (corresponding to a value of C of 41 for a grain size of 3 mm or 45 for a grain size of 1mm), although values from 1^14 m to 4 1^1 m are also considered. Equation of state for intergranular tension The intergranular tension sa is described by the equation sa ¼ s ss a ss þ s sl a sl ð7aþ where s ss and s sl are the surface energy at the solid^solid interfaces and the solid^liquid interfaces, respectively, and a ss and a sl are the area per unit volume of each type of interface. Basaltic melt in contact with olivine and pyroxene develops a surface energy s sl which is approximately half that at a solid^solid interface s ss (VonBargen & Waff, 1986). In this case, sa is a decreasing function of melt concentration f (i.e. there is an energetic advantage for the liquid to wet grain boundaries). Such variations depend both on the ratio of s ss and s sl (which controls the dihedral angle y at the melt^solid^solid triple junctions) and on the size a of the solid grains. In the case of basaltic melt in contact with olivine, y ¼38, and sa is of the form p sa ¼ A B ffiffiffi f : ð7bþ When a ¼15 mm, values for A and B fitted to the equations of VonBargen & Waff (1986) are 534 Pa and 15 Pa, respectively, these values being inversely proportional to grain size. Equation of state for excess pressure The excess pressure p is not strictly linearly dependent on the density contrast dr between solid r m and liquid r f, because the pressure release as a result of intergranular melt flow reduces its amplitude. As a result a correct estimation of p in a partially molten plastic mantle requires explicit resolution of the two-phase equations. During the 198s, different sets of two-phase flow equations were proposed (Sleep, 1978; McKenzie, 1984; Scott & Stevenson, 1984, 1986). The most popular of these was formulated by McKenzie (1984), who extended the equations of movement of a solid^liquid suspension (Drew & Segel, 1971) to model movement of a consolidated mush. The mush was assumed to be viscous and compressible, such that its rheology depends on two viscosities, the shear viscosity, Z, and the bulk viscosity. Furthermore, it is important to note that in this formalism the pressure in the solid and liquid fractions are assumed to be the same [see McKenzie (1984) equations (A1) and (A14), p. 753]. In this case the excess pressure field p is zero and cannot be deduced from the two-phase equations. In contrast, Scott & Stevenson (1984) considered a partially molten system in which the proportions of solid and liquid are fixed, with an excess pressure field, p, related to df/dt. In detail, f, Z and p were proposed to be related by the equation Z df p ¼ fð1 fþ dt : ð8aþ In this latter case Z represents the shear viscosity of the solid fraction, and not that of the solid plus liquid mixture, in contrast to the formalism of McKenzie (1984). Equation (8a) indicates how melt migrates; that is: (1) where p4 melt concentration increases; () where p5, the solid expels melt and its concentration locally decreases; (3) at the points where p4 and reaches a local maximum, melt concentration is a maximum and has an amplitude which increases with time. As a result, it is at these points that melt accumulates. This equation was found to be exact in the case of water flow inside a partially molten ice-cap (Fowler, 1984). However, equation (8a) does not take into account the effects of surface tension, nor the addition of liquid as a result of partial melting. To take account of these latter effects we use the excess pressure equation of Bercovici et al. (1a): p ¼ dðsaþ df þ fð1 fþ df dt : ð8bþ This equation assumes that the excess pressure that results from the volume increase during the transformation of solid into melt is instantaneously released, as discussed above. It should be noted that equation (8b) differs from that subsequently suggested by Bercovici & Ricard (3) in that the latter assumes conservation of mass and variable entropy of the mush, whereas the former assumes conservation of volume and constant entropy of the mush (i.e. it is a simpler case in which there is no effect of shear heating, thus no local variations in temperature and pressure). 178 Downloaded from on May 18

9 RABINOWICZ & TOPLIS MELT SEGREGATION IN MANTLE Equation (8b) may be expanded and simplified as follows. First, according to equation (7b), the first term on the right-hand side of equation (8b) may be written dðsaþ df ¼ p B ffiffiffi : ð9aþ f Second, we consider a frame moving uniformly with the local value of the mean velocity field of the mush, C, an assumption justified by the fact that melt segregation occurs on time-scales that are short in comparison with the transit time of the mantle through the zone of partial melting (see below). Third, given that the principal physical processes considered here develop when f 1 (see below), we replace (1 ^ f) by 1 in equation (3d), ¼ ~r:ðsþþ ð9bþ and p ¼ p B ffiffiffi Z r:ðsþ: f f ~ ð9cþ Simplified set of equations to describe melt migration during plastic deformation Once equations for the principal physical parameters have been determined, the next step is to simplify equation (4). For this purpose we assess which are the dominant terms in that equation. We begin by considering the last term on the left-hand side because, depending on how the energy partitioning coefficient between melt and solid, o, is defined, this term may be more or less important. In this respect we note that two definitions of o have been proposed. The simpler is that of Bercovici et al. (1): o ¼ f: ð1aþ In this case, the overall formalism is a straightforward generalization of that developed by McKenzie (1984), because when the bulk viscosity is equal to Z/f, and the surface tension is negligible, both formalisms lead to the same equation of evolution of melt concentration f (Bercovici et al., 1b). On the other hand, Bercovici & Ricard (3) proposed an alternative form for the energy partitioning coefficient: m o ¼ f mf þ Zð1 fþ : ð1bþ Given that the viscosity of liquid in the mantle m is at least 15 orders of magnitude lower than that of the solid Z, the partitioning coefficient o is close to zero when defined according to equation (1b). If o vanishes, the last term on the left-hand side of equation (4) is finite, and resolution of the two-phase equations including this term poses serious difficulties because it non-linearly couples S or p to f. Furthermore, this term renders the formalisms of Bercovici & Ricard (3) and McKenzie (1984) incompatible. For this reason we therefore employ equation (1a) proposed by Bercovici et al. (1), a choice consistent with our definition of excess pressure [equation (8b)]. In this case, the last term on the left-hand side of equation (4) is eliminated. Turning to the other terms in that equation, we assess the dependence of each on f. This exercise shows that the second and third terms on the left-hand side have a linear dependence on f: ~r:ð1 fþzð ~rs þð~rsþ t Þ ~r 3 ð1 fþz ~r:s f : ð1cþ Consideration of equation (9c) shows that for the fourth term of equation (4)! ~rðpþ ~r p B ffiffiffi Z r:s f f ~ B 4f 3= þ Z ~ f r:s B Max f 3=, Z f f 3= ð1dþ whereas, for the first term m kðfþ S m f 3 S f 3 : ð1eþ When f 1, the above relationships imply that m kðfþ S 3 ð1 fþ ~rðpþ 3 ~r:ð1 fþ Zðð ~rs þð~rsþ t Þ3 ~r 3 ð1 fþz ~r:s : ð1fþ In light of this exercise, terms that vary as f have been dropped, resulting in a modified version of equation (4): m kðfþ S ~r Z r:ðsþ f ~ ¼drg ~r:ð1 fþ Zð ~rc þð~rcþ t Þ B ð1gþ p 4f ffiffiffi ~rðfþ: f The validity of this simplification has been tested by running simulations of compaction or shear segregation including the f terms of equation (4) (Rabinowicz et al.,, and Rabinowicz & Vigneresse, 4, respectively). In light of the good agreement between the results obtained with and without those terms, equation (1g) has been used for all further simulations shown here. The terms on the right-hand side of equation (1g) represent: (1) the buoyancy driving compaction; () the gradient of stress driving shear segregation; (3) the gradients of surface tension, respectively. In the following, we resolve equation (1g) in a narrow region beneath and including the ridge plane. In this case, the strain tensor may be written ST=Z ð ~rc þð~rcþ t Þ¼@ ST=Z 1 A ð1hþ 179 Downloaded from on May 18

10 JOURNAL OF PETROLOGY VOLUME 5 NUMBER 6 JUNE 9 Fig. 1. Sketch of mantle flow and direction of compression beneath a mid-ocean ridge of viscosity 1 18 Pa s. The continuous flow lines and the direction of maximum compression ~s 1 directly beneath the ridge axis are indicated, based on the calculations of Scott & Stevenson (1989) and Rabinowicz & Ceuleneer (199). where ST/Z ¼ ^ST/Z when ~s 1 is horizontal, and ST/ Z ¼þST/Z when ~s 1 is vertical (Fig. 1). As suggested by Stevenson (1989), the strain tensor remains invariant during the early development of the segregation process, implying that flow is described by the equation ms kðfþ ~r Z f :ðsþ ¼ drg @f ðz Þ e e x B p 4f ffiffiffi ~rðfþ ð11þ f where x and z designate the horizontal and vertical directions, respectively. Dimensionless equations and reference values To make equations (9a^c) and (11) dimensionless, new scales for the variables are chosen (see Table 1). The simulations of this study start with a melt fraction ranging from 1 to 5%. Accordingly, the reference value f of melt concentration is taken to be either 1% or 5%, as appropriate. The bulk viscosity reference value Z is 1 18 Pa s and the reference value of deviatoric stresses ST ¼ 75 MPa (see above). Measured viscosities of melt formed near the anhydrous solidus of lherzolite range from 66 topas(kushiro, 1986). Considering a typical value of m ¼ 4 Pa s, the reference value for the ratio k(f )/m is 64 1^14 m /Pa/s (assuming k 5 ¼6 1^13 m, as discussed above). The reference value for distance is the compaction length L: sffiffiffiffiffiffiffiffiffi Z L ¼ k ð1aþ mf such that L ¼ 4 m or 1 m, when f ¼1% or 5%, respectively. The most appropriate choice for the excess pressure scale differs in a compaction and a shear segregation simulation. In a compaction experiment, the variation of pressure is proportional to the density contrast between melt and solid dr integrated over a height of one compaction length L. According to Stolper et al. (1981), between 6 and 8 km depth, dr ¼5 kg/m 3,afactorof two less than that expected at shallower depth (5 kg/ m 3 ). In this case, the reference value for pressure p is p ¼ drgl Pa or Pa ð1bþ when f ¼1% or 5%, respectively. In shear experiments, the appropriate scale for pressure ~p is defined as (Rabinowicz & Vigneresse, 4) ~p ¼ ST f Pa or Pa ð1cþ when f ¼1% or 5%, respectively. Because all models presented here concern shear segregation, we choose ~p to scale the excess pressure. The following stress ratio defines the dimensionless shear number N s (Rabinowicz & Vigneresse, 4): N s ¼ ~p p ¼ ST f drgl 1 8 : ð1dþ It should be noted that this latter value is independent of f because L is directly proportional to f,giventhatk in equation (1a) is proportional to f 3 [equation (6b)]. According to these choices, the time unit t is the relaxation time of the excess pressure ~p: t ¼ Z f ST 4 16 years or years ð1eþ when f ¼1% or 5%, respectively. The Darcy velocity unit V is L V ¼f t 51 4 mm=year or 71 mm=year ð1fþ 18 Downloaded from on May 18