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Journal of Petrology Volume 42 Number 5 Pages 963-998 2001
© Oxford University Press 2001
Calculation of Peridotite Partial Melting from Thermodynamic Models of Minerals and Melts, IV. Adiabatic Decompression and the Composition and Mean Properties of Mid-ocean Ridge Basalts
1DIVISION OF GEOLOGICAL AND PLANETARY SCIENCES, CALIFORNIA INSTITUTE OF TECHNOLOGY M/C 170-25, PASADENA, CA 91125, USA
2DEPARTMENT OF GEOLOGY AND GEOPHYSICS, UNIVERSITY OF MINNESOTA, 310 PILLSBURY DRIVE SE, MINNEAPOLIS, MN 55455-0219, USA
Received January 25, 2000; Revised typescript accepted August 16, 2000
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONISENTROPIC BATCH MELTINGINCREMENTALLY ADIABATIC...MEAN PROPERTIES OF MELTING...AGGREGATE PRIMARY MELT...EFFECTS OF SOURCE HETEROGENEITYCONCLUSIONSREFERENCES |
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Composition, mean pressure, mean melt fraction, and crustal thickness of model mid-ocean ridge basalts (MORBs) are calculated using MELTS. Polybaric, isentropic batch and fractional melts from ranges in source composition, potential temperature, and final melting pressure are integrated to represent idealized passive and active flow regimes. These MELTS-derived polybaric models are compared with other parameterizations; the results differ both in melt compositions, notably at small melt fractions, and in the solidus curve and melt productivity, as a result of the self-consistent energy balance in MELTS. MELTS predicts a maximum mean melt fraction (
0·08) and a limit to crustal thickness (
15 km) for passive flow. For melting to the base of the crust, MELTS requires an 200°C global potential temperature range to explain the range of oceanic crustal thickness; conversely, a global range of 60°C implies conductive cooling to 50 km. Low near-solidus productivity means that any given crustal thickness requires higher initial pressure in MELTS than in other models. MELTS cannot at present be used to model details of MORB chemistry because of errors in the calibration, particularly Na partitioning. Source heterogeneity cannot explain either global or local Na–Fe systematics or the FeO–K2O/TiO2 correlation but can confound any extent of melting signal in CaO/Al2O3. KEY WORDS: mantle melting; mid-ocean ridge basalt; peridotite composition; primary aggregate melt; thermodynamic calculations
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONISENTROPIC BATCH MELTINGINCREMENTALLY ADIABATIC...MEAN PROPERTIES OF MELTING...AGGREGATE PRIMARY MELT...EFFECTS OF SOURCE HETEROGENEITYCONCLUSIONSREFERENCES |
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Comparison of observed basalt compositions with the predictions of polybaric mantle melting models places important restrictions on melting processes in the mantle beneath mid-ocean ridges (Klein & Langmuir, 1987
; McKenzie & Bickle, 1988; Niu & Batiza, 1991, 1993; Kinzler & Grove, 1992b; Langmuir et al., 1992; Iwamori et al., 1995; Kinzler, 1997). It is now well accepted that mid-ocean ridge basalts (MORBs) represent mixtures of melts produced over a range of depths and that these melts separate from their sources at low melt fraction (McKenzie, 1984; von Bargen & Waff, 1986; Johnson et al., 1990; Langmuir et al., 1992). However, significant uncertainties remain regarding the depth of initial melting (related both to the range of mantle potential temperature and to the influence of minor incompatible components on the solidus), the depth of final melting (and hence the importance of spreading rate), the style of melt transport (e.g. the relative importance of fractional fusion vs equilibrium porous flow), and the effects of chemical heterogeneities in mantle sources. These uncertainties remain for a number of reasons (among them disagreements over the selection of data, the appropriate scale for averaging, and corrections for fractionation), but a key factor is that mantle melting algorithms have not been sufficiently accurate to evaluate quantitatively the consequences of competing hypotheses or to incorporate the complexities of the physics of melting and melt transport.
Langmuir et al. (1992) identified three functions that must be combined to create a forward model capable of quantitative prediction of the magmatic output of the mantle beneath mid-ocean ridges: a chemistry function, a melting function, and a mixing function. The chemistry function specifies the liquid composition as a function of pressure (P), temperature (T) (or, alternatively, of P and extent of melting, F), and source composition. The melting function specifies F for a single parcel of source as a function of P and T and the path (usually approximated as adiabatic) through (F, P, T) space. The mixing function is a representation of the 2D (or, in principle, 3D) form of the melting regime and specifies how the individual increments of liquid generated continuously over a range of depths and distances from the ridge axis are to be weighted to create an aggregate primary melt. These three functions are often constructed independently, but in fact the chemistry and melting functions are intimately dependent on one another, as both must satisfy mass and energy balance and both are controlled by the same thermodynamics of solid–liquid equilibrium. Furthermore, the possibility of reaction between melt and matrix during melt migration means that the mixing problem cannot be separated from the chemistry and melting functions (Spiegelman, 1996; Kelemen et al., 1997). There are several published chemistry, melting, and mixing functions based on parameterization of experimental peridotite melting data (Klein & Langmuir, 1987; McKenzie & Bickle, 1988; Niu & Batiza, 1991; Kinzler & Grove, 1992a, 1992b; Langmuir et al., 1992; Kinzler, 1997).
Over the past several years, we and our colleagues have been utilizing the MELTS algorithm (Ghiorso, 1994; Ghiorso & Sack, 1995) as a tool for trying to understand aspects of experiments on peridotite melting (Baker et al., 1995; Hirschmann et al., 1998b, 1999a, 1999b) and as a basis for forward models of polybaric mantle melting and of coupled melting and two-phase flow in upwelling mantle (Asimow, 1997; Asimow et al., 1997; Asimow & Stolper, 1999). The forward models of polybaric melting we utilize differ from other algorithms in that MELTS provides a self-consistent thermodynamic approach to the chemistry and melting functions. Although we must emphasize that MELTS is not sufficiently accurate to address in detail some key questions raised by the observed compositional variations of MORB magmas, it is nevertheless the first approach that allows a full and self-consistent integration of the thermodynamics and phase equilibria of partially molten peridotitic systems. For this reason, it is ideal for examining important issues such as the interplays between the depth-dependent productivity of upwelling mantle and the average composition and pressure of melting and between source heterogeneity and productivity.
In this paper, we present a set of polybaric calculations of mantle melting using MELTS. These calculations build on related isobaric calculations (Hirschmann et al., 1998b, 1999a, 1999b) and on polybaric, isentropic calculations (Asimow et al., 1995, 1997) and are used to illustrate both the potential and current limitations of the method. All calculations herein use the calibration of MELTS documented by Ghiorso & Sack (1995); this is the calibration underlying the widely distributed MELTS 2.0 package, and although we have modified the implementation for convenience, we have not altered the model in any way. This paper does not include results from newer calibrations such as pMELTS (Ghiorso & Hirschmann, in preparation). In general, MELTS predicts isobaric trends of composition vs melt fraction that are similar to those observed in experiments, but the calculated trends are frequently offset in T and in the concentrations of certain oxides. For example, at 1·0 GPa the best match in F between MELTS and experiments is obtained with an offset of 80°C in T and the resulting model liquids are 4% (absolute) too low in SiO2 and 2% too high in MgO (Baker et al., 1995). Comparison with experiments has also shown that MELTS yields too low a peridotite–liquid partition coefficient for Na (Hirschmann et al., 1998b). Given these inaccuracies in the current calibration of MELTS, we focus on using it as a tool for studying trends in relationships among variables rather than to predict the actual values of specific parameters. Accurate quantitative modeling of absolute values of compositional variables and phase proportions as functions of pressure, potential temperature, etc. is certainly possible with the approach we use here, but it will depend on improving or customizing the calibration.
The mixing function depends on geodynamic considerations such as the form of the solid flow field and melt extraction pathways. Our focus in this paper is on insights from thermodynamic modeling of melt composition and melting, so we limit our treatment to the simplest end-member mixing functions associated with either perfect active or perfect passive flow [we use standard definitions of mixing functions and mean properties from Plank et al. (1995); see details below in the section ‘Mean properties of melting regimes’]. Although all the calculations presented here use one of these two simple mixing functions, we will conclude in many places that neither function is adequate and that progress in modeling of ridges will depend on using physically based mixing functions.
One of the recurring issues in our modeling of melting at ridges will be an examination of the consequences of two competing views of the principal controls on variation of average magma compositions among ridge segments. Klein & Langmuir (1987) and others (McKenzie & Bickle, 1988; Klein & Langmuir, 1989; Langmuir et al., 1992; Plank & Langmuir, 1992) considered that, except perhaps at very slow-spreading rates, melting continues to a shallow depth, perhaps the base of the crust, at all ridge segments. Variations in average MORB composition on the world-wide ridge system, correlated with ridge topography and seismic velocity in the underlying mantle (Klein & Langmuir, 1987; Humler et al., 1993; Zhang et al., 1994), were then attributed primarily to variations in the potential temperature (TP) of the upwelling mantle (a range of 200–250°C, Klein & Langmuir, 1987; or 300°C, McKenzie & Bickle, 1988), which controls the intersection of the adiabat with the solidus and thus the initial pressure of melting (Po). We will refer in this work to this model of global variations as ‘variable-Po’ systematics. Shen & Forsyth (1995), on the other hand, attributed the variability in the compositions of MORBs primarily to the effectiveness of cooling from the surface and hence to the final pressure of melting (Pf). The total variation in potential temperature among non-hotspot-affected ridges was then estimated to be 60°C (Shen & Forsyth, 1995). A consistent model based on this second view generally includes a significant role for heterogeneous source compositions (Niu & Batiza, 1991; Shen & Forsyth, 1995) and implies a correlation of extent of melting with spreading rate (Niu & Batiza, 1993; Niu & Hékinian, 1997a, 1997b). We refer to this model of global variations as ‘variable-Pf’ systematics.
Evaluation of variability and correlations among compositions of MORB samples requires correcting for the effects of low-pressure fractionation so as to isolate the effects of magma generation processes in the mantle (Klein & Langmuir, 1987). In this work we restrict our attention mostly to model primary liquid compositions based on MELTS and compare them with other workers’ model primary aggregate liquids; i.e. we generally do not attempt to fractionate these model liquids to compare them with actual MORB data or with MORB data corrected for fractionation (i.e., we discuss mostly Na2O and FeO* rather than Na8 or Fe8, the equivalent values corrected for low-pressure fractionation to 8% MgO; Klein & Langmuir, 1987). We take this approach for several reasons. First, the 8% MgO standard obscures real variations in the MgO content of primary aggregate liquids; in particular, calculations herein are extended to the low potential temperature extreme where the primary liquid may have <8% MgO, which would require an artificial back-fractionation step. Second, the correction is difficult to perform quantitatively in most cases. The low SiO2 and high Na2O in liquids predicted by MELTS (Hirschmann et al., 1999b) results in primary aggregate liquids that are nepheline normative and do not follow tholeiitic fractionation paths, whether fractionated using MELTS or any other fractionation model. The model of Weaver & Langmuir (1990) is not calibrated on alkalic liquids, but yields a tholeiitic fractionation path for MELTS primary aggregate liquids that we use here with some caution. Moreover, the Langmuir et al. (1992) model does not predict CaO or Al2O3 in primary liquids, but these are essential components in determinations of pyroxene and plagioclase stability and the effects of these phases on liquid lines of descent, so comparison between our model primary liquids and those of Langmuir et al. (1992) is best performed without first fractionating them to 8% MgO. Nevertheless, much can be learned from examination of model primary liquids, particularly as the effects of fractionation are relatively minor for several of the elements of interest.
Recent efforts to interpret MORB compositions in terms of magma generation processes have, in addition to corrections for fractionation, distinguished variability among individual MORB samples within a region from variability on a global scale among regional averages of samples (Brodholt & Batiza, 1989; Klein & Langmuir, 1989). Correlations among regional averages are generally termed ‘global trends’ whereas correlations in variations among individual samples from a segment are termed ‘local trends’. Forward models of melting of the sort presented in this paper can be used to examine the possible spectrum of local variability by examining all the incremental melt compositions and partial mixtures among them that can be produced from a model melting regime (although the large range of partial mixing models that can be devised makes it difficult to specify a priori the particular local trend that will result from a given model melting regime).
In this paper, we begin by introducing phase diagrams in pressure–temperature and pressure–entropy space that set up the framework for all MELTS predictions of polybaric melting by showing where each phase assemblage is predicted to be stable and the contours of equal melt fraction above the solidus [the importance of which was discussed by Asimow et al. (1997)]. The coupling between the chemistry and melting functions predicted by MELTS is then illustrated using the SiO2 vs melt fraction (F) plot of Klein & Langmuir (1987). A brief discussion of the differences between isentropic batch melting and incrementally isentropic fractional melting (Asimow et al., 1995, 1997) introduces comparisons between mean properties of the melting regime (i.e. mean pressure, mean extent of melting, and crustal thickness) as functions of the initial and final pressures of melting. This is followed by predictions of correlations of these mean properties with compositional trends in primary aggregate liquids, particularly Na2O and FeO*. Finally, we consider the effects of variable source composition on primary aggregate liquid compositions, extending the discussion of Hirschmann et al. (1999b), which dealt only with calculations of isobaric melting.
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ISENTROPIC BATCH MELTING |
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TOPABSTRACTINTRODUCTIONISENTROPIC BATCH MELTINGINCREMENTALLY ADIABATIC...MEAN PROPERTIES OF MELTING...AGGREGATE PRIMARY MELT...EFFECTS OF SOURCE HETEROGENEITYCONCLUSIONSREFERENCES |
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Figure 1 shows maps of the stable phase assemblages for a model primitive mantle composition (Hart & Zindler, 1986
), with and without Cr2O3. The axes in Fig. 1a and c are P and specific entropy (S), or equivalently P and potential temperature [TP, the calculated temperature of the metastable solid assemblage at 1 bar with the given total S, allowing all solid reactions to reach equilibrium; see McKenzie & Bickle (1988)]. ‘Ordinary’ temperature (T ) is plotted vs P in Fig. 1b and d for reference, but P and S are the appropriate independent variables for reversible adiabatic (i.e. isentropic) melting (Verhoogen, 1965; McKenzie, 1984; Asimow et al., 1995, 1997). Any vertical line in a P–S diagram corresponds to a batch isentropic path. Contours of constant extent of melting by mass, F, are plotted in the supersolidus region of each map. The isentropic productivity of batch melting at any point is inversely proportional to the spacing of these contours as they intersect a vertical path. Hence, these diagrams show all possible isentropic batch melting paths for this source composition and the range of mantle potential temperatures likely to be seen by modern mid-ocean ridges. Although absolute temperatures at elevated pressure in the following discussion are subject to the errors noted by Baker et al. (1995) and Hirschmann et al. (1998b) and are likely to be 80°C hotter than the correct temperatures, potential temperature is defined at atmospheric pressure where MELTS is much more accurate. TP values obtained from MELTS can be compared directly with TP estimates from other models, with an uncertainty of perhaps 20°C.
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0·05 are shown dashed. (a) and (b) use the primitive upper-mantle composition of Hart & Zindler (1986)For TP < 1100–1120°C (boxed 1 in Fig. 1a and c), melt does not form at any pressure. Although this may be an artifact of MELTS and needs further investigation (see below), it is interesting that this minimum temperature, roughly the same for the Cr-bearing and Cr-absent cases, is not determined by the solidus temperature at 1 bar or at the base of the crust but rather by the location of the six-phase point olivine + orthopyroxene + clinopyroxene (cpx) + spinel + plagioclase + liquid; i.e. the point of the cusp on the solidus. This is the point on the MELTS-calculated solidus with the lowest specific entropy, and so it defines the lowest mantle potential temperature at which the adiabat intersects the solidus.
Melting paths with 1110°C < TP < 1225°C (boxed 1–3 in Fig. 1a and c) freeze completely as a result of the spinel–plagioclase transition (Asimow et al., 1995). These paths achieve peak melt fractions of F 0·025 in the spinel peridotite field. Those paths hotter than TP 1175°C (boxed 2 in Fig. 1a and c) would begin melting again in the plagioclase stability field if isentropic decompression continued all the way to 1 bar. All paths up to TP 1275°C with Cr2O3 (boxed 5 in Fig. 1c) have plagioclase in the residue for at least a small interval. TP 1250°C (boxed 4 in Fig. 1a and c) is the minimum for exhaustion of cpx from the residue. For the Cr-bearing case, TP > 1425°C (boxed 6 in Fig. 1a) is required for melting to begin in the garnet field; for the Cr-absent case garnet is calculated to be present on the solidus for TP > 1380°C (boxed 6 in Fig. 1c).
The shape of the MELTS-calculated solidus has two unusual features: the negative slope of the solidus at pressures just below that of the spinel–plagioclase transition and the substantial curvature of the solidus in the spinel and garnet stability fields, which leads to a maximum in TP on the solidus near 6·5 GPa (not shown in Fig. 1). Although simple considerations of phase diagram topology dictate that there must be a cusp on the solidus at the appearance of plagioclase (Presnall et al., 1979), the calculated result that there is actually a temperature drop with increasing pressure approaching the cusp is surprising. This predicted shape reflects primarily the solidus-lowering capacity of Na, which is enhanced in the spinel peridotite field relative to the plagioclase peridotite field by the greater incompatibility of Na in assemblages with less plagioclase. In this sense, the region of negative slope on the solidus is similar to that observed for amphibolite (Wyllie & Wolf, 1993) or amphibole-bearing peridotite (e.g. Green, 1973) where the breakdown of amphibole near 2·0–2·5 GPa converts water from a relatively compatible to an incompatible component. It is possible that the prediction of a minimum temperature at the cusp is an artifact of MELTS, as MELTS overestimates the incompatibility of Na in the spinel peridotite field (Hirschmann et al., 1998b), and, indeed, this shape has not been observed in experimental determinations of peridotite solidi. However, examination of the melt fraction contours in Fig. 1 shows that the shape as determined by experiment would be extremely sensitive to the minimum melt fraction required to identify melting in an experiment. For example, at F = 0·01, the temperature drop at the cusp is predicted to be only half as big as that on the solidus itself, and by F = 0·05, there is no temperature drop or negatively sloped region at all. Thus, experimental determinations of peridotite solidi, none of which have yet systematically explored such low melt fractions, are unlikely to have detected such behavior even if it does occur, as it is predicted to be confined to such small melt fractions.
Although MELTS as currently formulated was not intended to be used at pressures beyond 2 GPa (Ghiorso & Sack, 1995), the phase relations implicit in MELTS require that calculated melting begins deeper than this to model ridge segments with more than 4 km of crust. Although we have little confidence in specific predictions of the model (e.g. liquid compositions) above 3 GPa, the position and shape of the solidus are critical factors in forward modeling amounts of melt production and crustal thickness on a given isentropic path. The solidus of the KLB-1 peridotite composition (for which the most data are available at high P ) predicted by MELTS is compared in Fig. 2 with experimental brackets. This figure shows that the MELTS-calculated solidus for KLB-1 is similar (i.e. within 100°C) to all experimental brackets up to at least 8 GPa (Takahashi, 1986; Takahashi et al., 1993; Zhang & Herzberg, 1994). The MELTS solidus is also everywhere within 75°C of the solidus that McKenzie & Bickle (1988) fitted to peridotite solidus data (with only KLB-1 points above 4 GPa), but the MELTS solidus is more strongly curved and has a lower slope at very high pressure. MELTS tends to exaggerate the incompatibility of Na near the solidus, leading to too large a ‘freezing point depression’ (Hirschmann et al., 1998b), whereas at higher melt fractions it errs in the opposite direction by 80°C at 1 GPa (Baker et al., 1995), so the excellent correspondence between the calculated and measured solidus in Fig. 2 could be misleading. None the less, the similarity in the overall curvatures of the model and experimental determinations of the solidus in P–T space is useful in the context of forward models of melt production as a function of potential temperature. Although the magnitude of the calculated curvature may be exaggerated by errors in MELTS that grow worse with increasing pressure, it is particularly important that the MELTS calculations and experiments both display the flattening of the solidus expected as a result of the greater compressibility of liquid silicate components compared with minerals (Walker et al., 1988). Figure 2 also shows that the linear solidus with a slope of 130 K/GPa assumed by Langmuir et al. (1992) corresponds well to the more complex solidus calculated by MELTS and to the experimental determinations of the solidus up to a pressure of 5 GPa, beyond which it increasingly diverges from both the data and the MELTS calculation.
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As emphasized by Asimow et al. (1997), phase relations are typically more useful for understanding decompression melting when portrayed in S–P (or equivalently TP–P ) space than when portrayed in P–T space. There are no experimental measurements of entropy or potential temperature, of course, but when recast in these terms, mantle melting is more easily visualized, and insights can be obtained that would be difficult using P and T as the independent variables. In the case of MELTS calculations on the Hart & Zindler composition, with or without Cr, the solidus is actually predicted to have a vertical tangent at 6·4 GPa (off the tops of Fig. 1a–d) and 2000 K (TP 1500°C). If such a maximum on the solidus in S–P space exists, it would have some curious consequences, including progressive freezing of parcels of mantle as they decompress from higher pressures. Determination of whether such a maximum in entropy along the solidus actually exists for upper-mantle materials must await improved thermodynamic data on solids and liquids at high pressures, but from a practical standpoint, the fact that this feature is predicted by MELTS for fertile peridotite limits application of the standard relationships for triangular melting regimes (requiring a well-defined maximum melting pressure, Po; Plank et al., 1995) to MELTS calculations with TP < 1500°C. It is important to reemphasize that despite this limitation, the position of the solidus predicted by MELTS in both temperature and entropy (or potential temperature) space at pressures up to 6 GPa is not unreasonable, even though the equations of state used for minerals and liquids were not intended to extrapolate so far. We show calculations with TP up to this maximum for completeness and to further our understanding of the implications of this possible behavior of the solidus at high P, but the reader is cautioned that these calculations are extrapolated beyond any reasonable expectation of accurate prediction of actual phase relations.
The difficulty imposed by the inadequate treatment of Cr2O3 in MELTS (Hirschmann et al., 1998b) is illustrated by Fig. 1. In the present version of MELTS, spinel is the only mantle phase that accepts Cr2O3, whereas in natural systems, pyroxenes and garnet are significant Cr2O3 reservoirs. Hence when Cr2O3 is included in the calculation, spinel is stable under all subsolidus conditions and persists nearly to the liquidus (Fig. 1a and b). On the other hand, when Cr2O3 is excluded from the composition, as in Fig. 1c and d, spinel is insufficiently stable and hence the garnet–spinel and spinel–plagioclase transition regions are artificially narrow and spinel disappears from the residue before cpx, near 10% melting. We have tried to work around this problem using duplicate calculations in Cr-bearing and Cr-absent compositions. When similar behavior is observed in both cases, we infer that errors in spinel stability are not seriously affecting our results.
Previous attempts to estimate liquid compositions during isentropic batch melting have generally chosen a path through a chemistry function [i.e. liquid composition as a function of (P, F) or (P, T)] fitted to isobaric melting data, where the path is set by the independently estimated melting function or productivity, –dF/dP, or by an estimate of the thermal gradient during melting, dT/dP. For example, Klein & Langmuir (1987) illustrated the construction of such a model for SiO2 as a function of F and P using isobaric curves (fits to experiments expressed as SiO2 contents of partial melts of fertile peridotite vs F at constant P) and an estimated isentropic productivity of 1·2%/kbar (Fig. 3a; it should be noted that we retain units of %/kbar for consistency with previous work; 1%/kbar = 10%/GPa). In contrast, MELTS generates isentropic batch melting paths directly, without treating the chemistry and melting functions independently. The result (Fig. 3b) differs considerably from that of Klein & Langmuir (1987) (Fig. 3a) for two reasons. First, the isobaric melting curves are clearly different in that they show high SiO2 at low F (Baker et al., 1995), especially at the low-P end of the spinel peridotite field (Hirschmann et al., 1998a), a feature that was not apparent in experimental data as of 1987. Although this effect is clear in experiments of Baker et al. (1995), Kushiro (1996), and others, its magnitude is exaggerated by MELTS [for detailed comparison of MELTS compositions with isobaric experimental data, see Hirschmann et al. (1998b)]. Second, the isentropic productivity is not constant at 1·2%/kbar; instead, it systematically increases along each melting path from as little as 0·25%/kbar on the solidus to a maximum of 3%/kbar at the exhaustion of cpx from the residue (Hirschmann et al., 1994; Asimow et al., 1997). These two novel aspects combine to predict concave-up isentropic batch melting paths in Fig. 3b, in contrast to Klein & Langmuir’s concave-down paths. We emphasize, however, that although this figure is useful for building intuition in that it connects isobaric melting to the less familiar isentropic paths, it is probably not directly relevant to MORB petrogenesis, because the consequences of fractional melting on liquid and residue compositions are predicted to be significant (see below). Thus the idea of using batch melting isobars or isentropes to predict compositions from fractional fusion is less useful than was once thought (Klein & Langmuir, 1987; McKenzie & Bickle, 1988).
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INCREMENTALLY ADIABATIC FRACTIONAL MELTING |
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TOPABSTRACTINTRODUCTIONISENTROPIC BATCH MELTINGINCREMENTALLY ADIABATIC...MEAN PROPERTIES OF MELTING...AGGREGATE PRIMARY MELT...EFFECTS OF SOURCE HETEROGENEITYCONCLUSIONSREFERENCES |
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Fractional melting cannot be a locally isentropic process, in that escaping melts remove entropy from the system. Here we model fractional melting as an idealized process of infinitesimal isentropic batch melting steps followed by extraction of all liquid formed (see Asimow et al., 1995
, 1997). The composition and entropy of the residue of each step then serves as the reference for the next increment. The extension to continuous fusion, where some amount of melt remains behind after each step, is straightforward (Asimow et al., 1997).
Most previous attempts to construct models of polybaric fractional melting have been linked closely to melt compositions and melt fractions from batch melting experiments on a limited number of peridotite bulk compositions. Those that use compositions directly from batch melting experiments (Klein & Langmuir, 1987; McKenzie & Bickle, 1988; Watson & McKenzie, 1991; Iwamori et al., 1995) include none of the compositional effects of fractional melting and any differences between batch and fractional fusion reflect largely ad hoc estimates of the differences in productivity and P–T paths between batch and fractional melting. Other parameterizations use major element partition coefficients fitted to batch melting experiments (Niu & Batiza, 1991; Langmuir et al., 1992) or four-phase saturation surfaces (Kinzler & Grove, 1992a, 1992b; Kinzler, 1997): within the fitted range, these parameterizations try to account for the evolution of residue composition and variations in liquid composition with progressive fractional fusion. However, all these approaches have depended on poorly constrained (and largely non-thermodynamically grounded) estimates of productivity and P–T paths for fractional fusion. The incremental batch experimental approach of Hirose & Kushiro (1998) attempted to approximate the P–T–F path of incrementally adiabatic polybaric fractional fusion; although this approach is a promising one, it involves relatively large step sizes (i.e. the first increment is 6·5% melting), so it is not a good approximation to pure fractional fusion.
Figure 3c illustrates the likely magnitude of the differences between batch and fractional melting based on MELTS calculations. Conventional petrological wisdom holds that integrated fractional melts are similar to batch melts, but this is strictly true only for highly incompatible elements and only when partition coefficients are constant. In this polybaric case, however, where productivity is different for batch and fractional processes and where SiO2 partitioning depends strongly on pressure (O’Hara, 1968), batch and integrated fractional melts are very different, and the differences generally increase with F (i.e. with the pressure range from solidus to the pressure of comparison). The modeled differences shown in Fig. 3c are qualitatively similar to the results of Hirose & Kushiro (1998). Any melting model where the melt composition changes with pressure will yield such a difference between polybaric batch melting and integrated polybaric fractional fusion; such an effect is clear, for instance, in FeO* values in the model of Langmuir et al. (1992). The magnitude of the differences between batch and fractional melting from a given model, however, is sensitively dependent on the productivity function. As we will see below, in an adiabatic melting column, MELTS calculations produce the bulk of liquid mass over a smaller range of pressure than models with nearly linear productivity and hence predict smaller differences in SiO2 and FeO* concentrations between batch and accumulated fractional liquids.
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MEAN PROPERTIES OF MELTING REGIMES |
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TOPABSTRACTINTRODUCTIONISENTROPIC BATCH MELTINGINCREMENTALLY ADIABATIC...MEAN PROPERTIES OF MELTING...AGGREGATE PRIMARY MELT...EFFECTS OF SOURCE HETEROGENEITYCONCLUSIONSREFERENCES |
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Two-dimensional models of mid-ocean ridge melting can often be simply characterized by mean properties: e.g. the mean pressure of extraction (
), the mean extent of melting FB (see Plank et al., 1995), or the total crustal thickness (Zc, when given in units of kilometers, or Pc, the pressure at the base of the crust). For any model (e.g. active or passive, batch or fractional), the relationships between these average properties and the physical parameters of the model (e.g. Po or Pf) depend on the productivity and the form of its variations with F, Po, and source composition; i.e. the nonlinear melting function (or, equivalently, non-constant –dF/dP) predicted by thermodynamics (Asimow et al., 1997) results in nonlinear relationships among Po, Pf, , FB, and (Zc)1/2. We show below the relationships among all these variables according to the MELTS model and, for comparison, the models of Langmuir et al. (1992) and Kinzler (1997), all for the reference case of perfect fractional melting and passive flow for the Hart & Zindler (1986) source composition.
Formalisms for obtaining mean properties of melts produced by 2D model melting regimes using 1D melting models have been presented several times (Klein & Langmuir, 1987; McKenzie & Bickle, 1988; Plank & Langmuir, 1992; Richardson & McKenzie, 1994), but the issue requires clarification for a model such as ours with strongly varying productivity, as some of the prior treatments apply only to special cases. There is agreement that the mean melt composition, C, produced along each streamline or evaluated at each point along the exit boundary of the melting regime is obtained from a single integration,
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), but the meaning of c, defined as the composition of the melt added to increase the fraction of melt from F to F + dF, is obvious only for fractional melting, where it is the instantaneous melt composition produced by each increment of melting, dF. For batch melting and intermediate processes (e.g. ‘continuous’ or ‘dynamic’ melting with a retained porosity above which melts are fractionally removed; Johnson & Dick, 1992; Langmuir et al., 1992), this definition requires that c is the net transfer of components between solid and liquid, such that for batch melting C is the instantaneous liquid in equilibrium with the residue.
For end-member active flow, all streamlines and points on the exit boundary of the melting regime are the same (Plank et al., 1995), and equation (1) is all that is needed to compute the mean output of the melting column. Some models of aggregate MORB composition have considered only this column average (Niu & Batiza, 1991), but for flows with some 2D character to the exit boundary of the melting regime (e.g. the passive flow triangle), another step is required. McKenzie & Bickle (1988) defined the ‘point and depth average’, the mean composition of all melts exiting the melting regime, by integration with respect to depth z from the solidus (z = 0) to the height of the residual mantle column h:
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and Langmuir et al. (1992) averaged with respect to F:
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The relationship between (2) and (3) is revealed in the derivation of Plank & Langmuir (1992), who gave the more general equation
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; McKenzie & Bickle, 1988; McKenzie & O’Nions, 1991; Langmuir et al., 1992), although this is not the result of the simplest, constant-viscosity corner-flow model (Batchelor, 1967). If vx is held constant, it cancels out of (4), and it is clear that (4) and (2) are equivalent. Equation (3), however, is a special case for constant vx and dz/dF; constant dz/dF corresponds to constant spacing in depth of the melt fraction contours, or approximately to constant productivity, –dF/dP. The difficulty of using (4) in regions where productivity is zero (hence dz/dF is infinite), such as the shallow mantle above a variable-Pf melting regime, is clear. As z is not well characterized in our calculations (Asimow & Stolper, 1999), whereas P is known exactly as an independent variable, mean properties for all passive-flow models are calculated in this work according to
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For cases where melting stops at Pf > Pc, whether as a result of imposed cooling or productivity effects, the value of F at Pf, Fmax, applies throughout the interval Pf to Pc, which simulates a trapezoidal melting regime contained within a triangular corner-flow field,
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) and hence that even if liquids are removed from chemical equilibrium with the residue in the interior of the melting regime they are physically carried along solid-flow streamlines to the boundaries of the melting regime (with only a Boussinesq effect on the fluid dynamics). A truly rigorous model of mixing requires a mass-conservative calculation of the liquid and solid flow fields allowing for compaction (e.g. Spiegelman, 1996). The effects of productivity functions on extent of melting, mean extent of melting, and crustal thickness are explored in Figs 4–6. Figure 4 emphasizes the progress of F and its pressure derivative and integral along each adiabat from the solidus to the base of the crust, independent of the mixing function or the shape of the melting regime. Figure 5 illustrates variable-Po systematics by showing the final melt production by adiabats of differing potential temperature (or Po) where Pf is equal to the base of the crust Pc (i.e. the melting regime is triangular). Figure 6 illustrates variable-Pf systematics, showing the total output when melting stops at various values of Pf, but the integration continues at the final value of F all the way to the base of the crust (i.e. the melting regime is trapezoidal and the residual mantle column contains equally depleted material from the base of the crust all the way down to Pf).
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Productivity functions
In Fig. 4, the output of the MELTS model is compared with the parameterization of Langmuir et al. (1992), which has a slight decrease in productivity with decreasing pressure (a linear correction of 10 parts in 88 per GPa as a result of convergence of the liquidus and solidus) superimposed on a small (20%), discontinuous decrease in productivity at the depth of cpx exhaustion. Figure 4a and b shows the productivity functions of the two models vs pressure for melting paths with Po = 1·3, 1·7, 2·1, 2·7, and 4·4 GPa. The strongly increasing productivity leading up to cpx-out along each path in the MELTS model (Hirschmann et al., 1994; Asimow et al., 1997) is prominent in Fig. 4a, which also shows the following features of productivity predicted by MELTS: a doubling in productivity at the exhaustion of garnet at 3·1 GPa [this is a Cr-bearing composition; see Asimow et al. (1995)]; a moderate drop in productivity or a barren zone at the appearance of plagioclase on adiabats cold enough to form plagioclase (Asimow et al., 1995); and a large (50–60%) drop in productivity at cpx-out (Asimow et al., 1997). The inset in Fig. 4a showing productivity against F illustrates how F-dependent features (e.g. the low-productivity region at F < 0·03, the high-productivity region approaching cpx-out, and the drop at cpx-out) vary with Po. The low-productivity region extends to about the same F (0·03 for this source composition) for all Po. The extent of melting at cpx-out, however, decreases with increasing pressure as a result of changes in pyroxene composition. At equal F, the productivity decreases with increasing Po as a result of decreasing (
T/P )F (Asimow et al., 1997).
The productivity functions (–dF/dP) shown in Fig. 4a and b are integrated to produce the F vs P plots in Fig. 4c and d for the MELTS and Langmuir et al. (1992) models. The mean F and mean P for variable-Po systematics [i.e. using (6) and (7) and integrating to Pf = Pc] are shown by a filled circle along each F vs P curve. Comparing these plots shows three important differences between these models. First, because of the low productivity in the early stages of melting according to MELTS, there is a ‘tail’ 1–2·5 GPa wide in which F remains low. Second, as the shape of the F–P curve in the Langmuir et al. (1992) model is almost independent of Po, the mean melt fraction, FB, increases almost linearly with Po to values of at least 0·22. In the MELTS model, however, there is a maximum in FB at 0·08. This behavior is explored in more detail below. Third, the maximum F achieved by the Langmuir et al. (1992) model increases essentially without bound, whereas fractional melting with MELTS would require extraordinarily deep Po and high TP to reach Fmax more than a few percent higher than that achieved at cpx-out. Instead, in the MELTS model the lengthening of the tail, the decrease in F at cpx-out, and the overall lower productivity at equal F associated with increasing Po all combine to yield Fmax in the narrow range 0·18–0·22 at the base of the crust over the wide range in Po of 2·7–4·4 GPa (Fig. 4c).
The extent of melting functions shown in Fig. 4c and d are integrated to yield the 
FdP curves in Fig. 4e and f. For these plots, the upper limit of integration is varied along the P-axis from Po to Pc to generate the curves shown. For passive flow models that aggregate all liquids to form the crust, this integral is equal to the pressure at the base of the crust and so is related to crustal thickness by a simple density correction. Assuming a constant crustal density of 2·62 g/cm3, we obtain crustal thickness in kilometers according to the formalism of Klein & Langmuir (1987). These figures, like the F–P plots, show that any given crustal thickness is produced by a melting path with much higher Po according to MELTS than according to the Langmuir et al. (1992) model. For example, 7 km of crust results from an adiabat with Po 2·1 GPa in the Langmuir et al. (1992) model but requires Po 2·8 GPa according to MELTS. These figures also show that the Langmuir et al. (1992) model can readily generate crustal thickness of 30 km or more. MELTS, on the other hand, at least for the passive-flow case, never achieves values greater than 15 km within the range of Po limited by the TP maximum on the solidus discussed above (active flow and other means of generating more crust are discussed below).
Variable-Po systematics
Figure 5 presents the net melt production of melting regimes at the final pressure of melting, whereas Fig. 4 shows the evolution of melting with pressure through each melting regime. Results are shown for passive-flow fractional melting calculations with variable-Po systematics: Pf is adjusted to be equal to the base of the crust for each Po (i.e. melting stops in each column at
, any model with a linear solidus (in P–TP space) and constant productivity will yield linear relationships among TP, Po, , FB, and (Zc)1/2. The productivity variations in the model of Langmuir et al. (1992) are small enough that in Fig. 5 the relationships for this model are indistinguishable from straight lines for and FB and parabolas for Zc when plotted against both Po and TP. The model of Kinzler (1997) assumes constant productivity, also resulting in linear relations among Po, , FB, and (Zc)1/2 (Fig. 5d–f), despite predicting a mildly curved solidus that leads to weakly curved trends in these variables against TP (Fig. 5a–c). There is no simple way to combine plagioclase-, spinel-, and/or garnet-bearing calculations using the Kinzler (1997) model, so results are shown only for melting paths with residual spinel everywhere.
Results of MELTS calculations for both the Cr-bearing and Cr-absent source compositions are shown in Fig. 5; the results for both compositions are similar in all important respects (i.e. the following discussion is not sensitive to details of spinel stability). When plotted against potential temperature, the MELTS results and the models of Langmuir et al. (1992) and Kinzler (1997) agree in many respects in the ‘normal’ range of potential temperatures (i.e. 1300–1400°C), but differ for anomalously hot or cold mantle. When plotted against Po, MELTS predicts lower FB and Zc for all Po for reasons discussed above, but primarily because MELTS generates a low-productivity ‘tail’ near Po (Hirschmann et al., 1994; Asimow et al., 1997); hence for Po 1·5–3 GPa, MELTS calculations mimic a constant-productivity model with Po at least 0·5 GPa lower. The differences for abnormally cold conditions are not surprising given the novel behavior predicted by MELTS for low F (which shows up most strongly in integrated melts with low FB) and the influence of the spinel–plagioclase transition. At hotter than normal potential temperatures, the FB value attained by MELTS for increasing Po flattens and reaches a maximum for Po 3·5 GPa, which in turn leads to a decreasing slope of the crustal thickness vs solidus pressure curve beyond Po 3·0 GPa: that is, from simple 2D passive-flow fractional melting regimes of this type with a well-defined Po, MELTS cannot generate crustal thickness above 15 km, regardless of TP.
For solidus pressures greater than 1·7 GPa, i.e. potential temperatures higher than 1280°C (boxed 1 in Fig. 5), plagioclase does not appear in the residue during fractional melting (this value of TP is lower than the limit for plagioclase to appear on batch melting adiabats; boxed 5 in Fig. 1). As solidus pressure decreases from 1·7 GPa (TP < 1280°C, boxed 1 in Fig. 5), the spinel–plagioclase peridotite transition plays an increasingly important role in modifying the amount of melt produced and the pressure range over which melt production occurs. In a simple passive flow model, this transition first divides the melting region into two disconnected regions, an upper triangle and a lower trapezoid (Asimow et al., 1995). With falling potential temperature, the bottom of the triangle retreats upward and the top of the trapezoid retreats downwards, i.e. the transition shuts off melting at deeper levels and melting resumes at shallower levels (see solidus in Fig. 1) as Po and TP decrease. This effect is manifested in Fig. 5d as a turnaround in mean pressure of melting (i.e. the loss of the shallower parts of the melting region results in increasing mean pressures of melting with falling potential temperature or solidus pressure in this range). At a certain critical value of TP and Po (the kink labeled with boxed 2 in the Fig. 5 curves at TP 1200°C and Po 1·3 GPa), the upper triangle of the melting regime disappears; i.e. the pressure at which melting in the plagioclase field would resume becomes shallower than the minimum pressure of melting at the base of the crust. With further falls in TP, the lower trapezoid shrinks and finally disappears at TP 1100°C (the cusp on the solidus, point labeled with boxed 1 in Fig. 1); lower values of TP produce no melt. The details of this low-TP behavior depend on the particular near-solidus relations predicted by MELTS, but general features such as the division of a fractional melting regime into disconnected parts are a necessary consequence of the thermodynamics of plagioclase formation (Asimow et al., 1995).
For