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[1] This study explores the thermodynamics of adiabatic decompression melting of peridotitic mantle containing pyroxenite veins that have lower solidi than the peridotite. When a vein of lower solidus-temperature material melts adjacent to more refractory material, additional heat will flow into the melting region to increase its melting productivity. If pyroxenite veins have a solidus-depletion gradient ((∂T^{m}/∂F)_{P}) like that of olivine or peridotite, then the melting of the veins is enhanced by up to a factor of 4 by this heat. However, the solidus-depletion gradient of pyroxenites is apparently lower than that of peridotites; thus pyroxenite melting would be even more enhanced. If pyroxenite veins have a gentler solidus-pressure (T-P) dependence (i.e., lower (∂T^{m}/∂P)_{F}) than that of peridotite solidi, then although these veins will experience enhanced melting while they are the only melting assemblage, they will stop melting soon after their peridotite matrix begins to melt. During large-scale peridotite melting the material ascends along a T-P path close to that of the peridotite solidus, so that the mixture's temperature remains lower than the solidus of the residual pyroxenite, and pyroxenite melting ceases throughout the shallower sections of the melting column. If pyroxenitic material makes up a large fraction of the mantle mixture (∼20%), then the heat consumed by deep pyroxenite melting cools the ascending mantle mixture enough so that peridotite melting begins ∼5–10 km shallower than it would in the absence of precurser pyroxenite melting. After recycling into the mantle, the melt extraction residue will again melt if it is reheated to ambient mantle temperatures and rises again to shallow depths.

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[2] The primary cause of terrestrial volcanism is pressure-release melting of upwelling mantle beneath mid-ocean ridges and hotspots. At upwelling rates greater than ∼10 mm/yr the only significant heat flow into or out of the melting region is the heat carried by escaping melt. Because of this, melting can be reasonably approximated as occurring within a thermally isolated domain, within which melt production takes place as an adiabatic (isentropic) process.

[4] A recent addition to these genres of study has been Hirschmann, Asimow, Stolper, and coworkers' application of a thermodynamically based petrologic modeling tool, MELTS [Ghiorso and Sack, 1995], to the study of adiabatic pressure-release melting [Asimow et al., 1995, 1997; Hirschmann et al., 1999a, 1999b, 1998]. As a complement to studies using the MELTS program to study mantle melting, here I analyze the thermodynamics of pressure-release melting within a lithologically veined mantle (see Figure 1), building in particular upon prior work by Sleep [1984] and Hirschmann and Stolper [1996].

2. Mantle Source: A Veined Plum Pudding?

[5] Mid-ocean ridge and hotspot basalts are isotopically heterogeneous; thus their source must be too. This heterogeneity can be produced by subducting oceanic basalts and continental sediments into the mantle, and then stirring, folding, and mixing in these heterogeneities by mantle convection. The resulting marble cake or plum pudding mantle [Allegre and Turcotte, 1986; Polve and Allegre, 1980; Sleep, 1984; Zindler et al., 1984] can produce isotopically distinct magmas if various mantle components can be preferentially and variably tapped during melting [Phipps Morgan, 1999; Phipps Morgan and Morgan, 1999; Sleep, 1984; Zindler et al., 1984].

[6] Orogenic lherzolites such as the Ronda and Beni-Bousera peridotite massifs were the original geologic inspiration for a plum pudding mantle [Polve and Allegre, 1980]. Typically, these massifs have centimeter- to decimeter-thick isotopically enriched pyroxenite veins embedded within a mostly depleted peridotite matrix (see Figure 1) [Pearson et al., 1993; Polve and Allegre, 1980; Reisberg and Zindler, 1986]. While most commonly found in former continental collision zones, similar lherzolite/pyroxenite assemblages have since been found to outcrop at both ridge and transform segments of the Mid-Atlantic Ridge near St. Paul's Rock [Hekinian et al., 1998], which itself is an uplifted peridotite/pyroxenite assemblage. At present it seems plausible to infer that a veined lherzolite/pyroxenite assemblage is at least a good candidate to represent the lithologies residual to partial melting of the upper mantle. The matrix itself may be a layered package of variably depleted lherzolite–harburgite–dunite compositions.

[7] Vein compositions and abundances are likely to vary regionally within the mantle. Relics of subducted ocean island basalts, mid-ocean ridge basalts, and continental and marine sediments will likely have differing trace, isotope, and major element chemistries, as well as different mean ages and abundances within a given region of the mantle. Mantle “stirring” will be locally more complex and more efficient if plum and matrix lithologies have differing rheologies, so that weaker components act as loci for shear zones and “ductile cracks” in the evolving gneissic mixture. However, subsolidus convective stirring is unlikely to chemically homogenize the mantle mixture at a mineral scale [Hofmann and Hart, 1978], and thus convection is likely to lead to a veined peridotite/pyroxenite mantle containing fine-scale lithological variations in major element, trace element, and isotopic composition.)

3. Melting Assumptions

[8] How will a pyroxenite-veined lherzolite melt? Since its component pyroxenite and peridotite lithologies have different major element/mineral compositions, in general, they should melt at different P-T conditions (Figure 1) [Hirschmann and Stolper, 1996; Sleep, 1984]. This study develops a theoretical thermodynamic treatment for the adiabatic pressure-release melting of a finely layered plum pudding mantle. Fractional melting is considered; it is modeled as sequential small increments of melting followed by complete melt escape. Each melt increment is generated along a reversible adiabatic melting path.

[9] In contrast to many petrologic formulations, additional mantle is assumed to flow into the melting region to replace any mantle that is removed by partial melting and melt escape [Cordery and Phipps Morgan, 1993; Parmentier and Phipps Morgan, 1990]. This is a common assumption in numerical models of mantle flow and melting. It means that melt production is defined as the fractional melt per unit mass (or mole atoms) that melts, as opposed to per unit mass of the starting material.

[10] Plums and matrix are assumed to melt in chemical isolation and thermal equilibrium. This is compatible with but not required by existing field data on oceanic and orogenic peridotites. In the field, neighboring pyroxenite veins and peridotite matrix are often in isotopic disequilibrium, suggesting little chemical exchange between the veins and host rock [Pearson et al., 1993; Polve and Allegre, 1980; Reisberg and Zindler, 1986]. However, a “plum melt” in contact with surrounding peridotite could react corrosively with the peridotite to generate further peridotite melt. As long as this reaction occurs rapidly enough so that the bulk of the peridotite is still in disequilibrium with the melt, then the field data could still be satisfied. (In essence, plum melting + melt-peridotite reaction/assimilation would be chemically similar to the mechanical mixing of separate plum + peridotite melts.)

4. Length Scale of Thermal Equilibrium During Melt Extraction

[11] In what follows, frequent use will be made of the simplifying assumption that the mantle is “thinly” enough layered so that conductive lateral heat transport keeps all vein and matrix lithologies in thermal equilibrium during partial melt extraction. For plausible mantle ascent rates, if veins are less than ∼100–500 m thick, then they and their host matrix will remain in thermal equilibrium [Sleep, 1984]. Sleep [1984] showed this by solving numerical melting equations for a few idealized geometries of the veins and host matrix. A simple scaling argument corroborates this aspect of his numerical results and also shows the upwelling rate dependence for the length scale of local thermal equilibrium. Latent heat is consumed by melting. Thus, to maintain thermal equilibrium between melting veins and their host matrix, additional heat must flow laterally from the host rock into the veins. The length scale L for upwelling and latent-heat loss through melt production in a time dt is L = w · dt, where wis the ascent rate (in m/s). In the same time interval dt, heat conduction can transport heat over a characteristic length scale L^{2} = κ · dt, where κ is the thermal diffusivity (in m^{2}/s). Conductive heat transfer will maintain thermal equilibrium between the veins and host rock as long as the length scale of conductive heat transport is larger than that for advective heat transport. This length scale can be estimated by equating the two relations

or

For an ascent rate w = 30 mm/yr ≈10^{−9} m/s, and a typical mantle thermal diffusivity κ = 10^{−6}m^{2}/s, the length scale for lateral thermal equilibration by heat conduction is L ≈ 1 km. Sleep's [1984] explicit numerical calculations found for an ascent rate of 30 mm/yr, a 220-m diameter spherical “plum” remained in thermal equilibrium with its host-matrix, while a 2.2-km diameter spherical plum did not, in good agreement with this analysis. Upwelling 10 times faster decreases the lengthscale for thermal equilibrium by a factor of 10, and upwelling 100 times faster (w = 3 m/y) reduces the equilibration-length scale to ∼10 m. All of these length scales are significantly larger than the centimeter to meter thickness of pyroxenite veins in the mantle outcrops at St. Paul, Ronda and Beni-Bousera. Thus individual lithologies undergoing partial melting in this type of veined mantle are quite likely to remain in thermal equilibrium with their neighboring lithologies. (When this assumption is invalid, it will lead to an overestimate of the partial melting of the easy to melt lithologies and an underestimate of the melt extraction from more refractory lithologies.)

5. Thermodynamics of Pressure-Release Melting

[12] The following sections develop the thermodynamics of pressure-release melting of a veined mantle by exploring increasingly complex scenarios. We start with adiabatic pressure release without melting. This introduces several thermodynamic variables central to all following melting scenarios and also lets us assess the amount of adiabatic cooling present in all subsequent scenarios.

5.1. Thermodynamic Variables

[13] The independent variables that most simply determine the thermodynamic state of upwelling mantle are its pressure P and entropy S. Entropy is conserved during adiabatic ascent. In comparison to pressure, entropy is a more abstract variable. It often is introduced as a measure of the “disorder” of a system, which for minerals may have contributions from material defects, solid-solution substitution of atoms within crystal lattice sites, and thermal vibrations (here this last source of entropy will be sometimes called the “thermic entropy” of a material). If a mineral assemblage does not change composition during ascent (i.e., does not melt, does not undergo solid-solution transfer between minerals, and does not undergo a solid-state phase transition), then the “compositional” contributions to the entropy do not change and can thus be ignored. In this typical case, the equilibrium (or “reversible”) change in entropy δS is equal to the change in heat δq divided by its temperature T(°K), i.e., δS = δq/T. The change in the internal heat of a material is defined as the product of its heat capacity c_{P} (Jkg^{−1}°K^{−1} or Jmol^{−1}°K^{−1}), and its temperature change δT, δq = c_{P}δT. Thus the entropy change for a temperature change δT at constant pressure can also be expressed as δS = c_{P}δT/T, so that the units of entropy are those of heat capacity. The change in entropy with temperature at constant pressure is

where the parentheses around the partial derivative and subscript P are the notational convention used to show that the partial derivative is made keeping the subscripted variable P fixed. The change in entropy with pressure at constant temperature is

where α is the thermal expansivity (°K^{−1}, also called the “coefficient of thermal expansion”) and ρ is the density (kg/m^{3}).

5.2. Adiabatic Mantle Temperature Gradient

[14] These thermodynamic variables and relations are all that is needed to determine the adiabatic temperature gradient in upwelling mantle. Assume a parcel of mantle ascends adiabatically, which means that is undergoes a pressure drop dP without any heat flowing into or out of the parcel. Since its net entropy does not change, its thermal entropy contribution must decrease (i.e., its temperature drops by dT) to balance the entropy effect of the decrease in confining pressure:

Rearranging to solve for the adiabatic temperature–pressure gradient (dT/dP)_{S} yields

Often it is convenient to express the adiabatic temperature gradient in terms of depth instead of pressure. Substituting P = ρgz into (5), where ρ is the mantle volumetric density, g is the gravitational acceleration, and z is depth, yields the adiabatic temperature-depth gradient

For a typical coefficient of mantle thermal expansion α = 3 × 10^{−5}°C^{−1}, mantle temperature T ≈ 1600°K, mantle heat capacity c_{P} ≈ 1200 Jkg^{−1}°C^{−1}, and g = 9.8 m s^{−2}, the predicted mantle adiabatic temperature gradient is roughly 0.4°K/km or 12°K/GPa. (The mantle's heat capacity is very close to the Dulong–Petit value of 3R (Jmol^{−1}°C^{−1}), where the ideal gas constant R 8.314 Jmol^{−1}°C^{−1}. Evaluated for either the Pyrolite [White, 2001; Zindler and Hart, 1986] Ca_{0.06}Al_{0.11}Fe_{0.15}Mg_{1.3}SiO_{3.7} or Losimag [White, 2001; Zindler and Hart, 1986] Ca_{0.08}Al_{0.10}Fe_{0.14}Mg_{1.2}SiO_{3.6} mantle (molar) compositions, each of which has a molecular weight of ∼0.021 kg/mol, the mantle heat capacity would be 1187 Jkg^{−1}°K^{−1}.) In contrast to entropy, note that the energy of rising mantle is not locally conserved during adiabatic ascent. Instead, some energy goes into pressure–volume work as the rising parcel expands during decompression. This effect decreases the internal energy of the rising parcel while increasing the gravitational energy of the overlying mantle by a corresponding amount.

5.3. Adiabatic Partial Melting During Ascent of a Single Lithology

[15] Adiabatic partial melting of a single lithology has been recently studied by Asimow, Hirschmann, Stolper, and coworkers [Asimow et al., 1997; Hirschmann et al., 1999a, 1999b, 1998]. Next will follow an alternative derivation that is more easily generalized to the fractional melting of a veined plum pudding mantle. We start by assuming that the process may be approximated as reversible and entropy conserving. To solve for the thermodynamic behavior of this system, consider a single increment of partial melting. The entropy change associated with partial melting of a mass fraction dF needs to be added to the other terms in (4); i.e.,

The compact form of (7) hides many subtleties that are further discussed in the batch melting treatment of Asimow et al. [1997]. The following is a brief summary of the different effects that contribute to the dS/dF term. The configurational entropy difference between a solid lithology and its coexisting melt is the entropy of fusion or entropy of melting δS^{m}. It is closely related to the more familiar latent heat of melting (or heat of fusion) δH^{m}; for a pure substance δH^{m} = T^{m}δS^{m}. The proper generalization ofδS^{m} to solid solutions is ambiguous, because the transformation between solid and liquid states not only has a solid to melt entropy change but also a change in the solid-solution entropy of the coexisting solid and melt. Here I will define δS^{m} = δH^{m}/T^{m} to include both of these effects.

[16] For typical solid-solution lithologies the solidus temperature of the residue to melt extraction increases with progressive melt extraction and decreases with decreasing pressure. (The total amount of melt extraction from the solid residue is also referred to as its depletion.) The increase in a lithology's solidus temperature with increasing depletion/melt extraction F, (∂T^{m}/∂F)_{P}, will here be called its solidus-depletion gradient. The increase in solidus temperature with depletion leads to an increase in the thermal entropy of the residual solid:

[17] Solid-solution effects also change the configurational (or compositional) entropy of the residue from melt extraction. These effects are easy to formally include by adding analogous terms for the change in configurational entropy

However, solid-solution effects on compositional entropy will not included in the following analysis because these effects are much smaller than the effects that are included. (The relative unimportance of these effects on silicate melting are further discussed by Asimow et al. [1997] and Hirschmann et al. [1999a]).

[18] There is another thermodynamic constraint while a parcel of mantle undergoes partial melting: the change in the upwelling mantle temperature during partial melting is equal to the change in solidus temperature due to a drop in pressure dP and an increase in depletion dF:

Substituting this relation into (7) and considering the entropy of fusion and solidus-depletion effects leads to

This can be simplified for the productivity dF/dP:

Equation (12) can be substituted into (9) to find the temperature profile within the melting region:

5.4. Depth-Dependent Melt Productivity and Melt Production Rates

[19] In computational practice one often wishes to know the melt productivity as a function of depth (dF/dz) instead of pressure. This is found by substituting dP = ρgdz into (11), which yields

[20] The instantaneous melt production rate dF/dt is also frequently used in numerical experiments of mantle flow and melting. This quantity is related to the local upwelling rate w and the depth form of the melt productivity dF/dz by

5.5. Effect of the Solidus-Depletion Gradient on Melt Productivity

[21] The productivity dependence of adiabatic fractional partial melting is like that of a solid-state phase transformation (see Appendix C), plus one additional term. The additional term in the denominator of (11) contains the isobaric solidus-depletion gradient (dT^{m}/dF)_{P}. It reflects the increase in thermic entropy of the residue lithology due to the melt-extraction-induced increase in its solidus temperature. In comparison to a phase transformation that does not induce a solid-solution increase in the solidus temperature of the residue, this effect acts to effectively increase the latent heat of the solid-melt phase transformation.

[22] The solidus-depletion gradient (dT^{m}/dF)_{P} can be measured from isobaric phase diagrams of T versus solid-solution composition X. For example, Figure 2 shows the geometric construction used to evaluate (dT^{m}/dF)_{P} from a T−X phase diagram for the melting of olivine at 1-atm pressure. Assuming near-fractional melting, the lever rule [cf. Hess, 1989; White, 2001] relates the increment of melt extraction δF to the change in solid-solution composition δX of the solid phase and the local width δX of the solid–liquid phase loop: δF = δX/δX. (These terms are graphically shown on Figure 2.) The slope of the solidus dT^{m}/dX = δT^{m}/δX can therefore be rewritten in terms of the increment of melt extraction: dT^{m}/dX = δT^{m}/(δFδX) or δT^{m}/δF = (dT^{m}/dF)_{P} = δX(dT^{m}/dX). In words, the increase in solidus temperature with melt extraction/depletion (dT^{m}/dF)_{P} is equal to the product of the local slope of the solidus curve as a function of composition (dT^{m}/dX)_{P} and the compositional difference δX between coexisting melt and solid. Figure 2c shows the size of the solidus-depletion gradient of olivine as determined from the experimental measurements of Bowen and Schairer [1935].

[23] Note that the temperature difference between the solidus and liquidus at a given composition (the melting interval) is not a direct measure of the solidus-depletion gradient. These are equal only when the solidus and liquidus are subparallel between the tie compositions of solid and melt, as sketched in Figure B1. To compare the relative sizes of the solidus-depletion gradient and melting interval, Figure 2c also shows the experimentally determined melting interval for olivine. For Mg-rich olivine compositions characteristic of the mantle the melting interval is less than the solidus-depletion gradient. Since this is a general outcome for phase loops that narrow upward like that of olivine, I think that the solidus-depletion gradient of peridotites and pyroxenites at a given composition will most likely be greater than their melting interval at that composition. This inference will be later used to estimate the solidus-depletion gradient for pyroxenites, as so far only their melting interval has been measured experimentally.

[24] The appendix of Asimow et al. [1997] presents a different derivation of (11) where they introduce an analogous term 1/(dF/dT)_{P} in place of the solidus-depletion gradient (dT^{m}/dF)_{P}. (They call the reciprocal term (dF/dT)_{P} the “isobaric productivity.”) For fractional pressure-release melting I think that the above derivation presents a simpler and more intuitive explanation for the origin and physical importance of the solidus-depletion gradient during pressure-release melting. It stresses that a key physical effect in partial melting is that the increase in solidus temperature of the residue allows it to store more entropy as heat. The advantage of isobaric productivity formulation is that it also generalizes to batch melting, where there is another important, but completely different type of effect associated with changing the composition of the preexisting batch melt.

[25] In fractional partial melting the predominant entropy effects are all thermal in origin. Thus (11) can also be derived by directly considering the heat balance involved in partial melting. Heat available to drive melting during an ascent dP comes from the temperature difference between the adiabatic gradient and the (compositionally dependent) solidus gradient: δq = ρc_{p}(dT^{m}(F,P)/dP − αT/ρc_{p})dP. The (latent) heat consumed by an increment dF of partial melt generation is δq = −ρδH^{m}dF. The change in solidus as a function of pressure and temperature is dT^{m}(F,P)/dP = dT^{m}/dP + (dT^{m}/dF)dF/dP. Equating the heat available to drive melting with the heat consumed by melting will lead to (11).

[26] What are the relative sizes of the various effects contained in (13)? First, consider the numerator of the right-hand side. A typical temperature-depth gradient (∂T^{m}/∂z)_{F} of a peridotite solidus is ∼100°–130°C/GPa or 3°–4°C/km (see Figure 1 or Figure 17 of Hess [1989]). Adiabatic expansion during ascent reduces the thermic entropy available for melt production. The temperature gradient associated with adiabatic expansion, −αgT/c_{P}, is roughly 0.4°C/km as estimated in the discussion after (6). Thus adiabatic expansion reduces the effective mantle solidus temperature-depth gradient available for melt production by ∼10–15% (from 3°–4°C/km to 2.6°–3.6°C/km).

[27] Now consider the denominator. The heat of fusion or enthalpy of fusion TδS^{m} of peridotite is not very well known experimentally. Hess (cf. Table 8 of Hess [1989]) estimates the heat of fusion of peridotite to be ∼660 kJ/kg (or 160 cal/g or 140 J/mol). For a mantle heat capacity c_{P} ≈ 1200 Jkg^{−1}°K^{−1} (see discussion of this estimate after (6)), the term TδS^{m}/c_{P} is ∼550°C for partial melting of mantle peridotite. In general, silicates have an entropy of fusion of ∼R per mole atoms (e.g., δS^{m} = 1.00–1.03 R for fayalite, diopside, and enstatite at their 1-atm melting temperatures [Stebbins et al., 1984]), and a heat capacity of ∼3R per mole atoms (Dulong-Petit), where R is the ideal-gas constant. Thus the term TδS^{m}/c_{P} should be about one third, in good agreement with Hess's more detailed estimate.

[28] The second term in the denominator, the isobaric solidus-depletion gradient (∂T^{m}/∂F)_{P}, is even more poorly known than the heat of fusion. As noted above, we can estimate the size of this term from T-X phase diagrams. (For example, the 1-atm olivine solidus-depletion gradient is shown in Figure 2c. Typical mantle olivine compositions have a predicted solidus-depletion gradient of ∼200°–350°C.) The size of the solidus-depletion gradient can also be crudely estimated from the variation with composition of different solidi curves for peridotites that have been studied in laboratory experiments. If this effect were constant (which it will not be if it has a typical phase-loop geometry like that shown in Figure 2), and if the 40°–50°C difference between the KLB-1 and HK66 T-P solidus curves (cf. Figure 17 of Hess [1989]) is caused by a 10–20% difference in the relative depletion of these two peridotites, this leads to an estimate for (∂T^{m}/∂F)_{P} ≈ 200°C–500°C. Unfortunately, this estimate is extremely crude. Current MELTS-based determinations of the solidus-depletion gradient have only been published for a single incremental batch melting path for a relatively fertile peridotite composition at constant 1-GPa pressure, temperature increasing [Hirschmann et al., 1999a]. In this case the first percent of melt extraction has an extremely high (∂T^{m}/∂F)_{P} = 10,000°C, associated with the rapid loss of the “incompatible” major element sodium. At 5, 10, and 15% depletion, (∂T^{m}/∂F)_{P} = 350°, 185°, and 175°C, respectively. After 18% depletion, clinopyroxene is exhausted, the “lherzolite” becomes a “harzburgite,” and the solidus-depletion gradient jumps dramatically to ∼2000°C. In the primary melting interval (after the initial low-T melting precurser) the estimated range in solidus-depletion gradient is close to the ∼200–350°C range predicted for mantle olivine compositions.

[29] Even a low-range estimate of the solidus-depletion gradient, (∂T^{m}/∂F)_{P} ≈ 200°C, is a significant fraction of the enthalpy of fusion term TδS^{m}/c_{P} ≈ 550°C. This size solidus-depletion gradient would decrease melt productivity by ∼30% with respect to the productivity predicted by using only the enthalpy of fusion as a guide to the entropy change associated with partial melting of this lithology. To sum up, the solidus-depletion gradient is expected to have a much larger effect than adiabatic expansion in reducing pressure-release melt productivity.

[30] If we treat the olivine (Mg,Fe)_{2}SiO_{4} system as an ideal solid solution, then the enthalpy of fusion and solidus-depletion gradient have relatively simple analytical solutions. (The analytical treatment of this system is presented in Appendix A.) Figure 3 illustrates the relative size of these effects during the adiabatic pressure-release melting of an idealized olivine solid-solution system. For laboratory values of the heats of fusion of the pure Fe_{2}SiO_{4} (fayalite) and Mg_{2}SiO_{4} (forsterite) end-member compositions of olivine, the experimental T-X phase diagram is fairly well matched by the ideal-solid solution theory. For the parameters that best fit the laboratory data (cf. Figure 2b and 2c.) the solidus-depletion gradient is a maximum of ∼38% of the enthalpy of fusion at a solid solution mass composition (Mg–Ol_{0.57},Fe–Ol_{0.43}). For more refractory (Mg-rich) compositions than these, the progressive decrease in the solidus-depletion gradient with increasing depletion means that the denominator in the relation for melt productivity will progressively decrease with continued melt extraction (since the heat of fusion remains relatively constant [Hess, 1992]). This suggests that for compositions more Mg-rich than ∼(Mg–Ol_{0.57},Fe–Ol_{0.43}) (mass), the productivity of pressure-release melting of an olivine solid-solution would progressively increase with continued ascent and melt extraction. However, this effect is relatively small in the examples of “ideal olivine” pressure-release melting shown in Figure 3. Because the total degree of melt extraction (<10%) is small, the olivine's composition and solidus-depletion gradient change very little.

[31] As long as mantle melting takes place near the high-temperature (Mg) end of (Mg,Fe) peridotite solid solutions, similar behavior is anticipated to occur for any peridotite system. Volatile extraction can also lead to similar behavior. Rapid extraction of volatiles during early melting would rapidly raise the solidus temperature of the peridotite residues to melt extraction. A large solidus-depletion gradient in the depth range of initial peridotite melting would lower productivity at these depths in comparison to shallower peridotite melt-productivity. (Note that the (∂T^{m}/∂F)_{P} slope from volatile extraction is likely to be larger than the above estimates of (∂T^{m}/∂F)_{P} for volatile-poor solid-solution systems. For additional insights I especially recommend following Hirschmann and coworkers discussion of an idealized ternary system with an incompatible trace element [Hirschmann et al., 1999a].

[32] For Mg-rich solid-solution compositions the width of the solid-solution phase-loops (in T-X cross sections of T-P-X space) progressively decreases with melt extraction from the individual (>Mg_{0.8}) olivine and pyroxene mineral components of peridotite. This effect tends to progressively decrease the size of the solidus-depletion gradient during continued melt extraction, which leads to increasing melt productivity until the point where the exhaustion of a key melting phase causes a further increase in the solidus-depletion gradient. (The heat of fusion T^{m}δS^{m} is also expected to slightly decrease with progressive melting because the entropy change δS^{m}(P,T,X) remains relatively constant (∼R per mole atoms), while the peridotite's solidus temperature is decreasing with decreasing pressure. Hess's explicit calculations suggest that the heat of fusion of peridotite changes relatively little during progressive melt extraction [cf. Hess, 1989, equation (12)]. The ideal olivine examples (see Figure 3 and Appendix A) have only a very small change in the solidus-depletion gradient and heat of fusion during their progressive melting, because the ideal olivine changes composition by only ∼3% during its progressive melting history.)

[33] The amount of pressure-release productivity is limited more by the heat consumed by melting than by the depletion of the olivine. For example, assume that the residual olivine after ascent and melting is recycled back into the mantle and reheated before a second ascent. Melting during the second ascent starts at a shallower depth due to the increased depletion of the remelting material. However, its productivity during the second ascent is about identical to that of the first ascent (see Figure 4).

[34] Note that the above analysis and the similar analysis presented by Asimow et al. [1997] only apply to the melting of a single lithology. These results cannot be extrapolated to the melting of lower solidus-temperature lithologies embedded within a higher solidus-temperature peridotite matrix. In particular, the melting behavior of any low-solidus temperature lithology within a plum + peridotite mixture is not expected to rapidly decrease with melt extraction. Next let's explore some of these plum-melting scenarios.

5.6. Lower-Solidus Plum Melting Within a Nonmelting Matrix

[35]Sleep [1984] first noted the potential importance of heat flowing in from neighboring nonmelting lithologies to increase melt production of a low-solidus temperature plum vein. More recently, Hirschmann and Stolper [1996] have also explored this effect, which is likely to greatly enhance the productivity of plum melting, although the bulk (plum + matrix) productivity will still be less than the productivity of a pure plum lithology at comparable P-T conditions. To see this, consider a system where a fraction ϕ_{1} of the mantle (the “plum fraction”) ascends and crosses its solidus temperature while the rest of the ascending mantle ϕ_{2} = (1 − ϕ_{1}) (the “matrix” fraction) remains below its higher solidus temperature. Assuming the melting process to be reversible and entropy conserving implies

The first of the pair of equations in (15) is analogous to (7) with the additional term dQ_{21}/T reflecting the change of entropy in the plum lithology ϕ_{1} due to heat flowing from the peridotite lithology ϕ_{2}. The second equation states the entropy change in lithology ϕ_{2} including conductive heat and entropy loss into the melting plum lithology. Formally, this pair of equations describes the thermodynamic behavior of a specific reversible, entropy-conserving thermodynamic system. The two lithologies are assumed to be thinly enough interlayered so that the plum and nonmelting matrix lithologies stay at the same temperature during melting (thermal equilibrium). This criteria implies that the interlayering should be finer than ∼100 m as discussed earlier. The melt phase is also assumed to remain in chemical contact with only the plum lithology (reversible melting). Since the system ϕ_{1} + ϕ_{2} ascends adiabatically, its net entropy change is zero, i.e., dS_{1} + dS_{2} = 0, which leads to

The change in temperature of the system is governed by the solidus behavior of the melting plum lithology ϕ_{1};

which is just (9) rewritten for the melting lithology. Now, for simplicity, assume that the physical properties of the two lithologies differ only in that one has a lower solidus temperature than the other, i.e., that ∂S_{1}/∂T = ∂S_{2}/∂T = ∂S/∂T and ∂S_{1}/∂P = ∂S_{2}/∂P = ∂S/∂P. Equation (16) then simplifies to

which can be solved for dF_{1}/DP using the same algebraic steps as for (11):

The form of this solution is the same as that of (11) for the melting of a single lithology except that the heat of fusion term in the denominator is reduced by the fraction ϕ_{1} of the melting plum lithology. The effect of the surrounding nonmelting matrix fraction ϕ_{2} is to greatly increase the melting productivity of the plum lithology ϕ_{1}, because heat from the surrounding matrix flows in to add to the heat available for melt production.

[36] Note that for a small plum fractions,the rate-limiting term for melt production from the plum lithology depends only on the solidus-depletion gradient of the plum lithology, since the effect of the plum's heat of fusion is tiny compared to the heat available from the surrounding matrix (see Figure 5). If the plum has a solidus-depletion gradient like that of a peridotite or an olivine, then its solidus-depletion gradient is likely to be ∼1/3 to 1/2 as large as its heat of fusion (Figure 5), so that the productivity of the plum lithology would be ∼3–4 times faster than the typical productivity from pressure-release melting of peridotite. If the plum has a smaller solidus-depletion gradient than peridotite, then its productivity would be even higher (Figure 6). For small plum fractions the productivity of plum melting strongly depends on the plum's solidus-depletion gradient but only slightly depends on the actual plum abundance (see Figure 6). This suggests that small but variable-volume fractions of plum “veins” will experience similar melting behavior in similar T-P environments. Unfortunately, the solidus-depletion gradients of pyroxenites and eclogites are even more poorly known that that of peridotites. The small melting interval (∼100°C) of many eclogites [Yoder, 1976, p. 102] suggests that pyroxenites have a smaller solidus-depletion gradient than peridotites. Note that if the plum lithology has a solidus-depletion gradient like that of peridotite, then the plum is not expected to ever completely melt. Instead, it will melt at a high productivity until depths where the matrix exceeds the solidus temperature of the residual plum lithology.

[37] The bulk or aggregate productivity can also be found from (19), i.e.,

While the plum melts more rapidly than it would without the additional heat from the surrounding matrix, the aggregate productivity is actually less than if the entire assemblage were melting at these P-S conditions. (Compare (20) with (11).) This happens because the effect of increasing the plum's solidus temperature with progressive depletion allows the entire parcel (both plums and matrix) to absorb more entropy as heat, thus magnifying this inhibition to melt production. However, the plum residues remaining after pressure-release melt extraction are likely to have undergone much higher degrees of melt extraction than their surrounding peridotite.

[38] Finally, the temperature of the ascending two-lithology system can be calculated by substituting (19) into (17):

For small plum fractions the first (latent heat of melting) term in the denominator of (21) approaches zero, so that the P-T path approaches the adiabat. Examples of temperature-depth-productivity-aggregate productivity plots are shown in Figure 6b for the behavior of (19) with plum fractions from 1 to 10%.

5.7. Plum Melting Within a Melting Matrix

[39] The final idealized example that will be considered here is the simultaneous melting of both the plum and matrix lithologies of a plum pudding mantle. (At the end of this example this result will be extended to the more general n lithology plum pudding mantle.) Again, the important conceptual and model simplification is that lithologies ϕ_{1} and ϕ_{2} = 1 − ϕ_{1} remain in thermal equilibrium during melting: dT_{1} = dT_{2}, where the subscript refers to the property of either lithology 1 or lithology 2. (When multiple lithologies melt, it is most convenient to order them so that lithology 1 has the steepest solidus-pressure gradient, lithology 2 has the next-steepest solidus-pressure gradient, and so on.) In this case,

Since dT_{1} = dT_{2} (thermal equilibrium)

(Note that for the fractional melting considered here, dF_{2} cannot be negative. Instead, if dF_{2} were negative, then this lithology would simply not melt and instead would act as a “thermal bath” for the neighboring melting lithology as discussed in the previous section. A geometric criteria for this relation is that the temperature-pressure gradient of the system must be smaller than the solidus-pressure gradient of any melting lithology. If it were larger, then the lithology would not melt.) Considering the entropy changes of two lithologies undergoing melting leads to a straightforward extension to (15):

Once again, since the system ϕ_{1} + ϕ_{2} ascends adiabatically and reversibly, its net entropy change is zero, i.e., dS_{1} + dS_{2} = 0. This substitution into (24) leads to

which, with further substitution of (23) and rearranging in terms of dF_{1} and dP yields

where the 1 or 2 subscript or superscript refers to the first or second lithology. Equation (23) is used to solve for the other melt productivity function:

and the temperature gradient within the region where both lithologies are melting is given by

Note that the last relation in (26) is written in a form that can be readily generalized to n lithologies (with ϕ_{1} + ϕ_{2} + … + ϕ_{n} = 1). For example, if we assume that the n lithologies only differ in their physical properties related to melt extraction, i.e., that c_{P}^{1} = c_{P}^{2} = … = c_{P}^{n} = c_{P} and α_{1}/ρ_{1} = α_{2}/ρ_{2} = … = α_{n}/ρ_{n} = α/ρ, then (26) simplifies to

[40]Equations (26) and (29) have two apparent differences from the simpler (19) that describes the melting of a single plum lithology within a nonmelting host. The numerator of (26) and (29) has a positive latent-heat term reflecting the fact that the interval between the steepest solidus-pressure slope and the next steepest slope contains heat that is only available to enhance melting of the lithology with the steepest solidus-pressure slope. In the denominator of (26) and (29), the latent heat terms are multiplied by a ratio of solidus-depletion gradients. This ratio reflects the potential variations in melt productivity caused by differences in the solidus-depletion gradients of the melting lithologies.

5.8. Examples of Melting a Veined Peroxenite–Peridotite Mantle

[41] To better show possible consequences of pressure-release melting of a veined mantle, several illustrative calculations have been made using (29) for the experimentally determined peridotite and pyroxenite solidi shown in Figure 1b. The information given in Figure 1b is incomplete to describe the melting of a veined mantle, as the solidus-depletion gradient also needs to be known. In these examples the solidus-depletion gradient of peridotite is assumed to be a constant 250°C. Pyroxenites are assumed to have solidus-depletion gradients of 150°C except where explicitly noted. The peridotite and pyroxenite solidii shown in Figure 1 are given in Table 1 (the number given to each pyroxenite lithology is the number assigned by Hirschmann and Stolper [1996] in their original compilation of this data).

Table 1. Thermodynamic Properties of the Pyroxene and Peridotite Lithologies Used in the Example Calculationsa

Pyroxenite #

T^{m}(P = 0), °C

dT^{m}/dP, °C/GPa

dT^{m}/dF, °C

a

Pyroxenite numbers refer to the solidi listed in Figure 1b. Solidus temperature parameters are measured from Figure 1b. The solidus-depletion gradient for peridotite is estimated to be 250°C. This value is consistent with estimates made from ideal olivine and from experimental measurements of peridotite solidi as a function of depletion. The solidus-depletion gradient of pyroxenites is extremely poorly known but is assumed to be ∼50% larger than the measured melting interval of pyroxenite/eclogite materials. See text for further discussion.

7

867

200

150

5

1047

71

150

13

1091

122

150

14

1151

71

150

Peridotite (fertile)

1081

132

250

Peridotite (refractory)

1136

132

250

[42] The first example is shown in Figure 7. This is a comparison between the melting of a mixture of finely layered peridotite lithologies with different initial depletions from 0 to 22% (mean depletion 11%) and the melting of a single peridotite lithology of uniform 11% depletion. The least depleted lithologies of the peridotite mixture have the lowest solidi and are the first to melt. Their melting is enhanced by heat flow from the more refractory, nonmelting, lithologies. As the mixture ascends to pressures where other lithologies cross their solidi, the melt productivity of each melt lithology decreases since proportionally less heat is flowing in from the nonmelting lithologies. Once the entire peridotite is melting, each lithology melts with the same (average) productivity. Melting of the easier to melt lithologies cools the ascending mixture so that the 11% depleted lithology begins to melt at a shallower pressure than it would for pressure-release melting of a uniformly 11%-depleted peridotite. However, the total degree of melt extraction (or depletion) is almost exactly the same for the two cases: the increased deep productivity of the easier to melt peridotite lithologies is almost exactly counterbalanced by the reduction in the productivity of shallower melting due to the heat extraction by early deep melting.

[43] The second example (see Figure 8) is a comparison between melting the previous variably depleted peridotite and a similar peridotite hosting a 19% fraction of pyroxenite veins with a solidus like pyroxenite 14 in Figure 1 and a solidus-depletion gradient of 150°C. During ascent the pyroxenite lithology is the first to cross its solidus. It begins to extensively melt owing to heat flow from the adjacent peridotite. Heat loss to enhanced pyroxenite melting results in the peridotite-mixture crossing its solidus at a lower pressure than it would if no deep pyroxenite melting had occurred. Once a significant fraction (∼40%) of the peridotite is melting, then the temperature-pressure gradient becomes shallower than the solidus-pressure gradient of the pyroxenite so that pyroxenite melting stops. The remaining nonmelting pyroxenite now acts as a source of additional heat to enhance peridotite melting.

[44] In the third example shown in Figure 9 the 19% pyroxenite vein fraction is assumed to consist of a suite of different pyroxenites with the different solidi seen in Figure 1b. The pyroxenite mixture consists of pyroxenite 14 (10% of peridotite volume), pyroxenite 13 (5% of peridotite volume), pyroxenite 5 (3%), and pyroxenite 7 (1%). In this example both the pyroxenite 7 with the solidus-pressure gradient steeper than peridotite and the pyroxenite 5 with the extremely low solidus melt to exhaustion. The other pyroxenites undergo significant melting, while peridotite melting is reduced (relative to that of a pure peridotite mixture) as in the previous example. In this example, 49% of the pyroxenites and 25% of the peridotites melt, so that pyroxenitic material provides ∼27% of the aggregate pooled melt even though it makes up only 19% of the starting mixture.

[45] The final example shows the potential effects of the garnet–spinel peridotite and spinel–plagioclase peridotite phase transformations on pressure-release melting (compare Figures 9 and 10). In this example the same peridotite–pyroxenite mixture is used as in the third example, now also including the effects of solid–solid peridotite phase transitions described in Appendix C. These solid–solid phase transitions cool the mantle so that melting is suppressed in a narrow depth-interval above each phase transition (see Figure 10.) When pressure-release melting resumes, the steepest solidus-pressure lithologies are the first to resume melting. This effect could result in a wide compositional range of melts being produced near the depth interval of each solid–solid peridotite phase transition. (Note that if the aluminous minerals of the peridotite melt to near exhaustion before reaching the depth of the spinel–plagioclase peridotite phase transformation, then this phase transition may not be experienced by the peridotite fraction of the ascending mixture [Asimow et al., 1995]). Overall, solid-state peridotite phase transformations have little effect on the overall pressure-release productivity of the ascending mantle mixture. In this example they reduce the total melt extraction by less than 2%. Note that the peridotite-melt and solid-state peridotite phase boundaries are slightly inconsistent in this example. The T-P slope of the peridotite solidus should slightly decrease after crossing the depth of each solid-state phase transition. (See Figure 2–10 of Yoder [1976]) for an example of a correctly drawn peridotite phase diagram.) Although conceptually important, this effect is quantitatively so minor that it was not included in the peridotite solidi estimates summarized in Figure 1.

[46] Several points can be drawn from these examples. 1. Perhaps the most paradoxical implication of plum melting is that if the plum-solidus has a gentler T-P path than that of the matrix (e.g., case pyx-5 in Figure 7c), then the plum phase assemblage stops melting once the mantle ascends to depths where the peridotite begins to melt to a significant degree. This occurs because the T-P path of the melting peridotite has a steeper slope than that of the solidus of the residuum pyroxenite. So much heat is consumed by peridotite melting that the pyroxenite residuum remains below its solidus during further ascent. If the pyroxenite solidus-pressure gradient is only a slightly gentler than that of the peridotite, then pyroxenite melting will persist, but at a greatly reduced rate, once peridotite melting begins (see Figures 7–10.). 2. If lower solidus-temperature plum components make up a significant (>∼15%) fraction of the ascending mantle, then their melting consumes enough heat to noticeably reduce the initial depth of melting of the more refractory components (see Figures 7–10). In contrast, if the melting plum components make up only a few percent of the ascending mantle, then the mantle ascends essentially along its nonmelting adiabat until crossing the solidus of volumetrically important melting components (see the high-pressure melting region of Figures 9–10.). 3. The solid-state garnet–spinel and spinel–plagioclase peridotite phase transformations will have little effect on the overall pressure-release productivity of the mantle. They may have an observable effect by creating a wide spectrum of melt compositions if different lithologies sequentially resume melting after the narrow depth interval where melting is significantly reduced by a solid-state phase transition. 4. The melt productivity of a small fraction of melting plum material is significantly enhanced by additional heat conducted from neighboring nonmelting lithologies [Hirschmann and Stolper, 1996; Sleep, 1984]. This additional heat loss from enhanced low-solidus plum melting cools the ascending mixture so that melting of more refractory lithologies is inhibited in comparison with the case where no low-solidus plum lithology is available to melt.

6. Summary Discussion

[47] The above examples show that melt productivity is strongly enhanced in low-solidus plum lithologies within a mantle that is veined on a ∼500 m or smaller length scale. The additional heat from the surrounding matrix can lead to ∼3–4 times higher productivity for an embedded lower-solidus-temperature plum lithology if its solidus-depletion gradient is of order 250°C, similar to that of peridotite. The productivity of a small volume fraction of low-solidus plum material will be enhanced by up to approximately fivefold if its solidus-depletion gradient is of order ∼150°C, a value consistent with the ∼100°C melting interval of pyroxenites. It is intriguing that this simple analysis predicts a roughly three- to fivefold increase in plum productivity. A similar fourfold increase in productivity of embedded pyroxenite veins was previously estimated by Hirschmann and Stolper [1996], using Sleep's [1984] thermal analysis of a similar veined melting situation but with a different theoretical treatment for the melting behavior of pyroxenite. These estimates roughly coincide with the estimate from a recent evolution model for a plum pudding mantle that appears to grossly fit the present-day trace element and isotopic characteristics of ocean-island basalts, mid-ocean ridge basalts, and average continental crust [Phipps Morgan and Morgan, 1999]. This evolution model favors a roughly threefold higher productivity of plum melting with respect to peridotite melting. All of these results reaffirm Sleep's basic intuition that low-solidus plum melting will be significantly enhanced by conductive heat flow from neighboring nonmelting material. The same effect also enhances the productivity of the more fertile (low-solidus) fraction of the peridotite matrix.

[48] Within a veined mixture, pressure-release melting tends to most strongly melt the phases with the steepest solidus-pressure gradients. If a plum lithology has a gentler solidus-pressure gradient than the matrix lithology, then it is not only likely to cross its solidus and begin to melt at a deeper depth than its host matrix, but it is also likely to stop melting once the mantle reaches a depth where the matrix begins to melt to a significant degree. This effect should tend to concentrate pyroxenite melting within the deeper sections of the melting column beneath mid-ocean ridges or hotspots.

[49] The only plum lithologies likely to be entirely consumed by pressure-release melting are the following: (1) very low solidus lithologies and (2) very steep solidus-pressure gradient lithologies that have a steeper solidus-pressure gradient than that of the bulk of the peridotite matrix.

[50] The additional heat consumed by enhanced deep plum melting leads to a reduction in the productivity of shallower peridotite melting. If low-solidus plum lithologies make up ∼20% of the mantle, then the initial depth of melting of the peridotite fraction will be ∼5–10 km shallower than would be estimated for a pure peridotite lithology. The garnet–spinel–peridotite and spinel–plagioclase–peridotite phase transformations in the ascending mantle consume heat equivalent to ∼1–3% additional melt production and thus will somewhat further reduce the mantle's pressure-release productivity beneath a mid-ocean spreading center.

[51] The actual solidi and solidus-depletion gradients of possible mantle pyroxenite and peridotite lithologies are still not very well known. In particular, the solidus-depletion gradient has been largely neglected by laboratory study. Even if the simple melting assumptions used in this study are mostly valid, better experimental calibration of the depth and compositional dependence of these physical properties is needed to properly characterize the melting regime beneath hotspots and spreading centers. More experimental work on the pyroxenite and peridotite lithologies found in mantle outcrops (e.g., Ronda, Beni-Bousera) may also be a fruitful area for future study. If these outcrops are made from residues to partial melt extraction [Frey et al., 1985; Phipps Morgan, 1999; Phipps Morgan and Morgan, 1999; Suen and Frey, 1987], then the solidi of their pyroxenites and peridotites should be related. Are the solidus-pressure gradients of the pyroxenite layers larger or smaller than those of the peridotite lithologies? At low pressures, do the pyroxenite veins have a higher melting temperature than their peridotite matrix, as they would if they stopped melting once the peridotite matrix began to melt to a significant degree? At high pressures, do the pyroxenite veins typically have a lower solidus than their host peridotite?

[52] This study began from a simple-sounding question: what are the basic differences between the pressure-release melting of a homogeneous and heterogeneous (plum pudding) mantle? Thermodynamic principles proved to be extremely helpful in exploring the outlines of this question. Thermodynamics also showed its utility for transforming the results of laboratory melting measurements made at variable entropy, constant pressure conditions into useful constraints for isentropic pressure-release melting. The solidus-depletion gradient was seen to play a fundamental role in the pressure-release melting of various mantle lithologies. Fortunately, the pressure and compositional dependence of the solidus-depletion gradient can be determined from both standard T-X phase diagrams and MELTS-based calculations. With more complete measurements we should be able to much more precisely explore the chemical consequences of pressure-release melting of a veined plum pudding mantle.

Appendix A:: Thermodynamic Relations for and Idealized Olivine System

[53] Olivine is thought to be the most abundant mineral in the upper mantle. While (Mg,Fe)_{2}SiO_{4} olivines are one of nature's most common and straightforward examples of a mineral solid-solution, details of the (Mg,Fe) olivine solid solution are still poorly known, in part because the extremely high melting temperature of the Mg end-member (forsterite) makes this mineral difficult to access using standard calerometric techniques [Richet et al., 1993], in part because olivines crystallize so easily that it is extremely difficult to quench to a glassy (frozen liquid) state [Bowen and Schairer, 1935] and in part because the Fe–olivine end-member cannot be accurately measured using an iron-jacketed experimental charge [Morse, 1980].

[54]Bowen and Schairer [1935] discussed both the experimentally derived and theoretical thermodynamics of the olivine system in part of their comprehensive study of the olivine and pyroxene systems. This classic paper still is the primary experimental source for the 1-atm olivine phase diagram (e.g., it is the basis for the olivine phase diagrams in recent books [Hess, 1989; Maaloe, 1985; McBirney, 1993; Morse, 1980; Philpotts, 1990]. However, workers in the 1960s noted both a mathematical error and a conceptual error in Bowen and Schairer's derivation of the theoretical thermodynamics of the olivine system [Bradley, 1962; Saxena, 1973]. Next we will review the basic thermodynamics of this system while deriving the forms used to determine the melting behavior of an idealized olivine solid-solution phase.

[55] In the idealized (Mg,Fe) olivine solid-solution considered here, Mg and Fe atoms could substitute for each other at a crystal site without changing either the cell volume or the interatomic binding energy. In this case the only effect of (Mg,Fe) substitution would be on the configurational entropy of the system:

where m_{j} is the number of atoms in the jth crystallographic site and X_{i,j} is the mole fraction of the ith atom in the jth site. To a good first approximation, the M1 and M2 sites in olivine have identical Fe/Mg substitution. With 2 moles of Fe or Mg atoms per mole of olivine available for solid-solution substitution,

This relation assumes that the M1 and M2 cation sites in the olivine crystal have identical Mg/Fe substitution, while, in practice, a MgFeSiO_{4} olivine has 60% (instead of 50%) of its Mg in the M1 site [Putnis, 1992]. However, since the resulting compositional entropy differs by a maximum of ∼5% from the prediction of the much simpler formula, (A2) will be used instead of a more complicated formula that explicitly considers the M1 and M2 sites as separate contributors to the compositional entropy.

[56] How close is this ideal to the actual olivine (Mg,Fe) solid solution? One way to test for nonideality is to see how well this solid-solution theory fits the melting temperatures, heats of fusion, and specific heats of the actual Mg and Fe end-member phases. The best fit to Bowen and Schairer's experimental data occurs when the latent heats of melting of the Fe–Ol and Mg–Ol end-member compositions are the same. This parameter choice makes little theoretical sense because it predicts that the entropy of fusion should differ by ∼40% between the Fe–Ol and Mg–Ol end-members. More recent experimental measurements of the entropies of fusion of Fe–Ol and Mg–Ol have demonstrated that they do not differ by this magnitude [Richet et al., 1993] (cf. Table A1). For this ideal solid-solution the molar volume of the Fe–Ol and Mg–Ol phases should also be identical. Instead, they differ by ∼6% (see Table A1).

Table A1. Thermodynamic Properties of the (Fe,Mg) Olivine Solid-Solution End-Members and of the “Ideal Olivine” End-Members Used in the Example Calculations^{a}

[57] The derivation of the thermodynamic solid-melt equilibrium of this ideal solid solution is straightforward. It seems simplest to derive these relations using the chemical potential or molar version of the Gibbs free energy. The chemical potential of the solid and liquid phases is a function of composition, pressure, and temperature. The mixture with the lowest chemical potential is stable at a given P-T. Within the solid-solution melting interval the configurational entropy can trade-off against the entropy of fusion so that solid and melt phases of different composition coexist along a phase loop. The chemical potential will change with pressure and temperature. The pressure variation of the chemical potential is

where ν_{i0} is the molar volume (or inverse molar density) of the ith phase at a reference pressure P_{0}, P is the actual pressure, and β[Pa^{−1}] is its compressibility (or inverse bulk modulus K[Pa]). The temperature variation of the chemical potential is

where S_{i} is the entropy contribution of the ith phase.

[58] Within the melt-solid phase loop the solid and melt phases have the same chemical potential since the two phases exist in equilibrium. The entropy of each end-member component within the solid-solution has three contributions, a heat capacity-related contribution, a configurational entropy of the solid solution (equation (A2)), and an entropy component unrelated to the Fe–Mg solid solution. If we assume that the last entropy contribution is independent of temperature, then it is given by its value at the end-member's melting point at reference state (T_{⊕}^{m}, P_{⊕}^{m}). The thermic entropy contribution to the component is given by either of

depending on whether the phase is in its solid (s) or liquid (l) state. Thus the chemical potential of component i of the solution in either its liquid or solid state is given by

where the pressure dependence of the chemical potential was obtained by integrating (A3) assuming the compressibility to be constant with increasing pressure. Now at the end-member melting point, μ_{i}^{S} (T_{i⊕}, P_{⊕}) = μ_{i}^{l}(T_{i⊕},P_{⊕}). If we also assume that the heat capacity of the solid and liquid are identical, then = (For consistency with this last assumption we will also need to assume that this idealized olivine or “ideaolivine” has the same entropy change upon melting of each of its end-member phases.) In this case we can simply subtract the two equations of (A6) to find the relation between the solid-solution compositions of the ith component in the coexisting liquid and solid phases:

or

where δS_{i}^{l−S} = (S_{i}^{Xl} − S_{i}^{Xs}) is the entropy of melting of each end-member component. We can further define the free-energy change of melting as a function of pressure and temperature δG_{i}(T,P) to be given by the right-hand side in (A8):

in which case the solid and liquid compositions are related by

to which the application of mass conservation (X_{Mg–Ol}^{S} + X_{Fe–Ol}^{S} = 1; X_{Mg–Ol}^{l} + X_{Fendash;Ol}^{l} = 1) gives

With the last equation in (A12), the relations in (A11), and mass conservation, we can find analytical solutions for the mole fractions of each end-member component in the coexisting solid and liquid phases of the phase loop. Furthermore, when the solid and liquid compositions are equal, (A8) reduces to a Clapeyron equation (dT^{m} = (δV^{l − s}/δS^{l − s})dP) for the pressure dependence of the melting temperature of each solid-solution end-member:

Equations (A9), (A11), and (A13) completely express the P-T-X behavior of idealolivine, whose defining thermodynamic parameters are given in Table A1. In this study, the main utility of idealolivine is that it lets us use simple, analytically tractable, and self-consistent numerical melting “experiments” to explore the relative impact of the solidus-depletion gradient and the heat of fusion upon adiabatic pressure release melting. These expressions are also used, with the latent heats of Bowen and Schairer [1935], to parameterize an analytical expression that almost exactly fits their experimental data and which is therefore used to determine the solidus-depletion gradient of the experimentally measured olivine phase diagram shown in Figure 2c.

[60]Figures 3 and 4 illustrate some effects of pressure-release melting starting with a molar idealolivine composition of (Mg_{0.8}Fe_{0.2})_{2}SiO_{4}. In these examples the greatest potential effect on productivity variations during ascent is the change in solidus slope as a function of pressure. This change is caused by the decrease in the compressibility of the solid and liquid phases with increasing pressure. For the case where the volume change of melting is constant, the productivity shows only a very small variation linked to changes in the olivine's solid solution composition. Because these compositional changes are small, the solidus-depletion gradient changes little during ascent and progressive melt extraction.

Appendix B:: Comparison of Computational Expressions for Melt Production

[61] Previous computational codes have mostly used difference formulae to find the melt production rate; typically solving for the potentially time-variable depletion (accumulated extent of melt extraction) and then taking the time difference of the depletion along a flow-line to find the instantaneous melt-production rate [Parmentier and Phipps Morgan, 1990; Sotin and Parmentier, 1989]. This approach commonly involves solving an equation of state for the depletion f of the form

where τ = (T_{liquidus}[f ≡ 1] − T_{solidus}[f ≡ 0]) is the melting interval between the solidus and liquidus temperatures. Figure B1 shows the geometric construction used to solve for the melt productivity. (Note that this computational example ignores the adiabatic gradient, instead solving only for the nonadiabatic part of the temperature field while reducing the solidus P-T curve by the adiabatic gradient.) The computation assumes batch melting (i.e., a “differenced batch melting” formulation); yet with proper parameter choices it can be made to be mathematically identical to the scenario for fractional melting of a single lithology that was considered earlier in this study.

[62] Consider material upwelling from point 1 to point 2 shown in Figure B1. The melt fractions (or depletions) at points 1 and 2 are given by (subscripts 1 or 2 refer to the variable at point 1 or 2):

The solidus temperatures and are also related, i.e.,

This is commonly implemented into the computational form:

To find the implicit form of (B4) equate the specific heat change associated with the temperature change δT_{21} with the heat of fusion associated with the melt increment δf_{21}:

Equation (B7) becomes mathematically equivalent to (13) in the main text if we equate the melting interval τ = (T_{liquidus} − T_{solidus}[f = 0]) with the solidus-depletion gradient (dT^{m}/dF)_{P} term in (13). (Note that the solidus-temperature gradient in (B7) is already corrected for adiabatic effects; thus adiabatic terms do not formally appear in (B7).) The effective depletion-dependent solidus temperature implied by (B7) is given by

Using (B7), we can also solve for the melting interval τ that is self-consistent with a given productivity and solidus-depth gradient:

For a given depth productivity δf/δz = 0.033 km^{−1}(= 1% kbar^{−1}), enthalpy of fusion δH_{fus}/c_{P} = 600°C, and solidus-depth gradient (dT_{m}/dz)_{f = 0} = 3°K kbar^{−1}, then the self-consistent τ = 375°K.

[63] While these computational forms can be made equivalent to (13) for a constant solidus-depletion gradient, it appears that using (13) and (10) in a computational code provides better accuracy and stability than a calculation involving the finite difference form of (B4). This can be seen in a simple calculation assuming constant vertical upwelling flow on a 32-point mesh stencil in the vertical direction. Equivalent calculations using the forms (B4) and (13) to find the depth productivity are shown in Figure B2a and B2b , respectively. While the calculations are almost identical for the total depletion and temperature profiles, the difference calculation for the productivity df/dz produces overshooting and undershooting effects that are avoided by a direct solution for df/dz (equation (13)) in place of a formulation expressed in finite difference derivatives of f (equation (B4)). Morever, (13) (and its sister pair of equations (21) and (29) describing the melting of simple kinds of veined plum pudding mantle) can be readily adapted to situations where the solidus-pressure and solidus-depletion gradients are not constant.

Appendix C:: Adiabatic Temperature Change of a Solid-State Phase Transition

[64] The effects of the ∼60-km-deep (or ∼2 Gpa) garnet–spinel lherzolite and ∼25-km-deep spinel–plagioclase lherzolite phase transitions beneath a mid-ocean ridge are to further decrease the mantle temperature during ascent through these phase transitions (For a graphical illustration of this effect, compare Figures 9 and 10.). Assume there are no solid-solution effects during an adiabatic solid-state phase transformation. To solve for the thermodynamic behavior of this system, the configurational entropy change dS/dX associated with a mass (or molar) fraction dX undergoing a phase transformation needs to be added to the other terms in (4); i.e.,

The entropy difference between the two solid lithologies is δS^{S − S}, which is positive for an exothermic phase transformation. (This quantity is closely related to the more familiar latent heat or enthalpy change of the phase transformation: δH^{S − S}: δS^{S − S} = δH^{S − S}/T^{S − S}, where T^{S − S} is the temperature of the phase transformation.) Another thermodynamic constraint during the phase transformation is that the temperature lies along the P-T boundary of the phase transformation:

(dT^{S − S}/dP is itself a function of the entropy and volume/density change during the phase transformation, the Clapeyron relation states dT^{S − S}/dP = δV^{S − S}/δS^{S − S}, where the specific volume V ≡ 1/ρ_{molar}. Substituting these relations into (B10) yields

after the further simplification that (∂S/∂T)_{P} and (∂S/∂P)_{T} are the same in the two lithologies. (If this lack of notational precision seems worrisome, it is also possible to view (∂S/∂T)_{P} as a shorthand notation for (X_{1}(∂S/∂T)_{P}^{1} + X_{2}(∂S/∂T)_{P}^{2}), and so on.) Equation (B12) can be further simplified to

Equation (C4) states that the pressure dependence of the solid-state phase transformation depends on the difference between the slope of the phase boundary and the slope of the local adiabat and inversely depends on the latent heat of the phase transformation. If only a fraction φ of a mantle mixture is experiencing the phase change, then the latent heat term in the denominator is replaced by the latent heat of the fraction available for the phase transition φ(TδS^{S − S}/c_{P}). The pressure interval δP of the univariant phase transition is found by setting dX = 1 in (C4), and the temperature drop δT across the phase transition is determined by combining the expression for δP and (C2):

The pressure interval and temperature drop across the garnet–spinel peridodite and spinel–plagioclase peridotite phase transitions are relatively minor. For these phase transitions, (C5) is evaluated as follows (also see Phipps Morgan [1997, p. 219]). The latent heat term is found using a modified Clapeyron relation written in the following expanded form: δS/c_{P} = (δV/c_{P})/(dT^{S − S}/dP) = (V/c_{P})(δV/V)/(dT^{S − S}/dP). For most silicates, V/c_{P} ≈ 250°C/GPa [Broecker and Oversby, 1970]. The volume changes δV/V for the spinel–plagioclase peridotite and garnet–spinel peridotite are 2.1 and 1.5%, respectively [Ringwood, 1975], and the P-T slopes of these solid–solid phase transformations are 1425°C/GPa (spinel–plag.) and 1274°C/GPa (garnet–spinel) [Ringwood, 1975]. With these parameters, δS/c_{P} is 0.0029 for the garnet–spinel peridotite phase transition and 0.0037 for the spinel–plagioclase phase transition. For a typical mantle temperature of 1673°K the resulting estimate for the pressure interval of these phase transitions is extremely small, of order ∼50–100m (when they are assumed to be univariant). For typical upwelling rates this is so small that heat conduction would smooth the transition to a lengthscale of order ∼1 km (see the earlier discussion concerning the length scale of conductive equilibrium). Both solid-solution effects and compositional variation within the peridotite would also cause the phase transitions to occur over a wider depth interval [Asimow et al., 1995]. The resulting temperature change is also fairly small (∼5°–10°C) [Asimow et al., 1995; Phipps Morgan, 1997] but does have an effect in consuming heat that could otherwise go toward an additional 1–2% of melt production [Phipps Morgan, 1997]. The influence of these peridotite solid-state phase transitions is seen in the differences between the next to last and final examples of plum pudding melting shown in Figures 9 and 10. More discussion about these effects is given in the discussion of Figure 10 and by Asimow et al. [1995].

Acknowledgments

[65] This work was inspired by discussions with Marc Parmentier on the proper computational treatment of pressure release melting. He explained to me the alternative melting formulation (equation (B7)) discussed in Appendix B. Paul Asimow and W. Jason Morgan provided helpful comments on a early drafts of this manuscript, and Eric Hauri Marc Hirschmann, Hubert Staudigel, and Bill White provided very helpful reviews and editorial input. (Bill also condensed the abstract by a factor of 2, which greatly inspired my attempts to further compress an overwordy manuscript.) Paul Hess and Tom Duffy greatly helped my search for existing data and models of solid-solution effects on olivine melting. Fred Frey provided the photo shown in Figure 1a. Silke Schenck helped with the paper and electronic manuscript preparation. Work was partly supported by the National Science Foundation.