pet01$p044 - ASIjpet01.pdf

JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

PAGES 963–998

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

P. D. ASIMOW

∗

, M. M. HIRSCHMANN

AND E. M. STOLPER

DIVISION OF GEOLOGICAL AND PLANETARY SCIENCES, CALIFORNIA INSTITUTE OF TECHNOLOGY M/C 170-25,

PASADENA, CA 91125, USA

DEPARTMENT 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

Composition, mean pressure, mean melt fraction, and crustal thick-

INTRODUCTION

ness of model mid-ocean ridge basalts ( MORBs) are calculated

Comparison of observed basalt compositions with the

using MELTS. Polybaric, isentropic batch and fractional melts

predictions of polybaric mantle melting models places

from ranges in source composition, potential temperature, and final

important restrictions on melting processes in the mantle

melting pressure are integrated to represent idealized passive and

beneath mid-ocean ridges (Klein & Langmuir, 1987;

active flow regimes. These MELTS-derived polybaric models are

McKenzie & Bickle, 1988; Niu & Batiza, 1991, 1993;

compared with other parameterizations; the results di

ff

er both in

Kinzler & Grove, 1992

; Langmuir

et al.

, 1992; Iwamori

melt compositions, notably at small melt fractions, and in the

et al.

, 1995; Kinzler, 1997). It is now well accepted that

solidus curve and melt productivity, as a result of the self-consistent

mid-ocean ridge basalts ( MORBs) represent mixtures of

energy balance in MELTS. MELTS predicts a maximum mean

melts produced over a range of depths and that these

melt fraction (

0·08) and a limit to crustal thickness (

15 km)

melts separate from their sources at low melt fraction

for passive flow. For melting to the base of the crust, MELTS

(McKenzie, 1984; von Bargen & Wa

, 1986; Johnson

requires an

200

C global potential temperature range to explain

al.

, 1990; Langmuir

et al.

, 1992). However, significant

the range of oceanic crustal thickness; conversely, a global range of

uncertainties remain regarding the depth of initial melting

C implies conductive cooling to

50 km. Low near-solidus

(related both to the range of mantle potential temperature

productivity means that any given crustal thickness requires higher

and to the influence of minor incompatible components

initial pressure in MELTS than in other models. MELTS cannot

on the solidus), the depth of final melting (and hence the

at present be used to model details of MORB chemistry because

of errors in the calibration, particularly Na partitioning. Source

importance of spreading rate), the style of melt transport

heterogeneity cannot explain either global or local Na–Fe systematics

(e.g. the relative importance of fractional fusion vs equi-

or the FeO–K

O/TiO

correlation but can confound any extent of

librium porous flow), and the e

ects of chemical hetero-

melting signal in CaO/Al

geneities in mantle sources. These uncertainties remain

for a number of reasons (among them disagreements

over the selection of data, the appropriate scale for

KEY WORDS:

mantle melting; mid-ocean ridge basalt; peridotite com-

position; primary aggregate melt; thermodynamic calculations

averaging, and corrections for fractionation), but a key

∗

Corresponding author. Telephone: 626-395-4133. Fax: 626-568-0935.

E-mail: asimow@gps.caltech.edu



Oxford University Press 2001

JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

MAY 2001

factor is that mantle melting algorithms have not been

productivity of upwelling mantle and the average com-

position and pressure of melting and between source

ffi

ciently accurate to evaluate quantitatively the con-

sequences of competing hypotheses or to incorporate the

heterogeneity and productivity.

In this paper, we present a set of polybaric calculations

complexities of the physics of melting and melt transport.

Langmuir

et al.

(1992) identified three functions that

of mantle melting using MELTS. These calculations

build on related isobaric calculations (Hirschmann

must be combined to create a forward model capable of

quantitative prediction of the magmatic output of the

al.

, 1998

, 1999

) and on polybaric, isentropic

calculations (Asimow

et al.

, 1995, 1997) and are used to

mantle beneath mid-ocean ridges: a chemistry function,

a melting function, and a mixing function. The chemistry

illustrate both the potential and current limitations of the

method. All calculations herein use the calibration of

function specifies the liquid composition as a function of

pressure (

), temperature (

) (or, alternatively, of

and

MELTS documented by Ghiorso & Sack (1995); this is

the calibration underlying the widely distributed MELTS

extent of melting,

), and source composition. The

melting function specifies

for a single parcel of source

2.0 package, and although we have modified the im-

plementation for convenience, we have not altered the

as a function of

and

and the path (usually ap-

proximated as adiabatic) through (

P, T

) space. The

model in any way. This paper does not include results

from newer calibrations such as pMELTS (Ghiorso &

mixing function is a representation of the 2D (or, in

principle, 3D) form of the melting regime and specifies

Hirschmann, in preparation). In general, MELTS pre-

dicts isobaric trends of composition vs melt fraction that

how the individual increments of liquid generated con-

tinuously over a range of depths and distances from the

are similar to those observed in experiments, but the

calculated trends are frequently o

set in

and in the

ridge axis are to be weighted to create an aggregate

primary melt. These three functions are often constructed

concentrations of certain oxides. For example, at 1·0 GPa

the best match in

between MELTS and experiments is

independently, but in fact the chemistry and melting

functions are intimately dependent on one another, as

obtained with an o

set of 80

Cin

and the resulting

model liquids are

4% (absolute) too low in SiO

and

both must satisfy mass and energy balance and both are

controlled by the same thermodynamics of solid–liquid

2% too high in MgO (Baker

et al.

, 1995). Comparison

with experiments has also shown that MELTS yields

equilibrium. Furthermore, the possibility of reaction be-

tween melt and matrix during melt migration means

too low a peridotite–liquid partition coe

ffi

cient for Na

(Hirschmann

et al.

, 1998

). Given these inaccuracies in

that the mixing problem cannot be separated from the

chemistry and melting functions (Spiegelman, 1996; Kele-

the current calibration of MELTS, we focus on using it

as a tool for studying trends in relationships among

men

et al.

, 1997). There are several published chemistry,

melting, and mixing functions based on parameterization

variables rather than to predict the actual values of

specific parameters. Accurate quantitative modeling of

of experimental peridotite melting data (Klein & Lang-

muir, 1987; McKenzie & Bickle, 1988; Niu & Batiza,

absolute values of compositional variables and phase

proportions as functions of pressure, potential tem-

1991; Kinzler & Grove, 1992

, 1992

; Langmuir

et al.

1992; Kinzler, 1997).

perature, etc. is certainly possible with the approach we

use here, but it will depend on improving or customizing

Over the past several years, we and our colleagues

have been utilizing the MELTS algorithm (Ghiorso,

the calibration.

The mixing function depends on geodynamic con-

1994; Ghiorso & Sack, 1995) as a tool for trying to

understand aspects of experiments on peridotite melting

siderations such as the form of the solid flow field and

melt extraction pathways. Our focus in this paper is on

(Baker

et al.

, 1995; Hirschmann

et al.

, 1998

, 1999

1999

) and as a basis for forward models of polybaric

insights from thermodynamic modeling of melt com-

position and melting, so we limit our treatment to the

mantle melting and of coupled melting and two-phase

flow in upwelling mantle (Asimow, 1997; Asimow

et al.

simplest end-member mixing functions associated with

either perfect active or perfect passive flow [we use

1997; Asimow & Stolper, 1999). The forward models of

polybaric melting we utilize di

er from other algorithms

standard definitions of mixing functions and mean prop-

erties from Plank

et al.

(1995); see details below in the

in that MELTS provides a self-consistent thermodynamic

approach to the chemistry and melting functions. Al-

section ‘Mean properties of melting regimes’]. Although

all the calculations presented here use one of these two

though we must emphasize that MELTS is not su

ffi

ciently

accurate to address in detail some key questions raised

simple mixing functions, we will conclude in many places

that neither function is adequate and that progress in

by the observed compositional variations of MORB

magmas, it is nevertheless the first approach that allows a

modeling of ridges will depend on using physically based

mixing functions.

full and self-consistent integration of the thermodynamics

and phase equilibria of partially molten peridotitic sys-

One of the recurring issues in our modeling of melting

at ridges will be an examination of the consequences of

tems. For this reason, it is ideal for examining important

issues such as the interplays between the depth-dependent

two competing views of the principal controls on variation

964

ASIMOW

et al

MANTLE MELTING IV

of average magma compositions among ridge segments.

MELTS or any other fractionation model. The model

Klein & Langmuir (1987) and others (McKenzie & Bickle,

of Weaver & Langmuir (1990) is not calibrated on

1988; Klein & Langmuir, 1989; Langmuir

et al.

, 1992;

alkalic liquids, but yields a tholeiitic fractionation path

Plank & Langmuir, 1992) considered that, except perhaps

for MELTS primary aggregate liquids that we use here

at very slow-spreading rates, melting continues to a

with some caution. Moreover, the Langmuir

et al.

(1992)

shallow depth, perhaps the base of the crust, at all ridge

model does not predict CaO or Al

in primary liquids,

segments. Variations in average MORB composition

but these are essential components in determinations of

on the world-wide ridge system, correlated with ridge

pyroxene and plagioclase stability and the e

ects of these

topography and seismic velocity in the underlying mantle

phases on liquid lines of descent, so comparison between

(Klein & Langmuir, 1987; Humler

et al.

, 1993; Zhang

our model primary liquids and those of Langmuir

et al.

al.

, 1994), were then attributed primarily to variations in

(1992) is best performed without first fractionating them

the potential temperature (

) of the upwelling mantle

to 8% MgO. Nevertheless, much can be learned from

(a range of 200–250

C, Klein & Langmuir, 1987; or

examination of model primary liquids, particularly as the

300

C, McKenzie & Bickle, 1988), which controls the

ects of fractionation are relatively minor for several of

intersection of the adiabat with the solidus and thus the

the elements of interest.

initial pressure of melting (

). We will refer in this

Recent e

orts to interpret MORB compositions in

work to this model of global variations as ‘variable-

’

terms of magma generation processes have, in addition

systematics. Shen & Forsyth (1995), on the other hand,

to corrections for fractionation, distinguished variability

attributed the variability in the compositions of MORBs

among individual MORB samples within a region from

primarily to the e

ectiveness of cooling from the surface

variability on a global scale among regional averages of

and hence to the final pressure of melting (

). The total

samples (Brodholt & Batiza, 1989; Klein & Langmuir,

variation in potential temperature among non-hotspot-

1989). Correlations among regional averages are gen-

ected ridges was then estimated to be

C (Shen

erally termed ‘global trends’ whereas correlations in

& Forsyth, 1995). A consistent model based on this

variations among individual samples from a segment are

second view generally includes a significant role for

termed ‘local trends’. Forward models of melting of the

heterogeneous source compositions (Niu & Batiza, 1991;

sort presented in this paper can be used to examine the

Shen & Forsyth, 1995) and implies a correlation of extent

possible spectrum of local variability by examining all

of melting with spreading rate (Niu & Batiza, 1993; Niu

the incremental melt compositions and partial mixtures

&He

kinian, 1997

, 1997

). We refer to this model of

among them that can be produced from a model melting

global variations as ‘variable-

’ systematics.

regime (although the large range of partial mixing models

Evaluation of variability and correlations among com-

that can be devised makes it di

ffi

cult to specify

a priori

positions of MORB samples requires correcting for the

the particular local trend that will result from a given

ects of low-pressure fractionation so as to isolate the

model melting regime).

ects of magma generation processes in the mantle

In this paper, we begin by introducing phase diagrams

(Klein & Langmuir, 1987). In this work we restrict our

in pressure–temperature and pressure–entropy space

attention mostly to model primary liquid compositions

that set up the framework for all MELTS predictions

based on MELTS and compare them with other workers’

of polybaric melting by showing where each phase

model primary aggregate liquids; i.e. we generally do not

assemblage is predicted to be stable and the contours

attempt to fractionate these model liquids to compare

of equal melt fraction above the solidus [the importance

them with actual MORB data or with MORB data

of which was discussed by Asimow

et al.

(1997)]. The

corrected for fractionation (i.e., we discuss mostly Na

coupling between the chemistry and melting functions

and FeO

∗

rather than Na

or Fe

, the equivalent values

predicted by MELTS is then illustrated using the SiO

corrected for low-pressure fractionation to 8% MgO;

vs melt fraction (

) plot of Klein & Langmuir (1987).

Klein & Langmuir, 1987). We take this approach for

A brief discussion of the di

erences between isentropic

several reasons. First, the 8% MgO standard obscures

batch melting and incrementally isentropic fractional

real variations in the MgO content of primary aggregate

melting (Asimow

et al.

, 1995, 1997) introduces com-

liquids; in particular, calculations herein are extended to

parisons between mean properties of the melting regime

the low potential temperature extreme where the primary

(i.e. mean pressure, mean extent of melting, and crustal

liquid may have <8% MgO, which would require an

thickness) as functions of the initial and final pressures

artificial back-fractionation step. Second, the correction

of melting. This is followed by predictions of correlations

is di

ffi

cult to perform quantitatively in most cases. The

of these mean properties with compositional trends in

low SiO

and high Na

O in liquids predicted by MELTS

primary aggregate liquids, particularly Na

O and FeO

∗

(Hirschmann

et al.

, 1999

) results in primary aggregate

Finally, we consider the e

ects of variable source

liquids that are nepheline normative and do not follow

tholeiitic fractionation paths, whether fractionated using

composition on primary aggregate liquid compositions,

965

JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

MAY 2001

Fig. 1.

(a) and (b).

extending the discussion of Hirschmann

et al.

(1999

for this source composition and the range of mantle

potential temperatures likely to be seen by modern mid-

which dealt only with calculations of isobaric melting.

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.

ISENTROPIC BATCH MELTING

(1998

) and are likely to be

C hotter than the

Figure 1 shows maps of the stable phase assemblages for

correct temperatures, potential temperature is defined

a model primitive mantle composition (Hart & Zindler,

at atmospheric pressure where MELTS is much more

1986), with and without Cr

. The axes in Fig. 1a and

accurate.

values obtained from MELTS can be com-

c are

and specific entropy (

), or equivalently

and

pared directly with

estimates from other models, with

potential temperature [

, the calculated temperature of

an uncertainty of perhaps 20

the metastable solid assemblage at 1 bar with the given

For

<1100–1120

C (boxed 1 in Fig. 1a and c), melt

total

, allowing all solid reactions to reach equilibrium;

does not form at any pressure. Although this may be an

see McKenzie & Bickle (1988)]. ‘Ordinary’ temperature

artifact of MELTS and needs further investigation (see

(

) is plotted vs

in Fig. 1b and d for reference, but

below), it is interesting that this minimum temperature,

and

are the appropriate independent variables for

roughly the same for the Cr-bearing and Cr-absent cases,

reversible adiabatic (i.e. isentropic) melting (Verhoogen,

is not determined by the solidus temperature at 1 bar or

1965; McKenzie, 1984; Asimow

et al.

, 1995, 1997). Any

at the base of the crust but rather by the location of the

vertical line in a

P–S

diagram corresponds to a batch

six-phase point olivine

orthopyroxene

clino-

isentropic path. Contours of constant extent of melting

pyroxene (cpx)

spinel

plagioclase

liquid; i.e. the

by mass,

, are plotted in the supersolidus region of each

point of the cusp on the solidus. This is the point on

map. The isentropic productivity of batch melting at any

the MELTS-calculated solidus with the lowest specific

point is inversely proportional to the spacing of these

entropy, and so it defines the lowest mantle potential

contours as they intersect a vertical path. Hence, these

temperature at which the adiabat intersects the

solidus.

diagrams show all possible isentropic batch melting paths

966

ASIMOW

et al

MANTLE MELTING IV

Fig. 1.

(a)–(d) Maps of the stable phase assemblages predicted by MELTS for constant bulk compositions. In the region where liquid is present,

the mass fraction of liquid (

) is contoured. Contours at 1% intervals for

up to 0·04 are shown dotted. Contours at 5% intervals for

[

0·05

are shown dashed. (a) and (b) use the primitive upper-mantle composition of Hart & Zindler (1986); (c) and (d) use a Cr-free equivalent. The

axes in (a) and (c) are pressure (

) on the vertical axis and specific entropy (

) and potential temperature (

) on the bottom and top horizontal

axes. A vertical line on these diagrams is an isentropic batch melting path. In (b) and (d) the horizontal axis is temperature (

). Ol, olivine;

Opx, orthopyroxene; Cpx, clinopyroxene; Sp, spinel; Gt, garnet; Pl, plagioclase; Liq, liquid. Numbers in boxes refer to special points of interest

mentioned in the text.

Melting paths with

1110

<1225

C (boxed

pressures just below that of the spinel–plagioclase trans-

ition and the substantial curvature of the solidus in

1–3 in Fig. 1a and c) freeze completely as a result of the

spinel–plagioclase transition (Asimow

et al.

, 1995). These

the spinel and garnet stability fields, which leads to a

maximum in

on the solidus near 6·5 GPa (not shown in

paths achieve peak melt fractions of

0·025 in the

spinel peridotite field. Those paths hotter than

Fig. 1). Although simple considerations of phase diagram

topology dictate that there must be a cusp on the solidus

1175

C (boxed 2 in Fig. 1a and c) would begin melting

again in the plagioclase stability field if isentropic de-

at the appearance of plagioclase (Presnall

et al.

, 1979),

the calculated result that there is actually a temperature

compression continued all the way to 1 bar. All paths

up to

1275

C with Cr

(boxed 5 in Fig. 1c)

drop with increasing pressure approaching the cusp is

surprising. This predicted shape reflects primarily the

have plagioclase in the residue for at least a small interval.

1250

C (boxed 4 in Fig. 1a and c) is the minimum

solidus-lowering capacity of Na, which is enhanced in the

spinel peridotite field relative to the plagioclase peridotite

for exhaustion of cpx from the residue. For the Cr-

bearing case,

1425

C (boxed 6 in Fig. 1a) is

field by the greater incompatibility of Na in assemblages

with less plagioclase. In this sense, the region of negative

required for melting to begin in the garnet field; for the

Cr-absent case garnet is calculated to be present on the

slope on the solidus is similar to that observed for

amphibolite (Wyllie & Wolf, 1993) or amphibole-bearing

solidus for

1380

C (boxed 6 in Fig. 1c).

The shape of the MELTS-calculated solidus has two

peridotite (e.g. Green, 1973) where the breakdown of

amphibole near 2·0–2·5 GPa converts water from a

unusual features: the negative slope of the solidus at

967

JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

MAY 2001

relatively compatible to an incompatible component. It

flattening of the solidus expected as a result of the greater

compressibility of liquid silicate components compared

is possible that the prediction of a minimum temperature

at the cusp is an artifact of MELTS, as MELTS over-

with minerals (Walker

et al.

, 1988). Figure 2 also shows

that the linear solidus with a slope of 130 K/GPa assumed

estimates the incompatibility of Na in the spinel peridotite

field (Hirschmann

et al.

, 1998b), and, indeed, this shape

by Langmuir

et al.

(1992) corresponds well to the more

complex solidus calculated by MELTS and to the ex-

has not been observed in experimental determinations

of peridotite solidi. However, examination of the melt

perimental determinations of the solidus up to a pressure

5 GPa, beyond which it increasingly diverges from

fraction contours in Fig. 1 shows that the shape as

determined by experiment would be extremely sensitive

both the data and the MELTS calculation.

As emphasized by Asimow

et al.

(1997), phase relations

to the minimum melt fraction required to identify melting

in an experiment. For example, at

0·01, the tem-

are typically more useful for understanding de-

compression melting when portrayed in

–

(or equi-

perature drop at the cusp is predicted to be only half as

big as that on the solidus itself, and by

0·05, there

valently

–

) space than when portrayed in

–

space.

There are no experimental measurements of entropy or

is no temperature drop or negatively sloped region at

all. Thus, experimental determinations of peridotite sol-

potential temperature, of course, but when recast in these

terms, mantle melting is more easily visualized, and

idi, none of which have yet systematically explored such

low melt fractions, are unlikely to have detected such

insights can be obtained that would be di

ffi

cult using

and

as the independent variables. In the case of

behavior even if it does occur, as it is predicted to be

confined to such small melt fractions.

MELTS calculations on the Hart & Zindler composition,

with or without Cr, the solidus is actually predicted to

Although MELTS as currently formulated was not

intended to be used at pressures beyond

2 GPa (Ghiorso

have a vertical tangent at 6·4 GPa (o

the tops of Fig.

1a–d) and

2000 K (

1500

C). If such a maximum

& Sack, 1995), the phase relations implicit in MELTS

require that calculated melting begins deeper than this

on the solidus in

–

space exists, it would have some

curious consequences, including progressive freezing of

to model ridge segments with more than

4 km of crust.

Although we have little confidence in specific predictions

parcels of mantle as they decompress from higher pres-

sures. Determination of whether such a maximum in

of the model (e.g. liquid compositions) above

3 GPa,

the position and shape of the solidus are critical factors

entropy along the solidus actually exists for upper-mantle

materials must await improved thermodynamic data on

in forward modeling amounts of melt production and

crustal thickness on a given isentropic path. The solidus

solids and liquids at high pressures, but from a practical

standpoint, the fact that this feature is predicted by

of the KLB-1 peridotite composition (for which the most

data are available at high

) predicted by MELTS is

MELTS for fertile peridotite limits application of the

standard relationships for triangular melting regimes (re-

compared in Fig. 2 with experimental brackets. This

figure shows that the MELTS-calculated solidus for KLB-

quiring a well-defined maximum melting pressure,

;

Plank

et al.

, 1995) to MELTS calculations with

1 is similar (i.e. within 100

C) to all experimental brackets

up to at least 8 GPa (Takahashi, 1986; Takahashi

et al.

<1500

C. It is important to reemphasize that despite

this limitation, the position of the solidus predicted by

1993; Zhang & Herzberg, 1994). The MELTS solidus is

also everywhere within 75

C of the solidus that McKenzie

MELTS in both temperature and entropy (or potential

temperature) space at pressures up to

6 GPa is not

& Bickle (1988) fitted to peridotite solidus data (with only

KLB-1 points above 4 GPa), but the MELTS solidus is

unreasonable, even though the equations of state used

for minerals and liquids were not intended to extrapolate

more strongly curved and has a lower slope at very high

pressure. MELTS tends to exaggerate the incompatibility

so far. We show calculations with

up to this maximum

for completeness and to further our understanding of the

of Na near the solidus, leading to too large a ‘freezing

point depression’ (Hirschmann

et al.

, 1998

), whereas at

implications of this possible behavior of the solidus at

high

, but the reader is cautioned that these calculations

higher melt fractions it errs in the opposite direction by

C at 1 GPa (Baker

et al.

, 1995), so the excellent

are extrapolated beyond any reasonable expectation of

accurate prediction of actual phase relations.

correspondence between the calculated and measured

solidus in Fig. 2 could be misleading. None the less, the

The di

ffi

culty imposed by the inadequate treatment of

in MELTS (Hirschmann

et al.

, 1998b) is illustrated

similarity in the overall curvatures of the model and

experimental determinations of the solidus in

–

space

by Fig. 1. In the present version of MELTS, spinel is the

only mantle phase that accepts Cr

, whereas in natural

is useful in the context of forward models of melt pro-

duction as a function of potential temperature. Although

systems, pyroxenes and garnet are significant Cr

reservoirs. Hence when Cr

is included in the cal-

the magnitude of the calculated curvature may be ex-

aggerated by errors in MELTS that grow worse with

culation, spinel is stable under all subsolidus conditions

and persists nearly to the liquidus ( Fig. 1a and b).

increasing pressure, it is particularly important that the

MELTS calculations and experiments both display the

On the other hand, when Cr

is excluded from the

968

ASIMOW

et al

MANTLE MELTING IV

at the low-

end of the spinel peridotite field (Hirschmann

et al.

, 1998

), a feature that was not apparent in ex-

perimental data as of 1987. Although this e

ect 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 iso-

baric experimental data, see Hirschmann

et al.

(1998

)].

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

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

Fig. 2.

Comparison of the solidus predicted by MELTS for composition

KLB-1 (Takahashi, 1986) with model peridotite solidi of Langmuir

consequences of fractional melting on liquid and residue

al.

(1992) and McKenzie & Bickle (1988) and experimental brackets

compositions are predicted to be significant (see below).

on the solidus of KLB-1 (Takahashi, 1986; Takahashi

et al.

, 1993;

Thus the idea of using batch melting isobars or isentropes

Zhang & Herzberg, 1994). Right arrows are liquid-free experiments,

left arrows are liquid-bearing experiments, and paired brackets are

to predict compositions from fractional fusion is less

linked by continuous lines.

useful than was once thought (Klein & Langmuir, 1987;

McKenzie & Bickle, 1988).

composition, as in Fig. 1c and d, spinel is insu

ffi

ciently

stable and hence the garnet–spinel and spinel–plagioclase

transition regions are artificially narrow and spinel dis-

INCREMENTALLY ADIABATIC

appears from the residue before cpx, near 10% melting.

FRACTIONAL MELTING

We have tried to work around this problem using du-

plicate calculations in Cr-bearing and Cr-absent com-

Fractional melting cannot be a locally isentropic process,

in that escaping melts remove entropy from the system.

positions. When similar behavior is observed in both

cases, we infer that errors in spinel stability are not

Here we model fractional melting as an idealized process

of infinitesimal isentropic batch melting steps followed

seriously a

ecting our results.

Previous attempts to estimate liquid compositions dur-

by extraction of all liquid formed (see Asimow

et al.

1995, 1997). The composition and entropy of the residue

ing isentropic batch melting have generally chosen a path

through a chemistry function [i.e. liquid composition as

of each step then serves as the reference for the next

increment. The extension to continuous fusion, where

a function of (

)or(

)] fitted to isobaric melting

data, where the path is set by the independently estimated

some amount of melt remains behind after each step, is

straightforward (Asimow

et al.

, 1997).

melting function or productivity, –d

,orbyanes-

timate of the thermal gradient during melting, d

Most previous attempts to construct models of polybaric

fractional melting have been linked closely to melt com-

For example, Klein & Langmuir (1987) illustrated the

construction of such a model for SiO

as a function of

positions and melt fractions from batch melting ex-

periments on a limited number of peridotite bulk

and

using isobaric curves (fits to experiments expressed

as SiO

contents of partial melts of fertile peridotite vs

compositions. Those that use compositions directly from

batch melting experiments (Klein & Langmuir, 1987;

at constant

) and an estimated isentropic productivity

of 1·2%/kbar ( Fig. 3a; it should be noted that we retain

McKenzie & Bickle, 1988; Watson & McKenzie, 1991;

Iwamori

et al.

, 1995) include none of the compositional

units of %/kbar for consistency with previous work;

1%/kbar

10%/GPa). In contrast, MELTS generates

ects of fractional melting and any di

erences between

batch and fractional fusion reflect largely

ad hoc

estimates

isentropic batch melting paths directly, without treating

the chemistry and melting functions independently. The

of the di

erences in productivity and

–

paths between

batch and fractional melting. Other parameterizations

result ( Fig. 3b) di

ers considerably from that of Klein &

Langmuir (1987) ( Fig. 3a) for two reasons. First, the

use major element partition coe

ffi

cients fitted to batch

melting experiments (Niu & Batiza, 1991; Langmuir

isobaric melting curves are clearly di

erent in that they

show high SiO

at low

(Baker

et al.

, 1995), especially

al.

, 1992) or four-phase saturation surfaces (Kinzler &

969

JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

MAY 2001

Fig. 3.

SiO

in silicate liquids from melting of peridotite vs extent of melting,

. (a) Fits to isobaric batch melting data and estimated polybaric

paths from Klein & Langmuir (1987). Light curves are isobaric paths at the labeled pressures. Bold curves are isentropic paths beginning at the

labeled solidus intersection pressure

assuming a productivity, –d

, of 1·2%/kbar. (b) MELTS predictions for isentropic batch melting of

Cr-free Hart & Zindler (1986) composition (HZ noCr). (Note that the

1·4 GPa path intersects the spinel–plagioclase transition and freezes

completely before melting resumes in the plagioclase peridotite field.) Kinks on the polybaric paths occur at the garnet–spinel peridotite transiti

on the

3·0 and

4·0 GPa paths and at the exhaustion of spinel at

0·09 and the exhaustion of cpx at

0·18 on all paths. (c)

Adiabatic (incrementally isentropic) fractional melting according to MELTS: incremental melt compositions are shown as light continuous curves,

integrated fractional melts are shown as light dashed curves.

in all these cases is unity minus the mass fraction of the original solid remaining.

The batch melting paths from (b) are shown for comparison as bold continuous lines; the integrated fractional melts are substantially di

erent

from batch melting both in that lower melt fractions are achieved and SiO

content follows a di

erent path. Kinks correspond to phase

exhaustion as in (b).

Grove, 1992

, 1992

; Kinzler, 1997): within the fitted

erent, and the di

erences generally increase with

(i.e. with the pressure range from solidus to the pressure

range, these parameterizations try to account for the

of comparison). The modeled di

erences shown in Fig.

evolution of residue composition and variations in liquid

3c are qualitatively similar to the results of Hirose &

composition with progressive fractional fusion. However,

Kushiro (1998). Any melting model where the melt

all these approaches have depended on poorly con-

composition changes with pressure will yield such a

strained (and largely non-thermodynamically grounded)

erence between polybaric batch melting and in-

estimates of productivity and

–

paths for fractional

tegrated polybaric fractional fusion; such an e

ect is

fusion. The incremental batch experimental approach of

clear, for instance, in FeO

∗

values in the model of

Hirose & Kushiro (1998) attempted to approximate the

Langmuir

et al.

(1992). The magnitude of the di

erences

–

path of incrementally adiabatic polybaric frac-

between batch and fractional melting from a given model,

tional fusion; although this approach is a promising one,

however, is sensitively dependent on the productivity

it involves relatively large step sizes (i.e. the first increment

function. As we will see below, in an adiabatic melting

is 6·5% melting), so it is not a good approximation to

column, MELTS calculations produce the bulk of liquid

pure fractional fusion.

mass over a smaller range of pressure than models with

Figure 3c illustrates the likely magnitude of the di

er-

nearly linear productivity and hence predict smaller

ences between batch and fractional melting based on

erences in SiO

and FeO

∗

concentrations between

MELTS calculations. Conventional petrological wisdom

batch and accumulated fractional liquids.

holds that integrated fractional melts are similar to batch

melts, but this is strictly true only for highly incompatible

elements and only when partition coe

ffi

cients are con-

MEAN PROPERTIES OF MELTING

stant. In this polybaric case, however, where productivity

REGIMES

is di

erent for batch and fractional processes and where

SiO

partitioning depends strongly on pressure (O’Hara,

Two-dimensional models of mid-ocean ridge melting can

often be simply characterized by mean properties: e.g.

1968), batch and integrated fractional melts are very

970

ASIMOW

et al

MANTLE MELTING IV

the mean pressure of extraction (

), the mean extent of

triangle), another step is required. McKenzie & Bickle

melting

(see Plank

et al.

, 1995), or the total crustal

(1988) defined the ‘point and depth average’, the mean

thickness (

, when given in units of kilometers, or

composition of all melts exiting the melting regime, by

the pressure at the base of the crust). For any model (e.g.

integration with respect to depth

from the solidus (

active or passive, batch or fractional), the relationships

0) to the height of the residual mantle column

between these average properties and the physical para-

meters of the model (e.g.

) depend on the

productivity and the form of its variations with

and source composition; i.e. the nonlinear melting func-

tion (or, equivalently, non-constant –d

) predicted by

(2)

thermodynamics (Asimow

et al.

, 1997) results in nonlinear

relationships among

, and (

)

1/2

. We show

below the relationships among all these variables ac-

cording 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

where

is the ‘point average’ from equation (1); in

passive flow for the Hart & Zindler (1986) source com-

contrast, Klein & Langmuir (1987) and Langmuir

et al.

position.

(1992) averaged with respect to

Formalisms for obtaining mean properties of melts

produced by 2D model melting regimes using 1D melting

models have been presented several times (Klein & Lang-

muir, 1987; McKenzie & Bickle, 1988; Plank & Lang-

muir, 1992; Richardson & McKenzie, 1994), but the

max

(3)

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,

, produced along each

streamline or evaluated at each point along the exit

boundary of the melting regime is obtained from a single

For calculations using discrete intervals equally spaced

integration,

or in

, or for the case of constant productivity,

these definitions produce identical results. For cal-

culations discretized in

where –d

is not constant,

′

(1)

however, they di

er. For example, for a variable-

melting regime, where

max

is achieved at some pressure

but corner flow continues to a lower pressure (perhaps

(McKenzie & Bickle, 1988), but the meaning of

, defined

), giving a trapezoidal melting regime, (3) cannot des-

as the composition of the melt added to increase the

cribe the part of the residual mantle column between

fraction of melt from

, is obvious only for

and

that is characterized throughout by

max

fractional melting, where it is the instantaneous melt

The relationship between (2) and (3) is revealed in the

composition produced by each increment of melting,

derivation of Plank & Langmuir (1992), who gave the

. For batch melting and intermediate processes (e.g.

more general equation

‘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

is the net transfer of components between solid

and liquid, such that for batch melting

is the in-

stantaneous liquid in equilibrium with the residue.

max

CFv

max

(4)

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

where

is the horizontal velocity of the residue at a

considered only this column average (Niu & Batiza,

given depth in the residual mantle column. The value of

1991), but for flows with some 2D character to the exit

boundary of the melting regime (e.g. the passive flow

is most simply assumed to be independent of depth

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JOURNAL OF PETROLOGY

VOLUME 42

NUMBER 5

MAY 2001

(e.g. Klein & Langmuir, 1987; 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

is held

max

(

−

)

−

(7)

constant, it cancels out of (4), and it is clear that (4) and

(2) are equivalent. Equation (3), however, is a special case

Likewise, the integrals for mean pressure and mean

for constant

and d

; constant d

corresponds to

composition when

use the values of P

′

and

that

constant spacing in depth of the melt fraction contours,

obtain at

for the entire interval

. It should be

or approximately to constant productivity, –d

. The

noted that all these forms are based on the assumption

ffi

culty of using (4) in regions where productivity is zero

that the flow is incompressible (they are therefore equi-

(hence d

is infinite), such as the shallow mantle

valent to integration with respect to the stream function;

above a variable-

melting regime, is clear. As

is not

Richardson & McKenzie, 1994) and hence that even if

well characterized in our calculations (Asimow & Stolper,

liquids are removed from chemical equilibrium with the

1999), whereas

is known exactly as an independent

residue in the interior of the melting regime they are

variable, mean properties for all passive-flow models are

physically carried along solid-flow streamlines to the

calculated in this work according to

boundaries of the melting regime (with only a Boussinesq

ect 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 e

ects of productivity functions on extent of melt-

(5)

ing, mean extent of melting, and crustal thickness are

explored in Figs 4–6. Figure 4 emphasizes the progress

and its pressure derivative and integral along each

adiabat from the solidus to the base of the crust, in-

dependent of the mixing function or the shape of the

melting regime. Figure 5 illustrates variable-

systematics

which assumes constant

and is in practice nearly

by showing the final melt production by adiabats of

identical to (2). The mean pressure is calculated similarly,

ering potential temperature (or

) where

is equal

replacing

in the numerator of (5) with

for batch

to the base of the crust

(i.e. the melting regime is

triangular). Figure 6 illustrates variable-

systematics,

melting and with the column or streamline average

showing the total output when melting stops at various

′=

′

for fractional melting. The mean melt

values of

, but the integration continues at the final

fraction is calculated from

value of

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

−

(6)

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

In equations (5) and (6) we use

as the upper limit of

as a result of convergence of the liquidus and solidus)

integration, reflecting the idea that the residual mantle

superimposed on a small (

20%), discontinuous decrease

column and the mantle corner flow extend to the base

in productivity at the depth of cpx exhaustion. Figure 4a

of the crust.

and b shows the productivity functions of the two models

For cases where melting stops at

, whether as

vs pressure for melting paths with

1·3, 1·7, 2·1,

a result of imposed cooling or productivity e

ects, the

2·7, and 4·4 GPa. The strongly increasing productivity

value of

max

, applies throughout the interval

leading up to cpx-out along each path in the MELTS

, which simulates a trapezoidal melting regime

model (Hirschmann

et al.

, 1994; Asimow

et al.

, 1997) is

prominent in Fig. 4a, which also shows the following

contained within a triangular corner-flow field,

972

ASIMOW

et al

MANTLE MELTING IV

Fig. 4.

The productivity, –d

, extent of melting,

, and integrated thickness of extracted melts (in pressure units) are compared for polybaric

fractional melting as predicted by MELTS and by the model of Langmuir

et al.

(1992) for the Hart & Zindler (1986) mantle composition

(including Cr, for MELTS). Each panel shows five paths that intersect their solidus and begin melting at

1·3, 1·7, 2·1, 2·7, and 4·4 GPa,

respectively; the weight of the curve increases with

and

. (a) and (b) plot productivity vs

for each path, with productivity vs

shown as

an inset. (c) and (d) show

for each path. The locations where cpx is exhausted from the residues and the limit imposed by crustal thickness

are indicated by light dashed curves. The large filled circles on each path are plotted at the mean pressure (

) and mean extent of melting (

)

for a passive-flow mixing model based on each melting path. (e) and (f ) show the integral of

from

as a function of

along each path.

For passive-flow melting regimes where melting stops at the base of the crust, the final pressure of melting and the crustal thickness are found

by setting

, the pressure at the base of the crust, equal to the value of this integral at

. (See text for further discussion of this figure.)

features of productivity predicted by MELTS: a doubling

low-productivity region at

<0·03, the high-productivity

region approaching cpx-out, and the drop at cpx-out)

in productivity at the exhaustion of garnet at 3·1 GPa

[this is a Cr-bearing composition; see Asimow

et al.

vary with

. The low-productivity region extends to

about the same

(

0·03 for this source composition)

(1995)]; a moderate drop in productivity or a barren

zone at the appearance of plagioclase on adiabats cold

for all

. The extent of melting at cpx-out, however,

decreases with increasing pressure as a result of changes

enough to form plagioclase (Asimow

et al.

, 1995); and a

large (50–60%) drop in productivity at cpx-out (Asimow

in pyroxene composition. At equal

, the productivity

decreases with increasing

as a result of decreasing

et al.

, 1997). The inset in Fig. 4a showing productivity

against

illustrates how

-dependent features (e.g. the

(

∂

)

(Asimow

et al.

, 1997).

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JOURNAL OF PETROLOGY

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NUMBER 5

MAY 2001

Fig. 5.

Mean properties of 2D integrated melting regimes as functions of potential temperature and initial pressure of melting assuming passive

flow, with the final pressure of melting,

, assumed to be at the base of the crust (i.e. variable-

systematics). MELTS results are compared

with the models of Langmuir

et al.

(1992; long-dashed line) and Kinzler (1997; dotted line) as representatives of a class of published models with

nearly constant productivity. The input proportions of all oxides considered by each model were set to the Hart & Zindler (1986) source

composition. MELTS calculations for both the Cr-bearing (heavy continuous curves) and Cr-free (heavy shaded curves) source compositions are

shown. (a) Mean pressure,

, vs potential temperature,

. (b) Mean melt fraction,

(see Plank

et al.

, 1995), vs

. (c) Crustal thickness,

calculated according to the formalism of Klein & Langmuir (1987), vs

. The normal oceanic crustal thickness range of 7

1 km is shown,

as is the global range of oceanic crustal thickness from a minimum of

3 km to a maximum at Iceland, where crustal thickness estimates range

from 14 to 25 km or more. (d)

vs solidus intersection pressure,

. (e)

.(f)

. Boxed 1 indicates the maximum

for

plagioclase to appear in the residue; 2 indicates the minimum

for melting to occur between the appearance of residual plagioclase and

the base of the crust.

974

ASIMOW

et al

MANTLE MELTING IV

Fig. 6.

Variations of mean properties of the melting regime with variations in the final pressure of melting (

) for passive flow and the Cr-

bearing Hart & Zindler (1986) source composition (HZ) according to MELTS and Langmuir

et al.

(1992). The same five potential temperatures

are shown as in Fig. 4, although the intent is to see how calculated mean properties correlate with

is constant. For MELTS output the

extreme case of

6·4 GPa (

1500

C) is also shown, dashed. The integrations underlying these curves assume a trapezoidal melting

regime, with the upper part of the residual mantle column from

all characterized by

max

. (a) and (b) show

. (c) and (d) show

crustal thickness,

,vs

; the ranges of

thought to occur in nature are indicated as in Fig. 5. The crossing of curves in (a) is related to the

maximum in

for variable-

melting regimes that have the productivity functions output by MELTS ( Fig. 5e): i.e. when

6·4 GPa, the

solid flux into the melting regime is much larger than when

2·7 GPa because the width of the base of the triangular melting regime

increases with

, but the limits on

(i.e. growth of the low-productivity tail into the garnet field; the increase in size of cpx-absent melting

regime; and the lower overall productivity with increasing

) lead to a melt-flux out of the hotter melting regime only slightly larger and hence

the unintuitive result that the ratio of melt-flux-out to solid-flux-in (

) can be smaller for the hotter melting regime.

The productivity functions (–d

) shown in Fig. 4a

maximum

achieved by the Langmuir

et al.

(1992) model

increases essentially without bound, whereas fractional

and b are integrated to produce the

plots in Fig.

melting with MELTS would require extraordinarily deep

4c and d for the MELTS and Langmuir

et al.

(1992)

and high

to reach

max

more than a few percent

models. The mean

and mean

for variable-

sys-

higher than that achieved at cpx-out. Instead, in the

tematics [i.e. using (6) and (7) and integrating to

MELTS model the lengthening of the tail, the decrease

] are shown by a filled circle along each

curve.

at cpx-out, and the overall lower productivity at

Comparing these plots shows three important di

erences

equal

associated with increasing

all combine to yield

between these models. First, because of the low pro-

max

in the narrow range 0·18–0·22 at the base of the

ductivity in the early stages of melting according to

crust over the wide range in

of 2·7–4·4 GPa ( Fig. 4c).

MELTS, there is a ‘tail’ 1–2·5 GPa wide in which

The extent of melting functions shown in Fig. 4c and

remains low. Second, as the shape of the

–

curve in

d are integrated to yield the

curves in Fig. 4e and

the Langmuir

et al.

(1992) model is almost independent

, the mean melt fraction,

, increases almost linearly

f. For these plots, the upper limit of integration is varied

with

to values of at least 0·22. In the MELTS model,

along the

-axis from

to generate the curves

however, there is a maximum in

0·08. This

shown. For passive flow models that aggregate all liquids

to form the crust, this integral is equal to the pressure at

behavior is explored in more detail below. Third, the

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JOURNAL OF PETROLOGY

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MAY 2001

the base of the crust and so is related to crustal thickness

the results for both compositions are similar in all im-

portant respects (i.e. the following discussion is not sens-

by a simple density correction. Assuming a constant

itive to details of spinel stability). When plotted against

crustal density of 2·62 g/cm

, we obtain crustal thickness

potential temperature, the MELTS results and the models

in kilometers according to the formalism of Klein &

of Langmuir

et al.

(1992) and Kinzler (1997) agree in

Langmuir (1987). These figures, like the

–

plots, show

many respects in the ‘normal’ range of potential tem-

that any given crustal thickness is produced by a melting

peratures (i.e. 1300–1400

C), but di

er for anomalously

path with much higher

according to MELTS than

hot or cold mantle. When plotted against

, MELTS

according to the Langmuir

et al.

(1992) model. For

predicts lower

and

for all

for reasons discussed

example, 7 km of crust results from an adiabat with

above, but primarily because MELTS generates a low-

2·1 GPa in the Langmuir

et al.

(1992) model but

productivity ‘tail’ near

(Hirschmann

et al.

, 1994;

requires

2·8 GPa according to MELTS. These

Asimow

et al.

, 1997); hence for

1·5–3 GPa, MELTS

figures also show that the Langmuir

et al.

(1992) model

calculations mimic a constant-productivity model with

can readily generate crustal thickness of 30 km or more.

at least 0·5 GPa lower. The di

erences for abnormally

MELTS, on the other hand, at least for the passive-flow

cold conditions are not surprising given the novel be-

case, never achieves values greater than

15 km within

havior predicted by MELTS for low

(which shows up

the range of

limited by the

maximum on the

most strongly in integrated melts with low

) and the

solidus discussed above (active flow and other means of

influence of the spinel–plagioclase transition. At hotter

generating more crust are discussed below).

than normal potential temperatures, the

value attained

by MELTS for increasing

flattens and reaches a

maximum for

3·5 GPa, which in turn leads to a

decreasing slope of the crustal thickness vs solidus pressure

Variable-

systematics

curve beyond

3·0 GPa: that is, from simple 2D

Figure 5 presents the net melt production of melting

passive-flow fractional melting regimes of this type with a

regimes at the final pressure of melting, whereas Fig. 4

well-defined

, MELTS cannot generate crustal thickness

shows the evolution of melting with pressure through

above 15 km, regardless of

each melting regime. Results are shown for passive-flow

For solidus pressures greater than

1·7 GPa, i.e.

fractional melting calculations with variable-

sys-

potential temperatures higher than

1280

C (boxed 1

tematics:

is adjusted to be equal to the base of the

in Fig. 5), plagioclase does not appear in the residue

crust for each

(i.e. melting stops in each column at

during fractional melting (this value of

is lower than

) and

varies with

. The relationships

the limit for plagioclase to appear on batch melting

among the plotted variables are controlled by the shape

adiabats; boxed 5 in Fig. 1). As solidus pressure decreases

of the solidus (i.e.

as a function of

; see Fig. 1a and

from 1·7 GPa (

<1280

C, boxed 1 in Fig. 5), the

c), variations of productivity among melting paths of

spinel–plagioclase peridotite transition plays an in-

erent potential temperature ( Fig. 4a and b), and

creasingly important role in modifying the amount of

variations of productivity with

along the melting path

melt produced and the pressure range over which melt

at a given potential temperature (insets in Fig. 4a and

production occurs. In a simple passive flow model, this

b). As shown by Klein & Langmuir (1987), any model

transition first divides the melting region into two dis-

with a linear solidus (in

–

space) and constant pro-

connected regions, an upper triangle and a lower trap-

ductivity will yield linear relationships among

ezoid (Asimow

et al.

, 1995). With falling potential

, and (

)

1/2

. The productivity variations in the model

temperature, the bottom of the triangle retreats upward

of Langmuir

et al.

(1992) are small enough that in Fig. 5

and the top of the trapezoid retreats downwards, i.e. the

the relationships for this model are indistinguishable from

transition shuts o

melting at deeper levels and melting

straight lines for

and

and parabolas for

when

resumes at shallower levels (see solidus in Fig. 1) as

plotted against both

and

. The model of Kinzler

and

decrease. This e

ect is manifested in Fig. 5d as

(1997) assumes constant productivity, also resulting in

a turnaround in mean pressure of melting (i.e. the loss

linear relations among

, and (

)

1/2

( Fig. 5d–f ),

of the shallower parts of the melting region results in

despite predicting a mildly curved solidus that leads to

increasing mean pressures of melting with falling potential

weakly curved trends in these variables against

( Fig.

temperature or solidus pressure in this range). At a certain

5a–c). There is no simple way to combine plagioclase-,

critical value of

and

(the kink labeled with boxed

spinel-, and/or garnet-bearing calculations using the

2 in the Fig. 5 curves at

1200

C and

1·3

Kinzler (1997) model, so results are shown only for

GPa), the upper triangle of the melting regime disappears;

melting paths with residual spinel everywhere.

i.e. the pressure at which melting in the plagioclase field

Results of MELTS calculations for both the Cr-bearing

would resume becomes shallower than the minimum

pressure of melting at the base of the crust. With further

and Cr-absent source compositions are shown in Fig. 5;

976