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226Ra/230Th–excess generated in the lower crust: Implications for

magma transport and storage time scales

Josef Dufek*

Kari M. Cooper*

Department of Earth and Space Sciences, Box 351310, University of Washington, Seattle,

Washington 98195, USA

1GSA Data Repository item 2005xxx, Model derivation, is available online at http://www.geosociety.org/pubs/ft2005.htm, or on request from editing@geosociety.org or Documents Secretary, GSA, P.O. Box 9140, Boulder, CO 80301-9140, USA. —————

*E-mails: dufek@u.washington.edu, kmcooper@u.washington.edu

ABSTRACT

226Ra-excesses in arc magmas have been interpreted to result from flux melting of the

mantle above subducting slabs followed by fast ascent rates of magma from slab to surface, up to

1000 m/yr. However, we demonstrate that incongruent melting of the lower crust could either

maintain or augment mantle-derived 226Ra-excesses and so reduce inferred vertical transport

rates. We developed an incongruent, continuous melting model, and both the incongruent

melting reaction and ingrowth effects contribute to the 226Ra-excess. In particular, we found that

dehydration melting of amphibolite can produce modeled 226Ra-excesses greater than 300%.

Mixtures of such amphibolite dehydration melts with mantle melts will likely retain a 238U-

excess (subducted slab) signature. This amphibolite dehydration melting process will also

produce elevated light rare earth element to heavy rare earth element ratios, similar to those

observed in several arc settings, that may distinguish these magmas from those with 226Ra-

excesses produced by slab dewatering alone.

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Keywords: time scales, melting, 226Ra, 238U, 230Th, lower crust.

INTRODUCTION Arc magmas have compositions that reflect an integrated history of melt generation

processes in the mantle and variable extents of polybaric assimilation and fractionation within

the crust, and many continental arc magmas in particular have trace-element and isotopic

signatures consistent with significant contributions from crustal material (Davidson et al., 2005).

Numerous lines of evidence are consistent with lower crustal assimilation by mantle-derived

magmas, including the observation of lower crustal outcrops that preserve evidence of crustal

melting (Pickett and Saleeby, 1993; Williams et al., 1995), the trace element signature of residual

garnet in some continental arc magmas (Hildreth and Moorbath, 1988), isotopic data consistent

with crustal assimilation (Hart et al., 2002), and thermal models that predict enhanced melting in

thickened continental crust (Dufek and Bergantz, 2005).

Disequilibria between intermediate daughter products in the decay series of 238U provide

a useful chronometer for magmatic processes and ascent rates of magma from depth, sensitive to

time scales of ca. 10 ka–350 ka (238U-230Th disequilibria) or ca. 100 a–10 ka (230Th-226Ra

disequilibria). A combination of (238U)/(230Th) >1 (238U-excess) and (226Ra)/(230Th) >1 (226Ra-

excess) is usually observed in arc settings, distinguishing arc lavas from mid-oceanic-ridge basalt

(MORB) and oceanic-island basalt (OIB) sources (e.g., Lundstrom, 2003). Typical arc lavas

have 226Ra-excesses of ~200%–300% [i.e., (226Ra)/(230Th) = 2–3], although extremely large

226Ra-excesses have been observed in a few island arc lavas [>600%; Turner et al., 2000]. 226Ra-

excesses in arc lavas have predominantly been explained by fluid-induced partial melting

involving Ra-enriched aqueous fluids deep in the mantle (e.g., Turner et al., 2003a). Where

interaction of melts with the crust beneath volcanic arcs has been considered, it is assumed to

decrease mantle-derived 226Ra-excesses due to decay of 226Ra during transport and storage. This

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interpretation implies that the magma has not stalled at any depth for long periods of time

relative to the half-life of 226Ra (t1/2 = 1.6 k.y.).

Other interpretations of 226Ra-excesses in arcs are emerging, for example diffusion-

induced disequilibria created in the mantle beneath arcs (Feineman and DePaolo, 2003), or a

combination of flux melting and daughter ingrowth during melting (Thomas et al., 2002).

However, these mechanisms still focus on processes operating within the mantle, and the degree

to which crustal melting and wallrock interactions can impact U-series disequilibria during

storage and ascent of magmas has yet to be fully explored. Recently Berlo et al. (2004) examined

the case of batch melting of lower-crustal material at high melt fraction (0.15–0.40) and

concluded that U-series activity ratios of these melts were near equilibrium. However, the

assumption of congruent, batch melting to high melt fraction may be inappropriate under many

crustal melting conditions (Rushmer, 1995). For example, mafic, amphibolitic lower crust at

pressures of ~10 kbar will undergo incongruent dehydration melting in which amphibole and

plagioclase react to form garnet, pyroxenes, and melt (Wolf and Wyllie, 1994).

With this geological motivation, we develop an incongruent, continuous melting model

for lower crustal conditions1 to further investigate the role of crustal processes on the U-series

nuclides. This model predicts significant 226Ra-excesses (>3), similar to those in the majority of

arc lavas, and modest 230Th-238U disequilibria [(230Th)/(238U) = 0.7–1.12] in melts produced

during dehydration melting of the lower crust. When mixed with mantle-derived magmas, the

226Ra-excesses generated in melts of deep continental crust may augment preexisting

disequilibria and can relax the ascent time scales inferred for arc magmas.

CONTINUOUS DEHYDRATION MELTING

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We extend the work of Zou and Reid (2001) and McKenzie (1985) to incongruent

melting with ingrowth of radioactive nuclides. Melt production is modeled as continuous: the

melting process occurs at a constant rate in a static (i.e., not upwelling) residue, and the melt

fraction below a critical porosity (Φ) is in equilibrium with the residue (Williams and Gill,

1989). Any melt in excess of this critical porosity is instantaneously extracted and aggregated .

This model is a more physically plausible representation of melting of a static lower crust that is

being fluxed with mafic intrusions than is dynamic (i.e., upwelling residue), batch, or fractional

melting. The full derivation of the incongruent continuous melting model for both stable

elements and radioactive nuclides is included in the supplemental material1.

We applied this model to lower-crustal melting using the specific phase proportions and

compositions from the 10 kbar amphibolite dehydration melting experiments of Wolf and Wyllie

(1994), appropriate for melting occurring at the base of ~30-km-thick crust. The general form of

the incongruent melting reaction is Amp + Plag → Opx + Cpx + Gt + Melt. We used partition

coefficients (Ds) relevant to the phases, pressures, and compositions in the lower crust (Table 1),

and we assess the sensitivity of changing anorthite content on the U-series disequilibria.

MODEL RESULTS

As incongruent melting proceeds, the production of phases in which radium is highly

incompatible (garnet and pyroxene) combined with the consumption of phases in which Ra is

only moderately incompatible (plagioclase and amphibole) act in concert to elevate

(226Ra)/(230Th) in the melt well beyond what would be predicted based on congruent melting

models (Fig. 1). A further increase over batch melting occurs due to ingrowth of 226Ra during the

melting process and partitioning of Ra into the liquid. A competing effect is dilution of Ra and

other incompatible elements in the aggregate liquid as melting proceeds. At high melting rates

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(>1 kg/m3yr) and low critical mass porosity (<0.001), the model disequilibria are comparable to

or higher than those produced during congruent batch melting with similar residual mineralogy

(cf., Berlo et al., 2004). At slower melting rates (approaching ~1.0 × 10–4) the degree of

disequilibrium becomes a weak function of melting rate and—as with dynamic melting

calculations applied to mantle upwelling—(226Ra)/(230Th) is principally determined by the

critical mass porosity, with ratios >5 produced at melting rates of 1.0 × 10–4 and critical mass

porosities of 0.01. Mixtures of mantle-derived magmas with lower-crustal melts can thus have

substantial 226Ra-excesses, even if transport times through the mantle wedge were significant.

Even the lowest mass porosities considered here are an order of magnitude larger than

those used in many mantle melting calculations (e.g., Lundstrom, 2003; Sims et al., 1999) but are

consistent with observations that melt segregation during lower-crustal melting experiments

occurs at porosities of a few percent to 15% (Rushmer, 1995). Note that if we apply bulk

partition coefficients and melting conditions appropriate for mantle melting to this model, we

recover the results of mantle melting calculations (e.g., Zou and Zindler, 2000).

Both uranium and thorium become less incompatible as the dehydration reaction

progresses and the critical porosity has a modest control on the degree of 230Th-238U

disequilibria. At small critical porosities (<0.001) and low extracted melt fractions (<0.02),

(230Th)/(238U) = ~0.75. As the dehydration reaction proceeds, the amount of residual garnet

increases and uranium becomes increasingly more compatible in the residue. This, combined

with ingrowth of 230Th during melting, drives (230Th)/(238U) toward higher values, and 230Th-

excesses (~12%) develop at high critical porosities and extracted melt fractions greater than 0.2.

Mixtures of crustal melts with mantle-derived melts will likely be weighted toward 238U-

excesses, unless extracted melt fractions are high.

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This model also predicts an enrichment of light rare earth elements (LREE) relative to

heavy rare earth elements (HREE) in the melt, due in large part to the formation of garnet, as

well as enrichment of barium and strontium in the melt due to the melting of plagioclase.

Amphibolites present in the lower crust are likely to have formed by stalling and solidification of

earlier intrusions of mantle-derived hydrous basalts. Hence we use primitive island arc basalts as

a model source composition (Fig. 2B), although the general trace-element behavior (Fig. 2A)

will remain the same for a reasonable range of compositions of amphibolite. The trace element

pattern follows the same general trend as many continental andesites and dacites (Fig. 2B). When

mixed with mantle-derived magmas, amphibolite melts will dominate the trace-element budget

of mixtures even when they represent a small mass fraction (see Fig. 2B). Some upper crustal

fractionation (involving plagioclase) of amphibolite-derived and/or mixed melts would be

required to account for the negative Eu anomalies observed in many continental arc magmas.

Any mixing model must also satisfy major element constraints. Melts from amphibolite

dehydration are andesitic to rhyodacitic. Melts of quartz amphibolites would have the highest

SiO2 concentration (Patiño-Douce and Beard, 1994), whereas amphibolites with more primitive

compositions produce melts that at melt fractions less than 0.2 have 60–64 wt% SiO2. Mixtures

of these crustal melts with mantle-derived basalts will produce intermediate magmas of basaltic

andesite to andesite composition.

DISCUSSION

Based on our model results, crustal processes need not act solely to decrease mantle-

derived 226Ra-excesses, and U-series disequilibria measured in arc lavas may reflect the

influence of multiple processes (Fig. 3). In particular, mixtures of mantle-derived basalt with

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amphibolite melts can have significant 226Ra-excesses and trace-element patterns similar to those

observed in arc lavas.

There are appreciable uncertainties in the melting rate of the crust; numerical models of

crustal melting predict that melting rates can vary locally from 1.0 × 10 kg/m3yr to 1.0 × 10–4

kg/m3yr depending on the flux of basaltic magma and style of intrusion in the crust (Annen and

Sparks, 2002; Dufek and Bergantz, 2005) [cf., mantle decompression melting rates of 10–3

kg/m3yr to 10–5 kg/m3yr (Bourdon and Sims, 2003)]. Even assuming instantaneous melting,

significant Ra-excesses can develop.

Due to crystal-chemical effects on partitioning, melting of anorthitic plagioclase produces

larger Ra-excesses than melting of low-An plagioclase. Aggregate fractional melting of

amphibolite with An50 plagioclase produces (226Ra)/(230Th) of 1.21–1.55 for the melting range of

0–0.15 melt fraction, similar to the modest excesses predicted by Berlo et al. (2004). However, if

melt extraction rates are less than ~10–2 kg/m3 yr, ingrowth effects will result in Ra-excesses

exceeding 2.0 at melt fractions of 0.15 even with these lower-An plagioclase compositions.

Stable element patterns can be used to constrain the overall degree of melting for the

model. As an example, the La/Yb for lavas from Mt. St. Helens are consistent with modeled melt

fractions of ~5%–10% (with Φ = 0.1), which yields initial (226Ra)/(230Th)–excesses of 3.0–4.0 if

melting is slow enough for ingrowth effects to develop. Age-corrected (226Ra)/(230Th) in recent

lavas from Mount St. Helens are 1.46–1.9, and the time to reach (226Ra)/(230Th) = 1.5 from an

initial disequilibria of 3.0–4.0 is ~3000–4000 yr, similar to plagioclase ages in these lavas

(Cooper and Reid, 2003). Assuming a crustal thickness of 35 km, this would imply average

crustal ascent rates of ~8–10 m/yr, compared to ascent rates exceeding an order of magnitude

greater if the disequilibria were generated entirely in the mantle (Turner et al., 2001). Moreover,

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since the modeled 226Ra-230Th disequilibrium is generated as a result of melting processes in the

lower crust, it is insensitive to the time spent traversing the mantle wedge. Therefore, the 226Ra-

excesses of arc magmas could potentially be decoupled in time from stable trace-element or

long-lived isotopic signatures of slab fluids that are carried by older mantle-derived magmas that

formed lower-crustal amphibolites.

CONCLUSIONS

Ra-excesses observed in island arcs and continental arcs need not be produced

exclusively as a result of slab dewatering and resulting flux melting. The exposed roots of arcs

show ample evidence for garnet amphibolite compositions that are capable of melting to produce

magmas with significant 226Ra-excesses and with either 230Th- or 238U-excesses. Mixtures of

mantle and crustal melts in the lower crust will most likely be weighted toward 238U-excesses.

These mixed magmas can have significant 226Ra-excesses regardless of the transit time through

the mantle wedge, and therefore extreme transport rates in arcs are required only in the case of

primitive arc basalts where (226Ra)/(230Th) in arc lavas exceeds 3–5.

ACKNOWLEDGMENTS

This study was supported in part by a National Aeronautics and Space Administration

(NASA) Earth Systems Science Fellowship (JD). Discussions and suggestions from G.W.

Bergantz and O. Bachmann greatly improved the manuscript. We thank J. Miller, C. Lundstrom,

J. Davidson, M. Williams, and an anonymous reviewer for their insightful reviews.

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Figure 1. Isosurfaces of (A) (226Ra)/(230Th) and (B) (230Th)/(238U) in the aggregate melt from the

continuous melting of amphibolite (Initial mode: 67% amphibole, 33% plagioclase). Activity

ratios are shown for a range of critical mass porosity (Φ), extracted melt fraction, and melting

rate. Slices at constant melting rate, 100 and 10-3 kg/m3yr, are also shown. Aggregate fractional

melting is approached at high melting rates and low critical porosity (extreme right corner of the

figure). At high melting rates dilution effects dominate but as the melting rates decrease to be

equal or less than the melting timescale ( )1ln( XM e

avgmelt −

−=

ρτ , where Me is the melt

extraction rate, ρavg is the average density, and X is the extracted melt fraction) ingrowth effects

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of the daughter nuclide become important1. Extracted melt fraction is related to the total melt

fraction as XF )1( Φ−+Φ= , where F is the total melt fraction and X is the extracted melt

fraction. At F~.23 amphibole and plagioclase are consumed and the melting reaction becomes

congruent melting of the residue resulting in a change in curvature on these plots1.

Figure 2. (A) Trace element enrichment and (B) trace element concentration relative to primitive

mantle for amphibolite dehydration at 0.1 critical melt fraction (Sun and McDonough, 1989).

Initial composition for the amphibolite is assumed to be equal to primitive island arc basalt

(Basaltic Volcanism Study Project, 1981). Trace element fields for island arc basalts (Basaltic

Volcanism Study Project, 1981) and continental arc dacites are shown in shaded regions (Conrey

et al., 2001; Trumbull et al., 1999). For reference, a 50-50 mixture of crustal melt (melt fraction

0.05) and island arc basalt is shown (dashed line with square symbols).

Figure 3. Conceptual model of U-series disequilibria produced by different arc processes. Slab

dehydration releases fluids enriched in uranium and radium relative to thorium. These fluids

initiate flux melting in the mantle. Melts that reach crustal levels can potentially stall, solidify

(producing amphibolite or pyroxenite lithologies), and/or initiate heating and remelting of

previous mafic intrusions (amphibolite dehydration), producing melts with either (238U)/(230Th)

<1 or (238U)/(230Th) >1 and (226Ra)/(230Th) >1.

TABLE 1. PARTITION COEFFICIENTS FOR AMPHIBOLITE MELTING CALCULATIONS Amp† Plag§ Opx# Cpx** Gt †† Ba 0.12 0.056 0.00074 0.0096 1.0 ×

10–5 Ra 0.0096 0.008 7.4 x

10–6 9.6 × 10–5

1.0 × 10–9

Th 0.017 0.003 0.021 0.82 0.10 U 0.008 0.0006 0.018 0.77 0.37 Nb 0.19 0.12 0.001 0.017 0.005 La 0.12 0.12 0.001 0.017 0.025 Ce 0.24 0.07 0.12 0.13 0.04 Sr 0.28 1.1 0.07 0.09 0.20 Nd 0.62 0.08 0.27 0.31 0.09 Sm 1.4 0.05 0.46 0.60 0.83 Zr 0.23 0.0001 0.027 0.23 0.52 Eu 1.1 0.07 0.35 0.69 0.74 Ti 2.0 0.02 0.24 1.3 0.79 Gd 1.5 0.03 0.01 0.82 1.1 Dy 1.8 0.02 0.02 0.99 4.4 Y 1.7 0.012 0.09 1.2 10.0 Er 1.5 0.02 0.1 1.1 9.4 Yb 1.2 0.004 0.17 0.84 14.0 Lu 1.1 0.009 0.18 0.92 16.0 *Ra partition coefficients determined using lattice strain parameters with Ba partition coefficients. † (Brenan et al., 1995; Klein et al., 1997). § (Bindeman and Davis, 2000; Bindeman et al., 1998; Blundy and Wood, 2003). # (Beattie, 1993; McDade et al., 2003). **

(Barth et al., 2002; Klein et al., 2000).

†† (Barth et al., 2002; Beattie, 2003; Klein et al., 2000;

van Westrenen et al., 2001)

0.050.10.150.20.25

−2

−1.5

−1

0.050.10.150.20.25

−2

−1.5

−1

0.050.10.150.20.25

−2

−1.5

−1

0.10.2Extracted melt fraction

Melting Rate

(kg/m3yr)

3.02.0

- 5.0

- 4.0

- 3.0

- 2.0

- 1.0

0.25 0.20 0.15 0.10 0.05 0.010-2

10-1

I

(226Ra)(230Th)2.0

Isosurfaces of (226Ra)/(230Th)A

Extracted melt fraction

- 1.1

- 1.0

- 0.9

- 0.8

- 0.7

(230Th) (238U)

B Isosurfaces of (230Th)/(238U)

10-4 10-3 10-2 10-1 10-2

10-1

I

0.25 0.20 0.15 0.10 0.05 0.010-2

10-1

I 3.04.0

1.5

Melting Rate = 100

Melting Rate = 10-3

0.10.2

10-2

10-1

I

10-4 10-3 10-2 10-1

Melting Rate

(kg/m3yr)

Extracted melt fraction

1.0

.9

0.050.10.150.20.25

−2

−1.5

−1

Melting Rate = 100

Melting Rate = 10-3

10-2

10-1

I

10-2

10-1

I0.25 0.20 0.15 0.10 0.05 0.0

0.25 0.20 0.15 0.10 0.05 0.00.9

0.9

1.0

Extracted melt fraction

Figure 1, Dufek and Cooper

Cf/C

0C

f/PM

Ba Th U Nb La Ce Sr NdSm Zr Eu Ti Gd Dy Y Er Yb Lu

0.1

1

10

100

1000

10

1

.15

.05

.25

Continental arc dacites

Island Arc Basalts

Trace element enrichment

50% Mixture bas. + .05 Crustal melt

Dufek and Cooper, 2005Figure 2

A

B

Dehydration of theSlab: (238U)/(230Th)>1 (226Ra)/230Th)>1

Dehydration melting ofamphibolite: (238U)/(230Th)<1 or >1(226Ra)/230Th)>1

Mantle flux melting

0 km

~35 km

~100 km

Lower crustalmelting ofprevious intrusions

Slab

Crust

Not to scale

Dufek and Cooper, 2005Figure 3

Supplemental Material 1. Incongruent Continuous Melting Calculations We follow the definition of McKenzie (1985) and Williams and Gill (1989), and

refer to continuous melting as the integrated processes of melting and removal of magma

from a static residue having a critical porosity. Continuous melting in this sense is similar

to dynamic melting (commonly applied in the case of adiabatic melting of upwelling

mantle), but with the difference that the solid residue in continuous melting is static (i.e.,

has no upwelling velocity). In the case of stable elements, dynamic melting and

continuous melting are mathematically equivalent (Williams and Gill, 1989), and

equations describing incongruent dynamic melting of stable elements have been

presented previously (Zou and Reid, 2001). In this section we present the derivation of

equations describing U-series nuclide behavior during incongruent, continuous melting.

We first derive equations describing the behavior of stable elements (section 1.1), which

are mathematically identical to those obtained by Zou and Reid (2001) although derived

in a somewhat different manner. We then expand these equations to account for the

effects of ingrowth and decay during the melting process (section 1.2). In this paper, we

apply these equations to modeling of 226Ra-230Th and 230Th-238U disequilibria during

incongruent melting of amphibolite, but the equations are general and can be applied to

any parent-daughter pair and to any incongruent melting reaction.

1.1 Stable Element Calculation

During continuous melting, initial melt fractions remain in equilibrium with the

solid residue (batch melting) until a critical porosity is reached. Any additional melt

produced beyond this critical porosity is instantaneously extracted and pooled elsewhere

and the extracted melt fraction is referred to as X. As the critical mass porosity ( )

approaches 0, the continuous melting process approaches incongruent fractional melting.

We derive in this section equations describing the concentration of stable trace elements

in the melt phase as a function of the extracted melt fraction and the critical porosity.

Φ

During incongruent melting some solid phases are consumed while others are

produced along with the melt. Those which are consumed or produced in different

proportions from their initial modal abundance are termed “incongruent” phases, whereas

those which contribute to the melting reaction in their initial modal proportions are

termed “congruent” phases. Hence the bulk partition coefficient of element i (Di)

becomes a function of the melt fraction. Phase proportions are governed by an

incongruent reaction which can be written as:

Melt+++→+++++ ......... 212121 ββθθαα , (1)

where iα are the incongruently melting phases, iθ are the congruently melting phases and

iβ are the incongruently produced phases (Zou and Reid, 2001). For the case of

amphibolite dehydration melting, only incongruently consumed and produced phases are

present.

A general conservation equation for a stable element i (neglecting advective and

diffusion terms) is given by:

Mcct

ctc i

fis

if

f

is

s )()1( −=∂∂

+∂∂

− φρρφ , (2)

where φ is the critical volume porosity, sρ is the solid residue density, fρ is the magma

density, is the concentration in the residue, is the concentration in the magma and M

is the melting rate, assumed to be constant. Note that the concentration of stable trace

elements depends on the melt fraction but not on the timescale of melting and therefore is

independent of the melting rate; we include it here to make the derivation parallel to that

of the derivation of equations relevant to radioactive nuclides (section 1.2). We can

replace time with extracted melt fraction as a variable using the following relationship

between extracted melt fraction X and time (McKenzie, 1985):

isc i

fc

−+−

−= tMXsf

e

)1(exp1

φρφρ, (3)

where is the melt extraction rate. As pointed out by Zou (1998) the melt extraction

rate is not strictly the melt production rate, but they are related as:

eM

Φ−= 1eM

M , (4)

where Φ is the critical mass porosity rather than the critical volume porosity and is

related to the critical volume porosity by:

)1( φρφρφρ

−+=Φ

sf

f . (5)

As the stable element concentration is ultimately related to reaction progress rather than

strictly to time, it is advantageous to write the differential equation in (2) in terms of

extracted melt fraction rather than time. Using (3) we can perform a change of variables

in (2) , and using (4) and (5) and noting also that we get the following

expression:

is

iif cDc =

)1()1()1()1)(1( −Φ−=∂∂

Φ−+∂

∂−Φ− ii

f

if

if

i

DcXc

XXcD

X . (6)

During incongruent melting Di is not constant and remains inside the differentiation

operator. To perform this differentiation we first need to express Di as a function of the

melt progress through the extracted melt fraction variable. Both Zou (2001) and Hertogen

and Gijbels (1975) give essentially the same expression for in incongruent melting

situations based upon mass balance arguments. Using the notation of Zou:

iD

[ FQDF

D iii001

1−

−= ], (7)

where

∑= ji

ji KxD 00 , (8)

and

l

ji

jji

i

t

KtKQ

∑−= β

0 . (9)

F is the total melt fraction (which is related to the extracted melt fraction by

), is the partition coefficient of i in the jth phase (here the partition

coefficient is assumed constant through the melting range), is the fraction of the jth

phase originally present, and is the fraction of the jth solid incongruently produced

phase and is the fraction of melt produced. is the initial bulk partition coefficient

and Q is a constant related to the rate of change of the partition coefficient as a result of

reaction progress as t phases are being produced. Note that (9) would have to be

modified if congruently melting phases were also present (Zou and Reid, 2001).

Substituting (9) in (7), and using the chain rule gives the simplified expression:

XF )1( Φ−+Φ=

lt

i0

jiK

j

jx0

jtiD0

[ ] XQQXQD

QQcc

iiii

ii

if

if ∂

−Φ+−−Φ+

−−Φ+=

∂)1()1(

1)1(

0000

00 (10)

Integrating and using the fact that )1( 00

00 ii

iif QD

cc−Φ+

= (Zou and Reid, 2001) gives us

the following expression for cif as a function of critical mass porosity and extracted melt

fraction:

1)1(

1

00

00

00

0 00

)1()1(1

)1(

−−Φ+

−Φ+−Φ+

−−Φ+

=ii QQ

ii

ii

ii

iif X

QDQQ

QDcc . (11)

To get the concentration of element i in the aggregate melt this expression must be

integrated over X and then divided by X to give:

−Φ+−Φ+

−−=−Φ+ )1(

1

00

000 00

)1()1(11

ii QQ

ii

iiiif X

QDQQ

Xcc . (12)

This is identical to the expression given in Zou and Reid (2001) eqn. 16 although derived

in a somewhat different manner.

1.2 Radioactive Nuclides

The same general solution procedure is followed for the U-series nuclides, although in

this case there is an added source term in the conservation equation that complicates the

calculation. The half-life of 238U is much longer than the melting timescale considered,

therefore it can be treated as a stable element and equations (11) and (12) can be used to

determine the concentration. However, for 230Th and 226Ra a new conservation equation

must be used:

[ ] [ φρφρλφρφρλφρρφ Dff

Dss

DPff

Pss

PDf

Ds

Df

f

Ds

s ccccMcct

ct

c+−−+−+−=

∂∂

+∂∂

− )1()1()()1( ]

(13)

Here the superscripts D and P refer to daughter and parent nuclide, respectively, and λ is

the decay constant. The first term on the right-hand side accounts for transfer of daughter

atoms from the solid to the liquid through melting, the second term accounts for ingrowth

of the daughter due to decay of the parent during the melting process, and the third term

accounts for decay of the daughter. Using (3) and changing the independent variable from

time to extracted melt fraction gives:

[ ] [ φρφρλφρφρλφρφρ

φρφρφρ

ρφ Dff

Dss

DPff

Pss

PDf

Ds

Df

sf

ef

Ds

sf

es ccccMcc

XcXM

XcXM

+−−+−+−=∂∂

−+−

+∂∂

−+−

− )1()1()()1(

)1()1(

)1()1( ]

(14)

We can express this equation in terms of the concentration of daughter in the magma by

dividing (14) through by M, using φρφρ

φρ

fs

s

eMM

+−−

=)1(

)1( , defining

φρφρρ fsavg +−= )1( , and using the partition coefficient expression

( [ ])1( 0DQ−Φ) 0

DQX +−1( 00DDD QDD −Φ+= ):

[ ]

[ ] [ ]

[ ]

−

−Φ+−−Φ+

=

−

−Φ+−−Φ++

−Φ−−−−Φ+−−Φ+

−−Φ−−

+

Φ−−Φ+−−Φ+

∂

∂

XQQXQD

Mc

XQQXQD

MXXQQXQD

XQD

c

QQXQDXc

PPPPPPf

DDDDD

avg

DDDD

avg

DDDf

avg

DDDDDf

1)1()1(

)1()1()1(

)1)(1()1()1()1(

)1)(1(

)1()1()1(

0000

0000000000

0000

λ

λρρ

ρ

(15)

To simplify the expression, we define and : )1( 00 QDA −Φ+= )1( 00 QQB −Φ+=

[ ]

[ ] [ ]

[ ]

−−

=

−−

+−Φ−

−−−−

−Φ−−

+

Φ−−

∂∂

XBXA

Mc

XBXA

MXXBXA

XQDc

BXAXc

PPPPf

DDD

avg

DD

avg

DDDf

avg

DDDf

1

)1()1)(1()1(

)1)(1(

)1(

00

λ

λρρ

ρ

(16)

This equation cannot be solved by separation of variables but can be written in the form,

LPcc Df

Df =+' (17)

where P and L are constants. The solution is:

)(1

0

DBDDIDf AcTec

−

+= − , (18)

where

dXLeT I∫= , (19)

and

. (20)

∫= PdXI

P and L can be simplified to:

)1()1(

)()1(

XMXBABP avg

D

DD

D

−Φ−

+−−

=ρλ

, (21)

and

))(1()1)((

XBAXMXBAc

L DD

PPavg

Ppf

−−Φ−−

=ρλ

. (22)

So,

MBB

DDIavg

D

D

D

XXBAeρλ )1(1

)1()(Φ−−−

−−= (23)

and,

∫ ∂−−−Φ−

=−

Φ−−−

XXXBAXBAcM

T MBDDPPpf

avgP avg

D

D 1)1(1

)1())(()1( ρλρλ

. (24)

pfc

D

is also a function of X. The abundance of 238U can be computed with (18), but since

there is no parent nuclide T=0, the solution simplifies to that obtained by separation of

variables. The abundance of 238U can then be used to compute the abundance of 230Th

using (18). The integral in T is difficult to solve analytically because in general PDP BBA ≠≠≠A , and in this study T was evaluated numerically using the trapezoidal

method. With this strongly non-linear function care must be taken that the integration

steps are sufficiently small so that errors are small. This requires iterative refinement until

activities ratios changed by less than 0.01. Once is determined as a function of X

we can use it as the parent in (18) to determine . Equation (18) gives the

concentration of the daughter nuclide in the melt at any point in the melting process (the

instantaneous melt), but the concentration in the aggregated extracted melt is more

relevant to this work. This is again calculated using:

Thfc230

Raf

226

c

∫ ∂= XcX

c if

if

1 . (26)

1.3 Amphibolite Dehydration Reaction

The phase proportions from the amphibolite dehydration experiments of Wolf and

Wyllie (1994) were used to form the incongruent melting reaction: .703 Amp+ .297 Plag

→.031 Opx + .378 Cpx + .355 Gt +.236 Melt. The exact coefficients of this reaction will

change as a function of solid solution and P-T conditions (Beard and Lofgren, 1991;

Rapp et al., 1991), but this reaction serves as a reasonable proxy for melting occurring at

the base of ~30 km thick crust. After the amphibole and plagioclase have been exhausted

the residue is assumed to melt congruently.

Beard, J.S., and Lofgren, G.E., 1991, Dehydration melting and water-saturated melting of basaltic and andesitic greenstones and amphibolites at 1, 3, and 6.9 kb: Journal of Petrology, v. 32, p. 365-401.

Hertogen, J., and Gijbels, R., 1975, Calculation of trace element fractionation during partial melting: Geochimica Et Cosmochimica Acta, v. 40, p. 313-322.

McKenzie, D., 1985, 230Th-238U disequilibrium and the melting processes beneath ridge axes: Earth and Planetary Science Letters, v. 72, p. 149-157.

Rapp, R.P., Watson, E.B., and Miller, C.F., 1991, Partial melting of amphibolite/eclogite and the origin of Archean trondhjemites and tonalites: Precambrian Research, v. 51, p. 1-25.

Williams, R.W., and Gill, J.B., 1989, Effects of partial melting on the uranium decay series: Geochimica Et Cosmochimica Acta, v. 53, p. 1607-1619.

Wolf, M.B., and Wyllie, P.J., 1994, Dehydration-melting of amphibolite at 10 kabr: the effects of temperature and time: Contributions to Mineralogy and Petrology, v. 115, p. 369-383.

Zou, H., 1998, Trace element fractionation during modal and nonmodal dynamic melting and open-system melting: A mathematical treatment: Geochimica Et Cosmochimica Acta, v. 62, p. 1937-1945.

Zou, H., and Reid, M., 2001, Quantitative modeling of trace element fractionation during incongruent melting: Geochimica Et Cosmochimica Acta, v. 65, p. 153-162.