Arc Basalt Simulator version 3 (ABS3) calculation scheme
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Auxiliary Material 2.
Arc Basalt Simulator version 3 (ABS3) calculation scheme
1. General calculation scheme
This Auxiliary Material describes the calculation scheme used in the Arc Basalt Simulator version 3
(ABS3), Microsoft® Excel® based spreadsheet. ABS3 is the successor of ABS2 [Kimura, et al.,
2009]. The major changes from the ABS2 model are the incorporation of slab melting and
melt-fluxed mantle melting functions (Figure 1).
The first slab melting function is an extension of the metamorphic phase diagrams formed
by Perple_X [Connolly and Kerrick, 1987; Connolly and Petrini, 2002]. The modifications are in
slab melting mode and use the experimental datasets of Grove, et al. (2006), Moyen (2009), Moyen
and Stevens (2006) and Schmidt, et al., (2004). The second melt-fluxed mantle melting function is
built by parameterizing the mantle melting mode and the generated basalt melt compositions using
the pMELTS thermodynamic model [Ghiorso, et al., 2002]. Melt-fluxed melting is simulated by
adding a felsic slab melt [Moyen and Stevens, 2006] to the depleted Mid Oceanic Ridge Basalt
(MORB) source mantle (DMM) composition [Workman and Hart, 2005]. The mantle melting mode
with addition of as much as 30 % of the slab melt fraction was calculated using pMELTS. These
modifications allow ABS3 to simulate the genesis of both primary high-Mg basalts and high-Mg
andesites. Additional differences between ABS2 to ABS3 are discussed below. For details of the
ABS2 model, readers should refer to the companion paper [Kimura, et al., 2009]. The new ABS3
calculation flow diagram is shown in Figure 1.
Figure 1. Flow diagram indicating calculation modules used in the ABS3 model. Intensive and
extensive parameters used as fixed parameters are shown in blue. Parameters used for fitting
variables are shown in red. Slab dehydration model is from Perple_X and experiments for
sub-solidus and supersolidus conditions, respectively. Mantle melting mode is from
parameterizations based on pMELTS. Symbols are explained in detail in the text.
Arc Basalt Simulator version 3 (ABS3) calculation scheme
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Figure 2. Phase diagrams and P-T trajectories for slab SED (terrigeneous sediment), AOC (altered
oceanic crust), and DMM (depleted MORB mantle) (top row panels) with changes in modal
composition of minerals (mid row panels), and H2O and melt fractions (bottom row panels). Phase
diagrams indicate mineral stability, water content, and degree of melting of SED, AOC, and DMM in
the subducting slab and mantle wedge. SST: trajectory of slab surface temperature calculated by the
geodynamic model for Izu arc. Water fraction, melt fraction, and phase proportion in SED, AOC,
and DMM are calculated along the SST. ABS3 calculations use mineral mode tables, while XH2O,
Xmelt are based on the phase diagrams. Pheng: phengite, SiO2: silica mineral, Gt: garnet, Cpx:
clinopyroxenes, Chl: chlorite, Amp: amphibole, Law: lawsonite, Serp: serpentine minerals, pale blue
field with number: water content (wt.%), red thin line with number: isopleth of degree of partial
melting (wt.%).
2. Slab dehydration and melting simulation
The new slab phase diagrams (Figure 2) were calculated for slab sediment (SED), altered oceanic
crust (AOC), and depleted MORB mantle (DMM). Due to the relatively sparse experimental results
available for silica-rich (chert) and silica-poor (clay) sediments, only an intermediate sediment
composition (corresponding to pelite to terrigeneous sediments) was used for SED calculations.
AOC and DMM are the same as those in ABS2. The slab mode composition of chert, terrigeneous,
Arc Basalt Simulator version 3 (ABS3) calculation scheme
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and clay differ only in quartz relative to other minerals (e.g., chlorite, amphibole, garnet,
clinopyroxenes, lawsonite). Although this does affect the partition coefficients between liquid and
SED, the effects are limited due to the extremely low partition coefficients in quartz to all
incompatible trace elements. Although ABS3 does not use chert and clay, the intermediate SED
composition is still a good estimate for the slab SED component.
2.1 Slab dehydration and wedge base mantle re-hydration model
Slab SED and AOC dehydrate during metamorphism. It is more complex at the base of the mantle
wedge, where DMM re-hydrates due to addition of the SED/AOC fluids, and dehydrates due to
metamorphism (Figure 3). Starting slab materials have compositions CDMM0, CSED0, and CAOC0.
Stepwise dehydration of the SED/AOC/DMM is calculated based on the Rayleigh distillation
equation, as in ABS2 (Figure 4). Progressive changes in residual slab compositions are due to the
release of slab fluids (Figure 4). DMM is, in contrast, fully metasomatized by the addition of
AOC/SED fluids with increasing volume of fully hydrated DMM (Figure 5). The compositions of
the fluids released from the metasomatized DMM are also calculated by applying the Rayleigh
distillation equation (Figure 5). Addition of slab AOC/SED melts is not considered. XH2O and
mineral modes used in the calculations for AOC/SED/DMM are from modified Perple_X phase
diagrams (Figure 2).
Figure 3. Schematic mass transport in the slab (AOC, denoted by A; and SED, denoted by S) and the
base of the mantle wedge (DMM, denoted by D).
Figure 4. Calculation scheme for slab dehydration in each calculation step and batch dehydration
equation used for the incremental calculations.
Arc Basalt Simulator version 3 (ABS3) calculation scheme
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Figure 5. Liquid composition calculation scheme for DMM. Starting composition of DMM is D0.
Additional fluid compositions from AOC and SED at step x are Ax and Sx calculated by the scheme
shown in Figure 4. With a flux of fluids at XH2OAx and XH2OSx, DMM composition at x step
becomes Dx’. Due to a resulting increase in the mass fraction, XDx becomes XDx’, then CDx’ is
recalculated to 100%. This assumes inflation of the DMM volume as it becomes fully saturated with
H2O. The second calculation is the batch dehydration equation using CDx’ as a new source DMM
composition. Finally, the total slab liquid composition (CliqSLABx) is calculated at various extents
of mixing between AOC, AED, and DMM liquids (see also Figure 3).
Figure 6. Examples of compositional change during prograde metamorphism of the slab (AOC,
SED), and base of mantle wedge (DMM) calculated by ABS3 model.
Liquid compositions, including slab fluid and slab melt (CliqSLABx), are defined by
variable mixing between AOC, SED, and DMM liquids with the fractions FliqAOC + FliqSED +
FliqDMM = 1 (Figure 3). This is assumed because the contributions of AOC and SED to slab liquids
and the relative contributions of DMM and the slab are unknown. Residual AOC and SED
compositions are calculated by progressive modification of the original compositions as a result of
stepwise fluid extraction (Figure 6). However, compositional changes in DMM due to dehydration
are not calculated as the degree of re-hydration – dehydration is unable to be estimated due to the
open system reaction of DMM.
Arc Basalt Simulator version 3 (ABS3) calculation scheme
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2.2 Slab melting calculations and treatment of H2O in the slab melt
For simplicity, several assumptions are made in the slab melting calculations. First, the slab melt
composition is calculated from batch melting of slab AOC and SED after each step of dehydration.
Second, the slab liquid composition is a mixture of fluid and melt in any combination (i.e., SED melt
and AOC fluid). Third, melting of the base of the mantle wedge DMM is not considered. Fourth, the
H2O contents of AOC/SED slab melts are based on the H2O fraction in the hydrous minerals within
the slab, and the fractions of these minerals consumed during melting (see equations below). Thus,
the final H2O content of the slab liquid is the total sum of H2O from AOC/SED/DMM at a given
(user defined) extent of mixing between the three liquid sources (see Figure 3). H2O in the slab
liquid may also vary due to the solubility of salts and is user defined, ranging from 100 % to 50%
[Manning, 2004].
Partition coefficients for the slab fluids are similar to those used in ABS2. Partition coefficients for
melts are simply 10 times the partition coefficients for fluids. Comparing these values to the
compiled experimental data [Moyen and Stevens, 2006] shows this assumption to be valid (see also
Figure 6 in the main text). To simulate the behavior of HFSE-bearing minerals, such as “Ti-bearing”
minerals (including titanite and zircon), these residual mineral modes are set to zero when slab
melting occurs because these minerals are highly soluble in melts and insoluble in fluids [Moyen and
Stevens, 2006], making it difficult to model HFSE abundances and isotopes [Tollstrup and Gill,
2005].
3. Zone refining reaction between slab liquid and mantle
Whereas ABS2 uses two zone refining stages (ZR1 and ZR2), ABS3 uses only one zone refining
stage, ZR. This is because ZR1 in ABS2 is fully simulated during re-hydration/dehydration of the
base of the mantle wedge DMM, where the reaction degree (n factor) of ZR1 is alternatively
simulated by the contribution of metasomatized DMM fluid to the mixture of slab liquids from AOC
and SED. We have confirmed that the effect of DMM dehydration in ABS3 is identical to that of
ZR1 in ABS2. Calculation equations used for ZR in ABS3 are shown below.
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4. Fluid-fluxed and melt-fluxed melting calculations of mantle peridotite
4.1 Mode and melt compositions for fluid-fluxed melting
Melting mode and melt composition parameterizations are performed for fluid fluxed melting using
stacked 4th polynomial regression equations for modal composition calculated by pMELTS. The
equations are in a function of pressure (P) and degree of melting (F):
Modal orthopyroxene (P, F) = (a1P4 + b1P3 + c1P2 + d1P + e1) * F4 + (a2P4 + b2P3 + c2P2 + d2P +
e2) * F3 + (a3P4 + b3P3 + c3P2 + d3P + e3) * F2 + (a4P4 + b4P3 + c4P2 + d4P + e4) * F + (a5P4 +
b5P3 + c5P2 + d5P + e5)
Modal olivine (P, F) = ....... .
…………………….
for primary basalt melt compositions, the same equations are used:
SiO2 (P, F) = (a1P4 + b1P3 + c1P2 + d1P + e1) * F4 + (a2P4 + b2P3 + c2P2 + d2P + e2) * F3 + (a3P4
+ b3P3 + c3P2 + d3P + e3) * F2 + (a4P4 + b4P3 + c4P2 + d4P + e4) * F + (a5P4 + b5P3 + c5P2 + d5P
+ e5)
Al2O3 (P, F) = ....... .
…………………….
Na2O and K2O are not calculated by this method as these are considered to be incompatible elements
in the ABS3 model. The effect of alkalis in increasing the degree of partial melting is small
[Hirschmann, 2000], and thus negligible at the level required for the ABS calculations. The effect of
H2O in increasing the degree of partial melting is introduced to ABS3 based on the parameterization
of Katz, et al. [2003]. The estimated degree of partial melting at a given H2O content (Fkz) replaces
F in the above equations for modal and melt composition calculations.
4.2 Mode and melt compositions for melt-fluxed melting
Modal compositions for melt fluxed melting at different melt fractions (Xmelt) in the mantle source
are calculated after additions of averaged felsic slab melt [Moyen and Stevens, 2006] in steps of
5wt.%, from 0 wt.% to 30 wt.%. This leads to equations as a function P, F, and Xmelt(x):
Modal olivine (P, F, Xmelt5) = (a1m5P4 + b1m5P3 + c1m5P2 + d1m5P + e1m5) * F4 + (a2m5P4 + b2m5P3 +
c2m5P2 + d2m5P + e2m5) * F3 + (a3m5P4 + b3m5P3 + c3m5P2 + d3m5P + e3m5) * F2 + (a4m5P4 + b4m5P3
+ c4m5P2 + d4m5P + e4m5) * F + (a5m5P4 + b5m5P3 + c5m5P2 + d5m5P + e5m1)
Modal olivine (P, F, Xmelt10) = (a1m10P4 + b1m10P3 + c1m10P2 + d1m10P + e1m10) * F4 + (a2m10P4 +
b2m10P3 + c2m10P2 + d2m10P + e2m10) * F3 + (a3m10P4 + b3m10P3 + c3m10P2 + d3m10P + e3m10) * F2 +
(a4m10P4 + b4m10P3 + c4m10P2 + d4m10P + e4m10) * F + (a5m10P4 + b5m10P3 + c5m10P2 + d5m10P +
e5m10)
…………………….
Modal olivine (P, F, Xmelt30) = .......
The values between each calculation step are interpolated using 3rd order polynomial equations.
Primary basalt melt compositions are also calculated based on the same scheme. The effect of the
addition of H2O addition to the degree of melting is introduced using the Fkz parameter, as with the
fluid fluxed melting. Figure 7 indicates the effect of the addition of H2O on the degree of partial
melting (left panel of Figure 7) and effect of felsic melt addition on the modal compositions in the
residual mantle peridotite (right panel of Figure 7).
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Figure 7. Effect of H2O on degree of melting calculated at 3GPa using Katz et al. [2003] (left panel).
Effect of slab melt addition to mantle peridotite causing change in mineralogy of residual solid at 3
GPa.
5. Liquid fluxed open system melting for trace element calculations
Finally, the calculated modal composition of the mantle, the degree of partial melting (Fkz), and the
slab liquid composition are introduced into an open system melting equation [Zou, 1998] along with
the partition coefficients (see ABS2 paper by Kimura et al. [2009]). The liquid fraction added to the
mantle melting region is calculated using Fliq =  / F [Ozawa and Shimizu, 1995]. This calculation
step is similar to that used in ABS2, but the H2O fraction in Fliq is the melting parameterization
from Katz et al. [2003] for both fluid- and melt-fluxed melting. The ABS3 model can deal with
primary high-Mg basalts and high-Mg andesites formed by slab fluid-fluxed melting and slab
melt-fluxed melting of up to 30% of the mantle fraction. The calculation scheme assumes that the
open system melting for melt-fluxed melting is exactly the same as that for fluid-fluxed melting
apart from the effect of the slab-melt on the residual mantle mineralogy [Shaw, 2000]. The mantle
mineralogy of melt-fluxed melting has been calculated by pMELTS and is incorporated in the ABS3
model as noted above. The equation is as follows:
where
CBAS : concentration of an element in the basalt
CPERID: concentration of an element in the source peridotite
D : bulk melt distribution coefficient of an element determined by
n
D   Xa  Da
1
n
 Xa  1
1
X : modal composition of a residual mineral phase
Da : melt distribution coefficient of a residual mineral phase
P : contribution of consumed mineral
Arc Basalt Simulator version 3 (ABS3) calculation scheme
n
P   Pa  Da
1
8
n
 Pa  1
1
Pa : modal composition of a consumed mineral
Da : melt distribution coefficient of consumed mineral phase
: interstitial melt weight volume determined by ＝
: porosity of melting system or residual melt mass fraction after melting
: mass fraction of fluid in generated melt (when fluid influx is constant)
F : degree of partial melting
Fc : defined by )
CliqSLAB’’ : fluid composition
6. Simulating natural arc magmas using AS3
The results of the ABS3 calculations include trace element and Sr-Nd-Pb isotope compositions based
on a series of forward mass balance calculations, as with ABS2 [Kimura, et al., 2009]. The slab
mode at the depth where the slab liquid is derived and the mantle mode at the depth of mantle
melting depth are useful values as these can be compared to the output from the mass balance
calculations. In ABS3, major element compositions (SiO2, TiO2, Al2O3, FeO, MnO, MgO, CaO, and
H2O) of arc primary magmas are also calculated. K2O and Na2O are not included in the calculations
so that they have the same value as in the target natural magma (Figure 8). Although the major
element composition can be used as a guide to test the melting depth (alkalic or tholeiitic) or silica
saturation (high-Mg basalt or high-Mg andesite), the results do not reflect mass balance calculations,
and these numbers are used only for reference (Figure 8-1).
Figure 8-1. Calculated trace element compositions plot on multi-element diagram. PM normalized.
Arc Basalt Simulator version 3 (ABS3) calculation scheme
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Figure 8-2. Calculated slab mode, mantle melting mode, and magma compositions.
7. Parameter fitting for solution of the ABS3 forward model
The basic function of the ABS3 model is to fit a magma composition calculated using the sequential
calculations of the model to the target natural arc magma (Figure 1). This is achieved by altering
intensive and extensive variables on the Control Panel. Pre-defined and adjustable parameters are
separated in order to save calculation steps and thus time (Figures 9-1 and 9-2). Several new
features are added to ABS3. First, a total of 46 P-T paths for arcs world-wide are now available
[Syracuse, et al., submitted], each is denoted by a separate number. Details are shown in the
[Slb_PT] worksheet. Second, liquid fractions for SED/AOC and DMM/SLAB are determined in
Fliq(SED) and Fliq(DMM) cells that define XliqAOC, XliqSED, and XliqDMM. H2O in the fluid can also
be set in order to simulate the effect of H2O in aqueous fluids. Reasonable values vary between 0.5
and 1 [Manning, 2004]. The T-factor, which is a pre-defined conditions explores the effect of slab
temperature (SST) by simply multiplying SST at all depth ranges. Slab melting mode is shown in the
fitting parameter panel by showing [% slb mlt.] for AOC and SED. When slab melting mode has
been entered (denoted by [Slb Melt]), all calculation functions enter slab melting mode. For example,
if SED is in slab melting mode but AOC is not, all AOC fluid is captured by SED melt and mixed for
further calculations.
Figure 9-1. Control panel of Arc Basalt Simulator version 3 (ABS3).
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Figure 9-2. Control panel of Arc Basalt Simulator version 3 (ABS3).
For automated fitting, fitting parameters are set in the fitting macro pre-set parameter panel (Figure
10). Calculations begin using the pre-defined conditions and calculate each step within the range
given in the fitting macro panel. Calculations commence on pressing the MACRO button in the
control panel and results are saved in the Summary worksheet. For details on other functions please
refer to [Kimura, et al., 2009].
Figure 10. Fitting parameter settings for automated fitting calculations.
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