Calculation of water-bearing primary basalt and estimation of source mantle conditions beneath arcs: PRIMACALC2 model for WINDOWS

RESEARCH ARTICLE10.1002/2014GC005329

Calculation of water-bearing primary basalt and estimation ofsource mantle conditions beneath arcs: PRIMACALC2 model forWINDOWSJun-Ichi Kimura1 and Alexey A. Ariskin2

1Institute for Research on Earth Evolution, Japan Agency for Marine-Earth Science and Technology, Yokosuka, Japan,2Vernadsky Institute, Russian Academy of Science, Moscow, Russia

Abstract We present a new method for estimating the composition of water-bearing primary arc basaltand its source mantle conditions. The PRIMACALC2 model uses a thermodynamic fractional crystallizationmodel COMAGMAT3.72 and runs with an Excel macro to examine the mantle equilibrium and trace elementcalculations of a primary basalt. COMAGMAT3.72 calculates magma fractionation in 0–10 kb at various com-positions, pressure, oxygen fugacity, and water content, but is only applicable for forward calculations. PRI-MACALC2 first calculates the provisional composition of a primary basalt from an observed magma. Thebasalt composition is then calculated by COMAGMAT3.72 for crystallization. Differences in elemental con-centrations between observed and the closest-match calculated magmas are then adjusted in the primarybasalt. Further iteration continues until the calculated magma composition converges with the observedmagma, resulting in the primary basalt composition. Once the fitting is satisfied, back calculations of traceelements are made using stepwise addition of fractionated minerals. Mantle equilibrium of the primarybasalt is tested using the Fo-NiO relationship of olivine in equilibrium with the primary basalt, and thus withthe source mantle. Source mantle pressure, temperature, and degree of melting are estimated using petro-genetic grids based on experimental data obtained in anhydrous systems. Mantle melting temperature in ahydrous system is computed by adjusting T with a parameterization for a water-bearing system. PRIMA-CALC2 can be used either in dry or water-bearing arc magmas and is also applicable to mid-ocean ridgebasalts and nonalkalic ocean island basalts.

1. Introduction

Because erupted magmas have been subjected to fractional crystallization and assimilation, the estimationof chemical composition of primary magmas is important for examining source mantle processes. However,even if a subordinate role of crustal assimilation is assumed, the reconstruction of fractionation conditionsof a primary magma is a complex procedure because of the various possible crystallization paths withinintracrustal magma chamber systems [Almeev et al., 2013a, 2013b; Grove and Kinzler, 1986].

Olivine maximum fractionation calculations have been used in the estimation of primary basalts [Danyush-evsky et al., 2000; Herzberg and Asimow, 2008; Herzberg et al., 2007] and have been proven to be useful forless-fractionated magmas. In many arc lavas, however, the role of fractionation is generally large because ofthe existence of a thick sialic crust and because fractionation includes crystals other than olivine [Grove andBaker, 1984; Tatsumi and Suzuki, 2009].

Once a fractionating magma reaches multiple saturation and begins to move along a cotectic, it is not pos-sible to trace which side of the cotectic the magma originated from, nor where it joined the cotectic [e.g.,Ariskin et al., 1993]. Moreover, arc magmas contain varying amounts of water [Plank et al., 2013] and volatiles[Wallace, 2005], and water in particular affects the liquid line of descent (LLD) [Almeev et al., 2013a, 2013b;Sisson and Grove, 1993]. Therefore, back calculation models are not able to identify the unique primarymagma composition that fractionated to an observed composition [Danyushevsky and Plecov, 2011], andonly a family of plausible candidates can be identified.

Although difficult to obtain, a reasonable estimate of a primary basalt composition is important, since it ena-bles better examination of the genetic conditions of the basalt in the source mantle using (1) a petrogeneticgrid for mantle melting (e.g., PRIMELT2 model) [Herzberg and Asimow, 2008; Herzberg et al., 2007; Niu, 1997;

Key Points:� Model calculation of primary arc

basalt magma� Thermodynamic fractional

crystallization model� Mantle equilibrium determined using

petrogenetic grids

Supporting Information:� Readme� Supporting Information S1-S3

Correspondence to:J.-I. Kimura,[email protected]

Citation:Kimura, J.-I., and A. A. Ariskin (2014),Calculation of water-bearing primarybasalt and estimation of source mantleconditions beneath arcs: PRIMACALC2model for WINDOWS, Geochem.Geophys. Geosyst., 15, 1494–1514,doi:10.1002/2014GC005329.

Received 7 MAR 2014

Accepted 7 APR 2014

Accepted article online 10 APR 2014

Published online 29 APR 2014

KIMURA AND ARISKIN VC 2014. American Geophysical Union. All Rights Reserved. 1494

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Till et al., 2012]; (2) thermodynamic phase equilibria model calculations (e.g., pMELTS) [Asimow et al., 2004;Ghiorso et al., 2002; Kuritani et al., 2014b]; or (3) forward mass balance calculations (e.g., Arc Basalt Simulator(ABS) 3 and 4 models) [Kimura et al., 2009, 2010, 2006].

To achieve the aforementioned goal, we developed an iterative algorithm designed to calculate the compo-sition of primary basalt in relation to 10 major and 26 trace elements from basalt to basaltic andesite arcmagmas, mid-ocean ridge magmas, and subalkalic ocean island magmas. PRIMACALC version 2 (hereafterreferred to as PRIMACALC2) is an Excel based WINDOWS application software which allows estimates of aprimary basalt using back calculations along the fractional crystallization path of a magma and the sourcemantle conditions in a dry to wet system. In this paper, we describe the calculation scheme within PRIMA-CALC2 and the applications of the model.

The PRIMACALC2 program package, containing a PRIMACALC_2.00(COM3.72).xls Excel spreadsheet and aCOMAGMAT folder containing COMAGMAT3.72 FORTRAN code and relevant files, can be found with a briefinstallation guide in supporting information S1. We have confirmed that PRIMACALC2 runs with WINDOWSXP, 7 and 8, and with Excel 2003, 2007, and 2013 using a 32 bit mode. A Mac PC can also run PRIMACALC2using WINDOWS OS mode. For 64 bit Excel users, PRIMACALC_2.00(COM3.72)w64.xls is available uponrequest to the corresponding author of this study.

2. Calculation Scheme

Figure 1 shows a screen shot of PRIMACALC2 with the calculation flow chart. PRIMACALC2 is built on anExcel spreadsheet consisting of CONTROL_PANEL, PRIMACALC1, PRIMELT2_MOD, PLOT, COM_Result,TRACECALC, PRIMACOT, TRIPLOT, CMAS_CALC, PREMELT2_MOD2, Katz, and INPUT1_file Worksheets (seePRIMACALC_2.00(COM3.72).xls in supporting information S1). In this chapter, we use a step-by-step expla-nation of the method used by PRIMACALC2 to calculate the composition of a primary basalt.

2.1. Source Data and TreatmentPRIMACALC2 uses 11 major (SiO2, TiO2, Al2O3, FeO, MnO, MgO, CaO, Na2O, K2O, P2O5, and H2O) and 26 traceelements (Ni, Rb, Ba, Th, U, Nb, Ta, La, Ce, Pb, Pr, Sr, Nd, Sm, Zr, Hf, Eu, Gd, Tb, Dy, Y, Ho, Er, Tm, Yb, and Lu)contained in natural basalt to basaltic andesite (see (1) in Figure 1). We use the major element, Ni, and H2Oin the composition of natural magmas for 1. calculations of the phase equilibria and melt/mineral composi-tions in fractional crystallization within the intracrustal magma chamber using COMAGMAT3.72 [Ariskin,1999; Ariskin and Barmina, 2004; Ariskin et al., 1993] to identify the primary basalt; 2. calculations of thesource mantle conditions in equilibrium with the primary basalt, using petrogenetic grids modified fromPRIMELT2 [Herzberg and Asimow, 2008] and by extending these to wet conditions by parameterization ofKatz et al. [2003]; 3. then, back calculations for all the trace element compositions of the primary basalt aremade, by back-tracking the fractionation sequence with mineral/melt partitioning (see the calculation flowin Figure 1).

H2O content in natural magmas is normally poorly constrained. PRIMACALC2 explores the effect of H2Oalong with the other parameters, P and fO2, in the calculation shown in step 1 (details of the calculationsare shown below).

2.2. First Back Calculation with PRIMACALC1In order to apply the forward model of COMAGMAT3.72, we need to initially estimate the provisional com-position of a primary basalt. With the major element composition of a natural magma, the PRIMACALC1Worksheet calculates the composition of a primary basalt as the initial forecast. To achieve this, we prepareda typical LLD template for an arc basalt. We chose a basaltic andesite composition from the averaged arcmagma proposed by Tatsumi and Suzuki [2009]. The basaltic andesite has an olivine, pyroxene, and plagio-clase multiple saturated condition in a shallow magma chamber (Figure 2). We added 10 wt % equilibratedolivine stepwise, thereby forcing the basaltic andesite to be equilibrated with a depleted mantle. The result-ant primary basalt composition contained MgO 5 14 wt %, Mg# (Mg/[Mg 1 Fe21]) 5 0.68 (see the red starsin Figure 2).

We then performed forward fractional crystallization calculations using COMAGMAT3.72 with the conditionof the primary basalt in the magma chamber as P 5 0.3 GPa, fO2 5 QFM12, at various H2O 5 0–2.5 wt %,

Geochemistry, Geophysics, Geosystems 10.1002/2014GC005329


which is believed to represent the typical condition of an arc magma chamber [Tatsumi and Suzuki, 2009].We tested the versatility of the COMAGMAT3.72 thermodynamic model by comparing the model outputsto the experimental results of Tatsumi and Suzuki [2009], performed under the same P-T-XH2O conditions,but with a slightly lower fO2 at QFM12 rather than QFM13 used in Tatsumi and Suzuki [2009]. The calcu-lated LLDs show excellent agreement with those by experiments as shown in Figure 2. The versatility ofCOMAGMAT3.72 has also been shown for use with arc magmas in Eastern Kamchatka and for MORB in theprevious studies [Almeev et al., 2012, 2013a, 2013b, 2007, 2008].

Water saturation in a closed magma chamber is considered in COMAGMAT3.72. When the system is watersaturated, excess H2O is isolated as a separate phase and does not affect phase equilibria. Figure 3a showsan example of 2.5 wt % water in the primary basalt, saturated at MgO 5�5 wt %. No further increase ofH2O is considered. Water content (Figure 3a), melt compositions (Figure 3b), mineral compositions (Figure3c), proportions of the fractionated minerals (Figure 3e), and magmatic temperature T, fO2, and Fe21/Fe*(total Fe as Fe*) (Figure 3f) are calculated for each 0.5 wt % incremental step of MgO in the host melt. The

PRIMACALC_2.00_20131224 CODE INPUT COMAGMAT 3.72 CODE EXAMPLE Powered by COMAGMAT_3.72 and PRIMELT2, programed by JIKCLYSTALIZATION MODE FRC 1 1: FRC Default Element INPUT NORM COM372 MODEL MODEL OUTPUTPRESSURE_CONDITION ISO 1 1: ISO 2: DEC Isobaric/Decomp. Sample VCH-02 VCH-02 Cs/Condi. OL.max P.Bas_1 P.Bas_2 Recalc. Next Cs Sample DIFF(T-C) RND_TXTSET_SET P_MAX P [kbar] 5 5 3 10 <10kb SiO2 47.63 48.29 48.10 47.78 47.22 47.22 48.29 47.32 SiO2 0.094 47.22SET_MAX P_STEP P [kbar] 11 11 2OiT1.011 1.15 1.16 1.15 1.08 1.01 1.01 1.16 1.01 TiO2 0.006 1.01SET_NUL/ MIN P [kbar] 0 0 3O2lA5.20 17.60 17.84 17.73 16.49 15.48 15.48 17.84 15.54 Al2O3 0.056 15.48MINERAL_CHEMISTRY 3 Unused FeO 9.09 9.21 9.60 9.37 9.57 9.57 9.21 9.37 FeO -0.195 9.57

1LEDOM_xP_aC-WOL 1 0: PIG 1: OPX Default MnO 0.17 0.17 0.16 0.17 0.17 0.17 0.17 0.17 MnO 0.004 0.17NUMBER OF COMPOSITION 1 Unused MgO 8.54 8.66 8.65 11.74 13.80 13.80 8.66 13.80 MgO 0.003 13.80OXYGEN_SYSTEM OPE 1 1: OPE Default CaO 10.22 10.36 10.33 9.59 9.00 9.00 10.36 9.02 CaO 0.015 9.00OXYGEN_BUFFER_CONDI CON 1 1: CON Default Na2O 2.72 2.76 2.73 2.55 2.40 2.40 2.76 2.41 Na2O 0.014 2.40INIT_OXYGEN_BUFFER NNO 2 1: QFM 2: NNO 3: IW 4: HM K2O 1.31 1.33 1.32 1.23 1.15 1.15 1.33 1.15 K2O 0.002 1.15LOG_UNIT_SHIFT_OXBUFFER 0 0 P2O5 0.23 0.23 0.23 - 0.20 0.20 0.23 0.20 P2O5 0.001 0.20FINAL_OXYGEN_BUFFER (VAR) Unused PRESS for START SUM 99.55 100.00 100.00 100.00 100.00 100.00 0.00 0.00 Cr2O3 8.991 0.00XSTALLIZATION_INCRIM.[%] 1 Default Rb 41.64 41.64 - 38.55 35.16 35.16 0.00 0.00 LOI 9.064 0.00XSTALLIZATION_RANGE_MAX 75 Default Ba 401.69 401.69 - 371.94 339.19 339.19 100.00 100.00 Column 17TEMP_CONVERGENCE 0.5 Default Th 2.58 2.58 - 2.39 2.18 2.18PHASE_COMP[mol%] 0.25 Default PRESS for START U 0.72 0.72 - 0.67 0.61 0.61# 0 Unused PMC2_CALC. CONDITIONS Nb 2.68 2.68 - 2.48 2.27 2.27H2O_IN_PRIMARY_MELT[wt.%] 2 2 <7wt.% 2 Precision (Diff / X) Ta 0.16 0.16 - 0.14 0.13 0.13OUTPUT_FILE_NAME 00000001 Default 4 Iteration run (no.#) K 10858 10858 - 10054 9170 9170MAJOR_COMP 47.13 1.00 15. Excel PMC2_MANTLE COMP. La 12.09 12.09 - 11.20 10.23 10.23TRACE_COMP (DUMMY) 0.0000 Unused 90.0 PMC1_OL.max Fo Ce 27.14 27.14 - 25.13 22.94 22.94# 0 Unused 8.0 PRIMC mante FeO* Pb 2.55 2.55 - 2.36 2.15 2.15# 0 Unused 38.0 PRIMC mantle MgO Pr 3.65 3.65 - 3.38 3.09 3.09# Null PMC2_OLIVINE D(Ni) Sr 608 608 - 563 514 514

L!!!SNOITISOPLLECRETLATONOD Li, Metz, Bea, Wang Nd 16.43 16.43 - 15.21 13.88 13.88Ni (wt.%) IN OLIVINE INCOMPATIBLE TRACE ELEMENT Sm 4.00 4.00 - 3.71 3.38 3.38

Zr 95.98 95.98 - 88.87 81.16 81.16Hf 2.54 2.54 - 2.35 2.14 2.14Eu 1.35 1.35 - 1.25 1.14 1.14Gd 4.16 4.16 - 3.85 3.52 3.52Tb 0.67 0.67 - 0.62 0.56 0.56Dy 4.09 4.09 - 3.78 3.46 3.46Y 23.01 23.01 - 21.31 19.47 19.47Ho 0.86 0.86 - 0.79 0.72 0.72 SOURCE ADD/SUB P(OAQ) T(MgO) F%(OAQ) Twet(Kaz) F%(Kaz)Er 2.48 2.48 - 2.30 2.10 2.10 Peri PB1 - 2.4 1436 10 1356 9Tm 0.36 0.36 - 0.34 0.31 0.31 Peri PB2 0 2.4 1437 10 1370 9Yb 2.42 2.42 - 2.24 2.05 2.05 Pxite PB2 2.5 <= P(CSMSA)Lu 0.36 0.36 - 0.34 0.31 0.31 T(C) vs Mineral assemblage/ Composition/ H2O(melt)Ni (ppm) 118 118 - 409 454 454Ni(ol)wt% - - 0.22 0.31 0.40 0.42

Fo(ol)% - - 86.2 90.0 90.7 90.0Mg# Bas - - 0.63 0.69 0.72 0.72

OaCsvOgM3O2lAsvOgM2OiSsvOgM Fe2+/Fe(t) - - 0.83 0.86 0.86 0.86H2O(wt%) - - 2.30 - 2.00 2.00TC(COM) - - 1175 - 1312 -TWC(Katz) - - - - 1356 1370TDC(Herz) - - - - 1436 1437PGPa - - 0.05 - 2.4 2.4F%(Herz) - - - - 9 9%Xfrac. - - - 8.0 17 17MgO PM - - - - - 38

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1. Data input of observed magma

2. First back calculation with PRIMACALC1 and obtain primary basalt composition

3. Parameter input for COMAGMAT3.72 and run

4. Forward calculations by COMAGMAT3.72 with primary basalt composition

5. Find the closest match result with the observed magma, calculate compositional difference (DCx)

Subtract DCx from primary basalt and replace with new values

Iteration

Iteration end

6. Display crystallization sequence (panels) and primary basalt composition (column)

7. Adjust T in hydrous mantle source with Katz module and adjust MgO in the source mantle with olivine Fo

8. Back calculate trace element composition and display results in column and plot

Figure 1. Schematic calculation flow and screenshot of PRIMACALC2. (1) Input natural magma composition (major element, incompatible trace element, and Ni). (2) First back calculationto primary basalt at MgO 5 14 wt % by COMAGMAT3.72 fractionation template using a typical arc magma of Tatsumi and Suzuki [2009] calculated at 3 kb and a given H2O. (3) Forwardcrystallization calculation conditions in COMAGMAT3.72, given H2O, P, and fO2 used with the calculated primary basalt in step 2. (4) Elemental differences found between the closest-match calculation result and the natural magma, and the adjusted composition of primary basalt. (5) Steps 3 and 4 are automatically repeated until the solution is stabilized and the pri-mary basalt composition is calculated. (6) Repeat steps 3–5 by checking that the NiO in the olivine within the primary magma satisfies the mantle-olivine Ni-Fo array of mantle equilib-rium. (7) Calculate mantle P, T, and F in dry conditions using PRIMELT2; source mantle MgO is also obtained iteratively. (8) Recalculate T in the water-bearing mantle iteratively by Katzet al. [2003] for a given H2O in the primary basalt, with T(dry), P, and F in the mantle. The calculations give chemical compositions of the primary basalt (10 major and 27 trace elements);conditions of magma fractionation (given initial H2O, P, and fO2): % crystallization, mineral mode, mineral composition, major element, and H2O content in the fractionated magmaswith Fe21/Fe*; and conditions of mantle for the primary basalt (given H2O): P, T, and F (wet) conditions of mantle and MgO (fertility) in the source mantle.



PRIMACALC1 Worksheet includes the calculated results of LLD with initial values of H2O 5 0–2.5 wt % in3.0–14.5 wt % MgO. All the data are stored in lookup tables in the cell area $AL$788–$DC$1412 for minerals,and in $CD$1–$DE$784 for melts in the PRIMACALC1 Worksheet. With the data table, back calculations of afractionated magma in the range 3.0–14.0 wt % MgO are available.

The first back calculation requires H2O wt % to be present in the primary magma because the initial watercontent largely affects the LLDs, and thus the fractionation sequence (Figures 2 and 3a) [Grove and Baker,1984; Tatsumi and Suzuki, 2009]. Cell $C$20 in the CONTROL_PANEL Worksheet defines initial H2O as userinput. The model LLD templates are for the H2O 5 0–2.5 wt % range, and any excess H2O is then forced tobe set at 2.5 wt % in PRIMACALC1. Note that PRIMACALC2 accepts up to 7 wt % water in the starting com-position of a primary basalt, but the first back calculations only use values up to 2.5 wt %. More water isvalid in the second step iteration run, and the first step assumption is compensated after the second run(see section 2.3).

With the given H2O, PRIMACALC1 chooses the designated LLD path and adds equilibrated minerals at a 0.5wt % MgO step from the natural magma composition, until the bulk rock MgO reaches 14 wt %. Because of

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CATH

Figure 2. Comparisons of liquid lines of descent of a typical arc basalt with different water content between COMAGMAT3.72 model calcu-lations and experimental results by Tatsumi and Suzuki [2009]. Numbers with COM and TS09 show H2O contents in the COMAGMAT3.72and experimental systems, respectively. TH (tholeiitic) and CA (cacl-alkaline) boundary from Miyashiro [1974].



the existing compositional difference between the model basalt and natural magma, it is possible that thefinal MgO may not perfectly match MgO 5 14 wt % (see cell area $D$41–$AA$53 in the PRIMACALC1 Work-sheet). Trace element compositions (Figure 3d) are also back calculated using the addition of minerals inthe cell area $D$55–$AA$80 of PRIMACALC1 Worksheet. For trace element partition coefficients (Ds)between melts and minerals (see section 2.7). The first estimate of a primary basalt includes the trace ele-ment composition, but this is only for reference and not used for further calculations. Only the major ele-ment composition is used for further calculation steps. The PRIMACALC1 results are found in cell area$D$43–$D$80 in the PRIMACALC1 Worksheet. The results also plot on the INCOMPATIBLE TRACE ELEMENTpanel in the CONTROL_PANEL Worksheet as PMC_1, along with other model results (Figure 1).

PRIMARY MAGMA CALCULATOR ver. 1.00 [PRIMACALC_1: 2012/08/01]

LACAMIRP[]SNOITALUCLACROFDESUSELBAIRAVDNASRETEMARAPLEDOMYALPSID[

]NOITANIMRETEDO2HXMETSYSROFPIHSNOITALERO2H.svOgM[

C_1]

H2O 2.5 wt.% in sample (if not available, use XH2O = 1.5 wt.% below)

1 SiO2 [1:SiO2, 2:TiO2, 3:Al2O3, 4:FeO*, 5:MnO, 6:CaO, 7:Na2O, 8:K2O]

MgO 8.5 wt.% in model sample

ModeElement

3 Cpx [1:Ol, 2:Plag, 3:Cpx, 4:Opx, 5:Mt, 6:Melt]lareniM

reffubyxo2+MFQdna)xif(erusserpmetsysCFrabk3PX

3 En(Cpx) [1:Fo,2:An,3:En(Cpx),4:Fs(Cpx),5:Wo(Cpx),6:En(Opx),7:Fs(Opx),8:Wo(Opx),9:Usp]

XH2O 0.0 primary XH2O: read from diagram below and input [0-2.5 wt.%]

Variables 1 T(C) [1:T(C), 2:fO2, 3:Fe2+/Fe*]

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Figure 3. View of the PRIMACALC1 Worksheet. (a) H2O versus MgO in melt, (b) SiO2 versus MgO in melt, (c) MgO(melt) versus En composition of clinopyroxene, (d) trace element composi-tions, (e) mineral mode, and (f) magma temperature for both experimental [Tatsumi and Suzuki, 2009] (colored circles) and COMAGMAT3.72 results (lines).



2.3. Problems in Back CalculationsIn principle, the primary basalt composition given by PRIMACALC1 represents a primary basalt before frac-tionation in a magma chamber with a fixed condition of 0.3 GPa, fO2 5 QFM12, and a given H2O followinga model template. There are two major uncertainties, one from the real primary melt composition that isdetermined by which side of the cotectic the liquidus plain that the primary basalt first touches, and theother from the values of P, T, and fO2, and H2O present in the real fractionation conditions that then deter-mine the true cotectic.

In theory, the first uncertainty is not possible to solve using back calculations by any thermodynamic modelunless the primary melt composition is given. Therefore, PRIMACALC1 uses an ‘‘average arc magma LLD’’template to track-back to the primary basalt. The template includes tholeiite to calc-alkaline series fractiona-tion trends by controlling the values of H2O [Tatsumi and Suzuki, 2009] (Figure 2). The near-dry solution canthen be used for fractionated mid-ocean ridge basalt (MORB) or sub to mildly alkaline (tholeiitic) oceanisland basalt (OIB), as multiple saturation compositions are more or less similar to those of dry tholeiitic arcbasalts in terms of the major element contents. Good reproducibility of the phase equilibria in MORB hasbeen confirmed with COMAGMAT3.72 [e.g., Almeev et al., 2008].

It is noted that our template LLD approach is not the only way to deal with the first uncertainty and any vec-tor (trend) extrapolation analyses of the fractionation path is able to do the same. Our PRIMACALC1 modeluses the averaged trend deduced from arc basalt data, but also uses back-ups using experimental and ther-modynamic considerations for a wide range of magma compositions ranging from tholeiitic to calc-alkaline(Figure 2). Note, however, that any sort of template/trend calculation does not warrant mantle equilibriumof an estimated primary magma. This problem will be discussed below in section 2.5.

The second uncertainly, a variable cotectic, is basically related to the control of P, T, fO2, and H2O in amagma chamber. However, although water content data are available from olivine melt inclusions, H2O innatural magma is not well constrained due to degassing [Plank et al., 2013; Wallace, 2005]. Therefore, thefirst step of the H2O assumption in PRIMACALC1 already contains large errors. The assumed magma cham-ber pressure at P 5 0.3 GPa and fO2 5 QFM12 may be a good estimate for the condition of arc magma frac-tionation [Tatsumi and Suzuki, 2009], but it also contains large unknowns in relation to any particularmagma of interest. How PRIMACALC2 works on this problem is given in section 2.4.

2.4. COMAGMAT3.72 Forward Calculations and IterationsTo minimize the second uncertainty, we simulate the fractionation process of the provisional primarymagma using COMAGMAT3.72 forward calculations. We use P, T, fO2, and H2O conditions as fitting parame-ters. Among these, fO2 of QFM11 (or NNO) may be appropriate for many arc magmas [Kelley and Cottrell,2012], alternatively, QFM (NNO-1) buffer may be appropriate for MORB or OIB sources [Cottrell and Kelley,2011]. Versatility of the assumptions with the PRIMACALC2 model will be examined in section 4.1. MagmaticT is calculated by the COMAGMAT3.72 thermodynamic model when the melt composition, P, fO2, and H2Oare given. Therefore, P and H2O in the magma chamber are the key variables that can be explored usingCOMAGMAT3.72.

Inverse calculations along a plagioclase-olivine LLD can be achieved using other petrological models, suchas Petrolog3 [Danyushevsky and Plecov, 2011]. However, there are difficulties in using the inverse modelalong a multiply saturated LLD, because Petrolog3 calculations usually halt at around the first multiphasesaturation. Alternatively, iterative forward calculations are possible using the alphaMELTS thermodynamicmodel, which uses an algorism (1) after being given an initial estimate of the parental melt compositionand then (2) varies the melt composition until isobaric forward fractionation yields a specified target. This isknown as the Amoeba routine [Cooper et al., 2004; P. Antoshechkina and P. D. Asimow, AlphaMELTS Soft-ware Manual, 2013, http://magmasource.caltech.edu/alphamelts/1/alphamelts_manual.pdf]. The PRIMA-CALC2 model uses the same approach but employs the COMAGMAT3.72 thermodynamic model ratherthan MELTS.

For the iteration, COMAGMAT3.72 first calculates the fractional crystallization sequence from the provisionalprimary basalt at a given fO2, P, and H2O. MgO in the calculated magmas and that in the natural magma arethen compared, and the model magma which has the closest MgO content with the natural magma is cho-sen. Differences in the other major elements between the model and the natural magma are calculated and



http://magmasource.caltech.edu/alphamelts/1/alphamelts_manual.pdf

then added to or subtracted from the provisional primary basalt. The new basalt composition is then usedas the new input for the following iteration cycles. These iterations were continued until the natural and themodel magma compositions converge (Figure 4a).

In this iteration, the MgO content of the natural magma (therefore closest-match calculated magma) heldconstant, and all the major element composition (including MgO) varied in the primitive basalt. In order toobtain a better convergence, the subtracted/added element concentrations from/to the primary basalt aregiven using half of the difference between the natural and the chosen calculated magmas. This simpleapproach prevents an overshoot of the iteration and provides results that are precise enough (e.g., <0.37%relative difference (%R.D.) in SiO2 and %R.D.<�5% in MgO; see major %R.D. in the CONTROL_PANEL). Theprecision of iteration (given by [difference]/X, where X 5 2 defining added/subtracted amounts to 50% ofthe R.Ds) and the number of the iteration are set in cells $E$20–$E$21 in the CONTROL_PANEL Worksheet.Four iteration runs give sufficient convergence when X 5 2.

Figure 4b shows a fractional crystallization sequence including (1) mineral phases (olivine, plagioclase, clino-pyroxene, orthopyroxene, magnetite, and ilmenite), (2) olivine Fo, (3) plagioclase An, (4) H2O content inmagma, and (5) the fractionated mass of the solid phases plotted against the magmatic temperature (T);they were all calculated by the last COMAGMAT3.72 iteration run. Two red vertical lines delimit the range offractionation between the primary basalt and the target natural magma. This interval is used for back addi-tion calculations in PRIMACALC2. The diagram is useful for observing which mineral phase(s) is incorporatedin the back calculations, defining whether or not H2O is saturated in the fractionated magma, and definingthe magma temperature in the shallow magma chamber.

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Figure 4. Calculation results in CONTROL_PANEL Worksheet of PRIMACALC2. (a) Forward fitting calculation results of COMAGMAT3.72(blue dots) and the best fit result for the calculated primary basalt and magma compositions (red squares), showing a good fit to the targetmagma composition (red circle). (b) Fractional crystallization sequence calculated by COMAGMAT3.72 showing mineral assemblage, min-eral composition, H2O in the melt, and % crystallization (Ol: olivine, Plag: plagioclase, Cpx: clinopyroxene, Opx: orthopyroxne, Mt: magne-tite, Ilm: ilmenite). (c) Calculated NiO content (wt %) and Fo in olivines in fractionation (yellow circles) and in the primary basalt (orangesquare), showing a good fit with the olivine-mantle Ni array (orange parallelogram) [Takahashi, 1986]. Second stage clinopyroxene field(green square) [Herzberg, 2011] is also shown for a pyroxenite source.



A full data set of the calculated major element compositions and the mineral mode and compositions areshown in the COM_Result and the PLOT Worksheets. The calculated major element composition in theclosest-match model magma is given in COM372 Cs/Condi. in cell area $L$2–$L$13 in the CONTROL_PANEL.Residues between the model and observed magmas are found in $L$216–$L$25. The compositions ofNi(olivine), Fo(olivine), Mg#(melt), Fe21/Fe*(melt), H2O(melt), and the conditions of T(melt) and P(melt) in the magmachamber are also in the same column $L$42–$L$50. These provide information related to the equilibrium ofthe observed natural magma in the magma chamber conditions. The data are refreshed during iterationsand finally becomes valid after iteration converges.

2.5. Identification of Mantle Equilibrium Using Ni in OlivineThe purpose of PRIMACALC2 is to estimate a reasonable primary basalt composition for an arc magma. Wetherefore need to consider equilibration of the calculated primary basalt with the mantle peridotite, andthere are a number of possible ways to estimate this.

1. The use of Fe12/Mg ratios of magma, which should be in equilibrium with the mantle olivine (Fo86–90) byKd 5 (Fe12/Mg)ol/Fe12/Mg)melt 5 0.3 [Roeder and Emslie, 1970]. If the mantle olivine is Fo86–90, primary mag-mas must be Mg# (Mg/Mg 1 Fe12) 5 0.68–0.75 or have a weight ratio of FeO/MgO 5 0.4–0.7 in this model.

2. More precisely, the use of partial melting (F)-dependent FeO21/MgO partitioning between basalt andperidotite [Herzberg and O’Hara, 2002], in which estimated equilibrium is also dependent on the FeO andMgO contents of the source peridotite.

3. The use of NiO wt % of olivine (0.3–0.42 wt % for Fo85–94) in equilibrium with the mantle peridotite[Herzberg, 2011; Takahashi, 1986].

In PRIMACALC2, we use approach 3 because this model is independent from FeO/MgO in the source man-tle. The compositional field of mantle olivine is shown by the orange parallelogram in the diagram showingNiO (wt %) versus Fo in olivine in Figures 1 (6) and 4c. The NiO value in this field is regarded as the targetNiO content.

The partition coefficient of Ni between olivine and the melt D(Ni) is critical for the back calculation of NiO inolivine. We examined various models ranging from T-independent/composition-dependent parameteriza-tion of [Beattie et al., 1991] supported by [Jones, 1984] and [Herzberg et al., 2013] to the T and composition-dependent parameterizations by Li and Ripley [2010], Matzen et al. [2013], and Wang and Gaetani [2008]. Weexplored the relatively low T range (typically 1000–1400�C for fractional crystallization) with the Fe21-Mg ofmelt, both calculated by the COMAGMAT3.72 model.

Beattie et al. [1991] model always shows a high D(Ni) in olivine, and therefore the back addition of olivinedoes not satisfy mantle equilibrium in the PRIMACALC2 model. This could be related to the effect of the Tdependence of (Ni) [Matzen et al., 2013]. Wang and Gaetani [2008] model also shows a high D(Ni) in a low Trange (e.g., <1200�C), although T dependence is taken into account in the model. However, the D(Ni) of Liand Ripley [2010] and Matzen et al. [2013] always delivers reasonable fits in the PRIMACALC2 model for awide range of target magma composition, and we therefore use the Li and Ripley [2010] model in this paper,while providing the choice of using the D(Ni) model. Cell $E$27 in the CONTROL_PANEL Worksheet allowsthe use of the D(Ni) models by typing; L: Li and Ripley [2010], M: Matzen et al. [2013], B: Beattie et al. [1991],and W: Wang and Gaetani [2008] for the designated model.

2.6. Effect of P and H2O to D(Ni) in Fractional CrystallizationAs shown in section 2.4, H2O and P are the key variables in the COMAGMAT3.72 forward calculations. Theseparameters control not only the fractionation mineralogy by altering the liquidus temperatures of minerals[Almeev et al., 2012, 2013a, 2007; Ariskin, 1999; Ariskin and Barmina, 2004] but also the magmatic tempera-ture in crystallization [Almeev et al., 2007], which does affect D(Ni) because of the T and compositiondependence of Ni partitioning between olivine and the melt. This eventually results in a different NiO(olivine)

in the primary basalt.

Figure 5 shows examples of the fractionation sequences of a primary basalt at different values of P (0.3 and1.0 GPa) and at a different initial H2O content (0 and 2 wt %). An increase in P increases the T of olivine crys-tallization due to the increase in the liquidus temperature of olivine [Katz et al., 2003] (Figure 5a), and incontrast an increase in H2O decreases the liquidus temperature of olivine [Almeev et al., 2007; Ariskin and



Barmina, 2004; Falloon and Danyushevsky, 2000](Figure 5a). Due to the change in the crystalliza-tion temperature, the Fe-Mg partitioning betweenolivine and the melt also changes [Roeder andEmslie, 1970]. In PRIMACALC2, the above changesare in accordance with the thermodynamic modelof COMAGMAT3.72 [Ariskin et al., 1993] (Figure5b). The NiO content of olivine at equilibrium islow, either in relation to a lower P or to the higherH2O content of a magma, and this is reflected inthe T and compositional dependence of D(Ni) inthe Li and Ripley [2010] model (Figure 5c). Noteagain that both the models of Li and Ripley [2010]and Matzen et al. [2013] are T and composition-dependent, and we consider that these modelsare internally consistent with COMAGMAT3.72 forthe temperature range of <1400�C in the PRIMA-CALC2 model.

By adjusting the P and H2O variables, we canestablish a reasonable equilibrium of the primarybasalt with the source mantle in terms of the Fo-NiO contents of olivine (see Figures 4b and 5c).These calculations are achieved by repeating iter-ations with different P and H2O values given incells $C$4 and $C$20 in the CONTROL_PANEL. Asit would then be necessary to introduce furtherconstraints of P and H2O because they give anopposite effect we have not considered any auto-mated searches for these values, and this prob-lem will be discussed in section 3 for applicationto PRIMACALC2.

2.7. Back Calculations for Trace ElementsUsing the optimal crystallization sequenceobtained by the iteration calculations shownabove, back calculations of trace elements in the

primary basalt can be made using the stepwise addition of equilibrated minerals (e.g., magnetite, orthopyr-oxene, clinopyroxene, plagioclase, and olivine). Both the PRIMACALC1 Worksheet for the first provisional pri-mary basalt and the TRACECALC Worksheet for the final primary basalt use the same algorithm, which isdescribed as below.

Trace element back calculations, including NiO in olivine (see details in section 2.6), are available by usingthe melt and mineral compositions calculated by COMAGMAT3.72 in the TRACECALC Worksheet. The parti-tion coefficients of fractionated minerals change with the P, T, and mineral/melt composition, and we there-fore use the compositional, P, and T-dependent partition coefficients whenever applicable. The partitioncoefficients are derived from the following sources: the empirical MgO(melt)-dependent parameterization ofB�edard [2005] for olivine; the empirical T and An-dependent parameterization by Bindeman [2007] andBindeman et al. [1998] for plagioclase; the T, P, and melt/mineral composition-dependent lattice strainmodel of Wood and Blundy [1997] for rare earth elements (REEs) and the experiment-based partitioning ofPilet et al. [2011] for other incompatible elements of clinopyroxene; the orthopyroxene/clinopyroxene parti-tioning derived from compilations of Pilet et al. [2011] and the clinopyroxene partitioning for orthopyrox-ene; and the compilations by Pilet et al. [2011] and Rollinson [1993] for magnetite.

The back calculation result is shown as P.Bas_1 in cell area $N$3–$N$52 in the CONTROL_PANEL Worksheet.The trace element composition also plots graphically on incompatible trace element and on the Ni (wt %)

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Figure 5. (a) Effects of olivine crystallization temperature by differ-ent pressures and H2O; (b) Fo composition of equilibrated olivine;and (c) NiO contents in olivine controlled by both temperature andcomposition.



IN OLIVINE panels (Figure 1 and supporting information S1). These panels show a primitive mantle (PM)[Sun and McDonough, 1989] normalized multielement plots of the primary magma and MgO versus NiOplots of the olivine composition, respectively.

In addition to the PRIMACALC2 results, the olivine maximum fractionation model of PRIMELT2 [Herzbergand Asimow, 2008; Herzberg et al., 2007] is also applied to the natural magma by using the P, Fe21/Fe*, andolivine Fo composition for the primary basalt from PRIMACALC2 (see the PRIMELT2_MOD Worksheet). Thisalso calculates the major and trace element compositions OL.max in the primary basalt for comparison withthose by PRIMACALC1 and P.BAS_1 by PRIMACALC2. The PRIMELT2 results are in cell area $M$3–$M$53OL.max in the CONTROL_PANEL Worksheet. Ni partitioning in olivine is calculated by Li and Ripley [2010] forPRIMACALC2, whereas the Beattie et al. [1991] model is used in the PRIMELT2_MOD Worksheet. The differ-ences between the two models are shown as NiO in olivine calculated for the primary basalts of OL.maxand P.Bas_1.

2.8. Source Mantle Conditions (P-F-T)The PRIMELT2_MOD2 Worksheet calculates the P, T, and F conditions of the mantle for the P.Bas_1 compo-sition using PRIMELT2. P and F are both estimated by the Ol-An-Qz (projected from Di) CMAS projectionwhereas T is estimated using the T-MgO(‘‘primary basalt’’) relationship [Herzberg et al., 2007]. The PRIMACOTWorksheet shows certain CMAS plots, including Ol-CaTs-Qz, MgO-CaO, and FeO21 versus MgO, which arekey to the PRIMELT2 model. The TRIPLOT and CMAS_CALC Worksheets provide background calculations.Only Cs-Ms-A (Ol) and Ol-An-Qz (Di) projections are shown in the CONTROL_PANEL.

Fractional and batch melting are two extreme cases of mantle melting, but fractional melting is more plausi-ble mechanism. It is, however, hard to introduce this into the model, as CMAS petrogenetic grids are formedbased on the experimental results, which represent batch melting. Herzberg and O’Hara [2002] and Herzbergand Asimow [2008] have introduced an accumulated fractional melt AFM (Accumulated Fractional Melt)model in PRIMELT2 for F-Mg/Fe(melt) and T-MgO(melt) relations, but PRIMACALC2 uses the batch model ofCMAS.

The PRIMELT2 model [Herzberg and Asimow, 2008; Herzberg et al., 2007] is built on anhydrous experimentaldata. We plotted the experimental results, including the water-bearing system from LEPR database [Hirsch-mann et al., 2008] onto the Ol-An-Qz (Di) projection, and found that the P relationship is maintained (seeFigure 6a). There are not many water-bearing experiments that report F information in existence. We exam-ined the experiments of Hirose and Kawamto [1995] at 1 GPa (results shown in Figures 6b and 6c) and foundthat the estimated F by Ol-An-Qz (Di) projection reproduces the reported F well, and a near one-to-one cor-relation line with a negligibly high zero intercept (y 5 0.9769x 1 0.0359; Figure 6b) is plotted.

The peridotite melting in water-bearing systems is known to provide higher F values at given T [Katz et al.,2003]. In fact, T estimated by T-MgO(melt) by PRIMELT2 shows an erroneously �10% higher T than those inthe experiments by Hirose and Kawamto [1995] (y 5 1.096x; Figure 6c). Therefore, we considered that themantle T should be recalculated with an estimated P and F and the presence of H2O in the primary basalt,and this is available using the parameterization of Katz et al. [2003]. The P and F estimates come from theOl-An-Qz (Di) projection, and H2O is initially given for the PRIMACALC2 calculations. Thus, the T during the‘‘wet’’ melting at the postulated H2O in the primary basalt was iteratively calculated using the Katz Work-sheet. This was achieved by altering T in the Katz Worksheet, which begins with T(dry), and then findingwhen H2O in the primary basalt becomes unity to a given value of H2O. Figure 6d shows a graphical exam-ple of this calculation (starting with 1340�C–dry–F 10%, to find 1240�C–3 wt % H2O–F 10%).

The calculations are associated with the VBA macro iteration calculations, and the results T(dry), P(Ol-An-Qz), and F(Ol-An-Qz) from PRIMELT2, with T(wet), and F(wet) from the Katz Worksheet, are shown in the cellarea $Q$37–$W$37 in the CONTROL_PANEL Worksheet in the P.Bas_1 column. Since F is calculated inde-pendently in the Katz et al. [2003] model, it is also given in $W$37 for comparison. We usually obtain coher-ent F between the dry and the wet models that are within a 1% error, providing the basis of reliable wetmantle T estimates in PRIMACALC2.

2.9. Source Mantle Fertility (MgO)The final exploration with PRIMACALC2 is the fertility of the source peridotite. PRIMACALC2 provides the pri-mary basalt composition in equilibrium with the source mantle (see the P.Bas_1 column in the



CONTROL_PANEL Worksheet). The estimated P.Bas_1 composition always displays slight discrepancies fromthe equilibrium parameters given by the Fe21-Mg relationship at the given P by Ol-An-Qz (Di) (see section2.8) in the PRIMELT2 algorithm [Herzberg et al., 2007] (see also FeO21 versus MgO plots in the PRIMACOTWorksheet). PRIMELT2 cautions addition/subtraction of olivine to/from P.Bas_1 and the amounts shown(see cell $C$11 in PREMELT2_MOD2 Worksheet). To accommodate this problem, PRIMACALC2 has an addi-tional function that can modify the source mantle FeO and MgO compositions to take the mantle fertilityinto account for equilibration [Herzberg and Asimow, 2008; Herzberg et al., 2007]. These mantle compositionsare given in cells $E$24 and $E$25 in the CONTROL_PANEL Worksheet.

FeO in the mantle peridotite is a canonical value and is fairly uniform at �8 wt %, whereas MgO varies from25 to 50 wt % [Bodinier and Godard, 2003]. We therefore set FeO 5 8 wt % and change values of MgO tomake the equilibration conditions from PRIMELT2 consistent with the primary basalt estimated by PRIMA-CALC2 P.Bas_1. The automated calculation forces a discrepancy in the Fe21O-MgO equilibria (shown in the[FeO(wo_Fe31)-MgO] panel in the PRIMACOT Worksheet) during iteration of the VBA run. The renewedMgO appears in cell $E$25 in the CONTROL_PANEL. The resultant MgO is usually between 25 and 50 wt %,possibly indicating source heterogeneity. The recalculated primary basalt composition using the new

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Figure 6. (a) Petrogenetic CMAS grid used for P and F estimates; (b) comparison of experimental F and calculated F values hydrous peridotite system; (c) comparison of experimental Tand T from thermometry for hydrous peridotite melts; and (d) the conversion scheme of T(dry) to T(wet) mantle melting. In Figure 6a, Ol-An-Qz (projected from Di) plot is originally fromHerzberg and O’Hara [2002] modified by the authors using additional experimental data from REPL database [Hirschmann et al., 2008], including dry and hydrous (red circles) experi-ments. Correlations between experimental F values and those calculated by Ol-An-Qz (Di) projection for the hydrous experiments of Hirose and Kawamto [1995] are shown in Figure 6b.Figure 6c shows T estimates by MgO(melt) by Herzberg and O’Hara [2002] for the same hydrous peridotite melts showing �10% higher T (C) in calculations. Figure 6d shows a schematicexample of T(dry) to T(wet) conversion. T(wet) 5 1240�C is obtained when T(dry) 5 1340�C, F 5 0.1, H2O 5 3 wt %, and P 5 1.5 GPa are known (figure modified from Katz et al. [2003]).



mantle MgO content with a further addition/subtraction of olivine with PRIMELT2 to create a new equilib-rium is shown by P.Bas_2 in cell area $O$3–$O$53. The original olivine NiO(olivine) and Fo compositions cal-culated primarily by PRIMELT2 are shown in cells $O$42 and $O$43 for comparison.

For this final calculation, Fo and NiO(olivine) are recalculated using the model of Beattie et al. [1991] and Jones[1984] of Ni partitioning with the P.Bas_2 composition. The Fo versus NiO composition of the finally equilibratedolivine for P.Bas_2 and the fractionation-accumulation trajectories of olivine by PRIMELT2 are in the Ni (wt %) inolivine panel (see the big orange square and the small gray crosses, respectively, in Figure 4c). In many cases,the Fo and NiO in olivine from the PRIMACALC2 (COMAGMAT3.72 and Li and Ripley [2010] Ni partitioning) inP.Bas_1 and the PRIMELT2 P.Bas_2 results agree with each other (see the example in Figure 4c).

However, a large discrepancy occurs when, for example, Hawaiian basalts are calculated. The trials showhigher values of NiO (plot in the field of the ‘‘second stage olivine in pyroxenite melting’’ in Figure 4b) byPRIMELT2, even though the NiO in olivine from PRIMACALC2 falls into the Fo-NiO mantle array. The highNiO in olivine reproduces the pyroxenite source model proposed by Herzberg [2011]. However, any prob-lems associated with this are not discussed here because the results for the low-K tholeiitic to medium-Kcalc-alkaline arc magmas examined here did not show the problem. A caveat here is that P.Bas_2 uses Foand NiO(olivine) from the PRIMELT2 [Beattie et al., 1991; Jones, 1984] model. If a large discrepancy is found inrelation to the results of P.Bas_1 and P.Bas_2 in PRIMALCALC2, the user should note which values are used.We use the P.Bas_2 model results throughout this paper. Validity of the estimation of source mantle fertilityis examined in section 4.

3. Calculations of Arc Magmas With PRIMACALC2

In this section, we give a brief outline of how to operate PRIMACALC2. First, users should download support-ing information S1. PRIMACALC2.zip folder, unzip this on the desktop, then copy the whole COMAGMATfolder to the C: root directory of the hard disk. Next, copy PRIMACALC_2.00(COM3.72).xls on the desktopand double click it, thereby opening PRIMACALC2 with Excel which is ready for use. Please note that anychanges made to the file names or cell formats will destroy the program links. We therefore recommendkeeping the original (zip) files and not changing the file names.

In PRIMACALC2, the cells color-coded in yellow with red bold text are the input areas (CONTROL_PANELWorksheet; Figure 1). Those in blue with red bold text are the output regions. It is possible to alter theseinput values manually. Please note that any changes in other cells will cause fatal failure of the data linksand calculations. The same applies to other Worksheets, and thus it is advisable not to alter any cells. Theoutput data can be copied using the COPY function of Excel. Copies via Excel do not alter the cells, so thatit is possible to copy any part of PRIMACALC2 and export these numbers using the PASTE-SPECIAL function.

3.1. Data InputData input, calculations, and data output are managed in cell area $J$53–$O$53 in the CONTROL_PANEL.The geochemical data of an observed magma should include 10 major elements, 26 incompatible trace ele-ments (including K in ppm), and Ni (see (1) in Figure 1).

3.2. Back Calculation and Iteration RunTo run the VBA macros, including the COMAGMAT3.72 shell program, the predefined calculation conditionsof COMAGMAT3.72 should be set in cell area $C$2–$C$26 in the CONTROL_PANEL (see (3) in Figure 1). Inthis paper, we use pressure mode 1: [ISO]baric (value is input in cell $3$C), with a magma chamber pressureof 0.1–10 kb (cell $C$4), and various oxygen buffers in the magma chamber 1: QFM or 2: NNO (cell $C$12)plus/minus X log unit (cell $C$13), and a water content in the primary basalt of 0–7 wt % (cell $C$20). Theoption to calculate a polybaric (decompression) crystallization sequence with a constantly decreasing pres-sure increment is also available [Almeev et al., 2013a, 2013b; Ariskin and Barmina, 2004], but we do not use itin this paper for the basic model test. Detailed documentation of the COMAGMAT3.72 code settings are inthe INPUT1_file Worksheet. Note that some function codes (e.g., EQU: equilibrium crystallization) do notwork with PRIMACALC2. Changeable codes are shown by yellow cells in the CONTROL_PANEL; otherwise,codes should be set by a default function.

Once the assumed magma chamber conditions are set, it is possible to run the COMAGMAT3.72 FORTRANshell program by pressing the COMAGMAT CALC. button at cell $F$16. The PRIMACALC2 iteration



calculations include (1) saving the COMAGMAT3.72 control codes and the major element compositionsfrom the CONTROL_PANEL in supporting information; (2) running COMAGMAT3.72, which reads the textfiles, calculates, and saves the output results in another text file; and (3) reading the output text file andstoring in the COM_Result Worksheet. All processes are coded in the Excel VBA macro. Additional conver-gent calculations for T(wet) using the Katz Worksheet, and MgO(peridotite) in the mantle using the PRIMELT2_-MOD2 and PRIMACOT Worksheets (see section 2.8) run with the iteration calculations. The calculationprecision (see section 2.4) and the number of the iteration run are set in cells $E$20 and $E$21 usually twoand four, respectively (see above section 2.4).

3.3. OutputThe MgO versus SiO2, MgO versus Al2O3, and MgO versus CaO panels graphically show the fitting results(see (4) and (5) in Figure 1 and Figure 4a). The mantle equilibrium of the olivines is tested by Ni (wt. %) INOLIVINE panel in the CONTROL_PANEL (Figures 1 and 4c), where the users can test the controlling parame-ters of COMAGMAT3.72 iteratively. The fractionation sequence including the mineral phases and composi-tions and the fractionated solid and H2O content in the magma plot are shown in one panel (T(C) versusmineral assemblage/composition/H2O(melt)) (Figures 1 and 4b). Two CMAS plots ((7) in Figure 1) and thetrace multielement plots ((8) in Figure 1) show the calculation results. Finally, users can extract the resultsfrom PRIMACALC2 using the copy function of Excel from the field $I$2–$O$53 of the CONTROL_PANEL.

4. Applications

We evaluate PRIMACALC2 by applying the software program to the Quaternary arc magmas from the Japa-nese Islands. Figure 7 shows the locations of volcanoes from which the basalts to basaltic andesites exam-ined in this study were reported. From the Izu collision zone, we examined the N-Izu volcanic front (VF) Izu-Oshima, Toshima; Udonejima and rear-arc (RA) Niijima lavas [Kimura et al., 2010]; and Fuji [Watanabe et al.,2006]. From the southern NE Japan arc, we examined Asama [Gust et al., 1997; Kaneko, 1995; Okamoto,1979], Takahara, and Nasu [Ban et al., 2013] lavas. From the northern NE Japan arc, we examined Funagata

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[Kimura and Yoshida, 2006], Iwate [Kuritani et al., 2014a], Hachmantai [Ban et al., 2013], Akitakoma [Ban et al.,2013], and Hakkoda and Osore [Ban et al., 2013] lavas. From the NE Japan rear-arc, we examined Sannome-gata [Kuritani et al., 2014b], Kampu [Ban et al., 2013], and Chokai [Kimura and Yoshida, 2006]. Low-K tholeiiticbasalt to basaltic andesite were inferred from lower crustal amphibolite melts at Zao [Tatsumi et al., 2008]and Azuma [Takahashi et al., 2012] volcanoes, and lavas with significant crustal assimilation such as Harunavolcano [Kobayashi and Nakamura, 2001] were omitted from the examinations.

4.1. Preset and Explored Variables for CalculationsCOMAGMAT3.72, and thus PRIMACALC2, requires P, fO2, and H2O to be preset values. For the fO2 in themagma chamber, we assumed QFM to QFM10.5. This may be slightly reduced condition for an arc magmawhich has Fe21/Fe(total) 5 0.78 of QFM11 equivalent [Kelley and Cottrell, 2012] at H2O 5 3 wt % [Kelley andCottrell, 2009]. However, the use of QFM11 oxygen buffer led to early crystallization of magnetite after oli-vine in COMAGMAT3.72, which is inconsistent with the observed crystallization sequence. Measured Fe21/Fe* in the natural basalt melt in olivine melt inclusions showed a wide scatter of 0.78–0.75 (QFM-QFM11.5equivalent) [Kelley and Cottrell, 2012]. Given these values, our assumption is still within the observed range.It is noted, however, that the spinel thermodynamic model formulated by the atmospheric pressure experi-ments [Ariskin and Barmina, 1999] may have errors in the high pressure range. The second author is workingon this issue, and the results will be presented elsewhere.

Using NiO-Fo in olivine in the primary basalt, we can explore mantle equilibrium by altering P and H2O inthe magma chamber (see section 2.6). We began calculations with H2O 5 4 wt % or more (which providedgreater NiO in olivine; see Figure 4c) and found the associated magma chamber P (lower P also providedgreater NiO) by observing the Ni-Fo constraint. If the shallowest P did not satisfy the NiO-Fo constraint, H2Owas reduced accordingly. The origin of using 4 wt % H2O in the primary arc magma at the start has beendiscussed elsewhere [Almeev et al., 2013a, 2013b; Hamada and Fujii, 2007; Kimura et al., 2010; Kuritani et al.,2014a, 2014b; Plank et al., 2013]. The results always show 2.0–4.0 wt % H2O when the Li and Ripley [2010]model D(Ni) is used, and water saturation was noted in some cases (Figure 8a), but always with a shallowmagma chamber (e.g., 0.001 GPa for Izu-Oshima; see supporting information S2 and Figure 8a). In these

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cases, the initial water content was considered not to be a significant constraint, and the magma chamberP was considered to be the fundamental control.

4.2. Back Calculations for Fractionated BasaltWe first investigate the role of olivine, plagioclase, and clinopyroxene fractionation. Figures 8a and 8bshows an example of the calculation for IO-6 basalt from the older stage volcanic edifice of Izu-Oshima[Kimura et al., 2010] (see supporting information S2).

In order to satisfy the NiO-Fo constraint for olivine, a total of 65 wt % fractionation of crystals is required(supporting information S2). In the order of onset of crystallization, the fractionated minerals are olivine, pla-gioclase, and clinopyroxene (see Figure 8a). This mineral assemblage is consistent with that observed in thebasalt lava. The depth of the magma chamber needs to be very shallow at 0.001 GPa, and although themagma at this depth accommodates only 0.5 wt % H2O, the initial setting of water was >4 wt % accordingto the previous estimates by experiments and the ABS3 model calculations [Hamada and Fujii, 2007; Kimuraet al., 2010]. This means that crystallization in the shallow magma chamber should proceed at H2O-satu-rated conditions, with the excess of the initial water released as a result of decompression. The COMAG-MAT3.72 model allows the user to account for such a possibility in the modelling fractionation process[Almeev et al., 2008], so that the initial water content in the primary magma is not reflected in the shallowmagma chamber differentiation.

Mineral compositions were calculated and resulted in Fo72 olivine and An79 plagioclase when the calcu-lated magma had the same chemical composition as the observed IO-6 (at 1140�C in Figure 8a). The olderstage basalt lavas had �Fo75 olivine [Hamada and Fujii, 2007], close to the calculated value. In contrast, pla-gioclase inclusions in the olivine had >An90, indicating disequilibrium crystallization with early crystallizedplagioclase (Hamada M. pers. com.). Crystallization of >An90 plagioclase was estimated in an early crystalli-zation phase by COMAGMAT3.72 (at �1200�C; see Figure 8a). This discrepancy could be derived from thecrystallization mode between the perfect fractional crystallization used in the calculations. This examinationshows that the crustal magma chamber processes should be explored individually. PRIMACALC2 may pro-vide some predictions, but petrographical and geochemical observations in the natural magma are the pri-mary constraint.

The trace element compositions in the estimated primary basalt of IO-6 are compared in Figure 8b, usingvarious methods in PRIMACALC2 (Ol.max, P.Bas_1, and P.Bas_2). When comparing a simple olivine maxi-mum fractionation calculation by PRIMELT2 Ol.max to that of PRIMACALC2 P.Bas_2 (supporting informationS2 and Figure 8b), PRIMELT2 resulted in a 35 wt % olivine fractionation whereas PRIMACALC2 requires 65wt % fractionation. The differences in the calculated incompatible trace elements were 40%–46% lower forPRIMACALC2 for large ion lithophile elements (LILE) and high field strength elements (HFSE), and 10%–30%lower for REE. The differences in LILE, HFSE, and REE are related to the back addition of clinopyroxene andplagioclase in PRIMACALC2, whereas PRIMELT2 uses Ol-monosaturated modelling [Herzberg and Asimow,2008]. Sr is, in contrast, 24% higher by PRIMACALC2, reflecting the back addition of Sr-rich plagioclase (sup-porting information S2 and Figure 8b). As shown in Figure 8a, the back addition region is shown in the [T(C)versus mineral assemblage/composition/H2O(melt)] panel. Users can identify whether or not mineral phasesother than olivine are added. This provides a caution for the results of PRIMELT2 calculations in which theolivine maximum fractionation is only considered.

The difference between the models may not be significant in relation to the trace element mass balancecalculation for arc basalts. However, in these types of calculations using, for example, ABS models, there arelarge uncertainties in the source slab and mantle compositions and in the elemental partitioning in slabdehydration/melting [Kimura et al., 2010], but improving the estimation of the primary arc basalt composi-tion will improve this problem.

4.3. Back Calculations for Near-Primary BasaltThe PRIMACALC2 result is also given for a Sannomegata basalt [Kuritani et al., 2014b] (supporting informa-tion S2 and Figures 8c and 8d). The results of PRIMELT2 (fractionated olivine 4 wt %) and PRIMACALC2 (frac-tionated olivine 10 wt %) (see supporting information S2 and Figures 8c and 8d) are almost identical. This isconsistent with the near-primary nature of the Sannomegata basalt, as shown by the presence of skeletalolivine alone in its phenocryst phase [Kuritani et al., 2014b], and thereby suggests a small role for olivine



fractionation. In the case of the Sannomegata basalt, the water content in the primary basalt is set at 4 wt% and the magma chamber pressure at 1 GPa in PRIMACALC2 (supporting information S2) which is requiredto satisfy NiO-Fo mantle equilibration of olivine.

Although water was estimated at �7 wt % in the primary Sannomegata basalt using the thermodynamicphase equilibria [Kuritani et al., 2014b] and at 4–8 wt % using the trace element mass balance by ABS4[Kimura and Nakajima, 2013], the water content estimated by PRIMACALC2 was not more than 5 wt %.Water content in the fractionated magmas increased dramatically to >10 wt % after the onset of clinopyr-oxene in a deep closed magma chamber for the Sannomegata basalt (Figure 8c). There are, however, largeerrors in the COMAGMAT3.72 calculations, and thus the effects of water on crystal fractionation should beinvestigated further, but it is worth noting that the fractional crystallization sequence in the crust shouldcontrol water by means of the emplacement depth of the magma chamber. COMAGMAT3.72 could be usedto explore such an effect by comparison of a crystal fractionation sequence to the observed magmas[Almeev et al., 2013a, 2013b, 2007]. PRIMACALC2 output reflected the model characteristics and enabledusing these to constrain the mantle to crustal processes. But, as has been shown in previous work, individ-ual magma chamber processes should be examined in detail using petrography, petrology, and geochemis-try [Kobayashi and Nakamura, 2001; Takahashi et al., 2012; Tatsumi and Suzuki, 2009].

4.4. Source Mantle ConditionsWe calculated source mantle conditions using PRIMACALC2, and the results for volcanoes in N-Izu, southernand northern NE Japan arcs are given in supporting information S3 and Figure 9. Important factors are theT, F, P conditions, and the source fertility expressed by MgO in the source peridotite (Figures 9a–9c). Watercontent in the primary basalt, which affects the mantle melting T, was determined using the NiO-Fo con-straint and resulted in 2–4 wt % in all primary basalts (supporting information S3). The overall agreement ofthe water content for an arc magma (�4 wt %) [Kimura and Nakajima, 2013; Plank et al., 2013] was achieved,although �70% of the calculation results gave rise to 2 wt % initial H2O in the primary basalts in the magmachamber (supporting information S3).

The estimated source conditions demonstrate that the primary basalts in VF volcanoes had greater valuesof F (15–26%) over the temperature range 1190–1340�C, and in contrast lower F values (4–17%) were seenwith the same T range for the RA volcanoes (Figure 9a). Although the T ranges overlap, the RA basalts showa higher P at a given T (Figure 9c). In the experiments, an increase in P leads to an increase in the solidus-liquidus temperature [e.g., Katz et al., 2003; Tatsumi et al., 1983], so that the calculated relation in the acrossthe arc setting is consistent with the petrological rule.

Source fertility shown by MgO ranges from 27–45 wt % up to 53 wt %, covering almost the entire composi-tional range of mantle peridotites [Bodinier and Godard, 2003]. This suggests a pyroxene-rich lherzorite toharzburgite-dunite lithology of the source mantle (Figure 9b). Fo in the olivine in mantle equilibrium andMg# (Mg# 5 Mg/Mg1Fe molar ratio calculated by the estimated mantle MgO and fixed FeO 5 8 wt %) ofthe source mantle showed a clear correlation Mg# 5 0.9906 3 Fo (r2 5 0.937) (not shown). This indicatesthat the MgO estimate by our method is internally consistent.

It is also consistent that a fertile mantle source shows a greater F within the designated pressure range for VFand RA (Figures 9a and 9b), but T shows a clearer correlation with F (Figure 9a). MgO inversely correlates with F,irrespective of the similar T range for VF and RA basalts. Therefore, the systematic difference in F between VFand RA is likely to be due to the difference in melting pressure (Figure 9c) by which increases in P decrease F atthe same temperature [Katz et al., 2003] rather than seeing the effects from changes in source fertility or T.

Pressure is the important factor in the chemical difference between the VF and RA primary basalts, with alower P 5 0.8–1.6 GPa in VF and a higher P 5 1.4–2.3 GPa in RA (Figure 9d). F is the secondary control inboth VF and RA mantle, which is basically controlled by the difference in temperature [Falloon et al., 2007],i.e., lower at shallow depths with T 5 1200�C and higher at greater depths with T 5 1350�C (Figures 9a and9d). An increase in T with depth increases F, although an increase in P can cause a counter effect. This com-mon feature for both VF and RA suggests a strong normal thermal gradient in the upper half of the mantlewedge asthenosphere (Figure 9d).

This thermal gradient further suggests that convection in the mantle wedge plays a key role in the thermalstructure. The P, F, and T estimates developed here consider the effect of water. The water content in the



source most significantly affects the T estimate, with subordinate effects to P and F (Figure 6). Therefore,the conclusions given here are robust. The mantle P-T structure has been inferred by ABS models [Kimuraand Yoshida, 2006; Kimura and Nakajima, 2013; Kimura et al., 2009, 2010] and thermodynamic phase equili-bria [Kuritani et al., 2014a, 2014b]. PRIMACALC2 shows similar results, with a completely different modelscheme.

4.5. Along-Arc Variation of the Source MantleSource mantle conditions beneath VF and RA in the N-Izu arc are characterized by a greater F 5 14–22%beneath VF and a lesser F 5 10% beneath RA with a relatively fertile mantle composition (MgO 5 28–35 wt%; Figures 9a and 9b). The Fuji volcano in the N-Izu collision zone has RA characteristics with lower F 5 5–16% and a wide range of T 5 1220–1320�C (Figure 9a). Mantle fertility beneath Fuji varied with MgO 5 27–42 wt %, suggesting a heterogeneous source (Figure 9b) located in an intermediate depth range P 5 1.4–1.5 GPa (Figure 9c). These are different from the RA basalts in N-Izu.

The results from the Fuji volcano raise a question as to whether or not the source mantle beneath a volcanois heterogeneous. Each sample that has passed through the whole melting process, melt migration system,

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Figure 9. Intensive-extensive parameters estimated by PRIMACALC2 for N-Izu, southern NE Japan, and northern NE Japan arcs. (a) Mantlemelting temperature (T) versus degree of partial melting (F). Thick lines in light red and blue show the average of volcanic front and rear-arc primary basalts, respectively. (b) MgO in the source mantle versus degree of melting. (c) Melting pressure (P) versus temperature. (d)Across arc schematic cross section of mantle wedge. Black rectangles show P and T ranges of primary basalt segregation estimated by PRI-MACALC2. Overall positive correlation between P and T suggests geotherm of the melting zone shown as 1200 and 1350�C contour lines.For slab dehydration and melting conditions, see Kimura and Nakajima [2013].



and crustal plumbing system must represent a mixture of melts derived from a range of sources. The appa-rent source composition derived from PRIMACALC2 must therefore represent some kind of mixture. Thisimplies that the spectrum of heterogeneity is even larger than that inferred, since mixing presumablyreduces the range of extremes. If so, the wide variation found between T-F-MgO would relate to the thermalgradient in the mantle that forms greater F in the high T region, leaving a high MgO residue in the source.The P estimate for Fuji is fairly constant at P 5 1.4–1.6 GPa (Figure 9c), and the Sr-Nd-Pb isotope composi-tions are fairly uniform [Watanabe et al., 2006]. Thus the thermal heterogeneity, if persistent, is significanteven in the root of one single volcano.

The basalts from the southern NE Japan VF, including Asama, showed relatively low F and T, with depletedsources (Figures 9a and 9b). There was a considerable overlap between the southern and northern VF in NEJapan. However, the northern VF basalts tended to show higher F and T in comparison to N-Izu VF but withsimilar depletion in the source (Figures 9a and 9b). The RA basalts from northern NE Japan has a highermantle T and F, with an intermediate mantle depletion similar to the depleted MORB source mantle [Work-man and Hart, 2005] (Figure 9c).

Spatial distributions of average values of MgO and F in the source mantle are shown in Figure 6. The uppermantle beneath the northern NE Japan VF is relatively fertile (27–37 wt %), but in southern NE Japan it isdepleted (41–46 wt % MgO). The mantle becomes fertile again in N-Izu VF (29–35 wt % MgO) (Figure 6). Incontrast, the RA mantle consistently shows intermediate depletion (32–39 wt % MgO) close to the DMMvalue (35 wt % MgO), with the exception of the very depleted mantle (50 wt % MgO) beneath Kampu vol-cano (Figure 6). F relates to T and P and may represent the production rate of the mantle. F is high (20%–26%) in northern NE Japan, low (14%–20%) in the southern NE Japan VF, and remains lower (14%–22%) inthe VF of the N-Izu arc (Figure 6). F is low (9%–15%) in the NE Japan RA and remains low and constant(�10%) through Fuji to the N-Izu RA. All of these observations suggest RA characteristics of Fuji primarybasalt similar to those noted in a previous report [Tani et al., 2011].

The greater mantle depletion, and the lower degree of melting beneath the southern NE Japan VF, couldbe related to the overlap of the Philippine Sea Plate slab onto the Pacific Plate slab that promoted depletionof the mantle by a greater water supply to the source mantle [Nakamura and Iwamori, 2009; Nakamuraet al., 2008] and a poor development of the mantle wedge corner flow. Alternatively, intensive magma pro-duction in the middle Miocene fore arc, due to an extremely high temperature mantle during the openingof the Sea of Japan, depleted the subarc mantle beneath the southern NE Japan VF [Yamamoto and Hoang,2009].

As shown above, PRIMACALC2 is a useful petrological tool for estimating primary basalt compositions,including those of major/trace elements and water contents, as well as the P, T, F, and MgO source mantleconditions, which generate the tholeiitic low-K to calc-alkaline medium-K primary basalts. Moreover, a com-bined use with ABS models [Kimura and Nakajima, 2013; Kimura et al., 2009, 2010] provides (1) a robust esti-mate of the primary basalt composition which is immediately used for ABS and (2) a unique test for theestimates of the source mantle conditions beneath arcs by back calculation modelling. Because COMAG-MAT3.72, and thus PRIMACALC2, can be applied to dry basalt systems with tholeiitic to mildly alkalic com-positions, PRIMACALC2 is useful for MORB and some OIB, including those with tholeiitic to transitionalcompositions.

5. Summary

We developed a numerical simulation model PRIMACALC2 for estimation of hydrous primary magma com-positions and their genetic conditions in subduction zones. PRIMACALC2 uses COMAGMAT3.72 for the for-ward calculation of crystal fractionation of hydrous magma in the intracrustal depth magma chamber atgiven P, T, fO2, and H2O conditions, and back calculates major and trace element compositions to a primarybasalt in mantle equilibrium using the NiO-Fo relationship. PRIMACALC2 also calculates the P, T, and F con-ditions of the source mantle, and the source mantle fertility expressed by MgO. Calculations were appliedto VF and RA basalts and to basaltic andesite magmas from N-Izu, southern NE Japan, and northern NEJapan arcs. The results argue that the mantle beneath VF is melted at shallow depths (0.8–1.6 GPa), whereasmelting occurs deeper (1.4–2.3 GPa) beneath RA. Although a similar range of melting temperatures wasfound beneath VF and RA, higher temperatures were observed from the deeper mantle. These results



suggest that the isotherm in the mantle is inclined toward RA, and that a normal mantle geotherm is main-tained in the upper half of the mantle beneath VF and RA. Water contents of the primary magmas wereconstrained by relationships between crystallization conditions and NiO contents in olivine in mantle equi-librium. Initial H2O in the primary basalt positively correlated with NiO in olivine, whereas the magma cham-ber P was negatively correlated, thus providing the petrologically useful constraints. Estimated watercontents in the primary magmas were 2–4 wt %, consistent with (or somewhat lower than) previous esti-mates. Thus, PRIMACALC2 is a new powerful tool for estimation of the conditions and compositions of pri-mary magmas generated in the arcs, as well as for decoding P, T, F, H2O parameters of their fractionation.

Notation

Important acronyms used in PRIMACALC2 are listed below.

Ni(ol)wt % NiO in olivine.Fo(ol)% Fo content in olivine.Mg# Bas Mg# 5 Mg/(Mg1Fe21) molar ratio in basalt.Fe21/Fe* ferric iron content over total iron in basalt.H2O(wt %) water content in basalt.P(COM) magma chamber pressure set for COMAGMAT3.72.TWC(KAZ) hydrous mantle temperature estimated by Katz et al. [2003].TDC(Herz) dry mantle temperature estimated by Herzberg et al. [2007].PGPa(COM/Herz) pressure at magma chamber given for COMAGMAT3.72 and that estimated for source

mantle by Herzberg et al. [2007].F%(Herz) degree of partial melting estimated by Herzberg et al. [2007].%Xfrac. total fractionated mineral weight in magma chamber.MgO PM estimated MgO in the source mantle by Herzberg et al. [2007].

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AcknowledgmentsThis project was greatly improvedthrough discussions with G. Barmina ofVernadsky Institute, H. Kawabata, M.Hamada, and T. Sakyuama of IFREE/JAMSTEC, J. B. Gill of the University ofCalifornia Santa Cruz, R. J. Stern of theUniversity of Texas at Dallas, and M.Carr and M. Feigenson of RutgersUniversity. We thank M. Ban, T.Takahashi, Y. Hirahara, T. Ohba, A.Fujinawa, S. Hayashi, T. Yoshida, T.Miyazaki, Q. Chang, R. Senda, and Y.Tatsumi for allowing us to use theunpublished major element and Nidata for Takahara, Nasu, Akitakoma,Hakkoda, Osore, and Kampuvolcanoes. Thanks are also extendedto Kirill Bychkov for providingdescriptions of the interface and thedetails of the FORTRAN program ofCOMAGMAT3.72. Review commentsby C. Herzberg, an anonymousreviewer, and the Associate Editor C.-T.Lee greatly improved this paper.



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Calculation of water-bearing primary basalt and estimation of source mantle conditions beneath arcs: PRIMACALC2 model for WINDOWS