PUBLICATIONS. Geochemistry, Geophysics, Geosystems

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: Adiabatic mantle melting is modeled using incompatible trace elements Effects of hydrous and pyroxenitebearing sources are examined The model explores source conditions of MORB, OIB, BAB, and LIP Trace element mass balance in hydrous adiabatic mantle melting: The Hydrous Adiabatic Mantle Melting Simulator version (HAMMS) Jun-Ichi Kimura and Hiroshi Kawabata 2 Department of Solid Earth Geochemistry, Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Yokosuka, Japan, 2 Research and Education Faculty, Multidisciplinary Science Cluster, Interdisciplinary Science Unit, Kochi University, Kochi, Japan Supporting Information: ReadMe Supporting Information Dynamic Content S Supporting Information 2, 3 Correspondence to: J.-I. Kimura, jkimura@jamstec.go.jp Citation: Kimura, J.-I., and H. Kawabata (24), Trace element mass balance in hydrous adiabatic mantle melting: The Hydrous Adiabatic Mantle Melting Simulator version (HAMMS), Geochem. Geophys. Geosyst., 5, , doi:. Received 3 MAR 24 Accepted 28 MAY 24 Accepted article online 2 JUN 24 Published online 6 JUN 24 Abstract A numerical mass balance calculation model for the adiabatic melting of a dry to hydrous peridotite has been programmed in order to simulate the trace element compositions of basalts from midocean ridges, back-arc basins, ocean islands, and large igneous provinces. The Excel spreadsheet-based calculator, Hydrous Adiabatic Mantle Melting Simulator version (HAMMS) uses () a thermodynamic model of fractional adiabatic melting of mantle peridotite, with (2) the parameterized experimental melting relationships of primitive to depleted mantle sources in terms of pressure, temperature, water content, and degree of partial melting. The trace element composition of the model basalt is calculated from the accumulated incremental melts within the adiabatic melting regime, with consideration for source depletion. The mineralogic mode in the primitive to depleted source mantle in adiabat is calculated using parameterized experimental results. Partition coefficients of the trace elements of mantle minerals are parameterized to melt temperature mostly from a lattice strain model and are tested using the latest compilations of experimental results. The parameters that control the composition of trace elements in the model are as follows: () mantle potential temperature, (2) water content in the source mantle, (3) depth of termination of adiabatic melting, and (4) source mantle depletion. HAMMS enables us to obtain the above controlling parameters using Monte Carlo fitting calculations and by comparing the calculated basalt compositions to primary basalt compositions. Additionally, HAMMS compares melting parameters with a major element model, which uses petrogenetic grids formulated from experimental results, thus providing better constraints on the source conditions.. Introduction Adiabatic melting of mantle peridotite is an important mechanism for the genesis of mid-ocean ridge basalt (MORB), ocean island basalt (OIB), back-arc basin basalt (BAB), and basalt from a large igneous province (LIP) [Herzberg and Asimow, 28; McKenzie and O Nions, 995; Taylor and Martinez, 23]. Melting conditions of the mantle, such as the onset of melting, termination of melting, and degree of melting, are fundamentally controlled by the mantle potential temperature in the adiabatically rising mantle peridotite [McKenzie and Bickle, 988]. An additional controlling factor is the water content in the source mantle, which affects the melting regime [Asimow et al., 24; Katz et al., 23]. Thus, these parameters control the compositions of major and trace elements of the basalt [Asimow et al., 24]. Recent studies have dealt with the melting of multiple source mantle lithologies, e.g., peridotite and pyroxenite, which form a complex melting regime because of the different solidi of the sources [Phipps Morgan, 2]. In such a case, deep metasomatism of the peridotite can occur by melting of the pyroxenite due to its lower solidus [Yaxley, 2]. Melting of variously metasomatized (hybridized) peridotite would thus follow at a shallow depth to generate a basalt [Herzberg, 2]. Major, trace, and isotopic compositions of a basalt from such a metasomatized mantle should vary owing to the complex melt generation/coalescence; nevertheless, a simple mixing model between the melts from multiple source lithologies has been examined [Herzberg, 2; Ito and Mahoney, 25; Kawabata et al., 2; Phipps Morgan, 999]. The roles of water and metasomatic melting in peridotite sources have been rigorously discussed in the process of adiabatic melting; it is not easy to incorporate (or distinguish) the two factors in melting calculations (or in observations of natural rock). However, it is considered necessary to examine the combined effect of KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2467

2 water [Asimow et al., 24; Katz et al., 23] and metasomatic melts, such as silicic melts from pyroxenite [Grove et al., 23; Herzberg, 2; Ito and Mahoney, 25], slab-derived fluids/melts [Kimura et al., 2], or bulk solid sediment [Behn et al., 2] in relation to OIBs, subduction zone basalts, BABs, and also in LIP basalts [Gurenko and Kamenetsky, 2] or in MORBs [Ingle et al., 2]. In order to examine the roles of water and metasomatic melt in the peridotite source, we have developed a simple calculation model for hydrous adiabatic mantle melting that is applicable to both dry and hydrous and weakly melt-metasomatized peridotite sources. We have built the model using the incompatible trace element mass balance between the source peridotite and the estimated primary basalt. Incompatible trace elements are extremely sensitive in both the metasomatism and melting of peridotite and, therefore, have the potential to solve many petrological/geochemical problems. A major element model is also useful for the identification of a peridotite/pyroxenite source [e.g., Grove et al., 23; Herzberg et al., 27]; thus, we examine the major element model along with the trace element model. An Excel spreadsheet, the Hydrous Adiabatic Mantle Melting Simulator version (HAMMS), was coded for the above purposes. HAMMS provides a unique opportunity to examine the mass balance between the calculated basalt and primary basalt composition using 26 incompatible trace elements (including major element K, expressed as ppm of the major elements) and major elements. This paper describes the details of the HAMMS model and shows its application to MORB, OIB, and LIP basalts. 2. Model Descriptions We coded an Excel spreadsheet-based calculator, HAMMS, using combinations of calculations from Worksheets and a VBA macro (HAMMS_..xls in supporting information dynamic content S and Figure ). The HAMMS_..xls Excel file contains the [SUMMARY], [THERMO], [DATA_INPUT], [Per_Dep], [Katz (2)], and [CMAS] Worksheets for this purpose. In this section, we describe the basic assumptions used in the HAMMS model. 2.. Parameters and Data Used in HAMMS HAMMS uses a petrogenetic model of fractional adiabatic melting with mass balance calculations of incompatible trace elements between a source peridotite and the generated basalt melt. First, we briefly describe the parameters and compositions used in HAMMS; the details are then given in the following sections. The melting parameters explored in the HAMMS model are as follows:. Mantle potential temperature: T p in C (cell $C$7 of the [THERMO] Worksheet in supporting information dynamic content S and Figure ). 2. Initial water content in the source mantle: H 2 O (i) in wt % (cell $C$8) 3. Depth of termination of adiabatic melting: Pmt in GPa (cell $C$9) 4. Source depletion: CsDep in wt % MORB melt extraction (cell $C$) 5. Fraction of source contamination: F cont in wt % (cell $C$) (not examined in this paper) Of these, we define the CsDep-related factors as follows:. Source Peri. for the choice of initial source composition assuming primitive mantle (PM) [Sun and McDonough, 989] and depleted MORB source mantle (DMM) [Workman and Hart, 25] (cell $E$4). 2. Mode (D/P) defining modal depletion in the source peridotite (cell $E$). 3. Cpx mode in %, defining the subsolidus clinopyroxene mode in the source peridotite (cell $E$). Userdefined parameters are in the cells color-coded yellow in the [THERMO] Worksheet in HAMMS_..xls (supporting information dynamic content S and Figure ); otherwise, they are calculated automatically by the model (e.g., Mode (D/P) and Cpx mode). The target basalt composition used in HAMMS should be equilibrated with mantle peridotite and includes major elements (SiO 2, TiO 2,Al 2 O 3, FeO, MnO, MgO, CaO, Na 2 O, K 2 O, and P 2 O 5 ) and 26 incompatible trace elements (Rb, Ba, Th, U, Nb, Ta, La, Ce, Pb, Pr, Sr, Nd, Sm, Zr, Hf, Eu, Gd, Tb, Dy, Y, Ho, Er, Tm, Yb, Lu, and K in KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2468

3 Figure..Schematic calculation flow and screenshot of the HAMMS [THERMO] Worksheet. Inputs of the primary magma composition ( major and 26 incompatible trace elements) are set in the [DATA_INPUT] Worksheet (not shown). () Selection of target magma composition. (2) Source mantle type (PM or DM). (3) Olivine and clinopyroxene mode (DM 5 4, PM 5 ). (4) Calculation pressure range (<7 GPa). (5) Parameters: Tp/ C mantle potential temperature; H 2 O (i) /% water content in the source mantle; Pmt/GPa depth of termination of melting; CsDep/% source depletion from the selected source as calculated by % melt extraction; F cont% /% fraction of contaminant in the source mantle mass. (6) Settings for minimum-maximum step values of the parameters used for the Monte Carlo calculations. (7) Acceptable limits of relative differences (R.D.) in percentages from the target basalt values used in the Monte Carlo calculations. (8) Radio button for Excel VBA macro runs. (9) Display panels of calculated P-T path of liquidus-solidus melt, basalt compositions, partition coefficients, residual mantle mineralogy, and H 2 O in the basalt. ppm of the major element). These are inputs in the [DATA_INPUT] Worksheet (supporting information dynamic content S). Ten individual samples are acceptable, and the choice of the target is available from cell $E$3 in the [THERMO] Worksheet by giving a sample identification number ( ) Model Strategy The HAMMS model primarily uses the incompatible trace elements mass balance based on the petrological forward model of adiabatic fractional mantle melting. Incompatible trace elements are particularly sensitive to mantle melting and have been examined in adiabatic melting regimes [Ito and Mahoney, 25; Kimura and Sano, 22; Liang and Peng, 2; McKenzie and O Nions, 99]. Incompatible trace elements are also useful for examining the source mantle composition because slight enrichment/depletion of the source results in a large difference in the generated basalt [Kimura et al., 29, 2; Pearce et al., 25]. With this background, the trace element mass balance calculations can estimate intensive/extensive parameters, such as melting mode (Xa), degree of melting (F), pressure (P), and temperature (T) [Kawabata et al., 2; Kelley et al., 26; Kimura et al., 29, 2; Pilet et al., 2]. Among these, HAMMS explores T p, H 2 O (i), Pmt, CsDep, and F cont, in order to reproduce the target basalt compositions. Simultaneously, magma temperature T m ( C) and water in the basalt H 2 O bas (wt %) are also obtained. The source mineralogy (Xa) is prescribed by the parameterized forward model (section 2.) and not examined as a fitting parameter. Major elements are also useful for this purpose. For example, TiO 2,K 2 O, and Na 2 O reflect the degree of melting (F) or pressure (P) in the source peridotite [Asimow et al., 24; Herzberg and O Hara, 22; Herzberg and KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2469

4 Asimow, 28; Kelley et al., 26]. FeO and MgO are also useful for F and T estimates in batch/adiabatic mantle melting [Asimow et al., 24; Herzberg and O Hara, 22; Herzberg and Asimow, 28]. Variation of Al 2 O 3 with the CaO/Al 2 O 3 ratio is a useful indicator of P [Rogers et al., 2; Sisson et al., 29]. Petrological phase relations using major elements, such as CMAS projection, are also useful to investigate F and P [Grove et al., 23; Herzberg and O Hara, 22; Herzberg et al., 27; Till et al., 22]. HAMMS uses a simple major element petrogenetic model of batch melting to independently estimate the P, T, and F conditions of a primary basalt finally equilibrated with the mantle, and the results are compared to those from the trace element model. Below, we describe the calculation background of HAMMS in the following order: () thermodynamic model, (2) mantle melting mineralogy, (3) mode of melt coalescence, (4) partition coefficient, (5) source peridotite composition, (6) major element model, and (7) fitting calculations Thermodynamic Model With Hydrous Melting Parameterization Outline of HAMMS Model The HAMMS model uses () a thermodynamic fractional adiabatic melting model of a mantle [McKenzie and Bickle, 988; Phipps Morgan, 2]; with (2) parameterized P-T-H 2 O-F melting relationships of mantle peridotite based on experiments [Katz et al., 23]. The latter model is incorporated into the adiabatic melting calculations using Katz et al. [23, equations (2) (23)], with the (@Tm/@P) F and (@Tm/@F) P relationships obtained from the P-T-H 2 O-F parameterization. Exactly, the same results are obtained in the single phase adiabatic melting model of Phipps Morgan [2], where the physical parameters of the thermal expansivity of a solid a s 5.28 ( C), heat capacity C P 5 (J kg 2 C 2 ), rock density q s 5 33 (kg/m 3 ), entropy difference between solid and liquid DS 5 3 (J kg 2 C 2 ), gravitational acceleration g (m/s 2 ), and the (@Tm/@P) F and (@Tm/@F) P relations from Katz et al. [23] are used ([THERMO] and [Katz (2)] Worksheets in supporting information dynamic content S and also see section 2.3.2). A nonanalytical solution for the P-T-H 2 O-F relation shown by Katz et al. [23, equation (9)] is obtained by computation with a minimum finder coded in Excel [Kimura et al., 29], which consists of line 73 and below in the [Katz (2)] Worksheet (supporting information dynamic content S). In this model, the effects of liquidus/solidus drops and changes in the degree of melting (F) between the liquidus-solidus intervals are calculated for various amounts of H 2 O and incorporated into the thermodynamic calculations via (@Tm/@P) F and (@Tm/@F) p. This is thus useful for studying the hydrous adiabatic melting [Katz et al., 23]. The calculated P- T paths of the mantle potential temperature (T p ), dry solidus and liquidus, wet solidus and liquidus, and melt are shown in the [P-T paths] panel in the [THERMO] Worksheet (see Figures and 2 and supporting information dynamic content S) Solidus-Depletion Gradient in Fractional Adiabatic Melting The P-T path (dt/dp) of the melting region and melt productivity (df/dp) of a peridotite essentially determine the thermodynamic model of fractional adiabatic melting, because solidus-depletion of a peridotite in adiabat immediately correlates to the heat balance and the melt productivity of the system [Phipps Morgan, 2]. The temperature profile of the melting region (T m ) in adiabatic fractional melting (Figures 2a and 2b) is expressed as dt dp m P n TDS m F 2 at P o () from Phipps Morgan [2, equation (2)]. Melt productivity in the adiabatic melting (df/dp) (Figures 2e and 2f) is also expressed as 2 df dp 5 2 agt Cp TDS m P (2) from Phipps Morgan [2, equation ()], where T is the mantle temperature, P is the pressure, T m is the temperature of melting region, a is the thermal expansivity of peridotite, DS m is the melting enthalpy, C p is the heat capacity, q is the density, and g is the gravitational acceleration. The subscripts F, P, and S indicate degree of melting, pressure, and entropy dependences, respectively. KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 247

5 T (C) ( Tm/ F)P Mode (fraction) A C E Tp = 45 (C), Dry Tp (C) Tm (C) Solidus Dry solidus Liquidus Dry liquidus Tp = 45 (C), H2O =.7% melting onset GPa melting onset 5 4 GPa 4 GPa cpx out 2 2 cpx out 2 cpx out Ol Gar Opx Cpx F T (C) ( Tm/ F)P Mode (fraction) B D F melting onset GPa melting onset 4 GPa 4 GPa cpx out cpx out 2 cpx out Figure 2. Melting regime and modal composition of a fertile mantle peridotite source (PM) in an adiabatic upwelling. Cases with a potential temperature (T p ) of 45 C (a, c, e) dry and (b, d, e) wet (H 2 O 5.7 wt % in the mantle source) are shown. Pressure-dependent solidus-depletion gradient (@T m /@F) P are shown for (c) dry and (d) wet cases. Addition of water results in a deeper onset of melting and a slightly greater degree of melting at the termination depth of melting. Modal compositions change in accordance with the changes in melting regimes. Examples show deeper consumption of (e) garnet and (f) clinopyroxene in the wet melting. The (@T m /@P) F and (@T m /@z) F are estimated to be 3 C/GPa or 3 4 C/km [Hess, 989]. The value of DS m varies between rock types but was estimated for a peridotite to be 3 (J kg 2 C 2 )[Phipps Morgan, 2]. A remaining important yet well-constrained thermodynamic parameter is (@T m /@F) P, which is the pressure-dependent solidus-depletion gradient. This term means that, while a parcel of mantle undergoes partial melting, the change in the upwelling mantle temperature during partial melting (equation ()) is equal to the change in the solidus temperature due to a drop in pressure dp and an increase in depletion by melt extraction df [Phipps Morgan, 2]. The same parameter is also effective for melt productivity (equation (2)). The size of the solidus-depletion gradient can be measured from the melting intervals between the solidus and liquidus of a peridotite, which are determinable experimentally. For this, we can use the parameterization of peridotite melting from Katz et al. [23] by stepwise calculations of (@T m /@F) P,ifdP/dT is sufficiently small in each calculation step. Examples of the calculated (@T m /@F) P profile in an adiabatic melting (Tp 5 45 C) are shown in Figures 2c and 2d for dry and.7 wt % of water-present cases. The value of (@T m /@F) P varies between 5 and 4. Large values of (@T m /@F) P can be seen at the beginning of melting (5 5 C; see melting onset arrows in Figures 2c and 2d) and at the transition from lherzorite to harzburgite residual lithologies (the value increases from 4 Cto2 C, then progressively decreases) (see cpx out arrows in Figures 2c and 2d). Similar variation was reported by MELTS calculations for a single incremental batch melting at GPa which gave (@T m /@F) P 5, C for the first percent of melting and followed by decreases from 35, through 85, to 75 C at 5,, and 5% melting, respectively. The value increases again to 2 C at 8% melting when clinopyroxene is totally consumed and then decreases progressively [Hirschmann et al., 998]. The similarity of the values and profiles in (@T m /@F) P between Katz et al. s [23] parameterization and MELTS calculations is encouraging, and we use the former in HAMMS. KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 247

6 The changes in m P in adiabatic melting strongly control the profile of melt productivity (shown by F in Figures 2e and 2f) and the P-T path of the melting region (shown by Figures 2a and 2b) via equations () and (2). This is consistent with Phipps Morgan s [2] suggestion that the solidus-depletion gradient is expected to have a much larger effect than adiabatic expansion in reducing pressure-release melt productivity. A large solidus-depletion gradient in the depth range of initial melting lowers both the melt productivity and the magnitude of its departure from the P-T trajectory of mantle adiabat (Figures 2a and 2b). Addition of H 2 O to a peridotite also enhances melt productivity because such addition decreases (@T m /@F) P, with the effect of earlier (deeper) onset of melting due to significant lowering of the solidus (Figures 2d and 2e). Because of these effects, total melt productivity in a hydrous system is slightly higher than that of a dry system at a given pressure (Figures 2e and 2f). The thermodynamic model described above, and used in HAMMS, is based on fractional melting with progressive solidus-depletion of a peridotite due to melt extraction from the system. One may pose a question as to the combination of the fractional adiabatic melting model and melting parameters derived from liquidus-solidus intervals determined by batch melting experiments. A different derivation of melt productivity in isentropic melting using isobaric melt productivity is thought to be valid for adiabatic melting applications [Asimow et al., 997]. We therefore think our approach is valid (also see the first paragraph in section 2.4.). The most important intensive variable examined in the HAMMS model is T p (Figure and cell $C$7 in the [THERMO] Worksheet), which controls the entire melting regime in the adiabat. The second important parameter is the initial water content in the peridotite, H 2 O (i) (Figure and cell $C$8 in the [THERMO] Worksheet). This parameterization is valid as deep as 7 GPa because of the experimental data set used for melting parameterization [Katz et al., 23] Mantle Melting Mineralogy Melting mineralogy (modal composition Xa of olivine, orthopyroxene, clinopyroxene, and garnet) in a mantle source is a fundamental parameter used in examinations of the behavior of incompatible trace elements during adiabatic melting [Asimow et al., 24; Ghiorso et al., 22; Ito and Mahoney, 25; Kawabata et al., 2; McKenzie and O Nions, 99]. Spinel is stable over a wide pressure range but is minor in composition (usually less than 2 wt %). Spinel also has low partition coefficients for most incompatible trace elements [Pilet et al., 2] and thus is disregarded in our model. We describe our parameterization method for the residual mantle mineralogy during melting in the following section Melting Mode and Evaluation of pmelts The residual mineralogic mode of a mantle during melting varies with P-T and F in adiabatic melting [Liang and Peng, 2]. The change in residual mineralogy differs slightly between batch and fractional melting modes. In fractional melting, once a mineral phase (e.g., clinopyroxene) is completely melted, no further melting can occur until the T of the system is raised to that of the binary eutectic (olivine and orthopyroxene), whereas continuous reactions take place in batch melting [Wilson, 989]. Fractional melting is preferred in the modeling of adiabatic melting, and modeling is available with a thermodynamic model, such as pmelts. However, we use pmelts partially to assist experimental parameterization for the reasons shown below. A parameterization of fractional melting is difficult because experiments inherently imply batch melting. However, we use experimental results to parameterize the melting mineralogy because the use of a batch model does not considerably affect the sequence of melting mineralogy in an adiabatic system in bulk [Liang and Peng, 2; Wilson, 989]. We tested the pmelts thermodynamic model [Ghiorso et al., 22] calculated at P 5. GPa, F 5 4% with compositions of a primitive mantle (PM) such as fertile MM3 [Baker and Stolper, 994; Falloon et al., 999], a MORB source peridotite DMM [Workman and Hart, 25], and a depleted DMM of DMM [Wasylenki et al., 23] (Figure 3). The calculated modes for both fertile MM3 and ultradepleted DMM reproduce the experimental results fairly well when the Xa (mineral) is plotted against F [Hirschmann et al., 998]. Although there is a discrepancy between the T-mode relationship produced by pmelts [Hirschmann et al., 998; Lambart et al., 22; Ueki and Iwamori, 23], the match for the F-mode relationship is encouraging. We thus estimated melting modes using pmelts for a MORB source mantle (DMM) at GPa because experimental results for this composition are not available (see Figures 3a 3c). KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2472

7 Mode (%) ~4% Olivine Orthopyroxene ~4% ~4% Clinopyroxene MM3 (PM) pmelts MM3 DMM (DDM) pmelts DMM pmelts WH5 (DMM) 4 Mode (%) F (%) F (%) F (%) GPa 3-4GPa 5-6GPa ~4% Olivine A B C GPa 2GPa F (%) F (%) Clinopyroxene Garnet 3GPa 4GPa Orthopyroxene D E F ~4% 5GPa 6GPa KLB- 5GPa F (%) Figure 3. Comparison of modal compositions between experiments and pmelts calculations at (a c) GPa, and results of parameterization using experimental results of a fertile peridotite (KR43 and MM3, near PM composition) for (d f) 7 GPa. In Figures 3a 3c, open symbols are from experimental results, and crosses and solid circles are from pmelts calculations. Figures 3d 3f exhibit comparisons between experimental results (solid symbols) and parameterization (crosses and pluses). Different symbol shapes are for different pressures. Parameterization is made for a PM composition. Differences in olivine and clinopyroxene modes between DM and PM at GPa and 5 GPa are shown by black arrows. KR43 and MM3 results are from Baker and Stolper [994], Falloon et al. [999], and Walter [998]. DMM (depleted DM) results are from Wasylenki et al. [23]. We performed the same test at 3 GPa with the PM-like fertile KR43 composition [Walter, 998]. However, the pmelts calculations did not reproduce the experimental results. There were problems in the significantly large orthopyroxene mode by compensation of the lesser olivine and clinopyroxene modes. We therefore did not apply the pmelts model to the high-pressure range and used it only for DM source calculations at GPa. Instead, we mostly base the parameterization of mantle melting on the experimental results. This also provides the basis for self-consistency with the experimental solidus-liquidus intervals coupled with the thermodynamic model in HAMMS (see section 2.3.2) F-Mode Parameterization for PM Source We parameterized the F-mode relationship using the experimental results of () a PM-like fertile peridotite KR43 in the 3 7 GPa range from Walter [998] and (2) MM3 at GPa [Baker and Stolper, 994; Falloon et al., 999]. KR43 and MM3 have almost identical major element compositions [Baker and Stolper, 994; Walter, 998]. The mineralogic modes of olivine, clinopyroxene, orthopyroxene, and garnet show quasi-linear relationships against F in the 3% melting range over the examined pressure range (see experimental examples for, 3, 4, 5, and 6 GPa in Figures 3d 3f). We used linear regressions for each mineral phase at a given pressure and obtained Xa ðmineral ; P Þ 5a ðmineral ; P Þ F b ðmineral ; P Þ (3) Relationships between P and coefficients a (mineral, P) and b (mineral, P) were then formulated by second polynomial regressions, resulting in the following relationships: KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2473

8 a ðmineral; P Þ 5a P 2 b Pc (4) b ðmineral; P Þ 5a 2 P 2 b 2 Pc 2 (5) In supporting information text S2, Table S shows the coefficients a, b, c, and a 2, b 2, c 2 for each mineral. This parameterization successfully reproduces the experimental results of the fertile PM-like mantle peridotite of KR43 and MM3 (see model lines in Figures 3d 3f). The calculated Xa at a given P-F was parameterized to 7..5 GPa and then incorporated into the calculations with the given values of P-F from the adiabatic melting equations [Katz et al., 23]. Figure 2 shows two examples for both melting P-T paths and modes for dry and hydrous mantle melting at T p 5 45 C for dry H 2 O (i) 5 (Figures 2a and 2b) and wet H 2 O (i) 5.7 wt % (Figures 2c and 2d) cases. In this model, the effect of H 2 O on the modal composition, such as the liquidus drop of olivine [Falloon and Danyushevsky, 2], is not considered as comprehensive data set is not available for hydrous experiments. However, modal composition in the liquidus-solidus interval does not differ considerably between dry and wet experiments, irrespective of significant changes in F at given P-T [Hirose and Kawamoto, 995; Kimura and Ariskin, 24]. We applied the same parameterization throughout this paper because our parameterization is essentially based on the F-Xa (mineral) relationship. We have used a similar parameterization scheme with pmelts calculations for 3 GPa for arc magma genesis in the Arc Basalt Simulator (ABS) models [Kimura et al., 29, 2, 24], which has proven useful in the estimation of mantle melting under various subarc conditions. Although orthopyroxene and olivine modes are not well reproduced by pmelts at 3 GPa (see section 2.4.), effects of slab melt contamination in the mantle source are included in ABS3 (or later) by pmelts parameterization for use with the large amount of felsic slab melt additions, which need to be considered for arcs [Kimura et al., 2, 24]. We use experimental parameterization in HAMMS for better estimation of its deep melting regime, but we cannot deal with a large amount of source contamination by silicic melts. In the case of basaltic melt metasomatism (fertilization) derived from pyroxenite melting in the peridotite source, the melting mode parameterization of HAMMS is still valid and will be discussed in a later section F-Mode Parameterization for DM Source Source mantle compositions may vary between PM and DMM or materials even more depleted than those in various tectonic settings of adiabatic melting (e.g., DMM for MORBs) [Workman and Hart, 25] and PMlike sources for ocean island basalts (OIBs) [Kimura et al., 26; Willbold and Stracke, 26]. F-mode relationships should also differ owing to source depletion/fertilization. Unlike PM (section 2.4.2), not much experimental evidence is available for DMM to cover a wide pressure range for adiabatic melting. Even at GPa, none of the DMM experiments provides the modal composition. We examined pmelts for DMM composition at GPa, because pmelts provides a reasonable F-mode relationship (see section 2.4.). As shown in Figures 3a and 3c, the olivine mode systematically increases by 4% from MM3 (PM) to DMM by compensation of reduced clinopyroxene at 4%. The orthopyroxene mode is identical between MM3 and DMM for F < 2% up to total consumption of clinopyroxene, but is systematically low at 4% afterward, for DMM (Figures 3b and 3c). A 4% increase in olivine continues in DMM by F 5 4%, at which total consumption of orthopyroxene occurs (Figure 3b). This relationship is true for more depleted DMM, where a 2% increase in olivine is compensated by decreases in clinopyroxene of 2% from MM3 for F < %, up to total consumption of clinopyroxene; this relation continues by a reduction of 2% of orthopyroxene to F 5 35%, where total consumption of orthopyroxene occurs (Figure 3b). At high pressure, there are no experimental modes for DMM apart from the near solidus experiment using KLB- at 5 GPa [Herzberg and Zhang, 996]. KLB- has a similar composition to DMM [Takahashi, 986]. The result shows an increase in olivine of 4% with a suppression of clinopyroxene by 4% relative to PM (KR43) (see Figure 3f). There are no changes in the orthopyroxene and garnet modes caused by the difference in the source composition between PM (KR43) and DMM (KLB-). Based on the observations above, we simply increase olivine by 4% in the PM mode for the entire melting range of DMM and subtract 4% clinopyroxene from the PM mode before total clinopyroxene consumption. KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2474

9 This treatment applies to all P ranges. As shown by DMM at GPa, the same parameterization can be valid for a more depleted source. This simple parameterization is useful for a wide range of source depletions from PM (%), through DMM (4%), to DMM (2%). The value of cell $E$ Mode (D/P) in the [THERMO] Worksheet controls the modal depletion in the source peridotite. Zero % for PM and 4% for DMM are automatically set following the choice of PM/DMM done by Source Peri. in the $E$4 cell Effect of Source Depletion in the Thermodynamic Model Katz et al. [23] incorporated the effects of the clinopyroxene mode in their hydrous peridotite melting parameterization. The melt productivity (@F/@T) P is greater until total consumption of clinopyroxene, whereas it is lower for the olivine-orthopyroxene (harzburgite) residue [Katz et al., 23; Takahashi, 986]. This effect is parameterized in the (@F/@T) P relationship by the clinopyroxene mode in the subsolidus source peridotite [Katz et al., 23]. As shown in Figure 3c, subsolidus clinopyroxene is 8% for PM and 4% for DMM at GPa. This clinopyroxene mode is set by Cpx mode in cell $E$ in the [THERMO] Worksheet, and the values are 8% for PM and 4% for DMM. HAMMS automatically makes this change by setting Source Peri. along with the change in Mode (D/P) (see section 2.4.3). The clinopyroxene mode at solidus increases with P. Similarly, the lherzorite-harzburgite transition in melting residue occurs at higher F under high P (Figures 3d 3f), both of which are from the expansion of clinopyroxene volume. The melting parameterization of Katz et al. [23] also considers the role of clinopyroxene volume that affects the (@F/@T) P slope during melting. Our mode parameterization is consistent with that of Katz et al. [23] in the pressure range <2 GPa. Figure 2 shows the coherent transition of clinopyroxene out (from the mode parameterization by this study) and a change in the melt productivity P slope (from Katz et al. s [23] parameterization model) (see black arrows in Figures 2e and 2f). The discrepancy between the two models becomes significant at >2 GPa (not shown in figures). Deep total consumption of clinopyroxene is not the case in many adiabatic melting provinces with T p < 5 C (see Figure 2 for T p 5 45 C). However, some hot igneous provinces may possess T p > 5 C[Herzberg et al., 27; Putirka et al., 27]; thus, the discrepancy can affect the melting model. To cope with this problem, we adjust Cpx mode in cell $E$ in relation to T p by the empirically determined equations This adjustment is done automatically in cell $E$. ½Cpx modeš DMM 5½Cpx modeš DMM :3886 T p 58:95 (6) ½Cpx modeš PM 5½Cpx modeš PM :374 T p 56:642 (7) Solidus-Depletion by Composition of Peridotite The dry solidus temperature (T solidus ) at GPa is 6 24 C for fertile MM3 (PM) and C for ultradepleted DMM, based on experiments and MELTS calculations [Hirschmann et al., 998]. The T solidus of DMM composition is thus 22 C at GPa, considering the magnitude of depletion in modal clinopyroxene (melt component) (Figure 3c). Katz et al. s [23] T solidus at GPa is 22 C, which is appropriate for the DMM source, but the T solidus for PM is C lower. The solidus variation of 3 C between PM and DMM affects the estimates of T p and Pmt by HAMMS calculations. We therefore use the original T solidus for DMM alone and adjust T solidus to C lower for PM. We further relate the source depletion (CsDep) factor (see section 2.7) at 3 C/ wt % melt extraction (calculated from 3 C solidus-depletion in wt % melt depletion) to the PM and DMM sources, respectively. This is shown by cell $E$8 on the [THERMO] Worksheet and reflected in the calculation matrix in the [Katz(2)] Worksheet Mode of Melt Coalescence Melt Coalescence in Adiabatic Melting In adiabatic melting, the mode of melt coalescence is another important model factor. Previous models have used the fractional accumulation of instantaneous melts, which means that an instantaneous melt is immediately separated from the solid and melts at each depth and is accumulated to generate the final melt composition [Asimow et al., 24; Herzberg et al., 27; McKenzie and Bickle, 988; Phipps Morgan, 2]. This assumption is reasonable, because melt generation causes gravitational compaction of the KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2475

10 peridotite mineral grains [McKenzie, 984], resulting in an effective release of the melts from the peridotite body. Observations of mantle peridotite also support this. Fractional melting best explains many residual mantle peridotites [Norman, 998; Raye et al., 2; Sano and Kimura, 27]. Although thermodynamic modeling of adiabatic melting requires the conservation of heat, the released and accumulated melts can be, or can be regarded to be, retained in the same mantle mass in terms of heat [Phipps Morgan, 2]. Based on this assumption, HAMMS follows the previous proposal and uses incremental batch melting for calculations [Shaw, 2]. The source mantle depletion in terms of trace element extraction along the adiabatic melting is also calculated Depth of Termination of Melting In adiabatic melting, the onset of melting depends on the relationship between the peridotite solidus with/ without water [Asimow et al., 24; Katz et al., 23] and the mantle adiabat [McKenzie and O Nions, 995] and is therefore a fixed number after determination of the adiabatic and source mantle depletion parameters (see [P-T paths] in Figures, 2a, and 2e). Another important factor is the termination depth of adiabatic melting. This is controlled by the increased solidus temperature due to consecutive mantle depletion [Phipps Morgan, 2], or is forced by the thickness of the lithosphere lid, particularly in the case of OIBs, arc basalts, BABs, and LIP basalts, where continental or oceanic lithosphere overlies the upwelling mantle [Ito and Mahoney, 25]. The HAMMS model uses a forced melt termination depth Pmt defined by pressure in GPa, which is set in cell $C$9 in the [THERMO] Worksheet (see Figure and supporting information dynamic content S) Partition Coefficients Modeling Strategy Sections provide the basis for () the degree of melting F controlled by P, T p, CsDep, and H 2 O (i), (2) the modal composition Xa, and (3) melt coalescence in the HAMMS model. The next problem to be tackled is the partition coefficient (D) used for the calculations of incompatible trace elements in the generated basalt melts and residual solids. Several proposed values for D have been used for mantle melting models [e.g., McKenzie and O Nions, 99, 995; Pilet et al., 2], and many of these were fixed Ds. However, adiabatic melting occurs over wide P-T ranges of 7 GPa and 6 3 C[Herzberg et al., 27; Putirka et al., 27] (see examples in Figure 2). For these, fixed Ds may not be realistic because of the P-T-X (minerals and melts) dependence of partition coefficients [Wood and Blundy, 25]. This is particularly important for garnet and clinopyroxene, because these minerals have Ds that are usually greater than. and even > for heavy rare earth elements (HREEs). Experimentally determined Ds between minerals and basaltic melts [GERM, 23, and references therein] are useful, but the data set does not systematically cover a wide range of P-T-X. Lattice strain models for clinopyroxene and garnet have been developed and proven to predict D values. We examined Wood and Blundy s [997] lattice strain model for clinopyroxene and Draper and van Westrenen s [27] thermodynamic lattice strain model for garnet. Recently, reliable lattice strain models for orthopyroxene have become available [Yao et al., 22] whereas models for olivine are less well constrained, so we use empirically parameterized Ds for olivine [Bedard, 25]. The HAMMS model is a simple thermodynamic model combined with parameterization based on experiments. HAMMS can provide P-T, but cannot provide compositions of minerals and melts, unlike pmelts or other thermodynamic models. Instead, the experimental results, including the major element compositions of minerals and melts reported by Baker and Stolper [994] and Walter [998] were used for the model parameterization. With the P, T, and composition of minerals and melts (X (minerals and melts) ) data from these experiments, we can calculate D(REEs) using the lattice strain models for clinopyroxene [Wood and Blundy, 997], orthopyroxene [Yao et al., 22], and garnet [Draper and van Westrenen, 27] over 7 GPa, C. Using the same experimental data set for both modes and Ds should guarantee internal consistency in the P-T-F-mode-D relationship. We validate the model calculations by comparing the experimentally determined Ds as follows Partition Coefficients of REEs Figure 4a shows the calculated results of D(REE)s in clinopyroxene by the lattice strain model of Wood and Blundy [997] using the major element compositions of clinopyroxenes and melts from Baker and Stolper [994] and Walter [998] at and 3 7 GPa, and 25 8 C. Irrespective of their wide pressure range, D(REE)s on a logarithmic scale correlate linearly with T ( C). The same is true for the calculated D(REE)s in KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2476

11 D(REE) Cpx.. A 8-9 (C) 7-8 (C) 6-7 (C) 5-6 (C) 4-5 (C) 3-4 (C) D(REE) Opx... B Clinopyroxene Orthopyroxene Garnet T(C) T(C) T(C) D Johnson and Kinzler (989) (25-35, ) Adam and Green (26) Hart and Dan (993) (38, 3) E Green et al. (2) F Tuff and Gibson (27) (4, 3-7) Salters and Longhi (999) van Westrenen (2) (54, 3) Green (994) compiled (5-38, -3). D(REE) Gar... C La Ce Pr Nd Sm Eu Gd Tb Dy Ho Er Tm Yb Lu D(REE) Cpx D(REE) Ol La Ce Pr NdPmSmEuGdTb Dy Ho Er TmYb Lu G Model: Wood and Blundy (997) REE Adam and Green (26) Colason et al. (988) Kennedy et al. (993) Model: Bedard (27) REE Clinopyroxene Olivine La Ce Pr NdPmSmEuGdTb DyHo Er TmYb Lu D(REE) Gar MgO (wt%) H Model: Yao et al. (22) REE MgO =.287 T Orthopyroxene La Ce Pr NdPmSmEuGdTb Dy Ho Er TmYb Lu P(GPa) D(REE) Gar 5 T(C) Model: Draper and van Westerenen (27) Calc. Exp. Source R8+252 (86, 7) A4 (6, 4.5) MP24 (34, 3) REE Garnet La Ce Pr NdPmSmEuGdTb Dy Ho Er TmYb Lu Figure 4. Parameterization of the trace element partition coefficients for clinopyroxene, garnet, orthopyroxene, and olivine. D(REE)s are calculated based on the melt/mineral compositions and P-T from the experiments over 7 GPa using the lattice strain models of (a) Wood and Blundy [997], (b) Yao et al. [22], and (c) Draper and van Westrenen [27]. (a) T- dependent Ds for clinopyroxene were reproduced by the model and by the experiments of Hart and Dunn [993] and Johnson and Kinzler [989]. (b) Thermodynamic D garnet [Draper and van Westrenen, 27] was used because of a better fit with the experimental results. (d) T-dependent Ds were parameterized using exponential regressions for clinopyroxene, (e) orthopyroxene, and (f) garnet. (g) Ds between olivine and melt were parameterized by the MgO melt -dependent Ds of Bedard [25]. (h) MgO in the melts was calculated by the T-MgO relationship (see the text). Experimental data in Figure 4f are from Draper et al. [23] for R8252, Draper et al. [26] for A4, and Pertermann et al. [24] for MP24. orthopyroxene and garnet. Figure 4b shows the results calculated by the lattice strain model of Yao et al. [22] for the major element compositions of orthopyroxenes and melts at and 3 7 GPa, and C[Baker and Stolper, 994; Walter, 998]. Figure 4c also shows the results when the major element compositions of garnets and melts at 3 7 GPa, C[Walter, 998] are calculated with the lattice strain model of Draper and van Westrenen [27] and plotted together with the D(REE)s measured by Draper et al. [23] at 85 C, 7 GPa, by [Draper et al., 26] at 6 C, 4.5 GPa, and by Pertermann et al. [24] at 34 C, 3 GPa. The above examinations indicate that D(REE)s are primarily dependent on T and subordinately on P and X in a basaltic system. The relationship is also consistent with the prediction by the lattice strain models that D is largely dependent on T [Wood and Blundy, 25]. We cannot isolate P (and X) dependence of Ds from the T dependence curves because of the quasi-linear correlation between P and T conditions in the KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2477

12 experimental data set (see Figure 4h, red symbols). Therefore, our T-dependent parameterization involves these factors. To confirm the T dependence of D(REE)s, Figure 4d compares the calculated D(REE)s at different Ts (see Figure 4a for the temperature color code) and the experimental results of Johnson and Kinzler [989] at C, GPa, Hart and Dunn [993] at 38 C, 3 GPa, Tuff and Gibson [27] at 4 C, 3 7 GPa, and van Westrenen et al. [2] at 54 C, 3 GPa for clinopyroxene. The calculated results are able to replicate the experimental Ds at given temperatures sufficiently well. Figure 4e shows results for orthopyroxene that are the same as the experimental results of Adam and Green [26], Green et al. [2], and Salters and Longhi [999]. Figure 4f shows the results for garnet with experimental results by Draper et al. [23] at 85 C, 7 GPa, Draper et al. [26] at 6 C, 4.5 GPa, and Pertermann et al. [24] at 34 C, 3 GPa. The calculated D(REE)s for clinopyroxene, orthopyroxene, and garnet reproduce the experimental results relatively well. We thus formulate D(REE)s as D ðmineral; T Þ 5a expðbtþ (8) where T is temperature ( C). The coefficients (a and b) are from the regression lines in Figures 4a 4c and are summarized in supporting information text S2 Table S2 for clinopyroxene, Table S3 for orthopyroxene, and Table S4 for garnet. For D(REE)s of olivine, we used Bedard s [25] empirical parameterization, which requires MgO in the host basalt melts for calculation. MgO in a basalt from peridotite is T-dependent [Herzberg and O Hara, 22]. The experimental data set [Baker and Stolper, 994; Walter, 998] used in this paper also shows this relationship (Figure 4h, green solid circles with regression line). Herzberg and O Hara [22] further formulated the P dependence of the basalt MgO as T ð CÞ554 MgO ðwt%þ=; 22 ðmgo =; Þ PðbarÞ2:37 PðbarÞ 2 (9) We invert this equation by fitting and obtain MgO 5½:662 P 2 2:37 P :23424Š3expð:349 TÞ () and use this equation for calculations of MgO used in the parameterization of Bedard [25]. The calculated D(REE)s for olivine (Figure 4g) compare well with experimental results [Adam and Green, 26; Colason et al., 988; Kennedy et al., 993]. In supporting information text S2, Table S5 lists the parameterization coefficients and the equation proposed by Bedard [25] Other Incompatible Elements We use 26 incompatible trace elements, including REEs, in HAMMS. We estimate Ds other than D(REE)s for clinopyroxene and orthopyroxene based on D(Rb, Ba, Th, U, Nb, Ta, K, Pb, Sr)/D(La), D(Zr, Hf)/D (average of Sm and Eu), and D(Y)/D(average of Dy, Ho) using Ds from Pilet et al. [2] (supporting information text S2, Table S5). We did the same for the Ds for garnet but used Ds from McKenzie and O Nions [99, 995] because of better fits with the experimental data (supporting information text S2, Table S5). D(La) and D(Sm, Eu, Dy, Ho) largely vary by T in clinopyroxene, so that Ds for other elements vary by a factor of 5 (Figure 5a). D(La) in garnet is almost unchanged over a wide range of T, thus D(Rb, Ba, Th, U, Nb, Ta, K, Pb, Sr) are also unchanged (Figure 5b). We calculated orthopyroxene Ds based on the same assumption used for clinopyroxene (Figure 5c). Olivine Ds are available for all 26 elements by Bedard s [25] parameterization (Figure 5d and supporting information text S2, Table S5). We compared the calculated Ds with experimentally determined Ds for clinopyroxene, garnet, orthopyroxene, and olivine. The results are plotted together in Figures 5a 5d. Although our assumptions are simple, the calculated results reproduce the experimental results considerably well. Large variations in experimental results are obvious for highly incompatible elements (Rb, Ba, Th, U, Nb. Ta, K, Pb, Sr) including large ion lithophile elements (LILEs) and high field strength elements (HFSEs) (see gray bars in Figure 5; data from GERM [23]). These are all D <.5, and the variations do not affect the calculated basalt compositions much unless the degree of partial melting is extremely low. KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2478

13 Adam and Green (26) 2, 3 Tuff and Gibson (27) 4-75, 3-7 Perterman et al. (24) 34, 3 Adam and Green (26) 2, 3 Tuff and Gibson (27) 4-75, 3-7 D Clinopyroxene D Garnet Clinopyroxene 2 Element A. Rb Th Nb K Ce Pr Nd Zr Eu Tb Y Er Yb Ba U Ta La Pb Sr Sm Hf Gd Dy Ho Tm Lu. Garnet Element B. Rb Th Nb K Ce Pr Nd Zr Eu Tb Y Er Yb Ba U Ta La Pb Sr Sm Hf Gd Dy Ho Tm Lu. Adam and Green (26) 2, 3 Green et al. (2) 2, 3. Adam and Green (26) 2, 3 D Orthopyroxene.. D Olivine... Orthopyroxene C -5 Rb Th Nb K Ce Pr Nd Zr Eu Tb Y Er Yb Ba U Ta La Pb Sr Sm Hf Gd Dy Ho Tm Lu. Olivine Element D -5 Rb Th Nb K Ce Pr Nd Zr Eu Tb Y Er Yb Ba U Ta La Pb Sr Sm Hf Gd Dy Ho Tm Lu Figure 5. Partition coefficients of incompatible trace elements used in HAMMS and their comparisons to the experimentally determined Ds. Ranges of experimental Ds for mafic to ultramafic lithologies are shown by shaded bars [GERM, 23]. Representative results from Adam and Green [26], Green et al. [2], Pertermann et al. [24], and Tuff and Gibson [27] are shown by open circles with temperatures ( C) and pressures (GPa) from these experiments Variation in the Source Peridotite and Source Depletion The source peridotite may be more depleted in composition than the model mantle (DMM or PM) in terms of trace elements. We also explore this in HAMMS. The [Per_Dep] Worksheet calculates the source depletion by extracting a basalt melt from a DMM or PM source. The basic scheme is to extract a basalt melt from the peridotite at a given degree of batch partial melting at GPa, based on the assumption that incompatible trace element fractionation commonly occurs beneath MORs, which is one of the most dominant element fractionation processes in the evolution of Earth s mantle [Workman and Hart, 25]. We use the batch melting equation of Shaw [2] with a melting mode calculated by pmelts at GPa for the intermediate composition between DMM and PM because we saw that the pmelts model is reliable for this purpose at GPa (section 2.3.). The partition coefficients used are the fixed Ds from McKenzie and O Nions [99, 995] with slight modifications based on the discussions in section 2.6. The degree of depletion is given by wt % melt extraction in cell $C$ CsDep in the [THERMO] Worksheet (supporting information dynamic content S). With this model, a 4% melt extraction from PM generates a DMM equivalent in incompatible trace elements. This corresponds to a 4% reduction of modal clinopyroxene from PM to generate DMM. This is also consistent with the modal depletion from PM to DMM (section and Figure 3c). As shown in the source peridotite mode (sections and 2.4.4), any depletion assumed in CsDep should affect the Mode (D/P) and Cpx mode parameters in the source peridotite. A % depletion in CsDep KIMURA AND KAWABATA VC 24. American Geophysical Union. All Rights Reserved. 2479