Journal of Petrology Advance Access originally published online on October 14, 2004
Journal of Petrology 2004 45(12):2459-2480; doi:10.1093/petrology/egh072
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Journal of Petrology 45(12) © Oxford University Press 2004; all rights reserved
Open-System Magma Chamber Evolution: an Energy-constrained Geochemical Model Incorporating the Effects of Concurrent Eruption, Recharge, Variable Assimilation and Fractional Crystallization (EC-E'RA
FC)
1 INSTITUTE FOR CRUSTAL STUDIES AND DEPARTMENT OF EARTH SCIENCES, UNIVERSITY OF CALIFORNIA, SANTA BARBARA, CA 93106, USA
2 DEPARTMENT OF GEOLOGICAL SCIENCES, CENTRAL WASHINGTON UNIVERSITY, ELLENSBURG, WA 98926, USA
RECEIVED NOVEMBER 7, 2003; ACCEPTED AUGUST 6, 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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Significant petrogenetic processes governing the geochemical evolution of magma bodies include magma Recharge (including formation of ‘quenched inclusions’ or enclaves), heating and concomitant partial melting of country rock with possible ‘contamination’ of the evolving magma body (Assimilation), and formation and separation of cumulates by Fractional Crystallization (RAFC). Although the importance of modeling such open-system magma chambers subject to energy conservation has been demonstrated, the effects of concurrent removal of magma by eruption and/or variable assimilation (involving imperfect extraction of anatectic melt from wall rock) have not been considered. In this study, we extend the EC-RAFC model to include the effects of Eruption and variable amounts of assimilation, A
. This model, called EC-E'RAFC, tracks the compositions (trace elements and isotopes), temperatures, and masses of magma body liquid (melt), eruptive magma, cumulates and enclaves within a composite magmatic system undergoing simultaneous eruption, recharge, assimilation and fractional crystallization. The model is formulated as a set of 4 + t + i + s coupled nonlinear differential equations, where the number of trace elements, radiogenic and stable isotope ratios modeled are t, i and s, respectively. Solution of the EC-E'RAFC equations provides values for the average temperature of wall rock (Ta), mass of melt within the magma body (Mm), masses of cumulates (Mct), enclaves (Men) and wall rock (
) and the masses of anatectic melt generated (
) and assimilated (
). In addition, t trace element concentrations and i + s isotopic ratios in melt and eruptive magma (Cm, 
m, 
m), cumulates (Cct, m, m), enclaves (Cen, 
, 
) and anatectic melt (Ca, 
, 
) as a function of magma temperature (Tm) are also computed. Input parameters include the (user-defined) equilibration temperature (Teq), a factor describing the efficiency of addition of anatectic melt () from country rock to host magma, the initial temperature and composition of pristine host melt (
, 
, 
, 
), recharge melt (
, 
, , ) and wall rock (
, 
, , ), distribution coefficients (Dm, Dr, Da) and their temperature dependences (
Hm, Hr, Ha), latent heats of transition (melting or crystallization) for wall rock (ha), pristine magma (hm) and recharge magma (hr) as well as the isobaric specific heat capacity of assimilant (Cp,a), pristine (Cp,m) and recharge (Cp,r) melts. The magma recharge mass and eruptive magma mass functions, Mr(Tm) and Me(Tm), respectively, are specified a priori. Mr(Tm) and Me(Tm) are modeled as either continuous or episodic (step-like) processes. Melt productivity functions, which prescribe the relationship between melt mass fraction and temperature, are defined for end-member bulk compositions characterizing the local geologic site. EC-E'RAFC has potential for addressing fundamental questions in igneous petrology such as: What are intrusive to extrusive ratios (I/E) for particular magmatic systems, and how does this factor relate to rates of crustal growth? How does I/E vary temporally at single, long-lived magmatic centers? What system characteristics are most profoundly influenced by eruption? What is the quantitative relationship between recharge and assimilation? In cases where the extraction efficiency can be shown to be less than unity, what geologic criteria are important and can these criteria be linked to field observations? A critical aspect of the energy-constrained approach is that it requires integration of field, geochronological, petrologic, and geochemical data, and, thus, the EC-ERAFC ‘systems’ approach provides a means for answering broad questions while unifying observations from a number of disciplines relevant to the study of igneous rocks. KEY WORDS: assimilation; energy conservation; eruption; open system; recharge
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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It is generally understood that magma bodies evolve as open systems exchanging material and heat with their surroundings under far-from-equilibrium conditions. This interaction gives rise to the formation of what nowadays are recognized as self-organized dissipative structures (Nicolis & Prigogine, 1977
) at a variety of spatiotemporal scales. In magmatic systems, these structures may include, for example, rhythmically layered cumulates, compositionally zoned melt bodies, epithermal ore deposits formed by reaction between magmatic–hydrothermal fluids and country rock, as well as complex compositional zoning profiles within single crystals (for detailed arguments spanning a range of spatial and temporal scales see, e.g. Korzhinskii, 1970; Taylor, 1974; McBirney & Noyes, 1979; Rose & Burt, 1979; Smith, 1979; McBirney, 1980; Brimhall & Crerar, 1987; Feldstein et al., 1994; Halden, 1996; L'Heureux & Fowler, 1996; Tepley et al., 2000). In magmatic environments, the largest source of entropy production is the transport of heat between magma and its surroundings; entropy generation by magma mixing and irreversibility associated with growth and nucleation kinetics are additional, but generally smaller, sources. The mechanisms of heat transport include heat conduction into wall rock, heat transport by hydrothermal convection and the advection of heat associated with both recharge of fresh magma into an existing magma body and eruption of magma from that body. The mechanisms of mixing include advective mixing driven by thermal and compositional buoyancy of melt, multi-phase convection (Bergantz & Ni, 1999), and, to a lesser degree, simple molecular diffusion (Trial & Spera, 1990).
Although the study of heat transfer in magmatic systems has a venerable history (e.g. Ingersoll et al., 1954; Shaw, 1965; Jaeger, 1968; Spera, 1979; Carrigan, 1988; Huppert & Sparks, 1988; Bergantz, 1989; Marsh, 1989), the explicit connection between heat transfer and the trace element and isotopic evolution of magma in open systems has been investigated less thoroughly despite the obvious coupling. As noted by Taylor (1980; Taylor & Sheppard, 1986; see also DePaolo, 1981), it was N. L. Bowen (Bowen, 1928) who pointed out that assimilation is not a simple two-component mixing process. Instead, at a minimum, it represents a three end-member problem involving the coupled energetics (both sensible and latent) among magma, country rock assimilant and cumulates. If one allows for magma replenishment (addition) and magma removal (eruption), it is evident that there are many degrees of freedom available to magma undergoing open-system evolution. The geochemical path followed by an open system is dictated by the efficacy and coupling among the processes of eruption, recharge, mixing, assimilation and crystallization, and is therefore highly contingent on inherent vagaries particular to a given magmatic system, no two of which are alike in most respects.
Although geochemists and petrologists have long appreciated, in principle, the concept of open-system behavior, the translation of the concept into a practical geochemical model applicable to natural systems has proved challenging. A prominent contribution to this subject is the collective works of M. J. O'Hara and co-workers over the past quarter-century (e.g. O'Hara, 1977, 1980; O'Hara & Mathews, 1981; O'Hara & Fry, 1996; O'Hara & Herzberg, 2002). O'Hara and collaborators have consistently argued that many magmas on Earth exhibit characteristics suggesting they are not primary (e.g. O'Hara, 1980, 2000); among the most important of these characteristics is evidence of low-pressure crystallization, implying residence in shallow-level magma reservoirs where processes such as magma mixing, contamination and periodic eruption can potentially modify mantle ‘signatures’. O'Hara has also examined the question of how basalts (and other magmas) acquire their chemical signatures by investigating the consequences of magma migration through the crust. At the heart of this is the ‘space problem’ (e.g. O'Hara 1998). O'Hara (1998) noted that prior to 1950, the ‘space problem’ was connected to the study of assimilation, contamination and hybridization of evolving calc-alkaline plutons, the large volumes of which made the ‘space problem’ a visible issue. Field evidence, primarily from the margins of plutons, provided support for the importance of assimilation and hybridization in the petrogenesis of magmas. As basalts became increasingly important in understanding magma petrogenesis and deciphering sub-solidus convective mixing in the mantle, the ‘space problem’ and the relevance of assimilation became issues of secondary importance. However, as O'Hara particularly noted in his 1998 contribution on the thermal and geochemical consequences of large-scale assimilation in ocean island development, magmas emplaced at crustal levels must make space for themselves, and one possible mechanism is assimilation of crustal material as the magma body migrates. Thermal considerations also provide support for crustal assimilation (O'Hara, 1998). Through numerous quantitative models and associated analysis, O'Hara and colleagues have provided a framework in which the impact of Eruption, Recharge, Assimilation and Fractional Crystallization (ERAFC) processes on major and trace element and isotopic characteristics of magmas can be evaluated.
A critical step in further quantifying the types of models that O'Hara and colleagues have generated is development of a model for open-system magmatic behavior using coupled mass, momentum, species and energy conservation. Although, in principle, this seems straightforward, complicating matters quickly arise when conservation principles are applied to natural systems. For example, rarely is anything known regarding the three-dimensional form of a magma body. Without such knowledge, rates of heat transport are impossible to quantify, and this uncertainty makes it impossible to estimate rates of assimilation, magma mixing and fractional crystallization for specific magmatic systems. Similarly, phase equilibria models including minor and trace element partitioning and activity–composition relations for many important phases are still lacking. Incomplete information on the transport properties of magmatic materials such as thermal conductivity and the kinetics of transport phenomena introduces additional uncertainty into the modeling of magmatic systems. Unknown thermophysical property variations in host rock environments, such as the three-dimensional structure of the permeability, which can influence the rate of heat transfer between magma and country rock, serve to further obfuscate attempts to model particular natural systems.
In light of these difficulties, is there any hope for developing a model for the trace element and isotopic evolution of natural (open) magmatic systems? We believe the answer is yes. The phenomena of eruption, recharge and the mixing of magmas, assimilation, and fractional crystallization (i.e. ERAFC processes) are intimately linked through energy conservation, independent of the complicating details of momentum transport. This linkage can be exploited to develop a thermodynamic model for the open-system geochemical evolution of magma undergoing ERAFC processes. This model can then serve as a preliminary ‘reference state’ from which to evaluate the full complexity of geochemical evolution. Although this approach cannot provide absolute temporal information and is not a model for the dynamical evolution of magma, it does provide a relative chronology and material inventory for the succession of melt and solid compositions produced during ERAFC evolution and can be applied to natural systems. Significantly, model predictions can be compared with observed compositions and masses or relative masses of melts and crystalline products from particular natural systems. Indeed, when combined with trace and isotope material balance expressions, energy conservation leads to a self-consistent algorithm for determination of the geochemical path followed by magma along the path to thermal equilibrium from its initial far-from-equilibrium state. In the EC-ERAFC model, the path refers to the progression of wall-rock temperatures (Ta), trace element concentrations and isotope ratios in anatectic melt, host melt, eruptive magma and crystallized solids as a function of the melt temperature, Tm. The solid products of ERAFC evolution include cumulate rocks produced by fractional crystallization as well as solids produced by ‘instantaneous’ chilling of a portion of recharge melt. The latter solids, which are non-equilibrium with respect to the magma body, are identified with the class of inclusions ubiquitous in plutons and lavas denoted by the term ‘enclave’ in the classical petrologic literature (Best & Christiansen, 2001).
The purpose of this study is to present a description of the open-system model. Details of earlier EC-AFC and EC-RAFC models forming the backbone of EC-E'RAFC have been presented elsewhere (Bohrson & Spera, 2001, 2003; Spera & Bohrson, 2001, 2002). In the current work, attention is focused on the novel elements introduced by allowance for eruption (E') and for variable addition of anatectic melt into the host magma body (A). In the EC-AFC model, all melt generated in country rock is added to the evolving magma body. In the present model, that condition is relaxed. EC-E'RAFC is valid for arbitrary eruptive mass, 
(the non-dimensional eruptive mass), in the limit of no addition of anatectic melt into the magma body, = 0 (i.e. no contamination) and 
(no recharge). It also is valid in the limit of 
(no eruption) with arbitrary recharge mass, 
, for all such that 0 
1. It should be noted that is the mass fraction of anatectic liquid generated in the country rock that is added to and mixes with pre-existing host magma. For arbitrary and , the EC-E'RAFC presented in this study is limited to 
, approximately. The superscript on the E' is included to remind the reader that the current model is not valid for scenarios in which exceeds approximately 0·4. A summary of all the variables in the model is provided in Table 1.
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QUALITATIVE DESCRIPTION OF EC-E'RAFC MODEL: A SYSTEMS APPROACH
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TOPABSTRACTINTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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Figure 1 illustrates a crustal-scale view of magma transport, thereby providing a context for the ERAFC magma evolution model. Not included in the illustration are the regions where magma is generated, perhaps by isentropic pressure-release melting (Verhoogen, 1954
; Asimow, 2000), by volatile-induced solidus depression (Bailey, 1970), or by segregation from residual source material by percolation. Once segregated from residuum, melt rises as a result of the collective action of favorable pressure and buoyancy forces. The form (e.g. isolated pods or plexus of propagating cracks) and transport rate of magma through the lithosphere depends on a complex interplay of factors including the distribution of magma pressure and buoyancy forces, the magnitude, orientation and spatial variation of the principal stresses, and the thermophysical properties of both magma and country rock (e.g. see Petford & Koenders, 1988; Hart, 1993; Rubin, 1995). The loss of heat from the magma body depends upon the local temperature and thermophysical properties of the country rock as well as the relative importance of porous medium thermal–salinity convection compared with heat conduction (e.g. Norton & Cathles, 1979; Norton & Taylor, 1979; Taylor, 1986; Carrigan, 1988; Cathles et al., 1997; Schoofs & Spera, 2003). At shallow depths where porosity–permeability relations are favorable, large-scale hydrothermal circulation systems may develop and allow relatively efficient transfer of magmatic heat into country rock. In this case, although a large volume of country rock is heated, the fraction of wall rock undergoing partial melting may be restricted because of the relatively rapid transport of heat away from the magma body. In contrast, at greater depths where fluids may be absent or the permeability very low, heat conduction dominates heat loss. Although a smaller mass of country rock is heated, the local rise in temperature may lead to significant country rock anatexis. Germane to ERAFC evolution is that magma may sometimes be ‘stored’ in finite-volume bodies at depth. Two illustrative storage depths, drawn from a large number of possibilities, are illustrated in Fig. 1. One is the Moho, where a sharp contrast in density between mantle and lower crust exists, and another is the brittle–ductile rheological transition, where country rock thermophysical properties and the state of stress might impede upward magma transport. Once a substantial volume of magma is stored within the crust, the EC-E'RAFC model may be applied.
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In EC-E'RA
FC, a composite system is envisioned that is isolated adiabatically from its environment (Fig. 2). The composite system comprises four sub-systems with boundaries that may be open, closed or semi-permeable with respect to mass, and adiabatic or diathermal with respect to energy. In general, a stipulation regarding both heat and matter exchange between each sub-system is required to define an EC-E'RAFC evolution. The four sub-systems in the ERAFC model include country rock, magma body, recharge reservoir and an effusive (eruptive) volume. Wall rock (synonymous with country rock) is separated from the magma body by diathermal, semi-permeable boundaries. That is, heat can freely pass across the boundary that is permeable to fraction of anatectic melt generated in the country rock by partial fusion. The mass of country rock involved in E'RAFC evolution is governed by an integral energy balance that enforces energy conservation among all four sub-systems along the path towards thermal equilibration. The magma body consists of host melt, cumulates and enclaves. Two additional sub-systems include a reservoir of recharge melt of arbitrary mass, specific enthalpy and composition, and an eruptive or extrusive reservoir formed by partial eruption of the magma body. During eruptive episodes, the boundary between the magma body and the eruptive reservoir is perfectly open with respect to both heat and material transport. During episodes of recharge, the boundary between the recharge reservoir and the magma body is also open with respect to matter and heat. Boundaries between the recharge and eruptive reservoirs and the country rock are closed and adiabatic.
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is emplaced into country rock of temperature 
. (b) Early stage of E'RAIn the most general formulation of EC-E'RA
FC, each of the four sub-systems can be viewed as individual composite systems. In principle, this allows one to model effects such as compositionally zoned magma bodies, recharge magma of varying specific enthalpy and composition, the imperfect extraction of anatectic melts and eruption of magma with a different solid to melt ratio compared with the magma body at the time of eruption. The EC-E'RAFC versions presented here and in previous work (EC-AFC, EC-RAFC) apply to homogeneous (non-zoned) magma bodies replenished by recharge magma of fixed composition and temperature (specific enthalpy). Eruptive magma is at the same temperature as host magma, and its composition reflects that of host melt and average cumulates at Tm. Crystallinity is approximated from that of pristine magma at Tm. In EC-E'RAFC, the anatectic melt extraction factor, , is set by the investigator in the range 0 1. For = 1, all anatectic melt generated by partial melting of wall rock is added to and homogenized within the magma body. Alternatively, for = 0, none of the wall-rock anatectic melt is allowed to enter the magma body, although the energy needed for its generation is accounted for (Petford & Gallagher, 2001; Petford, 2003). The earlier EC-AFC and EC-RAFC models (Bohrson & Spera, 2001, 2003; Spera & Bohrson, 2001, 2002) implicitly set = 1, and are, therefore, ‘maximal contamination’ RAFC scenarios. In EC-E'RAFC, magma removed during eruption is a mixture of crystals and melt present in the same ratio as that in the magma body during the eruption, ignoring the effects of recharge and assimilation. That is, no fractionation between solids and melt is permitted during eruption and the contribution to the ratio of crystals and melts owing to recharge and assimilation is ignored. The latter constraint is not strictly correct but is a good approximation provided 
e is less than 0·4. This limit was empirically determined by comparing the solid to melt ratio within the magma body between cases with = 1 and finite and with zero and = 0.
To uniquely define an E'RAFC event, an integral energy balance is invoked to provide a connection between the mass and thermal properties of all sub-systems, given a set of initial conditions and a chosen equilibration temperature (Teq). The integral equation allows determination of the mass of wall rock, 
, which thermally equilibrates with the magma body sub-system during the ERAFC event. Once has been determined, path-dependent parameters, such as trace element and isotope characteristics of melt and solids (cumulates and enclaves) are determined by solution of a set of differential equations. These differential equations express conservation of energy, mass, species and isotope balance as a function of melt temperature for 
. There can be a positive or negative correlation between the extent of anatexis and the addition of recharge magma depending upon the initial temperature and thermodynamic properties of recharge melt relative to Tm at the time of replenishment. The removal of heat as a result of eruption always decreases the heat available for partial melting of country rock.
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DETAILED DESCRIPTION OF EC-E'RAFC MODEL
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TOPABSTRACTINTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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The thermodynamic description of the composite system during E'RA
FC evolution (Fig. 2) illustrates the progression of states along the geochemical EC-E'RAFC path. The initial state (Fig. 2a) is a composite system composed of four sub-systems: country rock of mass at mean initial temperature 
, a pristine batch of chemically homogeneous isothermal melt of mass Mo and temperature 
, a compositionally distinct (but homogeneous) recharge melt reservoir of mass 
and temperature 
and a ‘virtual’ eruptive reservoir (initially empty). The trace element and isotopic compositions of country rock, pristine magma, and recharge magma are specified. The country rock–magma body boundary is diathermal and semi-permeable, allowing exchange of enthalpy and mass fraction of anatectic melt generated by partial melting in the country rock. Recharge melt is added during the E'RAFC event according to an ab initio (user-defined) prescription Mr = Mr(Tm), where Tm is the temperature of host melt. Eruptive magma is similarly removed during E'RAFC according to an ab initio (user-defined) prescription Me = Me(Tm). Figure 2b, c and d shows successive steps along the evolutionary path as Tm varies from , the initial temperature, to the final or equilibration temperature, Teq, specified by the user (Fig. 2d). Although time is not considered in the thermodynamic model (indeed, that is precisely why the model is of general utility), Tm serves as the progress variable in the differential equations defining the geochemical path (see Edwards & Russell, 1998; Hawkesworth et al., 2000). Tm monotonically falls from the initial value, , to the final or equilibration temperature, Teq, and points in the same direction as the ‘arrow of time’. At Teq (Fig. 2d) the composite system has reached thermal equilibrium and entropy production associated with heat transfer vanishes within the composite system. The profoundness of the thermal interaction between wall rock and the magma body is measured by the value chosen for Teq.
The thermal consequences of addition of recharge melt to the magma body depend on the initial temperature () and the melt fraction–temperature relationship [herein termed the melt productivity, fr(Tm)] of recharge melt as well as the host melt temperature Tm at the time of recharge addition. The initial temperature of recharge melt, , is set equal to its liquidus temperature, Tr,l. If is less than Tm, thermal energy required to warm recharge melt comes from sensible heat stored in the host melt and latent heat associated with in situ cumulate formation; once local thermal equilibrium is reached, the remaining recharge melt homogenizes with host magma. In contrast, if exceeds Tm, the fraction 1 – fr(Tm) of recharge melt ‘instantaneously’ solidifies upon injection into cooler host magma. The solid ‘quench’ products of this thermal interaction are termed enclaves in the E'RAFC formulation. Such enclaves are commonly found in granitic (sensu lato) plutons and their volcanic equivalents and are generally interpreted as the chilled remnants of mafic magma injected into cooler magma (see, e.g. Furman & Spera, 1985; Didier & Barbarin, 1991; Wiebe & Snyder, 1993; Wiebe, 1994; Wiebe & Adams, 1997; Snyder & Tait, 1998; Waight et al., 2001, and references therein).
During the course of E'RAFC evolution (Fig. 2b and c), magma is removed by eruption from the magma body sub-system according to the a priori prescription Me(Tm). Magma removed by eruption is in thermal equilibrium with host melt and consists of a mixture of melt and solids in the same proportion as would be present in the pristine magma body at Tm; this is an approximation of the melt–crystal state of a body that has undergone assimilation and recharge. EC-E'RAFC evolution is complete (Fig. 2d) when the temperature of the wall-rock restite (that part of country rock that remains solid) is equal to the melt temperature, Tm, which, in turn, is equal to Teq. That is, at the completion of the E'RAFC event, Ta = Tm = Teq. In cases where all the country rock involved in the ERAFC interaction melts, the equilibration condition is simply Tm = Teq. When Teq is reached, the sub-systems include: (1) mass Mm of host melt of homogeneous composition at temperature Teq; (2) mass Mct of cumulates of variable (but known) composition formed by fractional crystallization; (3) mass Men of enclaves also of variable (but known) composition formed by closed-system fractional crystallization of recharge melt; (4) mass 
of residual wall rock (restite) plus anatectic melt trapped in the country rock reservoir; (5) mass 
of eruptive magma of known composition and temperature.
In the EC-E'RAFC model, host melt and wall rock have unique trace element (
, 
) and isotopic (
, 
, 
, 
, where m is radiogenic and m is oxygen) initial compositions. Recharge melt, with distinct trace element (
) and isotopic (
, 
) compositions at initial temperature (), is added to host magma according to user-defined recharge mass function Mr(Tm). The recharge mass added to the magma body during E'RAFC evolution is , where 
. The extrusive mass is likewise specified a priori by definition of the eruptive mass function, Me(Tm), and its composition is that of the host magma (melt plus average crystals) at the time of eruption.
Results from both experimental phase equilibria and thermodynamic modeling (e.g. MELTS, Ghiorso, 1997) are used to constrain the thermodynamics of melting. These constraints include melt productivity functions for wall rock, pristine magma and recharge magma and the phase assemblages as a function of temperature for estimation of bulk partition coefficients. The solution of the conservation equations provides the mass of heated wall rock (), the amount of anatectic melt (
) generated in wall rock and added to host magma (
), the mass of melt in the chamber (Mm), and the mass of cumulates (Mct) and enclaves (Men) as a function of Tm along the path 
. In addition to masses, trace element concentrations and radiogenic and oxygen isotope ratios in melt, cumulates and enclaves at each temperature along the path are determined.
In the following sections, mathematical details of the EC-E'RAFC algorithm are presented. The first section details the parameterization of the mass eruptive function, Me(Tm). The non-linear melt productivity functions fa(T), fm(T) and fr(T) and the mass recharge function Mr(Tm) have been presented elsewhere (Spera & Bohrson, 2002) and are used but not derived here. The derivation of the EC-E'RAFC algorithm is given in two parts. The first is an integral energy conservation statement that provides a set of ordered pairs (Teq, ) for particular thermodynamic properties, initial conditions, mass eruption history, Me(Tm), and total recharge mass, . A particular choice of Teq is made based upon geologic knowledge of the magmatic system under study. Once Teq has been chosen, the second part of the calculation gives the solution to the path-dependent differential equations defining the chemical and thermal evolution of the melt, wall rock, cumulates and enclaves along the E'RAFC temperature trajectory .
Mathematical details of the E'RAFC algorithm
Magma eruptive function
In EC-E'RAFC, magma is removed by eruption from the magma body during the approach to thermal equilibrium. In all cases, erupted material includes melt and crystals in the proportion they would exist within the pristine magma reservoir at temperature Tm. Two forms are useful as ‘end-member’ models for the eruption of magma during RAFC evolution. The simplest eruptive function is the linear one,
 | (1) |
Me,i is removed from the host magma body. There may be an arbitrary number (Ne) of eruptive episodes; the current code allows for up to 20 such episodes in any single E'RAFC simulation. The logistic form gives the cumulative mass of eruptive magma for multiple eruption pulses:
 | (2) |
Me,i is the mass of the ith pulse. By use of (2), specification of eruptive episodes of arbitrary mass at specific temperatures in the interval is accomplished. An illustrative example is given in Fig. 3 for an E'RAFC event made up of three eruptive pulses.
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and Teq = 950°C. The sum of the three pulses is 
.Differentiation of (2) gives a differential equation for the variation of the eruptive mass with melt temperature Tm:
 | (3) |
FC path.
Integral enthalpy balance
The integral enthalpy balance provides a fundamental constraint on the geochemical path followed by sub-systems during E'RAFC evolution. The balance defines a relation among four quantities characterizing E'RAFC evolution: the mass of country rock, , the total mass of recharge added, , the eruptive mass function, Me(Tm), and the equilibration temperature, Teq. The integral enthalpy balance incorporates heating and partial melting of country rock, addition of recharge melt of arbitrary composition and temperature, eruption of magma, magma cooling, and heat exchange and solidification associated with enclaves and cumulates. The required thermodynamic parameters include the melt productivity functions for country rock, pristine host magma and recharge magma (fa, fm and fr, respectively), enthalpy of crystallization of host and recharge magmas (hm and hr), the fusion enthalpy of country rock (ha), and the average isobaric specific heat capacity of all compositions (Cp,a, Cp,m and Cp,r). The expression derived in the Appendix is
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 | (5a) |
 | (5b) |
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), the integrals [A] and [B] are integrated numerically. Once Me(Tm), , melt productivity functions and thermodynamic properties of all materials are specified, a unique relationship exists between Teq and .
Calculation of thermal and geochemical paths
To compute the thermal, trace element and isotopic paths of anatectic melt, cumulates, enclaves and homogenized host melt within all sub-systems, a set of coupled ordinary nonlinear differential equations expressing conservation of enthalpy, mass, species and isotopic composition is solved. The output of this computation includes the temperature trajectory of the country rock restite (Ta), the mass of partially molten assimilant produced (), the mass of partially molten assimilant incorporated in host melt (), the mass of homogenized melt (Mm) within the magma body, the mass of cumulates formed by fractional crystallization (Mct), the mass of enclaves formed by ‘quenching’ a portion of recharge magma (Men), the mass of erupted magma Me as well as the concentration of trace elements in both host melt and erupted melt (Cm), and solids [cumulates (Cct) and possible enclaves (Cen)]. In addition, the isotopic compositions (radiogenic, m and oxygen, m) in melt and crystalline solids are also determined. The independent variable is the melt temperature Tm, and the calculation ends when Teq, set a priori, is reached. We emphasize that Teq must be specified in order to compute the path in temperature–composition space because , the mass of country rock involved in E'RAFC, is a function of Teq given in equation (4).
The model consists of 4 + t + i + s differential equations where t is the number of trace elements, i the number of radiogenic isotopic ratios and s the number of stable isotopes considered in the calculation. There are no formal limitations on t, s or i except the patience of the geochemist in dealing with the tyranny of numbers. Pressure is accommodated in EC-E'RAFC approximately by adjustment of liquidi and the solidus temperatures (Tl,a, Tl,m, Tl,r and Ts) and melt properties, when such variations can be deduced and are critical to the analysis.
The first two differential equations are known a priori and are the differential forms for the mass recharge function and the mass eruptive function. The form of the recharge function has been given by Spera & Bohrson (2002) and is not repeated here. The eruptive mass function is either linear [equation (1)] or logistic [equation (2)]. The third differential equation expresses conservation of energy along the path as country rock heats up, partially melts and thermally equilibrates with host melt that simultaneously undergoes recharge and eruption. Energy conservation leads to a differential expression for the country rock temperature, Ta, as a function of melt temperature, Tm. Anatectic melt of mass generated by country rock partial fusion is assumed to rapidly mix with host melt. The mass of anatectic melt remaining trapped in country rock is 
. Along the equilibration path, restite {the sum of crystalline country rock of mass 
, and trapped anatectic melt of mass } is not immediately brought into thermal equilibrium with magma. In contrast, assimilated anatectic melt of mass attains local thermal equilibrium with host melt at each point along the equilibration path . Incomplete extraction of anatectic melt from country rock is allowed for by specification of the extraction efficiency factor, . In EC-E'RAFC, is a parameter of the simulation (0 1) set a priori by the investigator based on available geological evidence. In cases where Teq is less than Tl,a, the thermal equilibrium condition is Ta = Tm = Teq. When Teq > Tl,a, the equilibrium condition is Tm = Teq, as no crystalline restite remains (i.e. 
).
All recharge melt initially enters the magma body at temperature 
, but is thermally equilibrated at the local Tm before mixing with host magma. The process of thermal equilibration depends on the melt temperature, Tm, at the time of replenishment. If 
, then because recharge magma is assumed to be initially intruded at its liquidus temperature (Tl,r), a fraction of the recharge melt crystallizes to form enclaves. The fraction of recharge melt that solidifies is 1 – fr(Tm) and the resulting increment of mass of enclaves is dMen = [1 – fr(Tm)]dMr. Enclaves are not in isotopic or trace element equilibrium with host melt; they are distinct from cumulates formed by fractional crystallization of host melt. That fraction of recharge melt that does not ‘quench’ is assumed to mix with host magma (at Tm) and deliver its excess heat, which then becomes available for anatexis of country rock. In this way, there is an intimate connection between recharge and host rock heating and possible contamination, depending on the value of . If, on the other hand, 
at the time of intrusion, heat is extracted from host magma and added to the recharge melt. This can give rise to a ‘wave’ of cumulate formation and puts limits on the extent of anatexis. Finally, the heat removed associated with the eruption of magma must be accounted for. The differential equation expressing conservation of energy that incorporates all of the above features gives the derivative of the country rock temperature Ta with respect to Tm along the EC-E'RAFC path to thermal equilibrium at Teq:
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is computed from equation (4). A derivation of (7) is presented in the Appendix.
The fourth constraint on the geochemical path is conservation of mass and explicitly provides an expression for the variation of the mass of liquid (melt) within the magma body as a function of Tm. The derivative of the mass of melt (Mm) in the host magma body with respect to magma temperature Tm is expressed in terms of the melt productivity functions and their derivatives and the mass recharge function and its derivative with respect to Tm. The mass of melt in the magma body along the path is related to the amount of melt initially present (Mo), the amount added by assimilation of anatectic melt () and by recharge (Mr, but allowing for enclave formation), the amount removed by cumulate formation (Mct) and, finally, the melt removed by eruption (Me). The expression has the differential form
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Conservation of species provides the basis for determining trace element abundance in melt as a function of Tm. Initial trace element concentrations in country rock, host melt and recharge melt are , and , respectively. We assume that partial melting of country rock is described by fractional melting so that the concentration of a trace element in the anatectic melt is given by
 | (9) |
, appendix I)]. Additionally, a distinct bulk melt–solid partition coefficient Dm, also dependent upon temperature, is defined to account for fractionation of the trace element between cumulate and melt. In the enclaves, the distribution of trace element takes place by closed-system fractional crystallization of the recharge melt. Chilling of the recharge melt by the cooler host magma precludes significant chemical mixing. The trace element bulk distribution coefficient that describes the fractionation of a trace element between recharge melt and its associated solid enclave is Dr, which may also be temperature dependent. With these expressions, the species balance expression for the variations of the concentration of a trace element in the host melt within the well-mixed magma body as a function of Tm is
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The species balance equation accounts for the formation of enclaves and cumulates, the introduction of anatectic melt derived by fractional fusion of country rock (assimilant), and both recharge and eruption of magma. The composition of the recharge magma is an initial condition whereas the composition of melt removed during eruption is identical to melt within the host magma body at the moment of eruption.
For an isotopic ratio in the host melt m, the differential equation is
 | (11) |
, and m represent the isotopic ratio [e.g. m = (87Sr/86Sr)melt] of anatectic melt (identical to country rock because we assume there is no fractionation of isotopes during partial melting), recharge melt and host melt at Tm, respectively. It should be noted that the epsilons refer to isotope ratios rather than the conventional definitions. Radiogenic in-growth and temperature-dependent isotopic fractionation are neglected in (11). Isotopic equilibrium is assumed to prevail between cumulates and melt; enclaves, on the other hand, are assumed to be in isotopic equilibrium with recharge melt of initial isotopic composition .
Finally, the differential equation expressing the oxygen isotope balance in host melt is
 | (12) |
). In cases where magma and country rock have nearly the same oxygen isotopic ratio, temperature effects may be important, and (12) should be modified to include temperature-dependent oxygen isotope fractionation. The ratios 
and 
represent respectively, the mass fraction ratios of oxygen in assimilant to melt and recharge melt to host melt before ERAFC processes. The mass fraction of oxygen in most natural compositions is about 47% and varies relatively little.
In addition to the primary variables computed by solution of the differential equations, other quantities may be calculated. For example, in a natural system, compositional data are sometimes available for the crystalline products (cumulates and enclaves), anatectic melt and eruptive products. The EC-E'RAFC solution links all parts of the composite system to one another. When modeling natural systems, it is important to consider the composition of cumulates, enclaves, and anatectic melt as well as the composition of the evolving host melt so as to obtain as robust a solution as possible. The mass, trace element and isotopic compositions (path average and instantaneous) of all solids (cumulates and enclaves) and of anatectic melt along the thermal equilibration path are part of the E'RAFC solution. Expressions for these quantities are given in the Appendix.
The set of 4 + t + i + s coupled ordinary differential equations representing E'RAFC evolution is posed as an initial value problem with Tm as the independent variable. This set of differential equations is subject to the following initial conditions: at 
, 
, 
, Mm = Mo, 
, 
, and 
for t trace element species, 
, 
, and 
for i isotope species and 
, 
, and 
for oxygen. Once cast into dimensionless form, the system of equations are numerically solved by a fourth-order Runge–Kutta method. The input and output are presented in a code programmed in Visual Basic. A copy of the EC-E'RAFC code is available at http://magma.geol.ucsb.edu/ and on the Journal of Petrology web site at http://www.petrology.oupjournals.org.
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APPLICATIONS OF EC-E'RAFC
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The utility, generality and robustness of the EC-E'RA
FC model is best discovered by systematic application to a wide range of magmatic systems. This work has just begun [e.g. see Fowler et al. (2004) for application to a portion of the British Tertiary Igneous Province]. Here we provide a few examples to illustrate application of EC-E'RAFC to situations where extraction of wall-rock partial melt may be variably efficient and where eruption and recharge interact non-linearly to alter the geochemical path of the sequence of melts along the approach to thermal equilibrium. These examples are not intended to be exhaustive, only illustrative.
Geologic evidence suggests that in some cases not all of the wall-rock melt generated during an ERAFC interaction is assimilated into the host magma body (e.g. Grove & Kinzler, 1986; see also James, 1981). For example, migmatites probably represent crustal sections that have undergone partial melting, but where melt has not been fully extracted (Johannes & Gupta, 1982). It is therefore important to examine the consequences of imperfect extraction (and hence incomplete addition) of anatectic melt to an evolving magma body. There is also abundant evidence that suggests recharge and eruption may be linked in open systems (Sparks & Sigurdsson, 1977; Blake, 1981). The ability to accommodate distinct Me(Tm) vs Mr(Tm) paths in EC-E'RAFC allows us to investigate the chemical consequences of the relationships between these two processes. Below, we begin by illustrating selected geochemical traits of a lower-crustal magma body undergoing E'RAFC; we follow this with specific comparisons of systems that are variably affected by assimilation, recharge and eruption.
EC-E'RAFC models of intrusion of mafic magma into lower crust of mafic–intermediate composition
Figure 4a–d illustrates selected geochemical characteristics of a basaltic magma intruded into lower crust of mafic–intermediate composition initially at 600°C. The geochemical paths revealed in the figure simulate evolution from 1320°C (
) to an equilibration temperature of 1090°C (Teq). Four cases are shown: AFC where = 1; AFC where = 0·5; E'RAFC where Me and Mr each are 0·3, are continuous, and = 1; E'RAFC where Me and Mr each are 0·3 and are continuous, and = 0·5. Modeling parameters are summarized in Table 2.
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Figure 4a shows the [Sr] vs 87Sr/86Sr trajectory for the four cases. The flat trends near
for the two AFC cases reflect heating up of wall rock to its solidus. Over approximately the same temperature range, the decrease in 87Sr/86Sr for the two cases involving recharge and eruption reflects addition of recharge magma that has a less radiogenic Sr isotope value. As Tm continues to fall in all four cases, the degree of heterogeneity in Sr isotope values over a small range of [Sr] is a consequence of wall-rock fractional melting of an element that is incompatible (wall-rock DSr = 0·05); the initial, low-degree melting of the wall rock releases high concentrations of Sr into the host magma, yielding distinct changes in 87Sr/86Sr as well as an increase in [Sr] in the host magma (despite Sr being compatible in magma; DSr = 1·5). The flattening of all trends as Tm becomes close to Teq is also an outcome of fractional melting of an incompatible element; most of the Sr has been stripped from the wall rock, so while mass exchange is still continuing (i.e. assimilation is continuing), little Sr is being added from the wall rock to the host magma. Thus, little change in the isotope ratio occurs, although for the cases involving recharge, there is a slight decrease in 87Sr/86Sr because of the addition of the less radiogenic recharge magma. An interesting observation regarding all of the [Sr] vs 87Sr/86Sr trends is the large degree of isotopic heterogeneity evident over a fairly restricted, relatively elevated range of Sr. 87Sr/86Sr varies from 
0·705 to 0·7105 over a range of 550 to 700 ppm Sr. In the absence of consideration of the influence that shallow-level processes might have on such trends, this type of compositional variation might incorrectly be attributed to mantle heterogeneity or the mixing of two isotopically distinct primary liquids.
The marked differences in Sr paths between cases where = 1·0 and = 0·5 illustrate the sensitivity of geochemical parameters to the efficiency of extraction of wall-rock partial melt. For these cases, full extraction yields 87Sr/86Sr of 0·7105, whereas 50% extraction yields lower ratios near 0·7082.
More subtle distinctions in isotopic ratios are evident through comparison of the AFC and E'RAFC cases. In the cases illustrated here, compared with the AFC cases, continuous eruption and recharge (Mr = Me = 0·3) yield slightly more radiogenic 87Sr/86Sr values. Compared with the AFC cases, the slightly more crustal signatures associated with the E'RAFC cases are at least partly a consequence of their higher (at Teq: AFC, = 1, 
; E'RAFC, = 1, 
; and AFC, = 0·5, 
; E'RAFC, = 0·5, 
), which reflect the additional energy provided by recharge magma. In these cases, the net effect of recharge and eruption is to increase the amount of energy available for heating and melting of wall rock. Although the total masses of material continuously added by recharge and subtracted by eruption are equal (0·3), the recharge magma has 
, whereas magma is removed at each Tm as the simulation runs from . Thus, for these cases, the amount of energy added by recharge exceeds the amount removed by eruption.
The trajectories for Nd are similar in form to those for Sr. The AFC trends show an initial flat trend, followed at lower Tm by marked changes in 143Nd/144Nd over a relatively restricted range of [Nd]. The trajectories terminate with relatively flat trends as Nd decreases. The E'RAFC are similar, although recharge at relatively high Tm yields an increase in 143Nd/144Nd as a result of the radiogenic nature of recharge magma. As Tm approaches Teq, [Nd] decreases despite it being incompatible in all sub-systems (DNd = 0·25, Table 2). These decreases are a consequence of the ‘dilution effect’ that occurs when most of an incompatible element has been stripped from the wall rock by fractional melting. At lower Tm, although assimilant is still being added, there is so little Nd being contributed to the host melt that its concentration in the host is actually diluted. By comparison of the = 1·0 vs = 0·5 cases, it is evident that the degree of dilution is less in the cases where incomplete extraction occurs.
Figure 4c illustrates Sr isotope–O isotope trajectories for the four cases. For the cases modeled, at higher Tm, Sr isotope signatures vary markedly whereas the variations in 18O are relatively restricted. At lower Tm, distinct changes in 18O are accompanied by modest variations in 87Sr/86Sr. An instructive observation regarding these trends is the concavity of the curves. Based on previous AFC models, such concavity has been linked to ‘source’ contamination rather than ‘crustal’ contamination (e.g. James, 1981; Taylor & Sheppard, 1986). This figure therefore underscores the complex coupling among energy, mass, and species, and also emphasizes the importance of accommodating the compositional changes experienced by wall rock as partial melting occurs. EC-E'RAFC results such as these further emphasize the necessity of assessing shallow-level processes prior to discussing mantle characteristics.
Figure 4d provides an example of E'RAFC trends in element–element space. In this instance, the behavior of an incompatible element (Th) is compared with that of a compatible element (Ni). For the AFC cases, at higher Tm, [Th] initially increases, consistent with its incompatible behavior in all sub-systems. [Ni] decreases because of its compatible nature. However, at higher Tm, [Th] decreases whereas [Ni] increases. The reversals in the trends of these elemental concentrations are tied to melting in the wall rock. Th, being incompatible, is stripped from the wall rock, and at lower Tm, little is being added (i.e. dilution effect). In contrast, for fractional melting of a compatible element, higher degrees of fractional melting yield higher concentrations of the element. Thus, [Ni] increases. The effects of recharge are seen in the E'RAFC cases, where Ni initially increases because of its higher concentration in the recharge magma. A critical aspect of note for these trends is that EC-E'RAFC does not necessarily yield monotonic trends in element–element space; thus, the magma parcel that has experienced the most E'RAFC is not necessarily the one with the highest Th or the lowest Ni. Such results emphasize the need to use a multifaceted approach when attempting to identify open-system magmas.
Chemical consequences of continuous eruption and recharge
There is abundant evidence to suggest that the dynamics of recharge and eruption may be linked (e.g. Sparks & Sigurdsson, 1977; Blake, 1981). To highlight the compositional consequences of this coupling, Fig. 5 summarizes cases in which combinations of recharge and eruptive masses are varied. Four cases are illustrated: E'RAFC where Me = 0·3 and Mr = 0·3; E'AFC where Me = 0·3 and Mr = 0·0; RAFC where Me = 0·0 (no eruption) and Mr = 0·3; AFC where Me = 0·0 and Mr = 0·0. For all cases, recharge and eruptive masses are linear functions of temperature (i.e. continuous recharge and/or eruption) and = 1·0. All other parameters are the same as those listed in Table 2.
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At Teq, Sr isotopes (Fig. 5a) vary, although the four cases cluster into two groups—the two characterized by eruption (87Sr/86Sr
0·7108), and the two lacking eruption (87Sr/86Sr 0·7102). Sr concentrations at Teq are similar among the four cases. Examination of Fig. 5b, 87Sr/86Sr vs 18O, shows the same groupings, with the cases involving eruption having slightly more crust-like O and Sr isotopes. These results are indicative of the complex consequences of the energetic and mass coupling of the sub-systems. A predictable consequence of an eruption is that less energy is available for transfer to country rock. Indeed, for the E'AFC case, the normalized mass of wall rock melted and assimilated into the host chamber () is the smallest, whereas that for the RAFC case is the largest (: E'AFC = 0·49, RAFC = 0·77, E'RAFC = 0·67, and AFC = 0·59). The relatively strong signature of crustal contamination in the E'AFC case is a function of the coupling between 
and Mm, which is also predictably the smallest of the four cases (Mm: E'AFC = 0·58, RAFC = 0·94, E'RAFC = 0·80, and AFC = 0·72). Thus, at Teq, although the mass of melt assimilated is the smallest, the total mass of melt is also the smallest. Together with the other parameters of the simulation, these system characteristics yield the most contaminated signatures. A key point to appreciate is that because of the coupling, predictions about how open systems will behave as they evolve are not straightforward. Rules of thumb that have guided assessment of the importance of open-system processes (such as a correlation between geochemical signature and amount of contamination) therefore need to be abandoned because of the inherent non-linearity implied when the energetics of melting is coupled to trace element conservation linked through the processes of partial melting, recharge, incomplete contamination ( < 1) and eruption.
Chemical consequences of episodic eruption and recharge
How sensitive is magma composition to the timing of recharge and eruption? Figure 6 summarizes results of simulations involving episodic recharge and eruption (Mr = 0·3 at 1225°C and Me = 0·3 at 1180°C; Mr = 0·3 at 1180°C and Me = 0·3 at 1225°C). The first case, in which recharge occurs at a higher Tm than eruption, is probably geologically more likely, but both cases are shown for purposes of illustration. Two additional cases are also portrayed for comparison: continuous Me = Mr = 0·3 (continuous recharge and eruption), and Me = Mr = 0·0 (no recharge or eruption). To better discuss the effects of episodic recharge and eruption, for these cases, was set equal to zero. This means that the evolving magma body is not contaminated by introduction of anatectic melts. That is, partial melts necessarily generated in the country rock remain there throughout the E'RFC evolution. Input parameters for these models are listed in Table 2; where appropriate, parameters are the same as those for the cases shown in Figs 4 and 5.
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The model in which eruption occurs at a higher Tm than recharge yields the most extreme 87Sr/86Sr (i.e. that most like recharge magma). This is because at relatively high Tm, 30% of the original normalized mass of the magma body is erupted (Me = 0·3), and thus, the mass of Sr in the chamber is reduced. When recharge occurs at a lower Tm, its Sr, which is less radiogenic than that in the pristine magma, has a proportionally greater effect on the host magma, thereby yielding a less radiogenic 87Sr/86Sr as Tm
Teq. For the case in which recharge occurs at a higher Tm than eruption, at the Tm of the recharge event, 87Sr/86Sr of the host magma decreases because recharge magma has a less radiogenic signature than pristine magma. Because the mass of the magma chamber at this temperature is larger than that for the previous case, the lever effect from recharge is not as large, and therefore the Sr isotope signature is higher. For the parameters of these cases, continuous recharge and eruption yield 87Sr/86Sr between those of the two episodic cases.
Systems that undergo episodic recharge and eruption can develop distinctly different geochemical characteristics; thus, the geochemical evolution (i.e. Tm–geochemical path) of these systems may be very different. Because Tm is the E'RAFC progress variable, it can be considered as a proxy for time, and thus, in natural systems in which there is geologic evidence for episodic recharge and eruption, it may be possible to hypothesize about the relative timing and magnitude of such events.
Finally, the evolution of systems characterized by episodes of recharge and eruption (± assimilation) may be distinct from those that undergo continuous recharge and eruption (± assimilation). Distinguishing between these scenarios geochemically relies not only upon high-quality geochemical data, but also on the ability to place samples in an evolutionary framework. For these reasons, the collection of relative and absolute geochronological information on igneous systems is of paramount importance.
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SUMMARY |
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TOPABSTRACTINTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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EC-E'RA
FC is a thermodynamic model that enforces energy conservation and total mass, trace element and isotope material balance in the composite system composed of magma body, country rock, replenishment reservoir and extrusive (eruptive) reservoir. EC-E'RAFC represents an extension of the earlier EC-AFC and EC-RAFC models and includes the effects of: (1) episodic or continuous magma removal by eruption; (2) variable addition of anatectic melt generated by wall-rock partial fusion. Results provide information on the trace element and isotopic composition of host melt, eruptive magma and crystalline products (cumulates and non-equilibrium enclaves) and the average wall-rock temperature along the equilibration path. Input parameters are based on knowledge of the magmatic system of interest as well as relevant thermodynamic properties such as the isobaric heat capacity, enthalpies of transition and bulk partition coefficients for trace elements. In this study we have not focused attention on the predicted compositions of cumulates and enclaves, although the EC-E'RAFC model makes predictions that can be tested, provided samples are available.
Several forward models provide insight into the theoretical behavior of E'RAFC magmatic systems. The geochemical consequences of imperfect extraction of wall-rock melt into the host magma body can be profound. A fruitful area of research involves linking geochemical signatures of melt and associated solids with field evidence of imperfect extraction from wall rock. In addition, theoretical work that characterizes how melt migrates may reveal systematics regarding how transport processes work in different magmatic environments. Simulations that illustrate the effects of different recharge–eruption–temperature paths provide convincing evidence that isotope and trace element signatures can be dramatically affected by E'RAFC processes. Hence, it is clear that the crustal-level geochemical evolution of a magma body must be evaluated from a mass, species and energy context before assignment of mantle characteristics can reliably be made. In addition, we emphasize that documenting magma chamber processes is inextricably linked to placing samples in evolutionary context. To develop meaningful models of how magmas evolve, collection of high-quality geochemical and geochronological data is required.
E'RAFC represents the culmination of a number of years of model development. Based on modeling results from several natural systems (Bohrson & Spera, 2001, 2003), the approach holds promise for enhancing our understanding of the behavior of open-system magmatic processes. In particular, the most detailed application to date, by Fowler et al. (2004) on rocks of the British Tertiary Igneous Complex, illustrates the potential the model has for fingerprinting mantle vs crustal processes. Further exploration of issues such as this, and application of E'RAFC to other natural datasets, will continue to build on contributions that M. J. O'Hara has made to understanding open-system magma chambers over the last 25 years.
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SUPPLEMENTARY DATA |
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Supplementary data for this paper are available at Journal of Petrology online.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONQUALITATIVE DESCRIPTION OF EC...DETAILED DESCRIPTION OF EC...APPLICATIONS OF EC-E'RA{chi}FCSUMMARYSUPPLEMENTARY DATAAPPENDIXREFERENCES |
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Derivation of integral energy balance
The starting point in the derivation is the differential form for the heat available from cooling and crystallization of magma (havail) and the heat absorbed by country rock for heating and partial fusion (habs). The expressions are from Spera & Bohrson (2002)
with addition of a term associated with the removal of magma from the geochemically evolving magma body. The expressions for the available and absorbed enthalpy are
 | (A1) |
 | (A2) |
and Teq using integration by parts and allowing for nonlinear melting and eruptive mass functions, fm(Tm) and Me(Tm), respectively. This gives the amount of heat available for heating cool country rock (by convention a negative quantity) by the cooling and crystallization of magma allowing for heat exchange owing to the processes of recharge and eruption. The result after algebraic simplification is
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and Teq to give the heat absorbed (by convention a positive quantity) by country rock of mass initially at temperature to raise its temperature to Teq and partially melt it to the extent fa(Teq). The result is
 | (A4) |
 | (A5) |
of as a function of Teq may be found by combining (A3) and (A4) and invoking energy conservation. The result is
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| (A7) |
 | (A8) |
is independent of , as the heat absorbed by country rock depends only on the initial and final temperature of country rock ( and Teq, respectively) and the extent of partial melting.
Derivation of path energy conservation equation
The energy conservation principle is used to compute the mean temperature of country rock, of mass , involved in E'RAFC evolution as a function of the host magma temperature, Tm. The quantity is determined, as outlined above, by the integral energy balance and is needed for solution of the differential equation relating Ta to Tm.
Differentiation of equation (A5) with respect to Tm and use of the chain rule gives
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Average and instantaneous compositions of solids along ERAFC path
Once the concentration of trace element in host melt is known, trace element concentrations in solids (cumulates and enclaves) may be determined. E'RAFC computed compositions provide a relative compositional chronology of cumulates and possible enclaves.
The instantaneous concentration of a trace element in enclave (at T = Tm) is
 | (A11) |
found by integration of (A11) along the ERAFC path is
 | (A12) |
 | (A13) |
is calculated from equation (10) in the text. The average trace element composition in cumulates formed along the path is
 | (A14) |
) or sink (if ) of enthalpy (heat). Enclaves form in the former case as a result of chilling of recharge melt intruded into cooler host magma. When , heat is also available from non-equilibrium crystallization of enclaves liberating specific latent heat hr. The mass of enclaves is given by the integral
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 | (A16) |
For partial melting in wall rock, we assume fractional melting occurs. The concentration of trace element in anatectic liquid at temperature Ta is
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| (A17) |
is
 | (A18) |
When the initial temperature of recharge melt exceeds that of host magma (i.e. ) the fraction [1 – fr(Tm)]dMr of recharge chills to form solid enclaves. The initial recharge melt temperature () is assumed equal to its liquidus temperature, Tl,r. The concentration of trace element available for mixing into host magma differs from that in pristine recharge melt because of depletion or enrichment of trace element as a result of enclave crystallization. This process is modeled as closed-system fractional crystallization according to
 | (A19) |
is the concentration of trace element in pristine recharge melt and Dr is the partition coefficient between enclave solid and recharge melt. All trace element equilibrium constants are taken as functions of temperature:
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 | (A20b) |
 | (A20c) |
Hj j
{a, m, r} represent the effective enthalpies of the reactions governing bulk partitioning of trace element between anatectic melt and country rock restite, host melt and cumulates, and pristine recharge melt and enclaves, respectively. The Hj values are ‘effective’ values in the sense that the dependence of Dj on phase assemblage is parameterized implicitly using the fictive temperature dependence of Dj. That is, ‘effective’ values Hj are chosen by consideration of phase equilibria relevant to the bulk compositions and equilibrium phase assemblages of {a, m and r} along the temperature trajectory . If this auxiliary information is not available or poorly known, constant bulk partition coefficients Dj may be used by setting Hj equal to zero. |
ACKNOWLEDGEMENTS |
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Support from the National Science Foundation (Sonia Esperanca) to W.A.B. (NSF EAR-0073883) and F.J.S. (NSF EAR 0073932) is gratefully acknowledged. F.J.S. acknowledges additional support from the US Department of Energy Geosciences Program (Nick Woodward) DE-FG03-01ER15210. Without the tireless efforts of programmer Mr. Guy Brown, EC-ERAFC would not be the sleek product we perceive it to be. The incisive reviews by Drs George Bergantz, Mark Ghiorso, Calvin Miller, and editor Marjorie Wilson greatly improved the form and content of this work and are gratefully acknowledged.
* Corresponding author: Department of Geological Sciences, University of California, Santa Barbara, CA 93106-9630, USA. Telephone: 805-893-4880. Fax: 805-893-2314. E-mail: spera{at}geol.ucsb.edu
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