Open-System Magma Chamber Evolution: an Energy-constrained Geochemical Model Incorporating the Effects of Concurrent Eruption, Recharge, Variable Assimilation and Fractional Crystallization (EC-E′RAχFC) - SPERA, FRANK J.; BOHRSON, WENDY A. (Journal of petrology, 2004)
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, Aχ. This model, called EC-E′RAχFC, 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′RAχFC 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 (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(M_{a}^{o}\) \end{document}) and the masses of anatectic melt generated (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(M_{a}^{{*}}\) \end{document}) and assimilated (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(χM_{a}^{{*}}\) \end{document}). In addition, t trace element concentrations and i + s isotopic ratios in melt and eruptive magma (Cm, εm, δm), cumulates (Cct, εm, δm), enclaves (Cen, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(ε_{r}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(δ_{r}^{o}\) \end{document}) and anatectic melt (Ca, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(ε_{a}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(δ_{a}^{o}\) \end{document}) 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 (χ) from country rock to host magma, the initial temperature and composition of pristine host melt (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(T_{m}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(C_{m}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(ε_{m}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(δ_{m}^{o}\) \end{document}), recharge melt (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(T_{r}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(C_{r}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(ε_{r}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(δ_{r}^{o}\) \end{document}) and wall rock (\batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(T_{a}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(C_{a}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(ε_{a}^{o}\) \end{document}, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(δ_{a}^{o}\) \end{document}), distribution coefficients (Dm, Dr, Da) and their temperature dependences (ΔHm, ΔHr, ΔHa), latent heats of transition (melting or crystallization) for wall rock (Δha), pristine magma (Δhm) and recharge magma (Δhr) 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′RAχFC 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.