Mass balance equations for open magmatic systems: Trace element behavior and its application to open system melting in the upper mantle

A general mass conservation equation for either stable or radioactive chemical components in an open magmatic system is formulated. The equation is applied to trace elements by assuming melt‐mineral equilibrium defined by a partition coefficient. A derived analytic expression has very wide applicability, and most of the mass conservation equations in the literature for stable trace elements are obtained by choosing appropriate values for the controlling model parameters, such as material influx or separation rate and melt fraction in the system. The general equation is also modified to facilitate an application to multistage problems. It is demonstrated that the general open system mass conservation equation for melting processes expressed without melt fraction is mathematically identical to that for one‐dimensional steady state two‐phase flow with sink and source terms. This one‐dimensional steady state model is applied to depleted abyssal peridotites to constrain their melting processes in an ascending mantle beneath ridges. The rare earth element (REE) patterns of clinopyroxene are fitted by optimizing melting parameters, and the results indicate a continuous influx of light REE (LREE)‐enriched melt up to 3–11% in degree of melting, which amounts in total to less than 1% of the initial peridotite mass. The open system melting equation for trace elements is also applied to peridotites from the Hayachine‐Miyamori ophiolite, northeastern Japan. The REE contents in clinopyroxene and the modal abundances in four peridotite groups are fitted all together to estimate melting reaction stoichiometry and REE contents in the influxing material. The estimated REE pattern is characterized by a strong LREE enrichment, which is comparable to that of melt in equilibrium with hornblende in the most refractory harzburgite from the ophiolite.