Trace element fractionation during modal and nonmodal dynamic melting and open-system melting: a mathematical treatment

Abstract

Dynamic partial melting model has recently drawn considerable attention because this model may explain the fractionation of some strongly incompatible nuclides in the uranium decay series (McKenzie, 1985; Beattie, 1993) and may account for ultra-depleted melt inclusions in olivine grains from mid-ocean ridge basalts (Sobolev and Shimizu, 1993) and Icelandic picrites (Eggins, 1992). There are three subsystems for dynamic melting model: the residual melt, the extracted melt, and the residual solid. This paper systematically derives consistent equations for the residual melt, the extracted melt, and the residual solid in the context of both modal and nonmodal dynamic melting models. Previous available equations are also evaluated. The keys for the derivation or evaluation of equations for dynamic melting models are (1) the exact relationship between the melting rate and the melt extraction rate, (2) the exact relationship between the total melting degree and the fraction of extracted melt relative to the initial amount of source before melting, and (3) clear concepts of physical parameters. In addition, the equation for the residual melt during open-system (nonmodal dynamic) melting (Ozawa and Shimizu, 1995) with material influx in the melting region is corrected and an equation for the extracted melt in the context of open-system melting is proposed here. Since melting and melt extraction processes are often coupled with slab-derived material influx in the melting region in the arc environments, these two new equations for the residual melt and the extracted melt during open-system melting are very useful in modeling arc magmatism.