To develop the model, we solve for the change in concentration of a trace element in the melt:
\(\frac{dC_{M}}{dM_{C}}=\ \frac{C_{M}}{M_{M}}\left(1-D\right)\)(4)
Here, CM is the concentration of a particular trace element in the melt, MC is the mass of crystallized solid, MM is the mass of melt, and D is the bulk partition coefficient (see Table 1 for symbols and definitions). Eq. (1) describes the change in concentration dCM in the melt caused by an increment of crystallization, dMC. Similarly we solve for the change in concentration of a trace element in the solid, which, in the case where mass loss only occurs in the melt phase, evolves as:
\(\frac{dC_{C,b}}{dM_{C}}=\ \frac{DC_{M}-C_{C,b}}{M_{C}}\) (5)
CC,b refers to the concentration in the bulk solid, rather than that which is instantaneously crystallized. Finally, the change in concentration in the lost melt phase is calculated as:
\(\frac{dC_{L}}{dM_{C}}=\ \frac{\left(C_{M}-C_{L}\right)}{M_{L}}\frac{dM_{L}}{dM_{C}}\)(6)
where CL is the concentration in the lost melt phase and ML is the mass of melt lost. This model only accounts for melt loss; however, many SMB samples are enriched in incompatible trace elements, suggesting they may have accumulated melt (Fig. 4 and 5). When mass loss only occurs by removing mass from the solid, the change in concentration of a particular element in the solid instead evolves as:
\(\frac{dC_{C,b}}{dM_{C}}=\ \frac{DC_{M}}{M_{C}}-\ \frac{C_{C,b}}{M_{C}}\left[\frac{dM_{\text{CL}}}{dM_{C}}+1\right]\)(7)
where MCL is the mass of lost solid.
The group of ordinary differential equations above includes terms that account for the change in concentration associated with a small amount of crystallization (dMC), as well as melt (dML) and crystal (dMCL) loss. Here, melt loss is relevant to crystal accumulation whereas crystal loss is relevant to melt accumulation; however, the box model assumes no physical process by which these occur. The full derivation is included in the supplements. Crucially, the equations are derived with respect to MC, which is the mass of crystallized solids. This means that if the functional forms of the mass of melt (ML) and of lost crystals (MCL) as function of mass crystallized are determined, then MM can be solved for and the equations are completely resolved. We argue that in the limit that D >> 1, the exact functional form of ML and MCL is of little importance (supplementary Fig. 1). The expressions for ML and MCL used in trace element calculations in this manuscript depend on MLF and MCLF, which are the final amount of melt or crystal loss as crystallization terminates. These parameters are solved for each SMB sample assuming a parental magma composition tabulated in Table 2.
Table 2. Composition of BC04 (assumed parental magma composition). Oxides are reported as wt % and elements as parts per million (ppm). Total Fe expressed as FeO.