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.