Modal melting

Modal melting. Solid phases melt in the same modal proportion in which they occur in the initial solid, so that the modal composition of the residual solid remains unchanged throughout the process. 

During modal melting, the melting proportion is the same as the mineral source proportion and thus x is constant throughout the melting process x =x q. When K is also constant (K = ), then the bulk partition... 

Modal melting normally does not happen. Due to preferential melting of some minerals in the source rock, x must change. For example, a mantle rock may have 10% cpx, but the cpx may contribute 25% of the melt as a result of preferential melting of cpx, resulting the decrease of as a function of F during melting. 

Note that for modal melting, melting proportion is the same as the initial mineral proportion in the source (p =Xq). In this case, Eq. (1.10) reduces to... 

Another proposed measure is the change in the modal melting temperature, i.e. the Amode (Brownell, 1983 Sarich et al. 1989). This may be useful under certain circumstances, however as a general measure it is probably not as accurate as ATj. One problem with this measure is it requires extrapolation based on certain assumptions. Also, unlike the HAP method, the TEACL method produces a cumulative melting curve which would require statistical manipulation in order to extract a mode. 

Equation (1.16) is the famous non-modal batch melting equation from Shaw (1970). 

Table 1.2. Summary of variations of mineral proportions and bulk partition coefficients during modal, eutectic and incongruent melting.

Modal Batch Melting Modal BM w/ constant Modal BM w/ linear K Modal BM w/ linear K and linear q ... 

Zou, H.B., 1998. Trace element fractionation during modal and nonmodal dynamic melting and open-system melting A mathematical treatment. Geochim. Cosmochim. Acta, 62 1937-1945. 

The previous four chapters deal with the fractionation of stable trace elements during partial melting. In this chapter, we study the behaviors of radioactive uranium decay series during partial melting. Since quantitative models for uranium-series disequilibria need to include additional parameters in decay constants and are thus more complicated, for simplicity, we assume that the partition coefficients remain constant during partial melting. Thus, we only present modal dynamic melting. 

Problem. Demonstrate low-degree (F] )/high-degree (F ) concentration ratio (Q) of the same element decreases with increasing partition coefficients (Dq) during modal batch melting. 

In the context of non-modal (eutectic) batch melting, the variation in the concentration of a trace element in a whole-rock residue,, is given by Shaw (1970) as... 

The linear regression method, or slope-intercept method (Zou, 1997), for whole-rock residues can only be applied to modal batch melting. For elements a (highly incompatible) and b (not-so-highly incompatible), their concentrations in the residue during modal batch melting, and, respectively, are given by... 

Alternatively, a linear relationship between l/C and l/Qj, for modal batch melting can also be obtained from Eqs. (7.51) and (7.52) as follows ... 

The basic equations describing the variation in concentration of a trace element in cpx of non-modal batch melting model can be obtained as follows (Johnson et al., 1990) ... 

In contrast to the case for whole rocks, the linear regression method for residual clinopyroxenes can be applied to non-modal batch melting. The basic equation for the variation of a trace element in cpx during non-modal batch melting has been given in Eq. (7.56). We can obtain the following linear relationship from Eq. (7.56) (Zou, 1997) ... 

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