『Abstract
　Positive correlation between Sr/Ca and Mg/Ca in cave dripwaters and speleothem is often attributed to ‘prior calcite precipitation’ (PCP), where exclusion of Mg and Sr from calcite precipitated upstream of the dripwater/speleothem site results in enrichment in the Mg/Ca and Sr/Ca ratios of residual effluent. Modelling predicts that PCP will produce a straight line correlation in graphs of ln(Sr/Ca) vs. ln(Mg/Ca) with a theoretical slope that is given by (KdSr - 1)/(KdMg - 1) (calculated to be 〜0.88±0.13) regardless of rock or absolute dripwater composition (Sinclair et al., submitted for publication). While this makes the correlation slope potentially diagnostic for PCP, Mg and Sr are also preferentially released from calcite during incongruent calcite dissolution (ICD), and it is possible that ICD produces a similar correlation between ln(Sr/Ca) and ln(Mg/Ca). To examine this possibility, mathematical formulations for two different mechanisms of ICD are presented. Model 1 simulates net dissolution as a dynamic kinetic process where forward and reverse reactions compete: dissolution is congruent but ions are fractionated by a simultaneous reverse reaction where new calcite forms with Sr/Ca and Mg/Ca ratios that are determined by the solution composition and KdSr and KdMg. Model 2 simulates leaching of Mg and Sr from a fresh calcite surface by tracking the dissolution of a thin layer where high ion mobility allows the calcite to maintain a homogeneous composition controlled by KdSr and KdMg. In both models, generalised kinetic equations are used in the derivations meaning that the results are not specific to a particular kinetic model.
　In both models of ICD, the mathematics predict that under conditions that may prevail during dissolution of limestone, the correlation between ln(Sr/Ca) and ln(Mg/Ca) is approximately linear with a slope given by (KdSr - 1)/(KdMg - 1). Thus, this property is not specific to just PCP. However, for both PCP and ICD, this slope is universal - expected to occur regardless of kinetics and the absolute composition of solution or host rock. A correlation slope of 〜0.88 in either karst waters or speleothem is therefore diagnostic for calcite-water interaction.

Keywords: Incongruent calcite dissolution; Speleothem; Cave drip water; Mathematical model; Magnesium; Strontium』

1. Introduction
2. Mathematical formulations
　2.1. Model 1 - dissolution/re-precipitation of calcite
　　2.1.1. Overview
　　2.1.2. Summary of equations
　2.2. Model 2 - ion leaching from fresh surfaces
　　2.2.1. Overview
　　2.2.2. Summary of equations
3. Model summary and discussion
　3.1. Model 1
　　3.1.1. Description of the model behaviour
　　3.1.2. Model predictions
　3.2. Model 2
　　3.2.1. Description of the model behaviour
　　3.2.2. Model predictions
4. Conclusions
Acknowledgements
Appendix A. Model 1 - proving the general case
Appendix B. Model 1 - deriving equations for a simple kinetic model
Appendix C. Model 2 - ion leaching from fresh surfaces
References