## 1 Introduction

[2] Thermal history calculations show that the Earth's solid inner core has been growing as the liquid outer core freezes from the bottom up [see *Nimmo* 2007 for a review]. The density difference between solid and liquid cores is too large to be explained solely by freezing, and the liquid is supposed to contain more light elements than the solid. Cosmochemical arguments favor oxygen, sulfur, and silicon as candidate light elements, and ab initio calculations of binary mixtures (Fe-O, Fe-S, Fe-Si) show that O partitions almost completely into the liquid on freezing, S partitions almost equally with a slightly higher proportion in the liquid, and Si partitions equally between the two phases to the accuracy of the calculations [*Alfè et al*., 2002]. This partitioning is crucial for powering the geodynamo in the outer core that gives rise to Earth's magnetic field; in this paper we show that it may also be crucial for the dynamics of the inner core.

[3] Seismic studies have shown the inner core to be both heterogeneous and anisotropic, with a pronounced east-west hemispheric structure [*Souriau*, 2007; *Irving and Deuss*, 2011]. There is also evidence of a different structure in the innermost 300 km of the inner core [*Ishii and Dziewoński*, 2002] and a ~100 km thick isotropic layer below the inner core boundary (ICB) [*Waszek and Deuss*, 2011]. To explain the observations, much work has focused on inner core convection, driven either thermally [*Jeanloz and Wenk*, 1988; *Buffett*, 2009] or in combination with compositional gradients [*Deguen and Cardin*, 2011; *Cottaar and Buffett*, 2012]. The hemispheric structure has been proposed to arise from a translational convective mode involving a uniform drift of the inner core material [*Monnereau et al*., 2010; *Alboussière et al*., 2010], while termination of convection at an early stage has been suggested as the cause of the innermost inner core and the outermost isotropic layer [*Deguen and Cardin*, 2009]. While the results are encouraging, the proposition of inner core convection rests on one highly uncertain hypothesis: the inner core is assumed to be unstably stratified.

[4] The aforementioned models all assume a thermal origin for the unstable inner core stratification, implying that the inner core temperature gradient exceeds the adiabatic gradient at the relevant pressure-temperature conditions. Previous studies have found that this may be the case at present and was more likely in the past [*Buffett*, 2009; *Deguen and Cardin*, 2009, 2011], but the results depend critically on the thermal conductivity of inner core material. Recent work found the thermal conductivity at the base of the outer core to be over three times larger than previous estimates [*Pozzo et al*., 2012] and this value must be further increased when applied to the solid to account for the lower concentration of light elements in the inner core. Such high values for the thermal conductivity make it highly unlikely that thermal convection can arise in the inner core [*Buffett*, 2012].

[5] When compositional effects have been incorporated into inner core convection models, they have been treated as neutral or stabilizing [*Deguen and Cardin*, 2011; *Cottaar and Buffett*, 2012]. This reasoning is based on the assumption that the partition coefficients (the solid-liquid concentration ratio) do not change with time. The concentration of light elements in the liquid increases as the outer core shrinks [*Stacey*, 1995]; a constant partition coefficient therefore implies that the concentration in the solid rises with that in the liquid causing a stabilizing density gradient. If this were the case, then inner core convection would not be viable because both thermal and compositional contributions to the inner core density gradient would be stabilizing.

[6] In this paper, we show that the partition coefficients for O, S, and Si are not constant but actually decrease with time. This important result is shown to arise as a direct consequence of the chemical potentials in the liquid being lower than in the solid and the drop in temperature at the ICB as the inner core grows due to the decrease in melting temperature with falling pressure. The decrease in partition coefficients with time has the opposite effect to the rising concentration in the liquid. A destabilizing gradient results if the partitioning effect is the strongest. The theory and equations are given in section 2; they are solved with appropriate numerical values in section 3 in the form of a Rayleigh number as a function of inner core radius. Conclusions and discussion are presented in section 4.