Electrical and Thermal Conductivity of Earth's Iron-enriched Basal Magma Ocean

TL;DR

The paper investigates whether iron-enriched basal magma oceans (BMO) could power a silicate dynamo in Earth's early history by computing electrical and thermal transport properties under BMO conditions. Using ab initio molecular dynamics and the Kubo–Greenwood approach, it derives σ_el and k_el as functions of Fe content X_Fe, pressure, and temperature, including spin-state populations to obtain equilibrium conductivities σ_el^eq and k_el^eq. Coupling these transport properties to a nonlinear thermal evolution model shows that iron enrichment elevates the magnetic Reynolds number (Rm) above the dynamo threshold of 40 for substantially longer, up to ~3.3 Gyr, compared with pyrolitic cases. The results suggest a viable silicate dynamo in the early Earth and imply that iron-enriched BMOs could be common dynamo hosts on other rocky planets lacking a core dynamo.

Abstract

The Earth's earliest magnetic field may have originated in a basal magma ocean, a layer of silicate melt surround the core that could have persisted for billions of years. Recent studies show that the electrical conductivity of liquid with a bulk silicate Earth composition exceeds 10000 S/m at basal magma ocean conditions, potentially surprising the threshold for dynamo activity. Over most of its history however, the basal magma ocean is more enriched in iron than the bulk silicate Earth, due to iron's incompatibility in the mineral assemblages of the lower mantle. Using ab-initio molecular dynamics calculations, we examine how iron content affects the silicate dynamo hypothesis. We investigate how the electrical conductivity of silicate liquid changes with iron enrichment, at pressures and temperatures relevant for Earth's basal magma ocean. We also compute the electronic contribution to the thermal conductivity , to evaluate convective instability of basal magma oceans. Finally, we apply our results to model the thermal and magnetic evolution of Earth's basal magma ocean over time.

Figures (5)

• Figure 1: Electronic contribution to electrical conductivity $\sigma_{\text{el}}$ of silicate liquid versus iron fraction $X_{\text{Fe}}$ = Fe/(Fe+Mg). Colors indicate different temperature and pressure conditions representative of Earth's basal magma ocean. Circles denote our conductivity results at equilibrium high spin fraction at three iron fractions, while dashed lines represent quadratic fits to these data points.
• Figure 2: Electronic density of states in the high-spin iron bearing silicate liquid at 6000 K and 100$\pm$10 GPa, shown for an iron-rich (top) and iron-poor (bottom) composition. Contributions from s (blue), p (green) and d (orange) states are shown separately, with up-spin and down-spin plotted as positive and negative, respectively. The black vertical line indicates the Fermi energy, $E_{F}$.
• Figure 3: Electronic contribution to thermal conductivity $k_{\text{el}}$ of silicate liquid versus iron fraction $X_{\text{Fe}}$ = Fe/(Fe+Mg). As in figure \ref{['fig:sigma_eq']}, colors indicate temperatures and pressures, while dashed lines represent quadratic fits our conductivity results at the equilibrium spin state (circles).
• Figure 4: Top: Time evolution of the magnetic Reynolds number, $R_{m}$, as calculated by our thermal evolution model. The red line represents a constant pyrolite composition ($X_{\text{Fe}}$ = 0.12), while the black line accounts for the effect of iron enrichment on electrical conductivity. The dotted line marks the threshold for a self-sustaining dynamo ($R_{\text{m}}>$ 40). Bottom: Parameter space illustrating the effect of basal magma ocean thickness and Fe-Mg fraction $X_{\text{Fe}}$ on the magnetic Reynolds number. The white contour ($R_{m} = 40$) represents the minimum BMO thickness required to sustain a dynamo for a given iron fraction. The dotted white line traces the thickness-$X_{\text{Fe}}$ relationship predicted by our thermal evolution model. The top axis indicates the corresponding liquidus temperature, $T_{\text{liq}}$, defined by $X_{\text{Fe}}$ (equation \ref{['evolve2']}). $R_{m}$ is calculated using a mixing length velocity scaling.
• Figure 5: Regime diagram illustrating the effect of electrical conductivity $\sigma$ and thermal conductivity $k$ on dynamo production. The boundary between the "dynamo" and "no dynamo" regimes is defined by the magnetic Reynold number $R_{m} = \mu_{0} v l \sigma = 40$, while no thermal dynamo will occur if $Q_{\text{cond}} = 4 \pi r^{2} k \nabla T_{\text{ad} } > Q_{\text{total}}$. The solid lines correspond to the Wiedemann-Franz law relation, $k_{\text{el}}/\sigma_{\text{el}} = \lambda_{0}T$, at 4000 K (white) and 6000 K (black). Top: basal magma ocean with a thickness of 300 km, total outward heat flux of $Q_{\text{total}} = 20$ TW, and adiabatic temperature gradient $\nabla T_{\text{ad}} = 0.6$ K/km stixrude2009thermodynamics. Circles are electronic conductivity values at 4000 K (white) and 6000 K (black) from figures \ref{['fig:sigma_eq']} and \ref{['fig:kel']}, with larger symbols indicating higher iron fraction $X_{\text{Fe}}$. Orange stars represent the time evolution of conductivity predicted by our thermal evolution model. Bottom: liquid core with thickness of 2260 km, $Q_{\text{total}} = 15$ TW, and $\nabla T_{\text{ad}} = 1$ K/km . Circles represent calculations of $\sigma_{\text{el}}$ and $k_{\text{el}}$ for Fe, Fe$_{7}$O, Fe$_{3}$O, Fe$_{7}$Si, and Fe$_{3}$Si liquid from reference de2012electrical. These calculations are also along 4000 K (white) and 6000 K (black) isotherms, with larger symbols again corresponding to higher iron content (or lower fraction of light elements).