<figure><img src="image_233.png" /></figure>

amplitudes of mean heat flux and heterogeneity in their dynamo models based on their analytical piles model as functions of pile height. In addition, they set the Rayleigh number based on the mean heat flux from their analytical piles model.

Figure 12 shows the non-dimensional reversal frequency as functions of pile height and mean outer boundary heat flux for the rotated $Y_2^0$ (Fig. 11) dynamos. The error bars correspond to $\sqrt{N}/\tau_d$ (where N is the reversal frequency and $\tau_d$ is the dipole free decay time), consistent with a Poisson distribution (Lhuillier et al.

2013). The reversal frequency increases approximately linearly beyond onset. There is a fairly small window separating the pile height that would produce magnetic superchrons and the pile height that would produce hyper-reversing dynamos. A relatively moderate increase of ~35% in CMB heat flux corresponds to a transition from dynamo models exhibiting superchrons to those in a hyper-reversing state. Amit and Olson (2015) concluded that pile growth may stimulate polarity reversals, whereas pile collapse may inhibit reversals.

## Localized mantle heating and the paleo dynamo of Mars

Here we review the papers by Sreenivasan and Jellinek (2012), Kuang et al. (2014), and Monteux et al. (2015). Sreenivasan and Jellinek (2012) simulated the localized CMB anomaly induced by the mantle plume that produced Tharsis by

$$q = q_{\min} \exp \left( - \left[ (\theta - \theta_p)^2 / \delta_{\theta}^2 + (\phi - \phi_p)^2 / \delta_{\phi}^2 \right] \right) \quad (17)$$

where $\theta$ and $\phi$ are latitude and longitude, $\theta_p$ and $\phi_p$ are the corresponding coordinates of the plume center, and $\delta_{\theta}^2$ and $\delta_{\phi}^2$ are the corresponding exponential decay coefficients, set to be 15° at the equator. The localized CMB plume-induced heat flux heterogeneity is visualized in the inlet of Fig. 13a.

The dynamo models of Sreenivasan and Jellinek (2012) are strongly supercritical (Table 1), requiring large heterogeneity amplitudes to terminate the dynamo (Fig. 13a). Increasing heterogeneity amplitude attenuates the field and eventually leads to the dynamo failure (Fig. 13b). In addition, it is easier to kill the dynamo if the mantle plume is centered at low-latitude (Fig. 13a).

The underlying mechanism for the demise of the dynamo due to the localized CMB heating is visualized in Fig. 14. The boundary heterogeneity weakens the axial flow and reverses its direction. As a result, the kinetic helicity also weakens significantly. The weakening of helicity reduces the field generation efficiency (Moffatt 1978) and leads to the demise of the dynamo (Sreenivasan and Jellinek 2012).

Kuang et al. (2014) also studied the possibility of Martian dynamo cessation due to localized CMB heating, but by a giant impact. The impact-induced thermal heterogeneity is modeled by shock wave heating emanating from the impact site (Pierazzo et al. 1997; Watters et al. 2009). The temperature rapidly decays from the impact center. In contrast to Sreenivasan and Jellinek (2012), the dynamo models of Kuang et al. (2014) are subcritical with moderate heterogeneity amplitudes of $\epsilon_l \le 0.3$ (see Table 1). These subcritical dynamos were found to be hemispheric (Kuang et al. 2008), in contrast to other studies that argued that under subcritical conditions helicity generation is
<figure><img src="image_234.png" /><figcaption>**Fig. 10** *Top*: Geomagnetic reversal frequency (yellow and *barcode below*) based on the Geomagnetic Polarity Time Scale (GPTS) database of Gradstein et al. (2012) vs. reversal frequency in dynamo models with time-dependent CMB heat flux (*black pluses*) and fixed present-day tomographic pattern (*red crosses*). The Cretaceous Normal Superchron (CNS), Kiaman Reversed Superchron (KRS), and Moyero Normal Superchron (MNS) are *grey shaded*. *Bottom*: Polarity bias, i.e., the difference between time spent in normal and reversed polarities normalized by their sum. From Olson et al. (2013)</figcaption></figure>

more efficient when the field is dipole-dominated (Hori and Wicht 2013; Sreenivasan and Jones 2011).

Kuang et al. (2014) also found that a giant impact at low-latitude and with a large heterogeneity amplitude is more likely to terminate the dynamo. Figure 15 shows that when the impacts where placed at latitude 60°N and higher the dynamos survived. The dynamos failed if the CMB heat flux heterogeneity amplitude was larger than a critical value. This critical value decreases with increasing latitude of the impact site. The dynamo failure occurs when the helicity is reduced in the hemisphere where most of the field is expected to be induced.

Localized mantle shock heating by a giant impact could have also altered the morphology of the Martian paleo

dynamo. Monteux et al. (2015) uniquely determined the CMB heat flux lateral variations as well as the reduction in the mean CMB heat flux by the size and geographic location of the impactor. Figure 16 shows the CMB heat flux patterns resulting from an impactor of 800 km radius falling on the north pole (left) or the equator (right) of Mars' surface. Because it is thought that there is no inner core in Mars (Breuer et al. 2010; Schubert and Spohn 1990), purely thermal convection was modeled, with a volumetric homogeneous heat source $\epsilon$ that compensated for the loss of heat through the outer boundary (Amit et al. 2011a; Dietrich and Wicht 2013). A small inner core corresponding to $r_i/r_o = 0.2$ was retained to avoid numerical instabilities, but the inner boundary was set to be convectively passive (Table 1). Both non-reversing and reversing dynamos were examined.

Figure 17 shows the time-average RMS radial field at the CMB and the time-average field intensity at the surface of Mars for a dynamo model with an impactor of radius 800 km falling on the north geographic pole (Fig. 16 left). The impactor yields an hemispheric field similar to those obtained with synthetic $Y_1^0$ CMB heat flux patterns (Amit et al. 2011a; Dietrich and Wicht 2013; Stanley et al. 2008). Monteux et al. (2015) found that a polar impactor leads to a north-south hemispheric magnetic dichotomy that is stronger than an east-west dichotomy created by an equatorial impactor (in agreement with Amit et al. 2011a, Dietrich and Wicht 2013).

Monteux et al. (2015) found that an impactor radius of about 1000 km could have recovered the magnetic dichotomy observed in the Martian crustal field.

<figure><img src="image_235.png" /><figcaption>**Fig. 11** CMB heat flux pattern in the dual superplume configuration. The mean heat flux is 100 mW/m² and the peak-to-peak variation is 60 mW/m². From Amit and Olson (2015)</figcaption></figure>
<figure><img src="image_236.png" /></figure>

However, strong enough magnetic dichotomies were obtained only for frequently reversing dynamos (as in Dietrich and Wicht 2013). Therefore, such dynamos may account for the observed field dichotomy only if very rapid post-impact magma cooling took place (Lebrun et al. 2013).

## Conclusions

### Discussion

The study of mantle control on planetary dynamos may be classified into three categories. In the first category, numerical dynamos with homogeneous boundary conditions have already reproduced successfully the most fundamental observed geodynamo properties, in particular the dipole dominance of the field (Glatzmaier and Roberts 1997) and its reversibility (Glatzmaier and Roberts 1995). Parametric studies of numerical dynamos show that increasing convection vigor (which is controlled by the mean CMB heat flux) causes a transition from dipole dominated non-reversing to multipolar reversing dynamos (Aubert et al. 2009; Christensen and Aubert 2006; Kutzner and Christensen 2002). The very different magnetic field geometries and amplitudes observed for the various planets and moons (Stevenson 1983) may represent different locations of their dynamos in parameter space with respect to this transition (Olson and Christensen 2006). In the second category, heterogeneous but simplified (i.e., not alternative) boundary conditions may help explain more refined dynamo properties, in particular for the Earth where detailed observations are available. In a third category, numerical dynamos with imposed alternative CMB heat flux can be considered as the next step to approach the recovery of more refined observations using more realistic CMB conditions.

Persistent morphological features of the present-day geodynamo were explained using alternative CMB heat flux models (Amit and Choblet 2009, 2012; Amit et al. in press). The modifications of the CMB heat flux patterns due to post-perovskite and thermal ridges are mild, and therefore their impact on the dynamo properties are also mild. Thus, in order to detect the differences between the reference dynamo models and those modified by the post-perovskite and thermal ridges effects (Table 2), long-term time-averaging is required (Amit and Choblet 2009, 2012). In contrast, the CMB heat flux pattern inferred from probabilistic tomography is dramatically distinctive from its reference (Amit et al. in press), and therefore provides a genuinely new insight into the possible control of the mantle heterogeneity on the geodynamo.

Dynamo models with imposed CMB heat flux inferred from both thermal ridges and probabilistic tomography were invoked to explain the presence of intense geomagnetic flux patches at low latitude (Amit and Choblet 2012; Amit et al. in press). In the latter models, enhanced low-latitude convective and magnetic activity is found, in particular during some occasional bursts during which the models exceed the observed relative low-latitude geomagnetic flux (Amit et al. in press). These robust low-latitude geomagnetic features (Jackson 2003) are more difficult to reproduce in dynamo models than high-latitude flux patches (Christensen et al. 1998), possibly due to flux dispersion by upwelling (Wicht et al. 2009) that reconnects with the downwellings near the tangent cylinder in the dynamo's meridional circulation (Aubert 2005). Intense low-latitude geomagnetic flux patches, which are indeed easier to recover with alternative CMB heat flux, have so far not been included explicitly as a morphological constraint for Earth-like dynamo models (Christensen et al. 2010).
<figure><img src="image_237.png" /><figcaption>**Fig. 13 a** Critical heat flux heterogeneity amplitudes required for dynamo failure vs. plume latitude for a mantle plume centered at 15°S (see CMB heat flux pattern in inlet). The three curves are based on three definitions of the heterogeneity amplitude: *f* (blue, Eq. (12)), *H*<sub>CMB</sub>/*H*<sub>B</sub> (solid black), and *H*<sub>plume</sub>/*H*<sub>B</sub> (dashed black). **b** Timeseries of normalized magnetic energy for a mantle plume centered at 45°S with *f* = 0 (blue), *f* = 4 (red), *f* = 10 (black), *f* = 25 (magenta), and *f* = 29 (green). From Sreenivasan and Jellinek (2012)</figcaption></figure>

Due to the very low viscosity and very large scale
of planetary outer cores, dynamo models operate in
a parametric regime very far from realistic conditions
(e.g., Christensen and Wicht 2007; Glatzmaier 2002). The
situation is more severe if long-term time-averaging is
required to detect the impact of alternative or any het-
erogeneous mantle control, because long simulations are
needed for adequate statistical convergence of the time-
average dynamo properties. An especially extreme case
concerns the study of reversal frequency which requires
very long simulations to obtain significant statistics of rare

events. Indeed the Ekman and magnetic Prandtl numbers
in the studies by Olson et al. (2013) and Amit and Olson
(2015) are very large (Table 1); the Ekman number is more
than 10 orders of magnitude larger than its geophysi-
cal estimate (e.g., Olson 2007b). Amit and Olson (2015)
explored some simulations with a lower Ekman num-
ber. They found qualitative agreement with their larger
Ekman number results, but computational limitations
prevented them from providing a confident quantitative
analysis, in this case of the reversal frequency vs. pile
height.
<figure><img src="image_238.png" /><figcaption>**Fig. 14** Zonal profiles of non-dimensional axial velocity **a** and non-dimensional axial kinetic helicity **b** in dynamo models without (**left**) and with (**right**) a plume. The minimum and maximum axial velocity values are [−40.4, 34.4] (**left**) and [−19.3, 18.3] (**right**); the minimum and maximum axial kinetic helicity values are [−174.6, 137.8] (**left**) and [−23.6, 24.8] (**right**). From Sreenivasan and Jellinek (2012)</figcaption></figure>

<figure><img src="image_239.png" /><figcaption>**Fig. 15** Survivability of dynamos as a function of CMB heat flux heterogeneity amplitude $\epsilon_i$ (radial distance) and the co-latitude of the impact $\theta_i$ (angle from vertical axis). From Kuang et al. (2014)</figcaption></figure>

The variability in the paleomagnetic reversal frequency was explained by alternative time-dependent CMB heat flux models (Amit and Olson 2015; Olson et al. 2013). The larger temporal variability in reversal frequency in the dynamo models with time-dependent CMB heat flux compared to those with a fixed tomographic pattern, in particular the presence of superchrons in the former, supports a time-dependent mantle convection structure over the past ~500 Myr (Olson et al. 2013). It also shows that core evolution by its own cannot account for the large variability in the paleomagnetic reversal frequency. The

HF1 model (Zhang and Zhong 2011) exhibits temporal variations in the mean CMB heat flux, its pattern and the amplitude of lateral heterogeneity (Fig. 9 and Table 1), all of which may affect reversal frequency (see Olson and Amit 2014, and references therein).

According to the HF1 model, the CMB heat flux pattern evolved from degree-1 dominance during Pangea to degree-2 at present. This transition is reflected in the time-average magnetic field of the dynamo models with prescribed heat flux pattern inferred from HF1. Olson et al. (2013) showed that at present, the dipole is very axial, whereas with increasing age the dipole axis substantially deviated from the geographic pole. The symmetry of the early time-average magnetic field highly correlates with the symmetry of the CMB heat flux heterogeneity, exhibiting magnetic field concentrations aligned with the axis of the large CMB heat flux.

Dziewonski et al. (2010) proposed that the CMB heat flux is quasi-stationary with two stable states that may each persist for hundreds of Myr. When the plates are organized in one supercontinent at Earth's surface, the CMB heat flux may be characterized by a rotated $Y_1^0$ pattern; when two continents are present, the CMB heat flux is expressed by a rotated $Y_2^0$ pattern. Amit and Olson (2015) showed that in both scenarios increasing pile height increases the mean CMB heat flux, the amplitude of its heterogeneity, and the resulting reversal frequency. For a given pile height, the rotated $Y_1^0$ pattern yields larger mean heat flux, heterogeneity amplitude and reversal frequency than the rotated $Y_2^0$ pattern (Amit and Olson
<figure><img src="image_240.png" /><figcaption>**Fig. 16** Non-dimensional CMB heat flux resulting from an impactor of radius 800 km falling on the north pole (**left**) or the equator (**right**). The same pattern was used for different $q^*$ values (see Table 1), hence the absence of a color bar. Red/blue denotes positive/negative heat flux anomalies, respectively. From Monteux et al. (2015)</figcaption></figure>

2015), suggesting higher sensitivity of the paleomagnetic reversal frequency to changing mantle conditions prior to the KRS.

The dependence of the reversal frequency on the heterogeneity amplitude is not trivial. In particular, the rotated $Y_2^0$ case contains polar cooling that according to most studies may suppress reversals (Glatzmaier et al. 1999; Kutzner and Christensen 2004). Polar cooling results in axial-dipole dominance due to flux concentration at high-latitudes (Amit et al. 2010b) and therefore indeed suppresses reversals (a state termed geographic control by Olson and Amit 2014). However, when the dynamo is far from the onset of reversals, Olson and Amit (2014) found that the opposite occurs and polar cooling increases reversal frequency by producing more turbulent conditions (which they termed inertial control). In the

piles model of Amit and Olson (2015), the effects of variable mean CMB heat flux and heterogeneity amplitude are non-separable. However, if indeed the rotated $Y_2^0$ pattern works in unison with the mean CMB heat flux to trigger reversals, it is suggestive of inertial rather than geographic control (Olson and Amit 2014).

The demise of the Martian dynamo was explained by a localized CMB heating (Kuang et al. 2014; Sreenivasan and Jellinek 2012). Sreenivasan and Jellinek (2012) imposed plume-induced thermal anomalies with large heterogeneity amplitudes on highly supercritical numerical dynamos, whereas Kuang et al. (2014) imposed impact-induced thermal anomalies with moderate heterogeneity amplitudes on subcritical dynamos. Under supercritical conditions, the CMB anomaly causes the dynamo to decay relatively slow—the field drops by one order of

<figure><img src="image_241.png" /><figcaption>**Fig. 17** Time-average magnetic field properties on the CMB (**left**) and on the surface of Mars (**right**) in a dynamo model with imposed outer boundary heat flux due to a polar impactor as in Fig. 16. From Monteux et al. (2015)</figcaption></figure>
magnitude in ~1.7 magnetic diffusion times (Sreenivasan and Jellinek 2012), whereas under subcritical conditions the termination is more sudden—the field decays ~4 times faster (Kuang et al. 2014). The large heterogeneity amplitudes in the dynamo models of Sreenivasan and Jellinek (2012) imply that below the mantle plume the heat is flowing unrealistically into the core. Another problem is that under vigorous convection the dynamo is expected to restart once the mantle anomaly vanishes. Note that reducing dynamo efficiency by heterogeneous CMB heat flux is not trivial; another pattern may have the opposite effect, as was shown for $Y_2^2$ heterogeneity which enhances helical core flow and hence improves dynamo efficiency (Sreenivasan 2009).

In both cases (Kuang et al. 2014; Sreenivasan and Jellinek 2012) it was found that low-latitude localized heating is favorable for terminating the dynamo, despite the distinctive role of convection in these two types of dynamo models. Sreenivasan and Jellinek (2012) argued that the critical latitude for plume location in order to kill the dynamo depends on the lowest latitude where axial columns driven by rapid rotation (Busse 1970) impinge the CMB (see Fig. 14a). Overall, these results may pose upper bounds on the amount of possible planet reorientation in the history of the Martian paleodynamo (Kuang et al. 2014; Sreenivasan and Jellinek 2012). The possibility of a major true polar wander (TPW) event on Mars is under debate. Significant TPW has been advocated for Early Noachian from studies of the crustal geoid associated to dynamical models for the formation of the crustal dichotomy (Roberts and Zhong 2007). TPW was also proposed for the subsequent 4 Gyr from studies of paleomagnetic poles (Hood et al. 2005) and paleoshorelines (Perron et al. 2007) or from theoretical models (Matsuyama et al. 2006). However, to our best knowledge, observed geological evidence for such processes (e.g., Grimm and Solomon 1986) are still lacking. Furthermore, models of lithospheric shell rotation associated with the apparent migration of Tharsis (e.g., Srámek and Zhong 2012) could lead to tectonic observations identical to that caused by TPW, although the two underlying physical processes are very different.

Some of the results obtained by Kuang et al. (2014) seem somewhat counter-intuitive, in particular the field morphology (see their Fig. 3). Their homogeneous CMB heat flux subcritical dynamo model indeed produces a hemispheric magnetic field (as previously reported by Kuang et al. 2008), with a peak at a specific latitude (~30°S). However, the field may reverse and this peak can migrate to different latitudes with time. It is therefore not clear why would the peak magnetic field of the heterogeneous dynamos almost coincide with that of the homogeneous case. It is also not intuitive that the field becomes spatially less heterogeneous with increasing

heterogeneity amplitude. In addition, there is no suggestion for a depleted field at the location of the impact, as may be expected. Overall, it seems that the effects of mantle control on subcritical dynamos are distinctive from those in supercritical dynamos.

Finally, Monteux et al. (2015) imposed a localized impact-induced CMB heat flux pattern on numerical dynamos to explain the observed hemispheric dichotomy in the Martian crustal field (Acuña et al. 1998; Langlais et al. 2004). The amplitude of the hemispheric magnetic dichotomy is mostly controlled by the horizontal Rayleigh number which represents the vigor of the convection driven by the lateral variations of the CMB heat flux. Monteux et al. 2015 showed that, for a given horizontal Rayleigh number, an impact-induced CMB heat flux heterogeneity is more efficient than a synthetic degree-1 heterogeneity in generating strong hemispheric magnetic dichotomies. However, even with the more efficient impact-induced CMB heat flux pattern, several important conditions should be met in order to accept the impact-induced hemispheric dynamo scenario as the origin of the crustal field dichotomy, in particular the need for a very rapid crustal cooling rate (Dietrich and Wicht 2013). Dynamo models that recover the observed magnetic hemisphericity have a short average chron time of about 18 kyr (Monteux et al. 2015). However, in the case of molten material induced by a giant impact a major part of the atmosphere can be eroded and crustal formation may occur within only 1000 years (Lebrun et al. 2013; Solomatov 2007), possibly enabling the recording of the magnetic hemisphericity in the crust.

## Future prospects

Because the CMB heat flux is an input to the dynamo models, understanding the dynamics of the lower mantle is a fundamental key to improve core dynamics models. In addition, progress in dynamo modeling is obviously relevant for all core-related problems, including the influence of heterogeneous (alternative or not) boundary conditions. Improved observations, in particular higher quality planetary magnetic field models, may provide tighter constraints on the dynamo models. Here we list some future prospects concerning alternative CMB heat flux and its influence on planetary dynamos. We also mention alternative scenarios that may also explain the non-trivial dynamo-related observations.

Knowledge of planetary magnetic fields will continue to grow in the coming years thanks to several new satellite missions that are currently flying or will soon be launched. Geomagnetic field and secular variation models will benefit from the successful Swarm ESA mission (Olsen et al. 2013). Three identical and magnetically dedicated satellites were launched simultaneously during the end of 2013 at two different altitudes to monitor with unprecedented
quality the Earth's magnetic field and its secular variation. Measurements of the NASA MESSENGER mission around Mercury revealed a weak, dipole-dominated, and hemispheric magnetic field (Anderson et al. 2012). However, these measurements have not yet been fully exploited, and it is anticipated that alternative modeling techniques (Oliveira et al. 2015) and/or the very low-altitude measurements at the end of the mission will bring new insights on the field morphology of Mercury. The ESA/JAXA BepiColombo mission that will be launched in 2017 will complete the geographical coverage of low altitudes of the planet and will lead to the first global view of the magnetic field of Mercury (Benkhoff et al. 2010). The description of the past Martian magnetic field and its interpretation in terms of past dynamo processes will be improved by the recently inserted-into-orbit MAVEN mission. These measurements will be completed by the upcoming InSight mission, which will study the internal structure of Mars and bring constraints on the size of the liquid core. The CASSINI mission to Saturn (Dougherty et al. 2004) is extended until 2016, and the forthcoming Juno (around Jupiter) and later JUICE (around Jupiter moons) missions (Grasset et al. 2013) will also bring new measurements around the giant gas planets and their moons. These new planetary magnetic field models will provide more reliable observational constraints on the processes generating the fields in the interior of the planets and moons.

The identification of the post-perovskite phase transition (Murakami et al. 2004) provided better constraints on the Earth's deepest mantle. Seismic observations with improved tomographic images as well as sophisticated analyses of seismic waves were obtained. A better description of lower mantle materials with laboratory experiments reaching the appropriate high pressure range as well as time-consuming ab initio quantum chemistry computations became possible. As can be expected, the emerging picture bears a much greater complexity which includes the probable role of compositional variations and possible contributions of mantle melting and spin transitions of iron. While the last decade opened new questions such as the complex effect of composition on the post-perovskite phase transition, the precise nature of the melting curve and the points of dissent that remain between laboratory experiments and computational mineralogy, the establishment over the coming decade of an integrative framework (Shim and Lay 2014) would offer a much clearer interpretation of seismic observations and allow to better extract the purely thermal contribution from tomographic models.

Progress in understanding the dynamics of the Earth mantle in the last hundreds of Myr ultimately requires a better description of plate motions in the framework of mantle convection. Promising research paths include

conceptual developments such as the understanding of lithospheric damage (Bercovici and Ricard 2014) as well as an improved treatment of plate-like behavior in spherical simulations of mantle dynamics (Bello et al. 2014; Rolf et al. 2012, 2014). In the meantime, improved reconstructions of plate velocities (Seton et al. 2012) can be prescribed to mantle convection models to investigate the role of specific ingredients such as mantle rheology, phase transitions and the nature of a possible dense basal layer. A more thorough investigation of mantle convection models than the pioneering attempt of Zhang and Zhong (2011) may establish more robust CMB heat flux evolution models than HF1.

One direct consequence of the results of the reversal frequency studies is the possibility to invert the record of paleomagnetic reversal frequency (Gradstein et al. 2012) for time-dependent core energetics. If the relation between pile height and CMB heat flux (mean and heterogeneity amplitude) is indeed approximately linear (Amit and Olson 2015; Olson and Amit 2014), the observed reversal frequency may be transformed into a relative CMB heat flux. This may provide a glimpse into the history of core convection.

As for the distant past of the Earth, prospects for other planetary objects mostly rely on modeling progress. Specific planetary scenarios that involve large impacts will most probably remain attractive in the next decade, concerning the ancient Martian magnetic field as well as other planets. Better descriptions of such events than the simplified approaches mentioned above can be simulated (Marinova et al. 2008, 2011). In terms of CMB heat flux, the geometry of impact heating with both a mantle contribution and a possible core contribution (Roberts and Arkani-Hamed 2014) could be described by such numerical tools. As mentioned in Monteux et al. (2015), the presence of impact-induced molten material is key in recording the impact-influenced magnetic field, and the interplay between the cooling time of this material and the transient dynamo properties will ultimately dictate the nature of remnant magnetism. In this regard, impact simulations could also supply patterns and volumes of ejecta deposits on the planetary surface. Describing the dynamics of the bulk molten material could involve on longer timescale phenomena such as isostatic adjustment of the retained melt region and lateral spreading as a gravity current (Reese et al. 2011).

Applications to other planets (apart from the Earth and Mars) may also be considered by invoking alternative CMB heat flux models. A $Y_2$ CMB heat flux pattern combined with internal heating may trigger an unstable odd convective mode which may explain the hemispheric field of Mercury (Cao et al. 2014; Wicht and Heyner 2014) as observed by MESSENGER (Anderson et al. 2012). While their dynamical scenario is convincing, the justification
for a $Y_2^0$ thermal heterogeneity at the mantle of Mercury is somewhat vague. Examination of gravity and topographic heterogeneities in Mercury may provide a more realistic (alternative) model for the CMB heat flux which can be invoked to recover MESSENGER's magnetic observations.

Due to computational limitations, dynamo models operate in a parametric regime very far from that of planetary core conditions (Glatzmaier 2002). This is especially restrictive for long simulations, as is the case when the long-term impact of an heterogeneous mantle is investigated. One of the most challenging problems in terms of computer power is that of modeling strongly time-variable magnetic reversal frequency, because adequate statistics require extremely long simulations. For example, the Ekman numbers used by Olson et al. (2013) and Amit and Olson (2015) are about four orders of magnitudes off the lowest values that were reported for numerical dynamos (Sakuraba and Roberts 2009). Improved numerical techniques which will allow running simulations with more challenging parameters, in particular lower Ekman and magnetic Prandtl numbers, are therefore of great interest for studies of mantle control on planetary dynamos. More efficient, massively parallelized dynamo codes are currently under development (http://geodynamics.org/cig/working-groups/geodynamo/).

More accurate incorporation of physical phenomena in dynamo models will also improve their predictions. In particular, correct implementation of thermochemical convection is necessary. Currently, most studies use the co-density formulation (Braginsky and Roberts 1995). Apart from neglecting double-diffusive convection, this approach fails to accurately prescribe the inner boundary condition. Some models have already incorporated thermal and chemical convection with appropriate boundary conditions in numerical dynamos (Manglik et al. 2010; Takahashi 2014; Trümper et al. 2012). Tracer methods which are commonly used in mantle convection simulations (e.g., Tackley and King 2003) may be applied to chemical convection in dynamo codes for more accurate numerical schemes. Other physical complications such as stable stratification in some regions of the outer core (Gubbins and Davies 2013) and compressibility may also be incorporated into dynamo models.

Observed dynamo properties that require boundary heterogeneity may alternatively be explained by lateral anomalies at the inner core boundary. Olson and Deguen (2012) showed that the dipole eccentricity in archeomagnetic field models can be reproduced by dynamo simulations with heterogeneous $Y_1^1$ inner boundary buoyancy flux pattern, corresponding to the east-west seismic anomalies at the upper inner core (Tanaka and Hamaguchi 1997). Aubert et al. (2013) showed that the localization of the westward drift of the geomagnetic field at the

Atlantic hemisphere is controlled by the same heterogeneous inner boundary buoyancy flux. In these studies, the inner boundary heterogeneity amplitude was set to be significantly larger than that of the outer boundary, therefore the role of the CMB heterogeneity becomes secondary. It would be interesting to explore dynamo models with comparable amplitude heterogeneities on both boundaries to investigate competing effects.

Progress in understanding mantle dynamics will lead to better description of boundary conditions for the dynamo and thus improved knowledge of core dynamics. On the other hand, magnetic observations may be used to constrain mantle convection scenarios. Overall, a multidisciplinary approach with interactive exchange of information among the different envelopes may advance the study of planetary deep interiors.

**Abbreviations**

CMB: core-mantle boundary; LLSVPs: large low shear-wave velocity provinces; ULVZs: ultra low velocity zones; KRS: Kiaman reversed superchron; CNS: cretaceous normal superchron; ESA: European Space Agency; NASA: National Aeronautics and Space Administration; MESSENGER: MERCURY surface: Space EnVironment: GEochemistry: and ranging; JAXA: Japan Aerospace eXploration Agency; MAVEN: Mars atmosphere and volatile evolution; InSight: Interior exploration using seismic investigations: Geodesy and Heat Transport; JUICE: JUpiter ICy moon explorer.

**Competing interests**

The authors declare that they have no competing interests.

**Authors' contributions**

The idea to this paper was in fact conceived following an invitation of HA by Takashi Nakagawa to give a talk on the topic of this paper in SEDI 2014. HA and GC wrote the first draft. GC, FD, and JM designed the illustrative Fig. 1. All authors discussed and improved the draft. All authors read and approved the final manuscript.

**Acknowledgements**

We thank Takashi Nakagawa, Binod Sreenivasan, and Weijia Kuang for providing material from their papers and for helpful discussions. H. Amit, G. Choblet, J. Monteux, and B. Langlais were partly supported by ANR project MARMITE (ANR-13-BS05-0012). J. Monteux was partly funded by ANR project Oxydeep (ANR-13-BS06-0008). F. Deschamps was supported by Academia Sinica grant 102-CDA-M02. We are grateful to James Roberts and Johannes Wicht for detailed reviews that significantly improved this review paper.

**Author details**

¹CNRS UMR 6112, Université de Nantes, Laboratoire de Planétologie et de Géodynamique, 2 rue de la Houssinière, Nantes, F-44000, France. ²Department of Earth and Planetary Sciences, Johns Hopkins University, 21218 Baltimore, MD, USA. ³Laboratoire Magmas et Volcans, Université Blaise Pascal - CNRS - IRD, OPGC, 5 rue Kessler, 63038 Clermont Ferrand, France. ⁴Institute of Earth Sciences, Academia Sinica, 128 Academia Road Sec. 2, Nangang, Taipei 11529, Taiwan.

Received: 10 March 2015 Accepted: 18 August 2015

Published online: 09 September 2015

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# 5 Couplage manteau-noyau et évolution long terme des dynamos planétaires

Une magnétisation rémanente et des champs magnétiques actifs ont été détectés pour plusieurs corps planétaires telluriques du Système Solaire (Terre, Mercure, Lune, Mars), ce qui suggère la présence de dynamos actives aux premiers stades de la formation de la planète et de durées de vie variables. Le champ magnétique de ces objets est la conséquence de mouvements convectifs vigoureux du fer dans la partie liquide de leur noyau (Jones, 2015). Cependant, la barrière chimique avec le manteau silicaté empêche la chaleur d'être transférée au manteau environnant par des mouvements advectifs. Les transferts de chaleur se font donc par conduction au travers des couches limites thermiques situées de part et d'autre de la frontière manteau-noyau. Le flux de chaleur conductif varie selon $k\Delta T/\delta$ avec $k$ la conductivité du matériau, $\Delta T$ la différence de température de part et d'autre de la couche limite thermique et $\delta$ l'épaisseur de la couche limite thermique. Lorsque la convection est très vigoureuse comme c'est le cas dans le noyau métallique liquide et dans les manteaux primitifs, les couches limites sont très fines et le flux de chaleur conductif est très important. Par conséquent, les conductivités du noyau métallique et du manteau silicaté, la vigueur de la convection et la quantité d'énergie stockée dans le noyau exercent un contrôle primordial sur le flux de chaleur à la CMB. Ces paramètres contrôlent à la fois l'état thermique du noyau, le bilan thermique transféré dans le manteau au cours du temps et la possibilité de générer une dynamo.

## 5.1 Influence des conductivités thermiques : cas de Mercure

La mission MESSENGER a révélé que le champ magnétique de Mercure pourrait fonctionner depuis 3.7-3.9 Ga. La faible intensité du champ magnétique de Mercure a longtemps déconcerté la communauté scientifique planétaire. Diverses explications ont été proposées mais il n'y a pas de consensus sur la façon dont une dynamo intrinsèque avec une énergie aussi faible a pu exister sur Mercure pendant une période prolongée. Alors que le magnétisme intrinsèque suggère l'existence d'une dynamo active dans le noyau de Mercure, le mécanisme responsable du maintien de la dynamo pendant une période prolongée reste inconnu.

Parmi les facteurs qui contrôlent la possibilité de générer des dynamos internes, la dynamique du manteau silicaté entourant le noyau et ses propriétés thermiques associées sont crucialles. Le manteau régit l'évacuation de la chaleur du noyau et, par conséquent, la probabilité d'une dynamo thermique précoce. La capacité du manteau a évacuer de la chaleur est gouvernée par son nombre de Rayleigh :

$$Ra = \frac{\alpha \rho g \Delta T H^3}{\eta \kappa}, \quad (2)$$

avec $\rho$ sa densité, $\alpha$ son coefficient de dilatation thermique, $g$ la gravité, $\Delta T$ la différence
de temperature entre le haut et la bas du manteau, *H* l'épaisseur du manteau, *η* sa viscosité et *κ* sa conductivité. Dans le cas où l'épaisseur *H* du manteau d'une planète est importante (i.e. manteau associé à des nombres de Rayleigh potentiellement élevés), la chaleur du noyau est efficacement évacuée par la convection du manteau sus-jacent et des dynamos thermiques précoces sont favorisées. À l'inverse, les planètes dotées d'un manteau mince (associé à des nombres de Rayleigh potentiellement plus faibles) peuvent évacuer leur chaleur interne uniquement par diffusion, ce qui rend plus difficile la mise en place de dynamos thermiques précoces. Dans le Système Solaire, Mercure est un exemple potentiel d'un tel régime. Si la faible épaisseur de son manteau par rapport au rayon de la planète était inhérente à sa position proche du Soleil (Charlier and Namur, 2019), un tel contexte magnétique pourrait être omniprésent parmi les objets telluriques formés à proximité de leur étoile.

<figure><img src="image_242.png" /></figure>

FIGURE 17 – Flux de chaleur en fonction du rayon planétaire pour un manteau planétaire entièrement solide (gauche) ou partiellement fondu (droite). Les lignes noires et grises représentent le flux de chaleur critique nécessaire à une dynamo thermique pour une conductivité thermique du noyau de 40 et 4 W/m/K, respectivement. Les lignes vertes foncées et claires représentent le flux de chaleur du manteau dans un régime conductif pour la gamme de conductivité thermique des silicates obtenue expérimentalement. Les triangles verts représentent la taille critique de la planète pour laquelle $Ra = Ra_c$, ce qui correspond donc au flux de chaleur minimum à la CMB, pour une viscosité du manteau allant de $10^{18}$ à $10^{22}$ Pa.s. Les valeurs extrapolées à $R = R_{Mercure} = 2440$ km sont également représentées avec des cercles verts pour les transferts de chaleur conductifs et des carrés rouges et roses pour les cas convectifs avec les viscosités du manteau indiquées à proximité. La ligne pointillée rouge représente la valeur critique pour $Re_m = 100$. A gauche de cette ligne, une dynamo est peu probable. La zone probable pour un rayon de planète capable d'alimenter une dynamo est représentée par la zone ombrée en rose. À partir de Freitas et al. (2021).

Pour savoir si une dynamo générée par de la convection thermique dans les planètes de type Mercure (avec un rapport rayon du noyau sur rayon de la planète élevé) est possible ou non, nous avons caractérisé la conductivité thermique de divers échantillons silicatés, solides,
vitreux et fondus (Freitas et al., 2021). Les conductivités thermiques ont été estimées pour des péridotites solides et partiellement fondues, avec différentes fractions de fusion, ainsi que pour des verres et des fondus basaltiques et rhyolitiques à des pressions de 2 GPa et à des températures allant jusqu'à 1700 K. Notre étude démontre que les matériels fondus ont des propriétés thermiques similaires malgré une large gamme de compositions étudiées. Les produits de fusion révèlent des conductivités thermiques beaucoup plus faibles que les solides, avec une diminution de près d'un ordre de grandeur : 1.70 (±0.19) à 2.29 (±0.26) W/m/K contre 0.18 (±0.01) à 0.41 (±0.03) W/m/K pour les péridotites à haute température et divers matériaux fondus respectivement. Les échantillons partiellement fondus se situent entre les deux et plusieurs lois prédictives sont proposées en fonction de la fraction fondue et de la texture solide/fondue. En utilisant nos résultats dans des calculs de flux de chaleur nécessaires pour la génération de dynamos pour les planètes de type Mercure, nous avons quantifié l'effet de la fusion du manteau sur la propension des noyaux planétaires à générer des dynamos par convection thermique (Figure 17). La présence d'un manteau "mushy" et ainsi de fusion partielle pourrait réduire de manière significative la capacité du manteau à évacuer la chaleur du noyau et peut empêcher ou affecter la présence d'un champ magnétique planétaire. La flottabilité et le devenir de la matière fondue dans ces corps peuvent donc influencer l'histoire magnétique de la planète. Les futures observations de planètes semblables à Mercure accrétees près de leur étoile et les détections de leurs signatures magnétiques pourraient fournir des contraintes sur leur état interne et leur histoire en terme de fusion partielle.

Avec G. Manthilake, nous avons aussi travaillé sur le rôle de la conductivité thermique des alliages Fe-S, la phase dominante dans le noyau externe de la planète Mercure, à une pression de 8 GPa et à des températures allant jusqu'à 1700 K (Manthilake et al., 2019). Nous avons montré que la conductivité électrique des alliages Fe-S à 1500 K est d'environ $10^3$ S/m, soit deux ordres de grandeur de moins que la valeur précédemment supposée pour les calculs de la dynamo herméenne. La conductivité thermique a été ensuite estimée en utilisant la loi de Wiedemann-Franz. La conductivité thermique de l'alliage Fe-S est estimée à ~4 Wm/K dans les conditions régnant à la limite noyau-manteau de Mercure. La faible conductivité thermique obtenue dans notre étude suggère que l'extraction de chaleur du noyau de Mercure à travers la couche solide de Fe-S est un processus relativement inefficace mais suggère qu'une dynamo thermique fonctionnant sur Mercure est plus probable que prévu. En effet, malgré cette faible conductivité, le flux de chaleur à la CMB reste supérieur au flux adiabatique dans le noyau ce qui est synonyme de convection et donc de dynamo pour un noyau de la taille de celui de Mercure (Figure 18). Si la flottabilité thermique est couplée à des sources de flottabilité chimique, il est possible de maintenir une dynamo intrinsèque pendant des échelles de temps compatibles avec les observations de MESSENGER. Nos estimations suggèrent que le flux de chaleur du noyau ne produit que moins de 1 TW d'énergie. Le faible flux de chaleur empêcherait le noyau herméen de se solidifier rapidement, ce qui pourrait favoriser le maintient d'une dynamo intrinsèque à Mercure depuis ~3.9 Ga.
<figure><img src="image_243.png" /></figure>

FIGURE 18 – Flux de chaleur à la frontière entre le noyau et le manteau de Mercure. Les calculs sont basés sur une conductivité thermique totale moyenne (électrons + phonons) de 4 Wm/K pour le FeS pour diverses épaisseurs de couche limite thermique en fonction du contraste de température à travers cette couche. Le flux thermique adiabatique ($Q_{ad}$) est représenté par une ligne horizontale noire en pointillés (d'après Manthilake et al. (2019)).

## 5.2 Une température à la CMB quasi constante au cours des temps géologiques

Les comparaisons entre les observations sismiques et les données de physique minérale sur les propriétés des réservoirs profonds fournissent des contraintes sur l'état thermique actuel de la Terre profonde. En particulier, le caractère non ubiquiste des zones à très faible vitesse (ULVZ) situées au-dessus du CMB fournit une température d'ancrage majeure de 4000 (±200) K à la CMB, soit quelques centaines de degrés en dessous de la température de fusion du manteau pyrolitique. L'existence d'une température à la CMB aussi élevée que ~ 4000 K aujourd'hui soulève des questions sur la température à la CMB au début de l'histoire de la Terre. Le maintien de la géodynamo depuis les plus anciennes traces de champ paléomagnétique (il y a environ 4,2 Ga (Tarduno et al., 2010)) jusqu'à aujourd'hui par une combinaison de flottabilité thermique et de convection compositionnelle nécessite beaucoup d'énergie. Pour que cela soit possible, deux scénarios peuvent être envisagés : (a) une température initiale du
noyau très élevée diminuant lentement avec le temps ou (b) une forte concentration d'éléments radiogéniques dans le noyau, tels que le $ ^{40}K $. Les deux hypothèses impliqueraient une fusion importante dans le manteau inférieur au début de la Terre.

Comme discuté précédemment, nous avons vu qu’un impact géant pourrait être à l’origine de la formation du système Terre-Lune (voir section 2). Cet impact a pu entraîner la fusion totale du manteau terrestre (Nakajima and Stevenson, 2015). Nous y reviendrons plus tard plus en détails mais le refroidissement d’un océan magmatique qui concernerait l’intégralité du manteau terrestre est un processus extrêmement efficace. En effet, nos modèles ont montré qu’en moins d’un millions d’années après l’impact, le manteau pourrait passer de complètement fondu à un taux de fusion de 40% (Monteux et al., 2016a). Durant ce laps de temps, une grande quantité de chaleur a pu être évacuée du noyau par un couplage efficace avec le manteau sus-jacent. À tel point qu’il est envisageable que le profil de température à l’intérieur de la Terre il y a 4.4 Ga ait pu atteindre son état actuel très rapidement. Le noyau solide aurait ainsi pu se former à la même époque soit quelques millions d’années après l’impact géant. Si une grande partie de l’énergie est évacuée rapidement du noyau, il est difficile de comprendre comment un champ magnétique peut être généré dans le noyau pendant plus de 4 Ga (Tarduno et al., 2010).

Dans ce contexte, la présence de la Lune a pu jouer un rôle essentiel pour maintenir le champ magnétique terrestre pendant aussi longtemps (Andrault et al., 2016) (Figure 19). En effet, la Terre adopte une forme aplatie, tourne autour d'un axe incliné qui oscille autour des pôles et son manteau se déforme élastiquement par effet de marée dû à la Lune. Nous avons ont montré que cet effet pourrait stimuler continuellement les mouvements de l'alliage de fer liquide qui constitue le noyau externe, et générer en retour le champ magnétique terrestre. Une puissance de 3 700 milliards de watts est constamment fournie à la Terre par transfert des énergies gravitationnelle et de rotation du système Terre-Lune-Soleil, et jusqu'à plus de mille milliards de watts seraient disponibles pour provoquer ce type de mouvements dans le noyau externe. Cette énergie est suffisante pour générer le champ magnétique terrestre ce qui, avec la Lune, résout le paradoxe majeur du modèle classique. Un tel effet des forces gravitationnelles sur le champ magnétique d'une planète est déjà amplement documenté pour Io, Europe, deux satellites naturels de Jupiter, et de nombreuses exoplanètes. De plus un tel scénario a été proposé par Le Bars et al. (2011) pour expliquer la dynamo lunaire.

Comme ni la rotation de la Terre autour de son axe, ni l'orientation de cet axe, ni l'orbite de la lune ne sont parfaitement régulières, leur influence cumulée sur les mouvements dans le noyau est instable et peut faire fluctuer la géodynamo. Ce phénomène permet d'expliquer certains pulses de chaleur dans le noyau externe et à sa frontière avec le manteau terrestre. Historiquement, cela a pu conduire à des pics de fusion dans le manteau profond et à d'éventuels évènements volcaniques majeurs à la surface de la Terre. Ce nouveau modèle souligne que l'influence de la Lune sur la Terre dépasse donc largement le simple cas des marées.
<figure><img src="image_244.png" /></figure>

FIGURE 19 – Les effets gravitationnels associés à la présence de la Lune et du Soleil induisent sur Terre la déformation cyclique du manteau et des oscillations de l'axe de rotation. Ce forçage mécanique appliqué à toute la planète induit de forts courants dans le noyau externe constitué d'un alliage de fer de très faible viscosité. Ces courants sont suffisants pour générer le champ magnétique terrestre.

## 5.3 Perspectives : Conductivité thermique du fer et refroidissement séculaire de la frontière manteau-noyau

Les études théoriques et expérimentales proposent des valeurs de la conductivité thermique du fer ($k_{Fe}$) comprises entre ~25 et ~250 W/m/K dans les conditions P-T compatibles avec celles de la CMB soit 135 GPa et 4000 K. Les valeurs les plus basses (par exemple 28-29 W/mK dans les travaux de Stacey and Loper (2007)) permettraient de maintenir une géodynamo active pendant plusieurs Gy, et seraient également en accord avec l'état thermique actuel du noyau. En revanche, la conductivité thermique beaucoup plus élevée (80-100 W/m/K) signalée dans les calculs théoriques récents (Pozzo et al., 2012; Koker et al., 2012) et les expériences (Gomi et al., 2013; Ohta et al., 2016) donnent un flux thermique conducteur actuel de 10-15TW le long de l'adiabat du noyau, ce qui rend le bilan énergétique du noyau très difficile à établir. De manière surprenante, seuls les travaux expérimentaux de Konôpková et al. (2016) suggèrent une faible valeur de $k_{Fe}$, comprise entre 18 et 44 W/m/K. Le débat sur l'état thermique du noyau au cours de son histoire restera probablement intense jusqu'à ce que les
vraies valeurs de $k_{Fe}$ soient confirmées par d'autres études.

La plupart des études précédentes ont déterminé la conductivité électrique du Fe ou des alliages de Fe, en utilisant soit des expériences dans la cellule à enclume de diamant chauffée au laser, soit des calculs théoriques ab initio. La conductivité électrique donne une valeur indir ecte de $k_{Fe}$ grâce à la loi de Wiedemann-Franz, dont le principe de base est que les électrons sont le principal vecteur des courants électriques et thermiques dans les métaux. La seule me sure directe disponible dans les conditions P-T relatives au noyau de la Terre utilise la méthode des impulsions laser (Konôpková et al., 2016). Cette méthode consiste à mesurer le temps né cessaire pour qu'une perturbation de température très courte ($\sim 10^{-7}$ s) produite d'un côté de l'échantillon traverse le volume de l'échantillon. Le défi expérimental consiste à détecter la perturbation de température des deux côtés de l'échantillon. Pour ce faire, cette étude a mis en place une expérience complexe avec une caméra à balayage couplée à un spectrographe à réseau. Elle fournit une mesure résolue dans le temps de la température sur l'une ou l'autre face de l'échantillon, mais pas sur les deux simultanément. Cela pourrait induire des artefacts dus aux difficultés de synchronisation de la caméra à bandes avec les impulsions de chaleur de chaque côté de l'échantillon.

Dans le projet que nous développons avec D. Andrault et L. Pison, nous effectuons des mesures directes de $k_{Fe}$ en utilisant la méthode de l'impulsion laser, comme dans l'étude de Konôpková et al. (2016). Au lieu d'utiliser une caméra à bandes comme dans l'étude précé dente, nous utilisons des mesures spectroscopiques beaucoup plus simples, qui ne fournissent pas la température réelle de l'impulsion. Cependant, les balayages temporels sont enregist trés simultanément sur les deux faces de l'échantillon, ce qui permet de déterminer avec une grande précision la propagation temporelle de l'impulsion thermique à travers l'échantillon. Nous effectuons ces mesures directes de la conductivité thermique du fer à l'aide du système LH-DAC disponible au Laboratoire Magmas et Volcans. La puissance de nos deux lasers à fibre continus est contrôlée pour stabiliser la température de $\sim 1200$ K à plus de 3000 K, et la puissance d'un troisième laser à fibre pulsé est ajoutée pour produire l'impulsion de chaleur sur un côté de l'échantillon. Les mesures d'impulsion sont effectuées entre 20 et 130 GPa, et entre 1200 K et le point de fusion de l'échantillon de fer. Ces plages de P-T permettront de déterminer les effets croisés de la pression et de la température sur $k_{Fe}$. Grâce à notre nouveau dispositif, nous documenterons pour la première fois expérimentalement et à des pressions élevées la dépendance en température de $k_{Fe}$. Cependant, les mesures quantitatives nécessitent la détermination précise de l'épaisseur de l'échantillon (qui se situe entre 3 et 6 microns) dans toutes les expériences. Pour ce faire, nous devons découper la feuille de Fe récupérée lors des expériences, à l'aide d'un appareil à faisceau d'ions focalisé (FIB).

Pour obtenir la conductivité thermique de nos échantillons à partir des mesures expéri mentales de la propagation de l'impulsion thermique à travers l'échantillon, nous avons dé veloppées des modèles numériques de propagation thermique à l'aide du logiciel COMSOL
Multiphysics (Figure 20). Grâce à cet outil de modélisation en éléments finis, nous modélisons l'évolution de la température dans un échantillon micrométrique soumis à un pulse thermique. Nous reproduisons pour cela les mêmes conditions que les conditions expérimentales que ce soit en terme de géométrie, de propriétés des matériaux ou de conditions aux limites. Nous considérons par exemple les propriétés du diamant, du milieu transmetteur de pression (couche de KCl) entre les diamants et l'échantillon et du joint au rhénium. Nous pouvons notamment implémenter les propriétés (densité, conductivité et capacité thermique) de ces divers matériaux. Nous considérons une géométrie axisymmétrique cylindrique et notre maillage est calculé directement par le mailleur de COMSOL. Cette comparaison permet de contraindre la conductivité thermique de l'échantillon comme cela a été fait dans l'étude de Konôpková et al. (2016). À terme, cette valeur affinée de $k_{Fe}$ sera implémentée dans des modèles de refroidissement du noyau pour modéliser la variation séculaire de la température et le flux de chaleur à travers la CMB. Cela permettra d'affiner les conditions de température dans la couche D" depuis les premières étapes d'évolution de la Terre mais aussi des autres planètes telluriques mais aussi d'affiner la conductivité thermique $k_{Fe}$ dans les conditions P-T du noyau externe de la Terre.

<figure><img src="image_245.png" /><figcaption>FIGURE 20 – Panels A à E : Évolution temporelle de la temperature au sein d'un échantillon de Fer pris en sandwich entre deux couches de KCl et deux diamants et soumis à un pulse laser afin d'être chauffé. Panel F : Comparaison entre les évolutions temporelles de la température de part et d'autre de l'échantillon obtenues expérimentalement (courbes rouge et jaune) et les courbes obtenues par nos modèles numériques pour différentes values de $k_{Fe}$.</figcaption></figure>
Références personnelles

Andrault D., **Monteux, J.**, *Le Bars M.*, Samuel H., 2016, *The deep Earth may not be co-oling down. Earth and Planetary Science Letters vol.443, p.195-203. **LIEN HAL**

Manthilake G., Chantel J., **Monteux, J.**, Andrault D., Bouhifd A., Bolfan-Casanova N., Boulard E., Guignot N., King A., Itie J.P., 2019, Thermal Conductivity of FeS and Its Implications for Mercury's Long Sustaining Magnetic Field. Journal of Geophysical Research - Planets vol.124. **LIEN HAL**

Freitas D., **Monteux, J.**, Andrault D., Manthilake G., Mathieu A., Schiavi F., Cluzel N., 2021, Thermal conductivities of solid and molten silicates: Implications for dynamos in mercury-like proto-planets. Physics of the Earth and Planetary Interiors vol.312, p.106655. **LIEN HAL**
<figure><img src="image_246.png" /></figure>

ELSEVIER

<figure><img src="image_247.png" /></figure>

# The deep Earth may not be cooling down

Denis Andrault<sup>a,*</sup>, Julien Monteux<sup>a</sup>, Michael Le Bars<sup>b</sup>, Henri Samuel<sup>c</sup>

## ARTICLE INFO

**Article history:**

Received 28 July 2015

Received in revised form 7 March 2016

Accepted 9 March 2016

Available online xxxx

Editor: C. Sotin

**Keywords:**

temperature profile in the deep Earth
secular cooling of the Earth
generation of the geomagnetic field

## ABSTRACT

The Earth is a thermal engine generating the fundamental processes of geomagnetic field, plate tectonics and volcanism. Large amounts of heat are permanently lost at the surface yielding the classic view of the deep Earth continuously cooling down. Contrary to this conventional depiction, we propose that the temperature profile in the deep Earth has remained almost constant for the last ~4.3 billion years. The core-mantle boundary (CMB) has reached a temperature of ~4400 K in probably less than 1 million years after the Moon-forming impact, regardless the initial core temperature. This temperature corresponds to an abrupt increase in mantle viscosity atop the CMB, when ~60% of partial crystallization was achieved, accompanied with a major decrease in heat flow at the CMB. Then, the deep Earth underwent a very slow cooling until it reached ~4100 K today. This temperature at, or just below, the mantle solidus is suggested by seismological evidence of ultra-low velocity zones in the D"-layer. Such a steady thermal state of the CMB temperature excludes thermal buoyancy from being the predominant mechanism to power the geodynamo over geological time.

An alternative mechanism to sustain the geodynamo is mechanical forcing by tidal distortion and planetary precession. Motions in the outer core are generated by the conversion of gravitational and rotational energies of the Earth-Moon-Sun system. Mechanical forcing remains efficient to drive the geodynamo even for a sub-adiabatic temperature gradient in the outer core. Our thermal model of the deep Earth is compatible with an average CMB heat flow of 3.0 to 4.7 TW. Furthermore, the regime of core instabilities and/or secular changes in the astronomical forces could have supplied the lowermost mantle with a heat source of variable intensity through geological time. Episodic release of large amounts of heat could have remelted the lowermost mantle, thereby inducing the dramatic volcanic events that occurred during the Earth's history. In this scenario, because the Moon is a necessary ingredient to sustain the magnetic field, the habitability on Earth appears to require the existence of a large satellite.

© 2016 Elsevier B.V. All rights reserved.

## 1. Introduction

Our knowledge of the present-day thermal state of the deep Earth has largely improved based on comparisons between seismic observations and experimental and theoretical characterizations of the Earth's materials, including studies of phase transitions and melting curves. In addition, we now have a more precise idea of the early evolution of the deep Earth based on paleomagnetic and geochemistry data as well as numerical modeling. However, the path the deep Earth has followed from its early formation to its current state remains puzzling. In particular, the Earth's heat budget over the ~4.5 billion years (Gy) of its existence remains difficult to balance satisfactorily. In this article, we challenge the classical view that the Earth is continuously cooling. We propose

an alternative model for the thermal and magnetic evolution of the deep Earth. First, we review the early and the present thermal states of the deep Earth (sections 2 and 3), then we highlight the major paradoxes implied by the classical scenario based on secular cooling (section 4). Finally, we propose a steady thermal state scenario (section 5) involving an alternative source of energy to power the geodynamo: precession and tides (section 6). Section 7 presents the energy budget of the deep Earth in the framework of our model. The implications of this scenario are presented in the final section.

## 2. The early temperature evolution of the deep Earth

### 2.1. The primordial core temperature

The initial core temperature is related to a variety of processes but primarily results from the mechanism of core-mantle segregation (Stevenson, 1990). Metal/silicate separation is a rapid event

<footnote><sup>a</sup> Laboratoire Magmas et Volcans, CNRS-OPGC-IRD, Université Blaise Pascal, Clermont-Ferrand, France</footnote>

<footnote><sup>b</sup> CNRS, Aix-Marseille University, Ecole Centrale Marseille, IRPHE, UMR 7342, Marseille, France</footnote>

<footnote><sup>c</sup> Institut de Recherche en Astrophysique et Planétologie, CNRS, Université Paul Sabatier, Toulouse, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: denis.andrault@univ-bpclermont.fr (D. Andrault).</footnote>
(<60 My) contemporaneous with the Earth's accretion from many planetary embryos (Kleine et al., 2002; Rudge et al., 2010). After a meteoritic impact, the Fe-droplets produced by fragmentation of the impactor's core descended and equilibrated thermally in a highly turbulent magma ocean (Deguen et al., 2011; Samuel, 2012; Wacheul et al., 2014). The resulting iron layer/pond that formed at the bottom of the magma ocean then descended through the underlying, more viscous mantle in the form of diapirs or, alternatively, through channels (Stevenson, 1990). In both cases, the corresponding gravitational potential energy released was dissipated into heat, but the heat partitioning between the iron and the silicate depends on the segregation mechanism (Rubie et al., 2015). The diapir mechanism tends to favor heat transfer to the viscous mantle (Monteux et al., 2009; Samuel et al., 2010), while channels favor a hotter core (Ke and Solomatov, 2009), yielding a wide range of plausible thermal states at the core-mantle boundary. If metal-silicate thermal equilibration had been efficient, the initial core temperature would be similar to that of the lowermost mantle. Taking into consideration the contributions of heating from large impacts and the decay of short-lived radionuclides, the early core was likely to be hotter than the mantle liquidus soon after its formation (Rubie et al., 2015).

## 2.2. Early cooling of the magma ocean

The giant Moon forming impact (MFI) that occurred ~60 million years (My) after the Earth's formation (Touboul et al., 2007), likely re-melted the entire mantle (Nakajima and Stevenson, 2015) and significantly heated up the Earth's core (Herzberg et al., 2010) (Fig. 1(a)). The MFI, through the release of energy induced by gravitational segregation of the impactor's core, could have potentially increased the core temperature further by 3500–4000 K (Rubie et al., 2015). This could have resulted in an initial CMB temperature on the order of 6000 K (Nakagawa and Tackley, 2010), which is well above the mantle solidus of ~4150 K (Andrault et al., 2011; Fiquet et al., 2010). Therefore, the MFI could have caused intensive melting in the lowermost mantle.

Upon cooling of this giant-impact induced magma ocean, a thin crust rapidly formed at the surface, within the cold upper thermal boundary layer (Solomatov, 2015) (Fig. 1(b)). Just below, the magma ocean is expected to have convected vigorously. Hence, its internal temperature would follow an adiabatic profile undergoing a progressive decrease in potential surface temperature with time (Abe, 1997; Solomatov, 2000). Due to the fact that the liquidus and solidus curves present P–T slopes steeper than the magma ocean adiabats for the chondritic-type composition (Andrault et al., 2011; Thomas and Asimow, 2013), the magma ocean should solidify from the bottom up (Fig. 1(b)). The heat flux at the surface could have been as high as ~$10^6$ W/m$^2$, which suggests crystallization of most of the magma ocean within ~$10^3$ yrs (Solomatov, 2015). However, the situation may have been complicated by physical and chemical processes such as suspension, turbulence, nucleation, and percolation or by the formation of an opaque atmosphere at the Earth's surface. These effects could have delayed the complete crystallization of the upper mantle up to $10^8$ yrs after the magma ocean began cooling (e.g. after the MFI) (Fig. 1(c) and 1(d)) (Lebrun et al., 2013; Sleep et al., 2014).

# 3. The present-day temperature profile in the deep Earth

## 3.1. Upper mantle, transition zone and the lower mantle

The temperature profile from the shallow mantle to a few hundred kilometers above the CMB is relatively well documented. The most robust constraints originate from the phase transformations at the 410, 520 and 660 km discontinuities; the depth of the

<figure><img src="image_248.png" /></figure>

**Fig. 1.** Schematic representation of the Earth's interior evolution from the Moon forming impact (MFI) to the present. Full crystallization of the molten Earth was probably very complex due to suspension, turbulence, nucleation, and percolation processes (Solomatov, 2015). (a) The MFI occurred 50–100 Myr after the formation of the Ca-Al rich inclusions (CAI, the oldest objects in the solar system). It left the Earth mostly molten with a core temperature potentially above 6000 K (Rubie et al., 2015). (b) In the magma ocean, high temperatures and turbulent state should efficiently cool down the deep mantle and the liquid core. This could favor a thermally driven geodynamo. Progressive decrease of the potential temperature below the viscous threshold of 60% crystallization took within $10^3$ to ~$10^6$ yrs, essentially depending on the magma ocean viscosity (Monteux et al., submitted for publication). During this time, a thin crust rapidly formed at the Earth's surface. Also, a basal magma ocean could have existed, however, its life time remains controversial (Labrosse et al., 2007; Monteux et al., submitted for publication). Large plumes of hot and/or partially-molten material could have migrated toward the Earth's surface. (c) The final step of mantle solidification could have taken longer, depending on the cooling efficiency at the Earth's surface (Lebrun et al., 2013; Sleep et al., 2014). This period would correspond to progressive decrease of the CMB temperature to the mantle solidus at ~4150 K (Fig. 2(a)), associated with the crystallization and the growth of the inner core (Fig. 2(b)). This would favor a compositionally driven geodynamo, an ingredient that could still today contribute to sustaining the geodynamo. (d) Later on, the long-term solid-state convection in the mantle, as we currently know it, started. At this period, the CMB temperature could have remained nearly constant for geological times. Because of moderate core cooling and growth of the inner core, a major ingredient to sustain the geodynamo could be mechanical forcing by astronomical forces. In fact, mechanical forcing could have started to induce core motions as soon as the moon was formed. Colors orange to gray (intermediate = yellow) correspond to the mantle encountering a degree of partial melting from 100% to 0% (intermediate = 40%), while black and white correspond to liquid (outer) and solid (inner) core, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

phase transformations must be compatible with the phase diagram of the major upper mantle minerals, mainly olivine, which is well constrained experimentally. When including the effect of entropy variations between the different polymorphs, a temperature discontinuity of a few hundred degrees is induced at the seismic discontinuities (Katsura et al., 2010; Stacey and Davis, 2008). Complications in the temperature determination using the olivine phase diagram may arise from the uncertainties in mantle concentrations of FeO, water and ferric iron (e.g. Frost and Dolejs, 2007), because they modify the pressure of the phase transitions slightly. Lateral temperature variations are also expected from colder temperatures in subduction zones to hotter temperatures in regions of upwelling mantle (ocean ridges, for example). Altogether, the
<figure><img src="image_249.png" /><figcaption>**Fig. 2.** Present-day temperature profile in the Earth's (a) mantle and (b) core inferred from several experimental arguments. Green circles correspond to the most likely temperature at the core–mantle boundary. (a) At the CMB pressure of 135 GPa, melting temperatures of 4180, 4150, 3800 and 3570 K were reported for peridotite (F-10, Fiquet et al., 2010), chondritic-type mantle (A-14, Andrault et al., 2014), mid-ocean ridge basalt (A-11, Andrault et al., 2011) and wet-pyrolite (N-14, Nomura et al., 2014), respectively. Dashed curves stand for adiabatic profiles (M-07, Matas et al., 2007, H-05, Hernlund et al., 2005, K-10, Katsura et al., 2010, BS-81, Brown and Shankland, 1981, B-01, Bunge et al., 2001). (b) Melting of pure Fe was reported at 4175 and 6230 K for pressure conditions of CMB (135 GPa) and ICB (330 GPa), respectively (A-13, Anzellini et al., 2013). A melting temperature depletion of ~650 K can account for the presence of light elements in the core (e.g. Morard et al., 2013). The CMB temperature is extrapolated from the ICB based on the adiabatic profile in the outer core. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

uncertainty is less than a couple of hundred degrees. Then, the temperature profile in the lower mantle is classically extrapolated from anchor points in the transition zone using an adiabatic gradient, which yields additional uncertainties. Slightly different temperature profiles can be obtained, depending on the equations of states used for the mantle (Brown and Shankland, 1981; Stacey and Davis, 2008). Other predictions give a significantly higher temperature profile when refining the seismic profiles (Vp, Vs, ρ) from the mineral equations of states (ρ, K, G) (Matas et al., 2007) (Fig. 2(a)).

## 3.2. The lowermost mantle

The thermal state of the lowermost mantle is not directly correlated to the surface potential temperature, but is rather tied to the temperature of the core and the heat flux at the CMB. The seismic observations of thermochemical heterogeneities

and partial melting in the D"-region provide additional information to anchor the CMB temperature (Herzberg et al., 2013; Rost et al., 2005; Wen and Helmberger, 1998). It has been argued that the thermochemical piles present in this region could be interpreted as patches of post-bridgmanite (PBg) embedded in bridgmanite (Bg), which would require a double crossing of the Bg-PBg phase transition (Hernlund et al., 2005). Using the P-T Clapeyron slope of the polymorphic transition, this situation is possible if a sharp temperature change occurs when approaching the CMB, for a CMB temperature higher than 4000 K. This method based on the phase diagram of an unrealistically pure MgSiO₃ end-member has been subsequently challenged by experiments performed on the geophysically relevant Al-bearing (Mg,Fe)SiO₃ Bg (Andrault et al., 2010; Catalli et al., 2009). Still, the argument for a double crossing may stand for a partial and progressive transition in Bg (Hernlund, 2010).

The CMB temperature can also be constrained using the melting curve of the silicate mantle. The non-ubiquitous character of the seismic features in the D"-layer forces the CMB temperature to be lower than the mantle solidus. Otherwise, there would be a continuous melting line below which the mantle would be partially molten, due to higher temperatures in the thermal boundary layer when approaching the CMB. The solidus of chondritic-type material plots at 4150 (±150) K at a CMB pressure of 135 GPa (Andrault et al., 2011), very close to that of the peridotitic-type mantle (Fiquet et al., 2010). This solidus temperature should actually remain valid for any reasonable mineralogical system composed of Bg, CaSiO₃-perovskite and (Mg,Fe)O-ferropericlase, because of the pseudo-eutectic behavior. In contrast, the solidus temperature could be lowered in the presence of a high FeO-content (Mao et al., 2005), high volatile contents (Nomura et al., 2014) or when the excess mantle ferropericlase is replaced by an excess SiO₂, e.g., for a basaltic composition (Andrault et al., 2014). Water can have a dramatic effect, lowering the solidus temperature to ~3570 K, but it is unlikely that the lower mantle contains a very high water content (Bolfan-Casanova et al., 2003). In contrast, the descent of slabs toward the CMB is clearly imaged by seismic tomography (Grand et al., 1997), and slabs may very well reach the CMB. The solidus temperature of a mid-ocean ridge basalt at the CMB was reported to be 3800 (±150) K (Andrault et al., 2014), which suggests that a CMB temperature of 4000 (±200) K would produce discontinuous regions of partial melt, in agreement with seismic observations (Fig. 2(a)).

## 3.3. The core

The melting curve of pure iron was a long-running source of controversy until recent experimental measurements using laser-heated diamond anvil cells fell in perfect agreement with shock-wave data and ab-initio calculations (Anzellini et al., 2013) (Fig. 2(b)). The originality of this experiment relies on fast heating, to prevent the sample pollution from C diffusing out of the diamond anvils, together with the *in situ* detection of sample melting using X-ray diffraction. It suggests a melting temperature of pure Fe at the inner core boundary (ICB) of 6230 (±500) K. The light elements present in the outer core at a level of 10 wt% should lower this melting point. The melting-temperature depletions induced by the presence of S, O and Si are 100, 50 or 30 K/wt%, respectively. Unfortunately, the nature and combination of light elements in the outer core remain subject to debate. A reasonable composition could be 2.5, 5.0 and 5.0 wt% of S, O and Si, respectively, in agreement with the geochemical constraints (Dreibus and Palme, 1996), the density jump at the ICB (Alfè et al., 2002) and the seismic profiles (Vp, ρ) in the outer core (Morard et al., 2013). For this Fe-alloy composition, the melting-temperature depletion
can be estimated to 650 (±100) K. This yields an ICB temperature of 5580 (±600) K.

When this anchor point is extrapolated to the CMB using the equation of state of Fe, it yields a CMB temperature of 4100 K, if we assume a constant Grüneisen parameter of 1.51 (Vocadlo et al., 2003). The relative changes in the cocktail of light elements would not drastically change this extrapolated CMB temperature. We note that regardless of the thermal model considered for the core, the heat flux at the CMB remains moderate. Thermal boundary layers with a large temperature jump are unlikely to develop inside the liquid outer core. Therefore, the CMB temperature is a good proxy to discuss the core temperatures, using a relevant adiabatic profile. Based on the melting diagram of the Fe-alloy, the inner core should disappear for CMB temperatures above ~4250 K. This temperature is only a couple hundred degrees above the current CMB temperature and is also just above the mantle solidus of ~4150 K. This indicates that the onset of the inner core crystallization is expected to happen before the lowermost mantle completely solidified.

## 4. Secular cooling of the deep Earth? Major unresolved paradoxes

The different lines of reasoning mentioned above converge to the remarkable conclusion that today the CMB temperature is precisely at, or just below, the solidus of the silicate mantle, at 4100 K (±200) K. If the Earth has been cooling for the last ~4.5 Gy, the early core would need to be significantly hotter in the past and, hence, overlaid by molten mantle, a primordial basal magma ocean (BMO) giving birth later to the D'-layer (Labrosse et al., 2007). In this article, we challenge this classic view of Earth's secular cooling based on three major paradoxes:

(i) Since when (and for how much additional time) the CMB temperature has (and will) remain *precisely* just below the solidus of the silicate mantle? It would be very unusual if this peculiar situation were a pure coincidence, since this temperature corresponds to a major change in the mantle state through a first-order phase transformation (the onset of melting at the solidus).

(ii) Producing the geodynamo by a combination of thermal buoyancy and compositional convection (Buffett, 2000) requires a heat flux through the CMB of possibly up to ~10–15 terawatts (TW), depending on the controversial values of electrical and thermal conductivities of the outer core (Pozzo et al., 2012; Zhang et al., 2015). Three ingredients have been advanced to explain the persistence of a significant CMB heat flux from the oldest (4.2 Gy ago) evidences of paleo-magnetic field (Tarduno et al., 2015) to the present day: (a) The first is a tremendous initial core temperature of 5000–7000 K (Davies et al., 2015; Labrosse, 2015; Nakagawa and Tackley, 2010), well above the mantle solidus and even above the liquidus. Such high temperatures would delay the crystallization of the inner core, possibly to a period as late as 0.7 Gy ago (Labrosse, 2015), even if such a late start is still controversial (Biggin et al., 2015). (b) The second ingredient is a high concentration of radiogenic K in the core. Experimental determination of the K partitioning between metal and silicate suggests a maximum of 250 ppm K in the core (Bouhifd et al., 2007), while values below 50 ppm K appear more likely (Corgne et al., 2007; Watanabe et al., 2014). However, models of core cooling with values as high as 400–800 ppm K are not capable of sustaining the geodynamo until the present (Nakagawa and Tackley, 2010). (c) It was suggested that layered structures in the lowermost mantle (Nakagawa and Tackley, 2014) or at the top of the core (Buffett, 2014) may help to retain heat within the core, but the density contrast needs to be large to maintain a gravitationally stable liquid layer in the highly turbulent flows expected in a very hot Earth. Such a global density stratification in the mantle or in the core has not been undoubtedly demonstrated yet seismically.

(iii) The large initial temperature required for the primordial core implies large fractions of melt in the lowermost mantle. However, it seems very difficult to maintain a basal magma ocean (BMO) at a temperature significantly above the mantle solidus (or, more precisely, above the viscous transition in the mantle, which correspond to ~40% of partial melting (Abe, 1997). This issue is detailed below) for a long period of time (see Monteux et al., submitted for publication). (a) The first reason for this is that the silicate melt viscosity is orders of magnitude lower than that of the solid mantle. To show its effect, let's assume a temperature jump of 1000 K in a super-adiabatic thermal boundary layer lying above the CMB. For a silicate melt, the boundary layer would be thinner than one meter (Solomatov, 2015), which would drive the heat flux to more than $10^6$ TW. This implies a negligible temperature jump at the CMB when the silicate is liquid above the CMB. A detailed calculation of the cooling of a hot primitive core with or without a molten layer above the CMB suggests core cooling $10^6$ times faster in the presence of a BMO (Monteux et al., 2011). (b) Another argument is based on the fact that the adiabatic profiles of solid, liquid and partially molten chondritic-type mantle present P–T slopes less steep than the mantle solidus (Andrault et al., 2011; Thomas and Asimow, 2013). This implies that the degree of mantle partial melting should increase with elevation from the CMB, not the contrary. For this reason, having a CMB temperature significantly above the mantle solidus would yield major mantle instabilities. If vertical chemical segregation would eventually produce a BMO with a composition different than the average mantle, the situation would not be drastically different. The temperature profile in the BMO would follow a quasi-adiabatic profile from the hot CMB to the interface between the BMO and the overlaying solid mantle. Then, all arguments raised above remain valid, but with a dominant interface for heat exchange being located between the BMO and the solid mantle, instead of exclusively at the CMB. (c) A last argument is that the hot melt could be unstable in the BMO because it is buoyant. It would travel through the mantle towards the Earth's surface. Unfortunately, this issue remains controversial (Andrault et al., 2012; Nomura et al., 2011).

## 5. A quasi-constant CMB temperature over geological time

The most likely thermal state after the mantle has achieved a stable character is that the BMO became significantly viscous, thus below the typical temperature threshold corresponding to 60% of crystallization. This happens at a CMB temperature of ~4400 K for a chondritic-type mantle (Andrault et al., 2011). We note that a peridotitic mantle would become viscous at a slightly higher temperature (Fiquet et al., 2010), however, this composition is much less relevant to the primitive mantle at the CMB. As discussed above, this final temperature could be achieved in less than $10^8$ yrs and it is weakly dependent on the initial CMB temperature (Lebrun et al., 2013; Sleep et al., 2014; Solomatov, 2015). This gives rise to a new paradox: while the CMB temperature reached ~4400 K early in the Earth's history, it is only a few hundred degrees below today, at a temperature of ~4100 K. We note that the uncertainty on the temperature difference of ~300 K is independent of the uncertainty on the experimental determination of the solidus temperature. Indeed, the early and present-day CMB temperatures (slightly above, and just below, respectively) are determined relative to a same reference that is the solidus of the average mantle at the CMB. The uncertainty on the ~300 K secular cooling of the CMB is estimated to ±100 K.

Such a stable CMB temperature is compatible with geological constraints on the time evolution of the mantle potential temperature (MPT, i.e. the extrapolation to the planetary surface of the mantle's adiabatic temperature profile). For example, petro-
logical analyses of Archean and Proterozoic basalts (between 1.5 and 3.5 Gy old) preserved at the Earth's surface show primary magma compositions compatible with an MPT only ~200 K greater than today (Herzberg et al., 2010). A similar temperature change is reported between Archean tonalite–trondhjemite–granodiorite associations of 4.0 to 2.5 Gy old (Martin and Moyen, 2002). We note that the CMB temperature and the MPT are not formally linked to each other, due to an adjustable temperature jump in the thermal boundary layer above the CMB. Still, they both refer to the thermal state of the deep Earth.

Because the CMB temperature is intimately linked to the core thermal state, a steady CMB temperature over billions of years excludes core cooling as a major ingredient for driving the geodynamo during this period. There are two alternative sources that can induce the turbulent fluid motion in the outer core needed to produce the geomagnetic field: (i) Chemical buoyancy occurs when light elements (mainly O, Alfe et al., 2002) are released at the ICB due to inner core growth. This effect becomes significant when the temperature drops below ~4250 K, thus at only ~150 K above the present-day CMB temperature (Fig. 2(b)). Previous work dedicated to the analysis of the relative effects of compositional and thermal convection suggests that the same magnetic field can be generated with approximately half the heat throughput needed if the geodynamo was purely thermally driven (Gubbins et al., 2004). Still, none of the recent studies suggest that chemical buoyancy could drive alone the geomagnetic field for billions of years (Davies et al., 2015; Labrosse, 2015). Alternatively, (ii) mechanical forcing induced by precession and tidal distortions of the CMB (Dwyer et al., 2011; Le Bars et al., 2015; Tilgner, 2005) could have been a major ingredient to maintain the geomagnetic field, since the formation of the Moon. It could still operate today.

**6. Precession and tides, an alternative mechanism to drive the geodynamo**

Precession and tidal distortions of a planet's CMB induced by gravitational interactions with a companion (e.g. Earth and Moon) are both capable of generating core turbulence and of sustaining a dynamo with critical magnetic Reynolds numbers comparable to thermal and compositional dynamos (Cebron and Hollerbach, 2014; Tilgner, 2005). Indeed, planetary cores, as any rotating fluid, permit eigenmodes of oscillation called “inertial modes”, whose restoring force is the Coriolis force. Precession and tides, seen from the mantle frame of reference as small periodic perturbations of the rotating fluid core, are capable of resonantly exciting those inertial modes, leading to fluid instabilities, turbulence and dynamo action. More specifically, two types of instabilities have been described in the literature, the same generic mechanisms working both for precession and tidal excitations (see details in Le Bars et al., 2015 and references therein): (1) the direct resonance of one given inertial mode, whose non-linear interactions produce a localized geostrophic shear layer, which can then destabilize and lead to turbulence (Malkus, 1968; Sauret et al., 2014); (2) the triadic resonance of two inertial modes with the harmonic forcing (Kerswell, 1993, 2002), which can either lead to sustained turbulence or to cycles of growth, saturation and collapse (Le Bars et al., 2010). In either case, it is important to recognize that a resonance is involved: even if the excitation amplitude is small, the resulting flows may be intense, draining their energy from the mechanism sustaining the excited waves, i.e. from the spin–orbit rotational energy of the considered system. Such mechanisms provide an appealing alternative explanation for planetary dynamos when the classical convective model does not apply. For instance, they have been proposed to explain the brief dynamos of the Moon and Mars (Stevenson, 2003), the size of these planets being insufficient to

sustain long-lived thermally-driven dynamos. The disappearance of Mars' orbiting companion after it collided into the young planet (Arkani-Hamed, 2009) and the recession of the Moon (Dwyer et al., 2011) accompanied by a decrease in the precession intensity, could explain the end of their magnetic histories.

The question then remains to determine (1) whether or not mechanically forced instabilities are present in the Earth's core, and (2) whether or not the generated flows are sufficiently powerful to explain the geomagnetic field. Regarding the first point, the recent literature indeed shows that the present Earth, in the absence of a convectively imposed magnetic field (as would be the case in our non-conventional model), is subject to both tidal and precession instabilities generating turbulence (see e.g. Cebron et al., 2012). The second point has been the subject of a long debate between the seminal work of Malkus, (1968, 1963) and the subsequent studies of Rochester et al. (1975) and Loper (1975). This debate was resolved by Kerswell (1996): even if the laminar flows considered by Rochester et al. (1975) and Loper (1975) are insufficient to sustain a dynamo, the expected turbulent states in the Earth's core are largely sufficient, and the huge amount of energy stored in the Earth–Moon–Sun system (spin and orbit) provides very large source of energy to sustain the magnetic field over geological time (see e.g. Le Bars et al., 2015).

**7. The energy budget to sustain the thermal steady state of the deep Earth**

**7.1. The core budget**

Additional arguments to support this proposal are provided by considering the energy budget of the Earth's rotational dynamics as a whole. Models supported by precise measurements coming from lunar laser ranging indicate that 3.7 TW is continuously injected from the Earth–Moon–Sun orbital system into the Earth system (Munk and Wunsch, 1998; Wunsch and Ferrari, 2004). Models also indicate that 0.2 TW is dissipated into the Earth's atmosphere and its mantle; direct satellite estimates show that 1 TW is lost to the deep ocean; and the most accurate models indicate additional tidal dissipation in shallow seas of up to 2 TW (Ferrari, 2015). Hence, 0.5 to 1 TW of the dissipated rotational power is still missing in the current energy budget: it may very well be continuously injected into the outer core, where it can fulfill the energy thirst of the geodynamo, estimated to range between 0.1 to 2 TW (Buffett, 2002; Christensen and Tilgner, 2004). The situation was probably even more favorable in the past, when the Moon was closer to the Earth and when the Earth was rotating faster. Indeed, tidal distortion was previously larger and dissipation measured by the Ekman number was smaller, both ingredients being favorable to instability and turbulence (e.g. Cebron et al., 2012). One can thus imagine that throughout the history of the Earth–Moon system, turbulent flows and dynamos have been excited by mechanical forcing, the energy dissipated by both ohmic and viscous dissipations participating into the Moon's recession and deceleration (e.g. Le Bars et al., 2010).

In addition to the 0.5 to 1 TW rotational power injected into the core for dynamo action, part of which is ultimately transformed into heat by viscous and Joule dissipation, three sources could significantly contribute to the heat budget: (i) Radioactive disintegration of potassium (⁴⁰K) could provide between 0.2 to 1.4 TW today (Bouhifd et al., 2007; Buffett, 2002; Corgne et al., 2007; Watanabe et al., 2014), (ii) the latent heat of inner core crystallization contributes 0.3 TW, assuming that its crystallization occurs over a period of ~4.3 Gy. (iii) Finally, we note that despite a global steady state, the core temperature may have decreased from ~4400 K originally to ~4100 K today (Fig. 3). Core cooling by ~300 K over 4.3 Gy would provide an average CMB heat flux of
<figure><img src="image_250.png" /></figure>

**Fig. 3.** Schematic evolution of the CMB temperature since the Earth's accretion. In the hypothesis of an initial CMB temperature above 6000 K (Rubie et al., 2015), rapid cooling is expected until the drastic increase in mantle viscosity at the CMB (see text and Monteux et al., submitted for publication). It corresponds to a degree of partial melting of ~40% (Abe, 1997), thus a temperature of ~4400 K for a primordial chondritic mantle (Andrault et al., 2011). Then, the complete mantle crystallization could have taken up to more than 1 Gy, as suggested by geodynamic modeling (e.g. Nakagawa and Tackley, 2010). After the mantle became significantly viscous, a purely thermally-driven dynamo becomes unlikely due to major slow-down of the CMB heat flux. As a result, the CMB temperature remained close to the mantle solidus, at ~4100 K, until today (see Fig. 2). At a period difficult to define precisely based on our model, the appearance of the inner core (indicating a CMB temperature below ~4250 K) provided buoyancy sources from the release of latent heat and light elements. Still, the sources of energy are insufficient to maintain the geodynamo from the first evidence of geomagnetic field, ~4.2 Gy ago (Tarduno et al., 2015) to present day. This strongly suggests that mechanical forcing induced by a combination of astronomical forces (see Le Bars et al., 2015) has been a major ingredient to maintain the geodynamo. Due to the intrinsically time-dependent character of the mechanical forcing, periods of growing instabilities and intense turbulent motions would alternate with cycles of relaxation associated with abrupt releases of large amounts of energy (e.g. Kerswell, 1993). Abrupt increases of the core temperature, triggering increases in partial mantle melting at the CMB (see inset), could be related to the geological evidence of periods of hot and intense volcanic eruptions (Arndt and Davaille, 2013; Martin et al., 2014).

~2 TW. Adding all contributions, the average heat flux coming out of the core could range between 3.0 and 4.7 TW.

On the other hand, the low range of values for the thermal conductivity of the outer core yields a heat flow along the outer-core adiabat between 1.7 to 3.6 TW (Buffett, 2002). For higher values of the conductivity, as suggested recently (Pozzo et al., 2012; Zhang et al., 2015), the adiabatic heat flow would be more than 10 TW (e.g. Labrosse, 2015). This range of values appears significantly higher than the 3.0 to 4.7 TW estimated using our model. We note, however, that the temperature profile in the outer core could very well be slightly sub-adiabatic. It would actually facilitate the vertical thermochemical stratification of the outer-core (Helffrich and Kaneshima, 2013). Dynamos excited by mechanical forcing do not require a super adiabatic temperature in the Earth's outer core: it has already been demonstrated that tidal and precession instabilities exist in a stratified environment, theoretically (Cebron et al., 2012), numerically (Cebron et al., 2010) and experimentally in a cylindrical geometry (Guimbard et al., 2010). Instability involves resonances of gravito-inertial waves rather than inertial waves, the main effect being to decrease the excited vertical wavelengths, with negligible or even positive effects on the instability threshold and growth. The same conclusion has been reached concerning other types of unstable flow, for instance Taylor-Couette flows (Le Bars and Le Gal, 2007): contrary to intuition, stratification is capable of increasing flow instability, and turbulence may develop while maintaining an overall global stratification. In addition to tides and precession, a dynamo driven by the solidification of the inner core could also lead to an overall subadiabatic core, as studied for the case of Mercury (Manglik et al., 2010).

## 7.2. The thermal boundary layer in the lowermost mantle

The way the CMB heat flux, estimated above in the range of 3.0 to 4.7 TW from the core energy budget, is accommodated in the

lowermost mantle depends on 3 major parameters: (i) the thickness of the thermal boundary layer ($e_{TBL}$) where conduction is the dominant mechanism of heat transfer. $e_{TBL}$ is generally assumed to be the thickness of the D"-layer, thus 100 to 300 km, as reported by seismological studies (e.g. Lay et al., 2004). However, we note that the seismic anomalies used to define $e_{TBL}$ are likely to be preferentially concentrated in the hottest (thus deepest) part of the TBL where the temperature approaches the mantle solidus. At larger distances from the CMB, the temperature profile could still be steeper than in the adiabatic profile (as expected within a TBL), which would however not produce detectable seismic anomalies because of the relatively lower temperatures. (ii) The thermal conductivity $k$ of the lowermost mantle, which remains subject to controversial reports. A recent study based on ab initio calculations proposed $k = 3.5$ W/m/K for bridgmanite-MgSiO$_3$ (Tang et al., 2014), a value that could be even lowered, by up to 50%, if accounting for the presence of Fe and Al in the Bg-lattice (Manthilake et al., 2011). On the other hand, values up to 16 W/m/K have been proposed for MgSiO$_3$ Bg and post-Bg, with a relatively higher (but anisotropic) $k$ value for post-Bg (Ohta et al., 2012). Also, intermediate values of 7–8 W/m/K were recently proposed, for the MgSiO$_3$ end-member again (Stackhouse et al., 2015). (iii) The difference ($\Delta T_{TBL}$) between the CMB temperature (4100 ± 200 K) and the mantle temperature a few hundred kilometers above the CMB as extrapolated from the adiabatic profile (2600 ± 200 K). The temperature jump in the TBL could be ~1500 K (Fig. 2).

Using reasonable values of $e_{TBL} = 200$ km, $k = 5$ W/m/K, and $\Delta T_{TBL} = 1500$ K, we calculate a heat flux at the CMB of ~3.6 TW. This value falls well within the validity limit of our model that is 3 to 4.7 TW. We acknowledge that lower $k$ values, or a thicker TLB, would induce a lower CMB heat flux. Considering all possible values of $e_{TBL}$ from 100 to 300 km and $k$ from 2 to 8 W/m/K results in plausible CMB heat fluxes ranging from 1.0 to 11.5 TW. Unfortunately, this broad range of uncertainties does not provide additional constraints to our model. We note that it has been sug-
gested that the possible presence of a dense viscous layer above the CMB could help reducing the CMB heat flux (Nakagawa and Tackley, 2014).

On the other hand, based on the heat carried by plumes ascending from the base of the mantle to the Earth's surface, the core heat loss has been estimated to be ~2.3 TW, or perhaps up to 3.5 TW if plumes loose significant amounts of heat during their ascent through the mantle (Davies, 2007). In such thermal budget, it is difficult to take into account the possible deep mantle complexities such as the cooling effect of plate tectonics, the insulating effect of a dense basal layer or also the importance of heat sources available in an enriched deep mantle, because the amplitude of these effects remain highly uncertain. One could argue that our estimated value of the CMB heat flux is much lower than that calculated in recent geodynamic models (e.g. Nakagawa and Tackley, 2014). However, such models generally assume a fully viscous mantle, even for an initial CMB temperature of 6000 K that is well above the mantle solidus and the viscous limit of 40% of partial melt. As acknowledged in Nakagawa and Tackley (2014), this artificially maintains a hot core for a long period of time, associated with a substantial CMB flux. In a recent study, it was instead shown that an extremely large CMB heat flux prevailed early in the Earth's history, until the viscous transition is reached in the lowermost mantle (Monteux et al., submitted for publication). It yields a CMB temperature of ~4400 K in less than ~1 My after the MFI, associated with a much lower CMB heat flux after the early fast cooling.

## 8. Implications for geodynamics and major geological events

In addition to generating the Earth's dynamo, turbulent motions excited by astronomical forcing can induce cycles of growth, saturation and abrupt relaxation of the hydrodynamic instabilities (Kerswell, 1993; Le Bars et al., 2010). The collapses could induce an abrupt release of energy, potentially up to $10^9$ TW (Kerswell, 1996) over short periods of time, in addition to the resonances in the Earth-Moon-Sun spin-orbit system (Greff-Lefftz and Legros, 1999). The pulse duration could vary from a couple periods of rotation (a couple of days) to several hundred years. This corresponds to a broad range of thermal energy release, which could induce core heating by a few to a hundred of degrees, depending on the integrated pulse amplitude. We note that the 0.5 to 1 TW currently dissipated into the Earth's outer core from the Earth-Moon-Sun orbital system cannot heat the core more than a couple hundred degrees. However, much larger heat pulses could have happened in the past when the Moon was closer to the Earth and when the Earth was rotating faster (Fig. 3).

Such fluctuations of the CMB temperature could have two major consequences. (a) Following the adiabatic temperature profile of the Fe-alloy from the CMB to the ICB, they should induce fluctuations in the size of the inner core (Fig. 2(b)): The abrupt release of hydrodynamic instabilities could reduce the size of the inner core and restore its capability to produce the geodynamo by chemical buoyancy, when the CMB temperature would eventually decrease again by cooling due to weaker dissipation by mechanical forcing. The possibility that an old inner core has undergone several changes in its size, with a rapid decrease and slow increase of its radius could be important for building the inner core anisotropy (Poupinet et al., 1983) as well as a mushy layer at the top of the inner core (Loper and Fearn, 1983). Indeed, both geophysical interpretation are closely related to the mechanism of inner core crystallization. (b) On the other hand, partial melting in the lowermost mantle could act as an efficient agent for transferring the excess heat of the core to the overlying mantle: Increasing the temperature above the mantle solidus at the CMB would result in an increase in the degree of partial melting in the lowermost

mantle, which in turn would induce a larger CMB heat flux. This mechanism could damp the fluctuations in heat production in the turbulent outer core yielding a stable CMB temperature, precisely at, or just below, the mantle solidus. This thermal state corresponds well to the present-day view of the D"-layer, where piles of partially molten silicate material (the ultra-low velocity zones) interact with mantle convection. Adding heat to the current lowermost mantle would certainly enhance partial melting and the thermal instabilities (inset in Fig. 3). As a result, one should expect an increase of the volcanic activity at the Earth's surface shortly after the influx of heat at the CMB (Greff-Lefftz and Legros, 1999). If the brutal energy influx is important, this could explain dramatic eruptions such as the Deccan Trapps (Courtillot and Fluteau, 2010), as well as the periodic growth of continents at the Earth's surface (Arndt and Davaille, 2013; Martin et al., 2014).

Finally, because the Moon appears to be a necessary ingredient to sustain the magnetic field, and because a magnetic field is needed to shield the Earth's atmosphere from erosion by solar wind (e.g. Dehant et al., 2007), the habitability of Earth-like planet may be subordinated to the existence of a large satellite. While more than 1000 exoplanets have already been observed, the detection of an accompanying exo-moon is rare (Bennett et al., 2014). Hence, our model could have major implications in future planetary missions as exoplanets with orbiting moons would more likely host extraterrestrial life.

## Acknowledgements

We thank D. Cébron, B. Favier, R. Ferrari, H. Martin and the anonymous reviewers for fruitful discussions and help. This work has gained value from previous collaborative works with N. Bolfan-Casanova, M.A. Bouhifd, M. Mezouar and G. Morard. This is an ANR-OxyDeep and ANR-Clervolc contribution No. 198.

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JGR Planets

RESEARCH ARTICLE

10.1029/2019JE005979

**Key Points:**

* Incorporation of S significantly reduces both electrical and thermal conductivities of FeS

* The low thermal conductivity of FeS may have prevented the Mercurian core from rapid solidification

* Mercury dynamo can be driven by both thermal and chemical buoyancy forces

**Correspondence to:**

G. Manthilake,
geeth.manthilake@uca.fr

**Citation:**

Manthilake, G., Chantel, J., Monteux, J., Andrault, D., Bouhifd, M. A., Bolfan Casanova, N., et al. (2019). Thermal conductivity of FeS and its implications for Mercury's long-sustaining magnetic field. *Journal of Geophysical Research: Planets*, 124. https://doi.org/10.1029/2019JE005979

Received 22 MAR 2019

Accepted 14 AUG 2019

Accepted article online 21 AUG 2019

**Author Contributions:**

**Conceptualization:** G. Manthilake

**Data curation:** G. Manthilake

**Formal analysis:** G. Manthilake, J.

Chantel, D. Andrault, M. A. Bouhifd, N. Bolfan Casanova, E. Boulard

**Funding acquisition:** G. Manthilake

**Investigation:** G. Manthilake, J.

Monteux, D. Andrault

**Methodology:** G. Manthilake, N.

Guignot, A. King, J. P. Itie

**Project administration:** G.

Manthilake

**Resources:** G. Manthilake, N. Guignot, A. King, J. P. Itie

**Writing - original draft:** G.

Manthilake, J. Monteux

**Writing - review & editing:** G.

Manthilake, J. Monteux

Thermal Conductivity of FeS and Its Implications
for Mercury's Long-Sustaining Magnetic Field

G. Manthilake¹ ID, J. Chantel² ID, J. Monteux¹ ID, D. Andrault¹, M. A. Bouhifd¹, N. Bolfan Casanova¹, E. Boulard³ ID, N. Guignot³, A. King³, and J. P. Itie³

¹Laboratoire Magmas et Volcans, Université Clermont Auvergne, CNRS, IRD, OPGC, Clermont-Ferrand, France, ²Unite Matériaux et Transformations, CNRS, Université de Lille, Villeneuve d'Ascq, France, ³Sorbonne Universite, Museum National d'Histoire Naturelle, UMR CNRS 7590, IRD, Institut de Mineralogie, Physique des Materiaux et Cosmochimie - IMPMC, Paris, France

**Abstract** The MESSENGER mission revealed that Mercury's magnetic field might have operated since 3.7–3.9 Ga. While the intrinsic magnetism suggests an active dynamo within Mercury's core, the mechanism that is responsible for sustaining the dynamo for prolonged period of time remains unknown. Here we investigated the electrical conductivity of Fe-S alloys at pressure of 8 GPa and temperatures up to 1,700 K. We show that the electrical conductivity of Fe-S alloys at 1,500 K is about 10³ S/m, 2 orders of magnitude lower than the previously assumed value for dynamo calculations. The thermal conductivity was estimated using the Wiedemann-Franz law. The total thermal conductivity of FeS is estimated to be ~4 Wm/K at the Mercurian core-mantle boundary conditions. The low thermal conductivity suggests that a thermally driven dynamo operating on Mercury is more likely than expected. If coupled with chemical buoyancy sources, it is possible to sustain an intrinsic dynamo during time scales compatible with the MESSENGER observations.

**Plain Language Summary** Mercury's weak magnetic field intensity that persisted over the last 3.9 billion years has long baffled the planetary science community. Various explanations have been proposed; nevertheless, there has been no consensus on how intrinsic dynamo with such weak energetics have existed in Mercury for prolonged period. In our submitted manuscript, we exclusively looked at the electrical and thermal conductivity in Fe-S alloys, the dominant phase in planet Mercury's outer core. Our results indicate that electrical conductivity of Fe-S alloys is 2 orders of magnitude lower than the previously assumed value for dynamo calculations. The low thermal conductivity obtained in this study suggests that the heat extraction from Mercury's core through the solid Fe-S layer is a highly inefficient process. Our estimations suggest the heat-flux from the core only produces less than 1 TW of energy. The low heat flux prevents the Mercurian core from rapid solidification, sustaining an intrinsic dynamo in Mercury since ~3.9 Ga.

**1. Introduction**

The planet Mercury currently exhibits a surface magnetic field with a *field* strength representing 1.1% that of the Earth's magnetic field (Anderson et al., 2011). The spin-aligned axisymmetric and offset dipole-dominated field is considered unique among planetary magnetic fields (Tian et al., 2015). The low-altitude magnetic field measurements by the MESSENGER mission have revealed a remnant magnetization in Mercury's crust with age of magnetization in the range of 3.7–3.9 Ga (Johnson et al., 2015). The presence of a magnetic field early in Mercury's history as early as 3.9 Ga and the extremely weak magnetic field strength in present-day Mercury suggest that Mercury's magnetic field may have been active for the last 3.7–3.9 Ga. While the observed intrinsic magnetism suggests an active dynamo generated in Mercury's liquid outer core, how Mercury's magnetic field sustained for a prolonged period of time remains puzzling even in the light of our current understanding of the Earth and planetary dynamos (Johnson et al., 2015).

It has been suggested that Mercury's dynamo is currently generated by chemical convection (Breuer et al., 2015; Cao et al., 2014; Chen et al., 2008; Dumberry & Rivoldini, 2015). The low magnetic field intensity of planet Mercury has been discussed using thermoelectric (Stevenson, 1987), thin shell (Stanley et al., 2005), thick shell (Heimpel et al., 2005), and feedback (Glassmeier et al., 2007) dynamo models. Recent dynamo studies linked the low intensity and the axisymmetry of the magnetic field to a stably stratified
layer with latitudinal heat-flux variations (Christensen & Wicht, 2008; Christensen, 2006; Schubert et al., 2004; Tian et al., 2015).

The chemical composition and the internal structure of Mercury remain poorly understood mainly due to the lack of observational constraints (Margot et al., 2018). Mercury's solid outer shell overlying the liquid core is estimated to be ~400 km thick and believed to consist mainly of silicate minerals (Hauck et al., 2013; Rivoldini & Van Hoolst, 2013). Recent gravity field analyses indicate that Mercury's internal structure consists of an Fe-S-Si liquid outer core (Chabot et al., 2014; Hauck et al., 2013; Knibbe & van Westrenen, 2015; Margot et al., 2018; Rivoldini & Van Hoolst, 2013; D. E. Smith et al., 2012). A substantial number of studies have considered S and Si as principal alloying elements in the liquid outer core (Chabot et al., 2014; Hauck et al., 2013; Malavergne et al., 2010; Namur et al., 2016; Rivoldini et al., 2009; Schubert et al., 1988; Stevenson et al., 1983). The presence of an inner core in Mercury has been debated (Margot et al., 2018). Different models for Mercury's interior suggest the presence of a solid inner core, which may have formed as a result of planetary cooling (Veasey & Dumberry, 2011). Recent estimations indicate that the diameter of the solid inner core would be 0.3-0.7 of the diameter of Mercury's core (Genova et al., 2019).

The elemental composition analyses during the MESSENGER mission indicate 1-4 wt % of Fe and S in Mercury's surface (Evans et al., 2012; Nittler et al., 2011). The high S abundance in the Mercury's mantle coupled with low surface abundance of Fe suggests that Mercury may have formed from highly reduced CB chondrites (Namur et al., 2016). The recent study based on S solubility in reduced mafic silicate melts suggest 7-11 wt % S in the mantle and <1.5 wt % S in the metallic inner core for a bulk S contents of 4 wt % in Mercury (Namur et al., 2016). Similarly, metal-silicate partitioning experiments indicate decrease of S contents in the core when Si contents increase (Chabot et al., 2014). These experimental results indicate that Mercury's core must contain Si if the S content in the core is <20 wt %. If Si contents in the core would be >10 wt %, the S contents in the core should be <2 wt % (Chabot et al., 2014).

The unusual feature in Mercury's internal structure models is the possibility of a solid FeS layer at the core-mantle boundary (D. E. Smith et al., 2012). The higher bulk density observed for Mercury's outer shell appears to be compatible with a silicate mantle coupled with an FeS layer with a thickness of ~90 km (Hauck et al., 2013; Namur et al., 2016). The immiscibility of Fe-S and Fe-Si liquids in the Fe-S-Si ternary system would result in segregation of Fe-S from Fe-S-Si liquids when the pressure is below 15 GPa (Morard & Katsura, 2010; Sanloup & Fei, 2004). The S contents up to 4 wt % in the silicate appear to fall within the liquid-liquid immiscibility field of Fe-S-Si system at the Mercury core-mantle boundary (CMB) pressure of about 8 GPa (Chabot et al., 2014; Namur et al., 2016). The origin of a FeS layer could also be explained by the crystallization of FeS in the binary Fe-FeS system from the segregated S-rich Fe liquids (Fei et al., 1997; Hauck et al., 2013). Due to the density contrast between solid FeS and Fe-S-Si liquid, the crystallized FeS is expected to float, forming a stable layer beneath the silicate mantle (Hauck et al., 2013). The possibility of a liquid FeS rich layer overlying an Fe-S-Si core has also been discussed in a recent study (Pommier et al., 2019). In this model, an insulating liquid FeS layer with the thickness >40 km is expected to control the heat flow from the core influencing the generation and the sustainability of Mercury's magnetic field (Pommier et al., 2019).

While a solid FeS layer at the CMB has been an important feature in Mercury's interior models, a number of recent studies have supported the low S contents or S absent conditions in Mercury's core, thereby questioning the stability of an FeS layer at the CMB (Chabot et al., 2014; Genova et al., 2019; Knibbe & van Westrenen, 2018; Margot et al., 2018). The silicate-metal partitioning experiments (Chabot et al., 2014) suggest S-free conditions, if the Si contents exceed 25 wt % in Mercury's core. However, such extreme partitioning of Si into the core would alter the major element ratios in the silicate mantle. Recent study investigating the density of FeS- and S-rich liquids (Knibbe & van Westrenen, 2015) reported higher density for FeS compared to the residual liquids, requiring a separate mechanism to stabilize a FeS layer at the base of Mercury's mantle. The recent geodetic constrains indicate a slightly higher value for gravitational potential Love number than previous estimations, supporting for a warm and weak mantle rather than rigid FeS layer at Mercury's CMB (Genova et al., 2019).

If the presence of a solidified FeS layer at the boundary between Mercury's liquid outer core and solid silicate mantle is assumed; this could have significant implications for the core dynamics. It has been suggested that the weak magnetic field strength observed for Mercury can be explained by the presence of conductive layer
<figure><img src="image_251.png" /><figcaption>**Figure 1.** Schematic cross section of the multianvil assembly. The two sets of thermocouples placed top and bottom of the sample serve as electrodes for electrical conductivity measurements. Inset denotes a schematic diagram showing the electrode connections for the Kelvin (four-wire) resistance measurement technique.</figcaption></figure>

overlying the convective liquid outer core (Ulrich R Christensen, 2006). Due to the stratification of the liquid core, the convection driven dynamo operates only in the deep part of the liquid core and the dynamo field may diffuse through the overlying conductive layers producing weak magnetic field strengths (Christensen, 2006). The heat flow across the CMB controls the solidification of the liquid core and the buoyancy sources available to generate a planetary dynamo (B. Buffett, 2003). The electrical and thermal conductivities of Fe-S compounds are therefore key physical properties to understand Mercury's core dynamics. In this study, we investigate the electrical and thermal conductivities of Fe-S alloys at high pressure and temperature, the principal constituent of Mercury's core-mantle boundary, with the aim of understanding the influence of Fe-S layer on the heat extraction from Mercury's core. We discuss how the outer core composition evolution influences the heat extraction from the core of Mercury and favor an intrinsic dynamo driven by thermal and chemical buoyancy forces and sustained for a prolonged period.

## 2. Methods

Three different FeSₓ samples with varying sulfur contents (x, in mol) 0.01, 0.02, and 0.05 were prepared mixing reagent grade Fe and S powders. For the FeS (x = 0.5) composition we used commercially available reagent grade FeS power. The powder mixtures were compressed to desired pressure and kept at 1,000 K for more than 1 hr to obtain solid cylindrical samples for electrical conductivity measurements in multianvil apparatus. The high-pressure and high-temperature experiments were conducted at 8 GPa up to 1,700 K using the 1,200-t DIA-type multianvil module at the PSICHE beamline in SOLEIL Synchrotron (France). Additional off-line experiments were performed using 1,500-t multianvil press at the Laboratoire Magmas et Volcans, France. For high-pressure generation, we used an octahedral MgO + Cr₂O₃ pressure medium in a 10/4 multianvil configuration. In order to electrically insulate the sample from the Re furnace during the measurements of electrical conductivity, we placed the polycrystalline Fe-S samples within a MgO capsule. The two iron (Fe) disks placed on the top and at the bottom of the cylindrical Fe-S sample served as electrodes for electrical conductivity measurements. Sample temperature was monitored using a tungsten-rhenium (W₉₅Re₅-W₇₄Re₂₆) thermocouple junction located at one end of the sample (Figure 1).

Energy-dispersive X-ray diffraction using a CAESAR-type diffractometer (Wang et al., 2004) was used to determine the sample pressure and verify the sample state. We use the pressure–volume–temperature
<figure><img src="image_252.png" /><figcaption>**Figure 2.** Electrical conductivity of Fe-S compounds. (a) As a function of reciprocal temperature for various sulfur contents. Note that the compositions are shown next to the fitting lines. Previous works on electrical conductivity of iron alloys are shown for comparison: A59 (Argyriades et al., 1959), D13 (Deng et al., 2013), P39 (Powell, 1939), P18 (Pommier, 2018), and P19 (Pommier et al., 2019) (b) Electrical conductivity before and after melting of FeS. The activation enthalpy changes with melting and the electrical conductivity of molten FeS depends weakly on temperature.</figcaption></figure>

equation of state of MgO to determine the sample pressure and the accuracy of pressure determination is estimated to be ~0.5 GPa. Diffraction peaks corresponding to Fe-S alloys are clearly visible in solid samples and the melting of Fe-S is characterized by a broad band of diffuse scattering as described previously elsewhere (Andrault et al., 2018). We also acquired radiographic images of the sample to determine the sample lengths, which were crucial for the accurate determination of electrical conductivity at each temperature. The focused beam configuration used in these experiments has the advantage to produce almost flat field corrected images, a feature usually not available when using multianvil modules (Andrault et al., 2018).

The electrical conductivity measurements were performed using the Kelvin resistance measurement technique (e.g., Deng et al., 2013). In this method, the digital multimeter measured the current going through the subject and the voltage drop across it, eliminating the lead and contact resistance from the measurement (Figure 1). This is an advantage for precise measurement of sample with low-resistance values such as metals. The electrical conductivity at each temperature can be calculated using the sample dimensions obtained using the X-ray radiograph using, $\sigma = l/RA$ where $\sigma$ is the electrical conductivity, $l$ is the sample length, $R$ is the measured sample resistance, and $A$ is the area of the cylindrical cross section. The temperature dependence of the electrical conductivity $\sigma$ can be described by the Arrhenius equation: $\sigma = \sigma_0e^{-\Delta H/k_B T}$, where $\sigma_0$ is the preexponential factor, $\Delta H$ is the activation enthalpy in eV, $k_B$ is the Boltzmann constant in eV/K, and $T$ is the absolute temperature in K.

Thermal conductivity of Fe-S alloys was estimated using the Wiedemann-Franz law $\kappa = L_0\sigma T$, an empirical law expressing the thermal conductivity ($k$) as a function of the electrical conductivity of a metal ($\sigma$) and the temperature ($T$) through a proportionality constant Lorenz ratio ($L$). The Lorenz ratio approaches the Sommerfeld derivation $L_0$, $2.44 \times 10^{-8}$ WK$^2$, provided that the thermal conductivity is purely electronic, the electron gas is highly degenerate, and the relaxation time is the same for electrical and thermal conduction (Klemens, 1989). The deviation of the Lorenz ratio has been observed for Fe alloys (de Koker et al., 2012; Secco, 2017). In metallic alloys such as Fe-S, the thermal transport is a complex process due to the coexistence of electron and phonon conduction. The total thermal conductivity ($\kappa_{tot}$) is the sum of its electronic component ($\kappa_e$), and its phononic component ($\kappa_{ph}$). In high-purity metals, the electronic component dominates the thermal conduction; however, in metallic alloys the heat conduction through phonon vibration becomes significant (Klemens & Williams, 1986; Konôpková et al., 2016).
<figure><img src="image_253.png" /><figcaption>**Figure 3.** X-ray diffraction patterns of FeS recorded at high temperature. (a) Evolution with increasing temperature of the diffraction pattern for the run performed at ~8 GPa. Red and blue dots correspond to diffraction peaks indexed based on FeS-V and Fe₃O₄ magnetite, respectively. The ratio of diffraction peak intensities (following the Rietveld approach) suggests presence of magnetite in the sample, which Fe₃O₄ could come from the unavoidable oxidation of FeS starting material. The general drift of diffraction peaks toward low energies is due to thermal expansion. Diffraction peaks get thinner at high temperatures due to the release of deviatoric stresses built upon compression at 300 K. The melting of the sample occurs between 1,400 and 1,500 K. (b–d) At 1,700 K after melting of FeS at two-theta diffraction angles of 4, 6, and 8°, respectively.</figcaption></figure>

### 3. Results

Our findings show that the electrical conductivity of Fe-S compounds decreases with increasing temperature characteristic of the electrical conduction in metals and metal alloys (Figure 2a). We observe that even a minor increase of sulfur contents reduces the electrical conductivity of Fe-S alloys (Figure 2a). At 1,000 K, the incorporation of 0.01 mol of S in Fe-S alloys decreases the electrical conductivity by 2 orders of magnitude to $10^4$ S/m compared to pure Fe ($10^6$ S/m; Powell, 1939). The electrical conductivity of FeS, corresponding to the sulfur content expected at the solid layer at the Mercury's core-mantle boundary is thus expected to be about $10^2$ S/m at 1,700 K. This value is about 3 orders of magnitude smaller than the value of $10^5$ S/m used in previous dynamo models (D. J. Stevenson, 2003). A change in activation enthalpy on melting of FeS (Figure 2b) suggests a change of conduction mechanism from electronic conduction in a solid to electrolytic conduction in FeS melt.

The melting of FeS is confirmed by the disappearance of peaks above 1,500 K (Figure 3), consistent with experimentally derived Fe-S phase relations (Urakawa et al., 2004). The electrical conductivity of molten FeS is about $10^2$ S/m at 1,700 K and remains relatively unchanged with increasing temperature.

Our measured electrical conductivities of Fe-S compositions at 8 GPa indicate significantly lower values compared to the previously reported values for Fe-S alloys (Pommier, 2018; Figure 4). While we could not determine the exact cause of the discrepancy among different laboratory measurements, it is possible that the high conductivity may originate from the presence of impurities in Fe-S alloys. For example, the experiments conducted on Al- and Sr-doped FeS₂ (Osuwa & Nwaokeorie, 2014) indicate that 0.02 M concentration of Al and Sr increases the electrical conductivity of FeS₂ by more than a factor of 10. However, calculations based on first-principle electronic band structure calculation of hcp Fe-based alloys at 40 GPa (Gomi & Yoshino, 2018) indicate that Si impurity concentrations up to 30 at. % is required to decrease the electrical conductivity by a factor of 100. Another possibility for high electrical conductivity could be the partial reaction of Fe and S to form Fe-S alloy with high Fe contents.

### 4. Discussion

In metallic alloys, the electrical conduction is controlled solely by the movements of electrons, while heat is carried by both electron vibrations and phonon collisions (Klemens & Williams, 1986). It is generally assumed that the phonon (lattice) thermal conductivity is negligible in metals and the electronic component is determined via the Sommerfeld value of the Lorentz ratio ($L_0$). The estimated electronic component of thermal conductivity varies from 0.5 Wm/K for FeS$_{0.01}$ to 0.006 Wm/K for FeS at 1,300 K (Figure 5). These values represent the lower bound of the thermal conductivity for Fe-S alloys (Pommier, 2018).

In metallic alloys however, scattering of electrons by solute atoms greatly reduces the electronic thermal conductivity, making heat transport via lattice vibrations a dominant factor (Klemens & Williams, 1986). Thus, for metallic alloys at high temperature, the thermal conductivity is described by the Smith-Palmer equation (C. S. Smith & Palmer, 1935), $\kappa = L_0\sigma T+D$, an empirical relation linking thermal conductivity to electrical conductivity via a constant $D$, which describes the temperature-dependent phonon thermal conductivity ($\kappa = L_0\sigma T+DT^{-a}$). Due to strong solute scattering, the phonon thermal conductivity of alloys are
<figure><img src="image_254.png" /><figcaption>**Figure 4.** Electrical conductivity as a function of sulfur content in Fe-S alloys. Electrical conductivity decreases with increasing sulfur contents. Previous electrical conductivity data of Fe-S alloys and Fe are shown for comparison. P18 (Pommier, 2018), A59 (Argyriades et al., 1959), D13 (Deng et al., 2013), and OK13 (Osuwa & Nwoakeorie, 2014).</figcaption></figure>

<figure><img src="image_255.png" /><figcaption>**Figure 5.** Thermal conductivity of Fe-S compounds. The electronic thermal conductivity estimated based on the Sommerfeld derivation of the Wiedemann-Franz law for our Fe-S compositions are shown in blue circles. The total thermal conductivity (electronic + phonon) of Fe, FeS₂, and S are shown in red circles. The thermal conductivity of FeS was estimated to be ~4 Wm/K at 1,000 K.</figcaption></figure>

expected to approach a constant value at high temperature (Klemens & Williams, 1986). For example, the lattice thermal conductivity Fe-Cr-Ni alloy systems converge into a narrow range of 4.6–6.0 W/m K at 1,000 K (Klemens & Williams, 1986).

The lattice thermal conductivity measurements of Fe-S alloys were limited to FeS₂ composition, which was reported at supercooled conditions in the temperature range 50–300 K (Popov et al., 2013). The thermal conductivity of FeS₂ at 300 K is estimated to be 42 ± 1 Wm/K. Extrapolation of this value to high temperature based on the theoretically predicted temperature dependence of phonon thermal conductivity of $T^{-a}$ (Klemens & Williams, 1986) indicates that the thermal conductivity of FeS₂ at 1,300 K would be about 3.6 W/m K. In order to estimate the total thermal conductivity of FeS (electronic + lattice), we have modeled the thermal conductivity of end-member Fe-S compositions together with data from the FeS₂ system (Figure 4). The thermal conductivity of Fe was calculated from electrical conductivity data of Powell (1939). Thermal conductivity of S at 300 K was obtained from Slack (1965). By fitting the total thermal conductivity of Fe, FeS₂, and S with powerlaw relation, we obtain a thermal conductivity ~4 Wm/K for the FeS at 1,300 K (Figure 5).

The temperature at Mercury's core mantle boundary has been investigated in numerous previous studies (Grott et al., 2011; Hauck et al., 2013; Tosi et al., 2013). The solidification of FeS below the CMB can be expected if the temperatures at the core mantle boundary decrease below 1,700 K (Breuer et al., 2015). We have obtained the thermal conductivity at 1,300 K, close to the eutectic temperature of Fe-FeS system at 10 GPa (Fei et al., 1997; Morard et al., 2007). Due to the weak temperature dependence of thermal conductivity at high temperature, a significant variation to the estimated phonon thermal conductivity could not be expected within 1,200–1,700-K temperature range.

Based on our estimation of the thermal conductivity of 4 Wm/K assumed for FeS at 1,300 K, we have computed the Mercurian core heat flow for a range of thermal boundary layer (TBL) thicknesses and possible temperature gradients across the TBL (Hauck et al., 2013; Rivoldini et al., 2009; Tosi et al., 2013; Figure 6). It is worth noting that our estimated value for the Fe-S thermal conductivity at the CMB conditions is close to the mantle thermal conductivity value usually assumed for Mercury's mantle convection models (Breuer et al., 2010; Tosi et al., 2013) meaning that the two reservoirs may be indistinct in terms of thermal conductivity. We have calculated the heat flow $Q$ based on the Fourier's law of thermal conduction:

$$Q = 4\pi R_c^2 k_{tot}(\Delta T/\delta)$$

with $R_c$ the core radius (=2,000 km for Mercury), $\Delta T$ the temperature difference across the thermal boundary layer, and $\delta$ the boundary layer thickness. The estimated heat flows computed from our thermal conductivity model indicate that the energy evacuated by core heat flow is ≈10 TW for a TBL with a thickness of 10 km and a temperature contrast of 500 K (Figure 4). The heat flow from the core is substantially reduced when increasing the TBL thickness, to ≈1 TW for a 100-km-thick TBL layer.

To drive a dynamo in a terrestrial planet, three conditions are necessary (e.g., Monteux et al., 2011): (1) the metallic core has to convect meaning
<figure><img src="image_256.png" /></figure>

**Figure 6.** Mercury's core-mantle boundary heat flow. The calculations are based on an average total thermal conductivity (electron + phonon) of 4 Wm/K for FeS for diverse thermal boundary layer thicknesses as a function of the temperature contrast across the TBL. The adiabatic heat flow ($Q_{ad} = 4\pi R_c^2 k_{tot} \alpha_c g_c T_c / C_p$) is represented with a black horizontal dashed line (considering $R_c = 2,000$ km, $\alpha_c = 7E-5$ K, $g_c = 4 \text{ m/s}^2$, $T_c = 1,800$ K, and $C_p = 850 \text{ J·kg}^{-1}·\text{K}^{-1}$).

that the heat flow out of the core needs to overcome the adiabatic heat flow (David J. Stevenson et al., 1983), (2) the rate of gravitational potential energy released by convection has to be much larger than the rate of ohmic dissipation (B. A. Buffett, 2002), and (3) the convective motions have to exhibit a complex structure to carry the magnetic field lines (leading to a critical magnetic Reynolds number; U. R. Christensen & Aubert, 2006). Figure 6 shows that the heat flow inferred from our study is larger than the adiabatic heat flow for a wide range of temperature contrasts and thermal boundary thicknesses meaning that thermally driven dynamo is a likely process on Mercury. Low values for both thermal and electrical conductivities remain puzzling to explain Mercury's magnetic field. In the Earth's core, values of the thermal conductivity range between 90 and 150 Wm/K (de Koker et al., 2012; Pozzo et al., 2012). Recent laboratory measurements suggested that the thermal conductivity of polycrystalline iron at Mercury's core conditions is 113–125 Wm/K (Deng et al., 2013). Large thermal conductivities increase the heat flux along the core adiabat and reduce the lifetime of a thermally driven dynamo (Breuer et al., 2015). The electrical conductivity of Fe-S at P/T conditions compatible with Mercury's core is about 2 orders of magnitude lower than for pure iron. To overcome a critical magnetic Reynolds number of 50 (Wicht et al., 2007), 20mm/s typical flow speed is required to maintain an active dynamo for Mercury, which is about 200 times stronger than that of the Earth's outer core. Moreover, if sulfur is the major light element in Mercury's core and if its weak magnetic field is related to a low electrical conductivity, our results would also suggest a very weak magnetic field for Ganymede, which contradicts Galileo magnetometer measurements (Kivelson et al., 1996).

It is worth noting that the heat flux from the core is controlled by the thermal evolution of the mantle. In the early history of Mercury, its core was likely to be liquid with Fe alloys (Fe-S, Fe-Si; Figure 7a). Consequently, the thermal conductivity of the core was large; the primitive heat was efficiently evacuated possibly enhanced by a surrounding partially molten silicate magma ocean. In this early regime, the dynamo was only thermally driven. Numerical models (Tosi et al., 2013) assuming a core thermal conductivity of 40 Wm/K indicate that a thermally driven dynamo would be feasible during the first few hundred million years of the evolution of Mercury. The value of the thermal conductivity inferred from our study would reduce the adiabatic core heat flux threshold for thermally driven dynamo from ≈20 mW/m² (Tosi et al., 2013) to ≈3 mW/m². According to Tosi et al. (2013), such a low-threshold heat flux value could be overcome during the last ≈2 Gyr of Mercury's history. Our estimations (Figure 6) indicate that such heat-flux values can be achieved even with a presence of a thin (<1 km) FeS layer.

A low thermal conductivity within the core can enhance thermal convection by reducing the threshold for thermally driven dynamo. In the case of a vigorous and thermally driven convection, a stratified outer core would be difficult to envision as well as a strong skin-effect attenuation (Ulrich R. Christensen et al., 2009). However, a low thermal conductivity of the core combined with an overlying solid mantle would also limit the heat evacuation and favor the formation of a stable layer below the CMB. For a thick Fe-S layer to form and to possibly affect the magnetic field of Mercury, a complementary process can be invoked such as the formation of FeS layer during the solidification of Mercury's core. The growth of the inner core leads to an increase of the S concentration in the outer core (Figure 7b). However, for crystallization to progress, the energy barrier for the formation of the first nucleus needs to be overcome (Huguet et al., 2018). In the classical view, either a critical supercooling of the core on the order of 1,000 K or transportation of solid metallic seeds from the mantle to the core when core reached liquidus temperature are possible mechanisms that may surpass the nucleation energy barrier. It has been proposed recently that such nucleation substrates can be present in the core during the early in the planet's history, leading to a slower inner core growth with smaller buoyancy flux (Huguet et al., 2018).
<figure><img src="image_257.png" /></figure>

**Figure 7.** Schematic diagram illustrating the evolution of Mercury's dynamo. The thermochemical evolution of Mercury's interior coupled with core-mantle dynamics. (a and b) The early stages of Mercury core evolution indicate largely a thermally driven dynamo in Mercury due to high heat flux from the core. The present Mercury should have a low heat flux due to the presence of FeS layer at the core-mantle boundary and the dynamo may be powered by both chemical- and thermal-driven processes.

As the Fe-S concentration increases, the thermal conductivity of the outer core should decrease (Pommier, 2018). The low thermal conductivity (4 Wm/K) inferred from our experiments means that dynamo action in Fe-S liquid core can be driven by thermal convection. At present, the second source of buoyancy force is likely to be related to chemical processes driven by crystallization in the iron-rich core (Breuer et al., 2015; Dumberry & Rivoldini, 2015) but the low thermal conductivity of Fe-S means that the dynamo can be both thermally and chemically driven (Figure 7c). The sum of these two processes may explain the longevity of the dynamo of Mercury. Moreover, by limiting the amount of heat flowing from the core, the solidification of the Fe-S layer may have prevented the Mercurian core (and potentially the Ganymede's core) from rapid solidification, sustaining an intrinsic dynamo by buoyancy forces in Mercury for time scales larger than 1 Gyr compatible with the MESSENGER mission observations.

## Acknowledgments

Discussions with David Stevenson and Ulrich R. Christensen help improved the manuscript. We thank the Editor Steven A. Hauck, II, A. Pommier, and the anonymous reviewer for their constructive comments. G.M. acknowledges funding from the French PNP program (INSU-CNRS). This research was financed by the French Government Laboratory of Excellence initiative ANR-10-LABX-0006, the Région Auvergne, and the European Regional Development Fund. This is ClerVolc contribution number 366. The authors confirm that the raw data supporting the findings of this study are available within the article (Figures 2–4).

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