Coupled fates of Earth’s mantle and core: Early sluggish-lid tectonics and a long-lived geodynamo

Abstract

Conventional Earth evolution models are unable to simultaneously reproduce two fundamental observations: the mantle’s secular temperature record and a long-lived geodynamo before inner core nucleation. Today, plate tectonics efficiently cools the mantle, but if assumed to operate throughout Earth’s history, past mantle temperature and plate motion become unrealistically high. Through coupled core-mantle modeling that self-consistently predicts multiple mantle convection regimes, we show that over most of the Precambrian, Earth likely operated in a distinct “sluggish-lid” tectonic mode, characterized by partial decoupling between the lithosphere and mantle. This dominant early regime is due to a hotter Earth and the presence of the asthenosphere. This mode regulates the core-mantle boundary heat flow, which powers the geodynamo before inner core nucleation. Both sluggish-lid tectonics and a long-lived dynamo demonstrate the inextricably connected paths of the core-mantle system. Moreover, our simulations simultaneously satisfy diverse geological observations and are consistent with emerging interpretations of such records.

In Precambrian tectonics, plates passively slide over a fast-moving mantle and exhibit a strong influence on core evolution.

INTRODUCTION

The modern style of tectonics on Earth, plate tectonics, is characterized by large, rigid plates, separated by weak zones with high deformation rates: subduction zones, mid-ocean ridges, and transform faults. Plates progressively cool and become dense. Once unstable, they subduct into the mantle, creating space for new mantle material to erupt to the surface, actively participating in mantle convection that ultimately cools our planet. They move relative to one another at a rate of ∼1 to 10 cm/year. The Earth is unique in this surface expression of mantle convection in the solar system; we do not observe plate tectonics on other silicate planetary bodies [but these can instead exhibit other forms of tectonics, see, e.g., (1)]. Why Earth exhibits plate tectonics and when it initiated is the focus of many studies across Earth and planetary sciences. A recently invigorated idea is that Earth did not always operate in the plate tectonics regime but transitioned into it from a different tectonic mode (1–5). However, there is disagreement on what exactly preceded plate tectonics and when the transition (or transitions) occurred (6–10).

The paleomagnetic record suggests the continual presence of relative and moderate plate motion as early as 3.25 Ga ago (Fig. 1A) (11–13), indicative of some form of surface mobility. Whether this mobility is driven by the negative buoyancy of the plates themselves (i.e., slab pull, as is characteristic of modern plate tectonics and what we term “active-lid” tectonics), or by the motion of the mantle below (i.e., “sluggish-lid” tectonics) (4, 14, 15), is a fundamental question in our understanding of planetary evolution. The history of cooling, degassing, atmospheric and oceanic evolution, magnetic field generation, as well as crustal generation and orogenesis are all shaped by the forces that drive plate motion (1). Hence, this calls for rigorous and systematic exploration into the cause of transitions between tectonics regimes (16).

Fig. 1. Compilation of geologic Earth evolution observations.

(A) Reconstructed plate velocities, U_p [pink lines, (11); blue lines, (12, 13); green lines, (62)]. (B) Mantle potential temperature, T_p, inferred from non-arc basalts and komatiites (63, 64). (C) Compilation of magnetic dipole strength (19). In (A) and (B), the shaded regions represents the envelope of successful histories with k_c = 100 W m⁻¹ K⁻¹, the same as fully successful simulations in Fig. 3.

Mantle cooling drives core cooling, which, along with the solidification of the inner core, drives convection in the liquid core. This convection produces Earth’s magnetic field through the geodynamo (17). It is well established that a substantial global magnetic field has existed through most of Earth history (Fig. 1B) (18, 19), and yet meeting this basic observation of our planet comes with difficulty. Today, both thermal and chemical convection act to power the geodynamo. The former convective source arises from the outward flow of heat to the mantle and the release of latent heat due to solidification of the inner core, while the latter convective source arises from the expulsion of light elements during the solidification of the inner core. Before inner core nucleation (ICN), the vigor of convection is lowered without sources of chemical buoyancy, substantially diminishing the power available to drive dynamo action. This is exacerbated, in two ways by the high thermal conductivity of the liquid core alloy (k_c): (i). Such values increase the estimates of the adiabatic core-mantle boundary (CMB) heat flow, and the associated entropy sink, which must be overcome to sustain convection and therefore the dynamo action (20, 21); (ii) they yield young ages of the inner core spanning ≤500 Ma to ≤1.5 Ga (21–23). A young inner core implies that over a substantial portion of Earth history, Earth’s long-lived dynamo must have been driven by thermal convection alone. While there is currently little agreement on the value of k_c, ranging from ~20 W m⁻¹ K⁻¹ (24, 25) to ~160 W m⁻¹ K⁻¹ (26), recent efforts to reconcile this range have converged to values between 50 and 100 W m⁻¹ K⁻¹ (27). The upper range of these values has been considered prohibitively high for geodynamo generation purely by thermal convection [e.g., (28)]. In light of this, there have been proposals of more exotic mechanisms that may have powered a dynamo before ICN, which we address in Discussion, but they been found insufficient to drive dynamo generation (29). Therefore, the presence of the magnetic field requires that the core cooling rate is high enough to overcome conduction and sustain convection in the core, i.e., the net entropy in the core (E_net) ≥ 0.

Because thermal convection in the fully liquid core is mainly driven by cooling to the mantle above, the evolution of one cannot be considered without the other. The mantle must accommodate the heat input from the core via vigorous convection and redistribution of the heat, ultimately cooling to space. In contrast to active-lid tectonics, a stagnant-lid regime—whereby the mantle convects completely decoupled from an immobile lithosphere—limits mantle cooling. This leads to the equilibration of core and mantle temperatures causing convection in the core to cease (30). Therefore, the paleomagnetic evidence for a substantial global magnetic field through most of Earth’s history (Fig. 1B) (18, 19) is often linked to the existence of plate tectonics across the same period of time (11). This conflation, however, presents yet another problem: If Earth’s modern plate tectonic regime was dominant in the Archean (4.0 to 2.5 Ga ago), then the predicted mantle temperature would be much higher than indicated by the geologic record (shown in Fig. 1C). The plates are also predicted to move and subduct at unrealistically high rates compared to the geologic record (Fig. 1B). This results in a rapidly cooling mantle that would drive the geodynamo for only a brief period of time before core convection inevitably ceases.

In this study, we explore the coupled evolution of the mantle’s mobile-lid tectonic regime (we do not consider the stagnant-lid phase of Earth’s evolution if it existed) and key properties of the core (the development of an inner core and magnetic field), dictated by the flow of heat between the two domains. We ground-truth our analysis in the context of geologic observations as guiding constraints. We base our mantle evolution on a semianalytical model that, without prescribing a priori, self-consistently predicts sluggish-lid and active-lid modes of convection (31, 32). We extend this theory by allowing transitions between the tectonic modes to naturally emerge and coupling it to a core evolution model [adapted from (23)]. We explore a wide parameter space and show that despite the large uncertainty in the environment of the Archean Earth, most parameter combinations result in sluggish-lid tectonics during the Archean, with plate speeds similar to (or slightly elevated from) present-day but are instead partially decoupled from the mantle by the presence of a low-viscosity layer. Furthermore, we show that this style of convection is capable of modulating the CMB heat flow in such a way that drives sufficiently vigorous thermal convection to power the dynamo all the way up to ICN, even when using the upper limit of the core thermal conductivity values (≥100 W m⁻¹ K⁻¹) derived from experimental and ab initio studies (27), which have been considered prohibitively high to sustain a thermal dynamo (28).

Model summary

We solve our coupled system many times to extensively explore the uncertain parameters in Earth’s initial conditions and structure (resulting in a suite of >10⁶ simulations; Table 1), and the most important of these are the mantle’s viscosity structure and core thermal conductivity. We consider four layers of the mantle: the lithosphere, asthenosphere, bulk mantle, and lowermost mantle. We hold the lithospheric viscosity constant throughout the model run at 3 × 10²³ Pa·s. Although this assumption is an oversimplification of the lithospheric evolution, it is sufficient for the aim this present study, and we return to it in Discussion below. The viscosities of the rest of the mantle layers are temperature dependent: The asthenospheric viscosity is a constant fraction of the bulk mantle viscosity [where the ratio of bulk mantle to asthenospheric viscosity, r_μ, is varied from 1 (with no asthenosphere) to 1000 (very weak asthenosphere)]. The lowermost mantle viscosity is also temperature dependent, calculated on the basis of the lower thermal boundary layer temperature. We vary the reference viscosity of the lowermost mantle independently from that of the bulk mantle, and, depending on our choice of reference lowermost mantle viscosity and the temperature evolution, the lowermost mantle may be more or less viscous than the bulk mantle. We also consider the case of an isoviscous mantle where the lowermost mantle viscosity is equal to the mantle viscosity. Although we do not consider the pressure dependency of viscosity explicitly, or the existence of a compositional layer in the lower mantle, setting the lowermost mantle reference viscosity as a free parameter allows us to investigate the effect of some viscosity variations with depth on the evolution. For details, the reader is referred to Materials and Methods. Each simulation spans 4.0 billion years, starting 500 Ma after Earth formation, i.e., after core-mantle differentiation and mantle solidification [we do not consider the evolution of a basal magma ocean, which may be important in the geodynamo evolution; see (33)].

Table 1. Parameters in simulations.

Values superscripted with “L” in left column are taken from (23).

Parameter	Symbol	Values		Number of Increments
Mantle initial potential temperature	T m,0	1650–2050 K		9
CMB initial temperature	T cmb,0	4700–5700 K		11
Ratio of bulk mantle to asthenospheric viscosties	r μ	1–1000		7
Lower mantle reference viscosity	μ0,lm	1022–1025 Pa·s		10
Core thermal conductivity	k c	50–120 W m−1 K−1		15
Lower mantle thermal conductivity	k lm	4, 10 W m−1 K−1		2
Core heat capacity	C P,c	715, 750, 840 J kg−1 K−1		3
Core model:		PREM (59)	MG (65)	2
compositional density jump across ICB	Δρξ	380 kg m−3	580 kg m−3
present-day light element concentration L	ξpd	4%	5.6%
present-day center melting temperature L	T L0,pd	5700 K	5500 K

We use the ratio, r_U, between the average mantle velocity below the lithosphere and lithosphere (or plate) velocity, to define the mantle convection mode. If r_U ∼ 1, we classify the mantle as operating in the active-lid regime, whereas r_U ≥ 10 signifies the sluggish-lid regime. Since these modes are defined by a velocity ratio, sluggish-lid tectonics may still result in substantial absolute plate velocities. We consider a simulation as successful if it ends with realistic present-day, t = t_f, features of: (I) mantle potential temperature [T_P(t = t_f) = 1375 ± 75°C]; (II) active-lid convection; (III) the inner core radius [r_ic(t = t_f) = 1220 ± 240 km]; and finally, (IV), that the geodynamo is in existence over the entire simulation (i.e., E_net > 0 for all simulation time).

RESULTS

We begin by examining the behavior of three different classes of evolution scenarios: active-lid only (simulations that operate in the active-lid mode throughout the model duration, shown in red envelopes in Fig. 2), sluggish-lid only (blue envelopes in Fig. 2), and transitioning regimes, which start in the sluggish-lid then gradually move toward the active-lid mode (a subset of which is shown in green envelopes in Fig. 2). These do not necessarily reach r_U ∼ 1 but have r_U < 10. The solid-colored curves in each envelope represent individual demonstrative examples of each behavior. In the active-lid evolution scenarios, the mantle cools quickly (as shown in red envelopes Fig. 2A) due to high initial plate velocities (Fig. 2B). Depending on the initial temperature of the core and the viscosity of the lowermost mantle, the inner core may either grow to be too large (in the case of independently varying lowermost mantle viscosity) or not grow at all (in the isoviscous mantle case) (Fig. 2C). In turn, this causes the CMB heat flow (Fig. 2E) and the net entropy production (Fig. 2D) to often fall below the required value to maintain dynamo action. This behavior is characteristic of models in which the initial mantle temperature and viscosity ratio are low. The models that always operate in the sluggish-lid mode (shown in blue envelopes), on the other hand, heat up initially, then modestly cool, resulting in present-day mantle temperatures that are too hot (Fig. 2A), due to low plate velocities (Fig. 2B). Not many (0.5%) simulations fall in this category (most simulations that begin in sluggish-lid tectonics cool to an extent that a transition to active-lid tectonics occurs), as it requires very high r_U, high initial mantle and core temperatures, and high mantle thermal conductivity. The fact that core temperature and lower mantle structure play a role in maintaining the sluggish-lid regime indicates that high heat input from the core (Fig. 2E) keeps the mantle hot enough to remain in the sluggish-lid mode. Although we do not explicitly model the stagnant-lid mode, the evolution of the temperature in the stagnant-lid would likely resemble that of the extreme cases of the sluggish-lid modes. Only 8% of these scenario are able to reproduce acceptable final inner core radius. However, those that do meet this criterion are also able to sustain the geodynamo. Last, the models that start in the sluggish-lid mode then transition into the active-lid ones, the subset of which that meets criteria I to IV is shown in the green envelopes in Fig. 2, have moderate initial plate velocities and therefore cause only moderate temperature cooling. Of the cases that transition from the sluggish-lid toward the active-one and which satisfy the inner core radius criterion, 95% of them are able to sustain a long-lived geodynamo. They require the existence of an asthenosphere and are the only models that are able to produce reasonable present-day mantle temperature, inner-core radius, and tectonic mode, while maintaining a dynamo throughout the model time. Examples for the velocity profiles at the beginning and end of the model duration for each of these scenarios is shown in Fig. 2 (F to H) for the active-lid, sluggish-lid, and transitioning case, respectively. For the interested reader, we show the evolution of the velocity ratio, the Urey ratio, the lowermost mantle viscosity, lower boundary layer thickness, and CMB temperature evolution in section S1.

Fig. 2. Evolution of key quantities in simulations (where k_c = 100 W m⁻¹ K⁻¹).

(A) to (E) show the mantle potential temperature (T_P), plate velocity (U_p), inner core radius (r_ic), net entropy production (E_net), and CMB heat flow (Q_cmb), respectively. In all panels, red-shaded envelopes span simulations that exhibit active-lid tectonics (r_U ∼ 1 at all times), blue-shaded envelopes span simulations that are always in the sluggish-lid mode (r_U > 10 at all times), and the green-shaded envelopes span those that satisfy all criteria (I) to (IV), which transition from the sluggish- to the active-lid mode. The solid curves of each color show individual examples of the three behaviors shown with shading. They have identical mantle and core structures and initial conditions, except with different mantle-to-asthenosphere viscosity ratio (r_μ), where in the red curve r_U = 1, the blue curve r_U = 1000, and the green curve r_U = 100. (F to H) show the horizontal velocity profiles (from the surface to the CMB) that correspond to the examples highlighted in (A) to (E) in the specified color. The dark colors in each of these panels is the velocity profile at the beginning of the simulation, while the light color is at the end. Note the different horizontal scale in each of the panels.

As noted, the presence of an asthenosphere is the most important parameter that governs the tectonic regime during this period, where, if present, an asthenosphere strongly correlates with sluggish-lid tectonics, which is in accord with numerical studies (34). Figure 3A illustrates this behavior where we plot the median r_U (r_U again representing the velocity ratio of the average mantle to the plate) over every simulation for all r_U tested. When r_U increases, the degree of sluggishness does too. The spread of results indicates that other factors, such as mantle temperature (color-scale of symbols), also contribute to tectonic mode. Other parameters that we fix in our model (e.g., asthenospheric thickness, convective cell size, and plate viscosity) can also influence convective mode, and we test their effect in the Sensitivity Test section (section S2) in the Supplementary Materials. Furthermore, to meet present-day mantle temperature constraints (criterion I), the mantle heat loss must be reduced, which is achieved if the Archean tectonic is sluggish-lid. Figure 3B illustrates this point, where most models that meet criterion I have a median velocity ratio > 1, and all (not shown) exhibit r_U > 1 for at least a portion of the Archean.

Fig. 3. The degree of sluggishness and its effect on mantle observations.

We use the median mantle-to-plate velocity ratio during each simulation, r_U, and asthenospheric-to-bulk mantle viscosity ratio, r_μ, as proxies for the degree of sluggishness. (A) Median r_U values (over each simulation) against r_μ for all simulations, where color-scale and inverse of size indicate initial mantle temperature, T_P(t = t₀). (B) Final mantle temperature, T_P(t = t_f), and its dependence on the median r_U in the simulation time. The gray-shaded region shows the acceptable range for the final mantle temperature (criterion I).

Having established that r_μ dictates mantle convective mode and temperature evolution, we consider the effect of the mantle’s tectonic mode on the core. Core temperature evolution (and thus r_ic) is governed mainly by the lowermost mantle viscosity, μ_0,lbl (and thus the lower boundary layer thickness; Eq. 9), as depicted in Fig. 4A, which shows the present-day inner core radius decreasing with increasing lower mantle reference viscosity. However, the degree of tectonic sluggishness (r_U) also plays an important a role in dictating the core evolution, extending to dynamo generation. If we select evolution scenarios in which all parameters except the initial mantle temperature and asthenospheric viscosity ratio (r_μ) are fixed, it is clear that r_μ plays an important role in the entropy budget (Fig. 4B) and can change whether a certain evolution scenario can (i.e., scenarios with positive entropy and high viscosity ratio) or cannot (i.e., scenarios with negative entropy and low viscosity ratio) sustain the geodynamo before ICN. We focus here on the time just before ICN because during this period, there is the least entropy production due to the decrease in cooling entropy and the unavailability of inner-core solidification–related sources (Eq. 17). In other words, this is the most difficult period of time to sustain the geodynamo.

Fig. 4. Meeting core success criteria.

(A) Present-day inner core radius, r_ic(t = t_f), as a function of lower boundary mantle viscosity. Color-scale shows initial core temperature [T_cmb(t = t₀)]. The departure from the trend for log(μ_0,lbl) = 23 is because this value refers to an isoviscous lower mantle [i.e. μ_lbl(T) = μ_m(T)] rather than using the lower mantle temperature to compute the lower mantle viscosity. (B) Net entropy production (E_net) just before ICN with r_μ for simulations that satisfy the inner core radius condition (criterion III) for two sets of parameters: group 1 with the parameters μ_0,lm = 10^22.5 Pa·s, T_cmb(t = t₀) = 5600°C, k_c = 100 W m⁻¹ K⁻¹; group 2 with an isoviscous lower bulk mantle, T_cmb(t = t₀) = 5000°C, k_c = 70 W m⁻¹ K⁻¹. Both groups have C_P,c = 750 J kg⁻¹ K⁻¹ and MG core model (Table 1) (65). (C) Histograms of fully successful simulations (meeting criteria I to IV) against core thermal conductivity, k_c. Solid black line histogram demonstrates cumulative effect of sustaining the dynamo, where for any given set of model parameters and a given $k_{\mathrm{c}}^{*}$ , all criteria are met for any $k_{\mathrm{c}}\le k_{\mathrm{c}}^{*}$ . The solid pink histogram removes this cumulative effect.

Our results indicate that it is possible to sustain a thermally driven geodynamo if the dominant tectonic regime for early Earth is sluggish-lid, even for high thermal conductivity values (>100 W m⁻¹ K⁻¹). Figure 4C shows histograms of the successful simulations against core thermal conductivity. Evolution scenarios that are able to maintain a thermally driven dynamo for a given $k_{\mathrm{c}}^{*}$ are also able to maintain it for all $k_{\mathrm{c}}\le k_{\mathrm{c}}^{*}$ . This cumulative nature is illustrated by the continuous histogram (solid black line, Fig. 4C). The pink histogram represents the differences between the populations within the cumulative distribution. It is important to stress that increasing k_c will always make sustaining the geodynamo more difficult, but Fig. 4C demonstrates that a nonnegligible portion of our simulations can do so even with k_c ≥ 100 W m⁻¹ K⁻¹ (∼9% of the differential distribution).

Collectively, fully successful simulations are characterized by initial condition ranges of 1450° to 1750°C and 4400° to 5400°C mantle and CMB temperatures, respectively, with the latter depending on core thermal conductivity, asthenospheric-to-bulk mantle viscosity ratios of 50 to 1000, core thermal conductivity values <120 W m⁻¹ K⁻¹, and inner core age ranges from 0.5 to 1.5 Ga, depending on k_c. Meeting the criteria also places some constraints on model parameters, shown in section S3.

While our success criteria focus on present-day constraints, our successful simulations are also in accord with geological inferences of plate velocity and mantle potential temperature, discussed earlier, and shown by overlaying successful simulations (where k_c = 100 W m⁻¹ K⁻¹) against such geological data (Fig. 1, A and C). Thus, an early mantle characterized by sluggish-lid tectonics is not only required to meet key present-day characteristics but is consistent with geological core and mantle constraints in Earth’s past.

DISCUSSION

Our results indicate that maintaining a thermally driven dynamo before inner core-nucleation with realistic mantle and core temperature evolution requires that Earth operates in the sluggish-lid mode early in its history, which in turn requires the existence of an asthenosphere. In our simulations, the possibility of a sluggish-lid tectonic regime arises because we assume that the dissipation in the plate is larger than the stresses caused by slab pull [Eq. 25 and (32)]. However, the relative strength of these terms may also evolve with time (14, 15), and if slab pull proceeds to a point at which it exceeds plate dissipation stresses, there may be more abrupt transitions than our models predict. We illustrate a simple example of this in section S4. In addition, as we show in section S2, other factors, such as the size of the convection cell and the effective plate strength, also play an important role in determining the convective regime, which may alter the successful parameter space (e.g. the range of successful viscosity ratios), but not the conclusions of this study. Nevertheless, the observed and likely trend of these factors in the Archean still favor the idea of early sluggish-lid tectonics. For example, we (section S2) and others (35) find that smaller plates promote sluggish-lid tectonics. Geologic observations of Archean plate margin-length (36), as well as geodynamic modeling (37), indicate that plates were likely smaller during this time. Last, as well as large plate dissipation, synonymous to early sluggish-lid tectonics is the existence of an early asthenosphere. While we do not explore specific mechanisms to produce this crucial weak layer, our work shows the importance of further exploring asthenospheric evolution (e.g., its expansion or reduction in thickness; section S4), its source (e.g., partial melt and water content), and its dynamics (16). This is an important direction for future studies of tectonic evolution.

We note that our treatment of the mantle is quite simplified, as we do not include processes that likely matter in the tectonic evolution, especially for the early Earth. For example, we do not include the evolution of the lithospheric viscosity with temperature, which results in smaller plate dissipation. However, the weakening of the lithosphere also result in lower slab pull stresses due to slab break off (15), possibly countering the effect of decreased dissipation. Partial melting and crustal production also likely play an important role in upper mantle dynamics. Crustal production can increase slab buoyancy, thickness (14), as well as thermal insulation (38) due to the partitioning of radiogenic elements in the melt (39). Partial melting can also alter the rheology of the residual mantle (40). Nevertheless, our model self-consistently captures transitions between multiple modes of convection within the explored parameter space and is thus sufficiently complete to address our hypotheses.

Mechanisms for generating an early dynamo

The lack of agreement between experimental and ab initio studies on k_c [e.g., (21, 41)] has spurned new approaches to reconcile the range of estimates, for example, by measuring k_c directly (24, 25), by measuring iron-light-element alloys rather than pure iron (42), or by accounting for missing processes in numerical calculations (27). The convergence toward high k_c values (50 to 100 W m⁻¹ K⁻¹) have motivated the search for alternative mechanisms to power the geodynamo before ICN, such as silicate convection in a basal magma ocean (43), chemical interaction between the core and the mantle (44–46), and tidally and/or precessionally driven convection (47). While these processes may indeed have contributed to the core entropy budget, they seem unable to power the geodynamo alone (29). Nevertheless, each mechanism carries a distinct prediction of the nature of the dynamo produced. In particular, some of these mechanisms may not predict a dipole-dominated geodynamo. In contrast, our proposal of a thermally driven dynamo inside the liquid core is expected to produce a dipole-dominated geodynamo (and thus similar to the geometry of the present-day field). Such a prediction may be used to test the viability of our proposed mechanism as more paleomagnetic data are gathered and, within that field, our finds have important implications on the longevity of the so-called geocentric axial dipole hypothesis.

We make several assumptions regarding the core structure in our model that are important to address. First, we assume the core to be isentropic (Eq. 3), and convection occurs in the entire liquid core. We do not include the evolution of a stably stratified layer (whether of thermal or chemical origins) (48, 49). While there is evidence for its existence today (50), a recent study that explored the evolution of both a thermal and chemical stratification layer found that long-lived magnetic field generation is mostly inconsistent with a long-lived stably stratified layer (51). However, some solutions are permissible if this layer is considerably thin (<75 km), which would allow for the isentropic assumption made in this study. Another important assumption in our model is that we do not consider ohmic dissipation in the core. This term represents an entropy sinks in the core and, when considered, increase the difficulty of maintaining a thermal geodynamo. However, the value of this dissipation term is highly uncertain, even for today’s magnetic field, because of the difficulty of estimating the magnetic power spectrum in the core and the magnitude of the toroidal magnetic field (52). By that token, we also do not consider other mechanisms that may help with geodynamo generation, such as chemical interactions between the core and the mantle (45), which depends on the core cooling rate and may provide substantial entropy in the early Earth.

Other indications of tectonic transitions

The onset of plate tectonics and transitions in tectonic regime cannot, of course, be considered without the evaluation of geologic data. In addition, while this is not the topic of this paper, we highlight a few observations that are consistent with the idea of tectonic transitions [for a review of this topic, see, e.g., (6, 53, 54)]. Much geological evidence suggests transitions in plate-mantle dynamics occurring toward the end of the Archean from a bifurcation in metamorphic style ~2.5 Ga (6). Although the interpretation of this trend is contentious (10, 55), and we do not claim this to be evidence for a sluggish- to active-lid transition, such a transition would no doubt lead to global changes in metamorphic style. While plate velocities do not substantially vary over time in our successful models, the underlying mantle velocities do, thus resulting in large changes in the stresses (both vertical and horizontal) imparted by the mantle on the plate. Similarly, the seeming lack of Archaen-aged blueschist rocks (indicative of modern-day subduction style) has been used to argue for a change in tectonic style (8, 54). Since the sluggish-lid mode is associated with higher mantle temperatures and lower lithospheric stresses (although does not preclude subduction), such geological data may be interpreted in the context of mantle transitions. Moreover, the nature of Earth’s metamorphic transition has been suggested to be gradual (56), which could be consistent with our smooth r_U evolution (e.g., green curves in Fig. 2). Put together, we propose that the tectonic regime of the Earth evolved smoothly from the sluggish-lid mode to the present-day active-lid one, as schematically depicted in Fig. 5. Our simulations connect the deepest part of Earth to the surface expression of mantle convection and have revealed critical interdependencies between large-scale systematics of core, mantle, and plate dynamics, able to address some basic Earth science questions.

Fig. 5. Schematic diagram depicting mantle convective modes.

Features of each mode are listed in the insets, where, if sluggish-lid dominated early in Earth history, as our simulations suggest, the early Earth would have been characterized by thicker plates due to slow plate motion (exaggerated here) and fast mantle flow that drive plate motion (see schematic horizontal velocity profile). In contrast, the later, active-lid mode would feature thinner plates fully coupled to the mantle. For illustrative purposes, arrow size and density indicates relative magnitude of flow velocity.

MATERIALS AND METHODS

Our goal is to evolve the energy balances of the core and the (plate-)mantle system over Earth history. We begin with stating the core energy and entropy balances and how they are calculated, followed by a similar section for the mantle system, considering both the mechanical and thermal energy balances. We then describe how each system is coupled and evolved and the details of our parameter sweep to produce our suite of simulations.

Core equations

In the following sections, we describe our approach to calculate the core’s thermal [Part (i)] and entropy [Part (ii)] budgets. We treat the core as spherically symmetric, i.e., parameters and variables vary only with radius. When appropriate, bulk averages of the entire core are assumed. Our philosophy in the core analysis is to capture the important physics that govern whether a geodynamo can or cannot exist. Note that, where necessary, we distinguish core and mantle quantities with the subscripts “c” and “m” but otherwise omit them for notational brevity.

(i) Core energy balance

The evolution of the core’s average temperature can be described using the simple energy balance that states that the secular cooling of the core is equal to the difference between the heat sources and sinks. While the energetics of the core is a topic covered in detail in many comprehensive papers (17, 57, 58), we summarize the relevant equations and assumptions here. The dominant energy sources that we consider in the core are latent heat and compositional (gravitational) energy due to inner-core freezing, while the sink is heat lost to the mantle through CMB heat flow. This balance is

$$Q_{\mathrm{s},\mathrm{c}}=Q_g+Q_{\mathrm{L}}-Q_{\text{cmb}}$$ (1)

where Q_s,c, Q_g, Q_L, and Q_cmb are the terms that represent secular cooling, the gravitational energy, the latent heat release, and the heat lost through the CMB, respectively. The secular cooling of the core is given by

$$Q_{\mathrm{s},\mathrm{c}}=-\underset{{\mathrm{V}}_{\mathrm{c}}}{\int }‍{\mathrm{\rho }}_{\mathrm{a}}(r)C_{\mathrm{P},\mathrm{c}}\frac{\mathrm{\partial }{T}_{\mathrm{a},\mathrm{c}}}{\mathrm{\partial }t} \mathrm{d}V$$ (2)

where t is time, T_a is the adiabatic temperature of the core that corresponds to ρ_a, the adiabatic density profile, and C_P,c is the heat capacity of the core, dV is the volume element, and V_c is the volume of the core. We use the fourth order polynomial fit from (23) to the adiabatic density profile derived from PREM (59), and its associated adiabatic temperature profile, given by

$$T_{\mathrm{a},\mathrm{c}}(r,t)=T_{\text{cmb}}(t){\left(\frac{{\mathrm{\rho }}_{\mathrm{a}}(r)}{{\mathrm{\rho }}_{\text{cmb}}}\right)}^{\mathrm{\gamma }}=T_{\text{cmb}}(t){\left(\frac{{\mathrm{\rho }}_0}{{\mathrm{\rho }}_{\text{cmb}}}\right)}^{\mathrm{\gamma }}{(1-\frac{r^2}{L_{\mathrm{\rho }}^2}-A_{\mathrm{\rho }}\frac{r^4}{L_{\mathrm{\rho }}^4})}^{\mathrm{\gamma }}$$ (3)

where, instead of using the inner-core boundary (ICB) as the reference point, as is done in (23), we use the CMB. ρ₀ and ρ_cmb are, respectively, the adiabatic density of the liquid core at the center and at the CMB, γ is the Grüneisen parameter, and L_ρ and A_ρ are the polynomial fit parameters. Performing the integral in Eq. 2 and only keeping the terms up to order 4 results in

$$Q_{\mathrm{s},\mathrm{c}}=-\frac{4}{3}\mathrm{\pi }{r}_{\mathrm{c}}^3{C}_{\mathrm{P},\mathrm{c}}{\mathrm{\rho }}_0\frac{\mathrm{\partial }{T}_{\text{cmb}}}{\mathrm{\partial }t}$$ (4)

The gravitational energy term that arises because of the redistribution of the core material when light elements are released at the ICB is

$$Q_g=\underset{{\mathrm{V}}_{\mathrm{c}}}{\int }{\mathrm{\rho }}_{\mathrm{a}}{\mathrm{\mu }}^{*}\frac{d\mathrm{\xi }}{\mathit{dt}}\text{dV}$$ (5)

where μ^* is the chemical potential difference across the ICB due to compositional differences and ξ is the concentration of light elements in the outer core, assuming that no light elements are incorporated into the inner core and that they are evenly distributed in the outer core. The exact expression for μ^* is in (23), but it is a function of β, the coefficient of chemical expansion, which relates to the density jump across the CMB due to compositional differences alone, Δ_ξρ (excluding the difference due to the phase transition between the liquid and solid states), and the concentration of light elements in the outer core, ξ [β = (Δ_ξρ)/(ρ₀ξ)]. The final expression for the gravitational heat is given by

$$Q_g=8{\mathrm{\pi }}^2{\mathrm{\xi }}_{0}G{\mathrm{\rho }}_0^2\mathrm{\beta } \frac{r_{\text{ic}}^2{L}_{\mathrm{\rho }}^2}{f_c(r_{\mathrm{c}}/L_{\mathrm{\rho }},0)}[f_{\mathrm{\chi }}\left(\frac{r_{\mathrm{c}}}{L_{\mathrm{\rho }}}\right)-f_{\mathrm{\chi }}\left(\frac{r_{\text{ic}}}{L_{\mathrm{\rho }}}\right)]\frac{{\text{dr}}_{\text{ic}}}{\text{dt}}$$ (6)

where ξ₀ is the initial concentration of light elements in the core, G is the gravitational constant, and f_c and f_χ are geometric functions that arise from integration over the core’s volume defined in (23) for notational convenience (see the Appendix therein). Q_L is the energy release associated with the freezing of the inner core, given by

$$Q_{\mathrm{L}}=4\mathrm{\pi }{r}_{\text{ic}}^2{\mathrm{\rho }}_{\mathrm{a}}(r_{\text{ic}}) T_{\mathrm{L},\mathrm{c}}(r_{\text{ic}})\mathrm{\Delta }S\frac{{\mathrm{d}r}_{\text{ic}}}{\text{dt}}$$ (7)

where ΔS is the entropy of crystallization per unit mass of solidified (inner) core, and T_L(r_ic) is the liquidus temperature of core material at the inner core boundary given in (23):

$$T_{\mathrm{L},\mathrm{c}}(r)=T_{\mathrm{L},0}+K_0{\left(\frac{\mathrm{\partial }{T}_{\mathrm{L},\mathrm{c}}}{\mathrm{\partial }P}\right)}_{\mathrm{\xi }}\frac{r_{\text{ic}}^2}{L_{\mathrm{\rho }}}+{\left(\frac{\mathrm{\partial }{T}_{\mathrm{L},\mathrm{c}}}{\mathrm{\partial \xi }}\right)}_P\frac{{\mathrm{\xi }}_0{r}_{\text{ic}}^3}{L_{\mathrm{\rho }}^3 f_{\mathrm{c}} (r_{\mathrm{c}}/L_{\mathrm{\rho }},0)}$$ (8)

where T_L,0 is the liquidus at the central pressure of the core. The partial derivatives represent the rate of change of the liquidus with pressure and light element concentration while holding the other variable constant, respectively, as they appear in the equation. Last, the heat flow across the CMB, Q_cmb, is given by Fourier’s law:

$$Q_{\text{cmb}}=4\mathrm{\pi }{r}_{\mathrm{c}}^2 \frac{k_{\text{lm}}\mathrm{\Delta }{T}_{\text{lm}}}{d_{\text{lm}}}$$ (9)

where k_lm is the thermal conductivity of the lowermost mantle, ΔT_lm is the temperature difference across the CMB (i.e., ΔT_lm = T_cmb − T_lm), and d_lm is the thickness of the lower thermal boundary layer. The expressions for ΔT_lm and d_lm are in the mantle section below. The growth of the inner core (r_ic) is determined by the intersection of the core adiabat and the liquidus given by Eqs. 3 and 8 above, and the rate of the growth, dr_ic/dt, can be expressed as (dr_ic/dT_cmb)(dT_cmb/dt); the first term is evaluated numerically, and the second is the only unknown variable left that we solve for.

(ii) Core entropy balance

Since we are not modeling heat transport within the core and are mainly concerned with the average evolution of the core, the thermal conductivity of the core does not enter the temperature evolution equation above. However, to evaluate the ability of any resulting core evolution to sustain a dynamo, an important consideration is the net generation of entropy, the main sink of which is heat conduction. The entropy may be considered as a proxy for whether a global magnetic field is generated. The analogous entropy balance to the energy balance (Eq. 1) is

$$E_{\mathrm{s},\mathrm{c}}+E_{\mathrm{L}}+E_g=E_k+E_{\mathrm{\Phi }}$$ (10)

where the left-hand side represents the sources of entropy. Subscripts are associated with those shown in the energy balance (Eq. 1). The right-hand side comprises the entropy sinks due to heat conduction (E_k) and magnetic and viscous diffusion (E_Φ). The entropy sources can be derived from the equivalent heat flow terms multiplied by a thermodynamic efficiency term (17, 58). This efficiency term is the difference of the inverse between the production temperature and the destruction temperature associated with that source. For example, secular cooling is produced throughout the core volume and depends on the integral of the core adiabat (T_eff) [see the Appendix in (23)] and is lost at the CMB at a temperature of T_cmb. Therefore, the entropy associated with cooling is defined as

$$E_{\mathrm{s},\mathrm{c}}=(\frac{1}{T_{\text{eff}}}-\frac{1}{T_{\text{cmb}}})Q_{\mathrm{s}\mathrm{c}}$$ (11)

Similarly, the entropy production term associated with the latent heat release at the ICB is

$$E_{\mathrm{L}}=(\frac{1}{T_{\text{icb}}}-\frac{1}{T_{\text{cmb}}})Q_{\mathrm{L}}$$ (12)

where T_icb is the temperature at the ICB, determined using the adiabatic temperature profile in Eq. 3. Because we assume no light element flux from the core to the mantle, the entropy production term associated with gravitation buoyancy has higher efficiency (17) and is given by

$$E_{\mathrm{g}}=\frac{1}{T_{\text{cmb}}}{Q}_{\mathrm{g}}$$ (13)

Moving onto the right-hand side terms, we first consider E_k, which represents a major entropy sink. E_k is

$$E_k=\underset{V_{\mathrm{c}}}{\int }k{\left(\frac{\mathrm{\Delta }{T}_{\mathrm{a},\mathrm{c}}}{T_{\mathrm{a},\mathrm{c}}}\right)}^2\text{dV}$$ (14)

where T_a,c is once more the adiabatic temperature of the core, given by Eq. 3. If we take the core conductivity to be constant, i.e., k(r) = k_c, and perform the integral over the volume of the core, we arrive at

$$E_k=16{\mathrm{\pi \gamma }}^2{k}_{\mathrm{c}}{L}_{\mathrm{\rho }}[f_{\mathrm{k}}\left(\frac{r_{\mathrm{c}}}{L_{\mathrm{\rho }}}\right)-f_{\mathrm{k}}\left(\frac{r_{\text{ic}}}{L_{\mathrm{\rho }}}\right)]$$ (15)

where f_k is another geometric integral. Last, to evaluate the ability of convection in the core to sustain a dynamo, the entropy sources must exceed the sinks, as a minimum requirement. Without making assumptions about the dynamic of the core convection, this means that

$$E_k+E_{\mathrm{\Phi }}>E_{\mathrm{g}}+E_{\mathrm{L}}+E_{\mathrm{c},\mathrm{s}}$$ (16)

or

$$E_{\mathrm{\Phi }}=E_{\mathrm{g}}+E_{\mathrm{L}}+E_{\mathrm{c},\mathrm{s}}-E_k>0$$ (17)

This is considered a minimum requirement because we have not included the entropy sink due to magnetic and viscous dissipation in the entropy budget. The magnetic dissipation is a function of the strength of the magnetic field and the wavelength of convection, and, while we may introduce errors due to its omission, this omission is commonly applied to avoid making assumptions about the dynamics within the core driving the dynamo (17). Similarly, viscous dissipation omission is common because of the low viscosity of liquid iron at core conditions. With Eqs. 11 to 13, 15, and 17, we can evaluate, at any given time in our evolution, whether the core produces a magnetic field or not. The relevant core parameters used are listed in table S1.

The plate-mantle system

The mechanical and thermal evolution of the plate and mantle are governed by conservation of momentum and conservation of energy, respectively. In analytical mantle convection models, since bulk average properties are considered, the momentum equation is often written in the form of conservation of mechanical energy. The two conservation equations (total and mechanical energy) are coupled through the plate velocity. The plate velocity is obtained by solving the mechanical energy equation and governs the heat loss through the surface in the total energy equation. In the following two sections, we outline the models we use to calculate the plate velocity (and the entire velocity profile of the mantle) from the mechanical energy equation. We then explain how the plate velocity is used to evolve the mantle temperature.

(i) Mechanical energy

Here, we outline the forms of the mechanical energy equation used in our model based on the formulation of Crowley and O’Connell [hereafter, COC12 (31)] with two important improvements. First, we remove the assumption of steady state convection from the mechanical energy equation. Although this assumption is often made in analytical models of mantle convection, it is strictly inconsistent with thermal evolution models, which are based on an imbalance between supplied and lost energies. In addition, embedded within this assumption is the removal of the hyteresis (or the path dependence) of the mechanical energy equation, which has been shown to be important in numerical models (60). Conceptually, path dependence means that the convective state of the mantle does not only depend on the thermal state at an instance in time but also on its prior evolution of previous states. This change, while practically straightforward to implement, represents a philosophical departure from many previous applications. Second, we introduce a lowermost mantle layer that represents the thermal boundary layer between the core and the mantle, which has its own mechanical properties (specifically, reference viscosity). This change is simple but is important for coupling the mantle with the core.

Following COC12, it can be shown that the conservation of mechanical energy equation in two dimensions (2D) can be written as

$${\mathrm{\Phi }}_{\mathrm{P}}=-\underset{\mathrm{S}}{\int }(u_{i}P{\mathrm{\delta }}_{\mathit{ij}}-u_i{\mathrm{\tau }}_{\mathit{ij}})\mathrm{d}{S}_j+{\mathrm{\Phi }}_{\mathrm{L}}+{\mathrm{\Phi }}_{\mathrm{M}}$$ (18)

which states the rate of change of potential energy in the mantle (Φ_P) is balanced by the work done by viscous forces in the mantle (Φ_M), the work done to bend or shear the lithosphere (Φ_L), and the work done of the surface forces of the system (represented by the terms within the surface integral). Here, u_i is the velocity vector, P is the pressure, δ_ij is the Kronecker delta function, and τ_ij is the deviatoric stress. The subscripts i, j follow the Einstein convention.

We model the mechanical energy of the mantle using 2D Cartesian geometry [see figure 1 in (32)], and we represent all the quantities per unit length along the z direction (along strike of the ridge/subduction boundaries). The rate of change of potential energy in the system is proportional to the depth-averaged vertically advected heat flow, <Q>_adv, and for high Rayleigh number systems, such as the mantle, advection dominates such that the advected heat flow is approximately equal to the total heat flow <Q>. In steady state systems, <Q>_ss is

$$<Q{>}_{\text{ss}}=(q_{\text{cmb}}+\frac{1}{2}\mathit{Hd})L=(q_{\text{surf}}-\frac{1}{2}\mathit{Hd})L$$ (19)

where q_cmb and q_surf are, respectively, the CMB and surface heat fluxes, and d and L are the depth and length of the system. H is the radiogenic heating per unit volume in the mantle (Q_rad in Eq. 28 divided by the mantle volume V_m). In the nonsteady state system, we aim to consider that Eq. 19 no longer holds because the change of internal energy of the mantle can act to change the total advected heat flow. When the mantle is heating/cooling, a part of the energy is not advected but is used to heat/cool the system. We must depart from the original balance stated in COC12 and modify the equation above to

$$<Q{>}_{\text{nss}}=(q_{\text{cmb}}+\frac{1}{2}\mathit{Hd}-\frac{1}{2}{\mathrm{\rho }}_{\mathrm{m}}{C}_{\mathrm{P},\mathrm{m}}\frac{{\mathit{dT}}_{\mathrm{m}}}{\mathit{dt}})L$$ (20)

to account for the nonsteady state (nss) balance of the advected heat flow. Here, T_m is the average mantle temperature, and ρ_m and C_P,m are the average mantle density and heat capacity, respectively. The geometry of these expressions are Cartesian (as formulated in COC12) such that the areas of the surface and the CMB are assumed to be equivalent. This overestimates the CMB heat flow and is inconsistent with the core model presented above. To account for this, we instead use the version of this equation consistent with spherical geometry provided in section S5.

Performing the integral in Eq. 18, we consider that the lithospheric system <Q>_L alone, and the whole mantle system, <Q>_nss, separately, yields

$$\frac{\mathrm{\alpha }{\mathit{gd}}_{\mathrm{l}}}{C_{\mathrm{P},\mathrm{m}}}<Q{>}_{\mathrm{L}}={\mathit{LP}}_{{\mathrm{x}}_{\mathrm{p}}}{d}_{\mathrm{l}}{U}_{\mathrm{p}}+L{\mathrm{\tau }}_{\mathrm{p}}{U}_{\mathrm{p}}+{\mathrm{\tau }}_{\mathrm{R}}{d}_{\mathrm{l}}{U}_{\mathrm{p}}$$ (21)

$$\frac{\mathrm{\alpha }\mathit{gd}}{C_{\mathrm{P},\mathrm{m}}}<Q{>}_{\text{nss}}={\mathrm{\tau }}_{\mathrm{F}}{d}_{\mathrm{l}}{U}_{\mathrm{p}}+{\mathrm{\tau }}_{\mathrm{Y}}{d}_{\mathrm{l}}{U}_{\mathrm{p}}+{\mathrm{\Phi }}_{\mathrm{M}}$$ (22)

Here, α is the coefficient of thermal expansion, g is the gravitational acceleration, and d_l is the thickness of the lithosphere at the subduction zone, for which we use the half-space cooling model

$$d_{\mathrm{l}}=c_0{\left(\frac{\mathrm{\kappa }L}{U_{\mathrm{p}}}\right)}^{1/2}$$ (23)

where c₀ is a constant, κ is the thermal diffusivity, and U_p is the plate velocity. <Q>_L is the depth-averaged advected heat flow through the lithosphere and is equal to half the surface heat flow (Q_surf), given by

$$Q_{\text{surf}}=2k_{\text{um}}\mathrm{\Delta }T{\left(\frac{U_{\mathrm{p}}L}{\mathrm{\pi \kappa }}\right)}^{1/2}$$ (24)

where k_um is the thermal conductivity of the upper mantle and ΔT = T_u − T_s is the temperature difference across the lithosphere (i.e., the difference between the upper mantle, T_u, and surface, T_s, temperatures; see below). Whether for the lithosphere alone or the entire mantle, Eqs. 21 and 22 state that the rate of change of potential energy in the system is balanced by the work done on the interior and boundary of the system.

For the lithospheric energy balance (Eq. 21), P_xp is the lateral pressure gradient evaluated at the base of the lithosphere, τ_p is the viscous stress at the base of the lithosphere, and τ_R is the net resistive stress of the lithosphere. It represents the difference between the stresses that hinder and drive lithospheric flow and is given by

$${\mathrm{\tau }}_{\mathrm{R}}={\mathrm{\tau }}_{\mathrm{Y}}+{\mathrm{\tau }}_{\mathrm{F}}-{\mathrm{\tau }}_{\text{SP}}$$ (25)

where τ_Y is the effective yield stress of the lithosphere, τ_F is the faulting stress, and τ_SP is the stress due to slab pull.

The work done on the surface of the lithosphere is concentrated at the lower boundary due to the lateral pressure gradient (first term on the right-hand side of Eq. 21) and the traction at the base of the lithosphere (second term on the right-hand side of Eq. 21) due the flow of the mantle beneath the plate, both of which are functions of the mantle’s flow pattern and viscosity structure. The vertical boundary at the subduction zone also contributes work in the form of faulting and slab pull. Within the volume of the lithosphere, work is done to bend the lithosphere (where our expressions make no assumption on the rheological model). The individual contributions of τ_Y, τ_F, and τ_SP do not matter in the lithospheric energy balance but rather the sum of these contributions, which has a considerable role is determining the mode of convection in the mantle. If slab pull is larger than the sum of bending and faulting stresses, τ_R is negative, it leads to a plate-mantle coupling state in active-lid mode. In this case, the plate moves because of its own negative buoyancy. However, when τ_R is positive, active and multiple sluggish-lid solutions are possible that represent different dominant balances in the energy equations (31, 32). While these stresses evolve with the evolution of the mantle, we set each of them as constant model parameters (values given in table S2) following the justification in COC12 and for the lack of analytical models that adequately describe their behavior with time. While the mechanical strengths are constant, the dissipation associated with these terms varies with lithospheric thickness and plate velocity. The rate of change of potential energy in the lithosphere is equivalent to the work commonly referred to as ridge push (61).

For the mantle energy balance (Eq. 22), the whole mantle’s change in potential energy is balanced by the work done on the boundaries and within the mantle. The only surface contribution is the faulting stress due to both boundaries being stress-free. Within the volume of the mantle, both deformation of the lithosphere (second term on the right-hand side of Eq. 22) and viscous deformation due to mantle flow contribute work on the mantle (third term on right-hand side of Eq. 22). Viscous dissipation is a function of the viscosity and flow structure of the mantle and has contributions due to lateral shearing and corner flow. While the expressions above make no assumption of such structure, in our model, the mantle is subdivided into the four major regions: the lithosphere, asthenosphere, bulk mantle, and lower mantle. The thickness and mechanical properties of these regions depend on their thermal state and therefore evolve with time.

We follow the approximations and method of solution outlined by COC12 (see their Appendix) to solve the mantle and lithospheric energy equations for the plate velocity. We present the equations consistent with our model, due to the addition of the lower thermal boundary layer, in section S7.

(ii) Thermal evolution

Upon determining the plate velocity, U_p, the surface heat flow is obtained (Eq. 24), and, with this, we are able to evolve the mantle temperature. The secular cooling of the mantle, similarly to Eq. 1 for the core, is determined by the difference between the energy sources and sinks in the mantle and is given by

$$Q_{\mathrm{s},\mathrm{m}}=Q_{\text{rad}}(t)+Q_{\text{cmb}}-Q_{\text{surf}}$$ (26)

where Q_s,m and Q_rad are the energy sources in the mantle and represent the mantle’s secular heat and radiogenic heating, respectively, given by

$$Q_{\mathrm{s},\mathrm{m}}=-V_{\mathrm{m}}{\mathrm{\rho }}_{\mathrm{m}}{C}_{\mathrm{P},\mathrm{m}}\frac{{\text{dT}}_{\mathrm{m}}}{\text{dt}}$$ (27)

and

$$Q_{\text{rad}}(t)=V_m{\mathrm{\rho }}_m\sum _i{C}_i{H}_i{e}^{t\text{ln}(2)/{\mathrm{\tau }}_i}$$ (28)

where V_m is the mantle volume and ρ_m is the average mantle density. Radiogenic heat is produced in the mantle by many radioactive elements, assumed to be uniformly distributed within the mantle (listed in table S3). The total heat produced is the sum of the individual elements i with concentration C_i, heat production (per unit mass) H_i, and half-life τ_i. Substituting Eq. 27 into Eq. 26 gives the temperature evolution of the mantle.

As noted, the CMB heat flux depends on the difference across the thermal boundary layer (with thickness d_lm), as well as the lowermost mantle viscosity, thermal conductivity, and thermal boundary layer thickness (see Eq. 9). The latter can be determined from consideration of local instability when the local Rayleigh number, Ra_lm, exceeds the critical Rayleigh number for convection, Ra_crit. The lower mantle Rayleigh number is defined by

$${\text{Ra}}_{\text{lm}}=\frac{{\mathrm{\alpha \rho }}_{\mathrm{m}}g\mathrm{\Delta }{T}_{\text{lm}}{d}_{\text{lm}}^3}{{\mathrm{\mu }}_{\text{lm}}\mathrm{\kappa }}$$ (29)

where α is the thermal expansion coefficient, κ is the thermal diffusivity, and μ_lm is the lower mantle viscosity. This viscosity is governed by the Arrhenius relationship

$${\mathrm{\mu }}_{\mathrm{R}}(T)={\mathrm{\mu }}_{\mathrm{R},0}{e}^{\frac{E}{R_{\mathrm{g}}}(\frac{1}{T_{\mathrm{R}}}-\frac{1}{T_{\mathrm{R},0}})}$$ (30)

where μ_R is the viscosity of any given region, R. In this specific case, R = lm, the lower mantle. Here, μ_0,R is the reference viscosity defined at T = T_0,R, T_R is the temperature of region R (=T_lm), E is the activation energy, and R_g is the ideal gas constant. The temperature of the lower mantle is determined by extrapolating down a linear adiabat given by

$$T_{\mathrm{a},\mathrm{m}}(y)=2T_{\mathrm{P}}({\mathrm{\eta }}_{\mathrm{m}}-1)\frac{(y-d_{\mathrm{l}})}{(d-d_{\mathrm{l}})}+T_{\mathrm{P}}$$ (31)

where y is the depth from the surface. This adiabatic profile is set such that the mantle potential (T_p) and average (T_m) mantle temperatures have the relationship T_m = η_mT_P. The mantle parameters we use are listed in table S2.

Coupling the core and mantle evolutions

The core and mantle systems are coupled via the expressions for heat flow at the CMB (Eq. 9), which appears in both the core energy balance (Eq. 1) and the mantle energy balance (Eq. 26) and depends on the temperature difference between the core and the mantle and thus both of their evolutions. Beginning with the initial conditions (t = t₀) of the mantle temperature (T_m,0) and the core temperature at the CMB (T_cmb,0), we determine the initial mantle viscosity structure (Eq. 30), the thickness of the lower thermal boundary layer (Eq. 29), and Q_cmb (Eq. 9). We then solve for the plate velocity numerically using the method described in the Supplementary Text and COC12. Last, we use the plate velocity solution to calculate the surface heat flow (Eq. 24) and can evaluate the rate of change of mantle (dT_m/dt) and core at the CMB (dT_cmb/dt) temperatures using Eqs. 26 and 1, respectively. With these two derivatives, the temperatures of both the mantle and core are advanced to the next time step using a first order Euler method, with a time step of 40 Ma. We tested several time steps, and since the evolution paths are smooth over our 4.0–billion year histories, our simple time-stepping scheme is efficient and accurate. We begin the evolution at time t₀ = 500 Ma to use initial conditions consistent with a differentiated Earth and solidified mantle.

Across different simulations, we vary the initial temperature (T_m,0), the initial core temperature at the CMB (T_cmb,0), the asthenospheric (μ_0,a), and lower mantle (μ_0,lm) reference viscosities. The range of values tested for each of these parameters is listed in Table 1, which result in ~10⁵ thermal histories. For each of these histories, we then calculate the entropy budget (Eq. 17) using different values of core thermal conductivity (k_c), also listed in Table 1, resulting in >10⁶ entropy histories. With these histories and tracked variables such as the plate and mantle velocities, the mantle temperature, the inner core radius, and the core entropy production, we are able to present results described in the Results section.

Supplementary Materials

This PDF file includes:

Sections S1 to S7

Figs. S1 to S5

Tables S1 to S3

References