The heat flux across the core–mantle boundary (CMB) is a fundamental variable for Earth evolution and internal dynamics. Seismic tomography provides access to seismic heterogeneities in the lower mantle, which can be related to present-day thermal heterogeneities. Alternatively, mantle convection models can be used to either infer past CMB heat flux or to produce statistically realistic CMB heat flux patterns in self-consistent models. Mantle dynamics modifies the inertia tensor of the Earth, which implies a rotation of the Earth with respect to its spin axis, a phenomenon called true polar wander (TPW). This rotation must be taken into account to link the dynamics of the mantle to the dynamics of the core. In this study, we explore the impact of TPW on the CMB heat flux over long timescales (

Temperature heterogeneities in the lower mantle impose a heterogeneous heat flux at the top of the core, across the core–mantle boundary (CMB). This CMB heat flux is an important variable of Earth's thermal evolution and dynamics, especially for core convection and the geodynamo. The mean CMB heat flux controls the core cooling rate, which determines the power available for the geodynamo. Both the CMB heat flux mean value and lateral variations affect dynamo behaviour in numerical simulations, with strong effects on magnetic reversal frequency and on the angle between spin and magnetic dipole axes

Because viscosity is much larger in Earth's mantle than in Earth's outer core, the CMB is an isothermal boundary for the mantle, while the core sees the CMB as an imposed laterally varying heat flux. This heat flux changes on mantle convection timescales, which are much larger than core dynamics timescales. Our understanding of the CMB heat flux, and notably its spatial distribution, depends on our knowledge of lower-mantle structure and dynamics. Seismic tomography offers a view of the lowermost mantle, revealing more and more complex structures

Records of eruption sites of hotspots suggest that LLVPs could have remained fixed for the past 300 Myr at least

These models are useful to explore the past of mantle convection. They are, however, limited by the accuracy of plate reconstructions, which remain poorly constrained before the assembly of Pangea

Earth's spin plays a crucial role in core dynamics. If we are to explore the impact of realistic CMB heat flux patterns on the geodynamo, it is essential that these patterns be produced in a reference frame that preserves the spin axis. Mantle convection simulations do not depend on the position of Earth's spin axis since rotational forces are negligible in the mantle, and surface and CMB boundary conditions are not affected by a global rotation of the mantle with respect to its spin axis. However, mass redistribution and boundary topographies caused by convection modify the moments of inertia of the mantle

This work aims at describing the CMB heat flux produced by two up-to-date mantle convection models in the reference frame relevant for core dynamics. One of these models is driven by a plate reconstruction

Our study rests upon an analysis of two published 3D mantle convection simulations. Both models simulate mantle convection in a 3D spherical shell including tectonic plates at the surface and chemical piles at the bottom. They differ, however, in the way plate tectonics is handled. The first model (similar to case NNR of

Model MF was computed using the code

This model predicts CMB heat flux based on a reconstruction of past positions of tectonic plates and associated mantle flow. The evolution of the thermochemical basal boundary layer is dictated by subducting slabs, which are introduced at the surface following the plate model. Heat flux patterns are thus directly related to the past 1 Gyr of mantle convection history, which depends on the plate tectonic reconstruction imposed as a surface boundary condition. Whilst the past 200 Myr are well constrained, notably from magnetic anomalies and hotspot tracks preserved in the oceanic crust, plate motions are more uncertain for earlier geological times. The advantages of model MF are that it generally matches the present-day structure of the mantle

Model MC reproduces in total 1131 Myr of mantle evolution using the code

The initial state of the simulation is an equilibrated mantle circulation obtained with two fixed antipodal 500 km thick chemical piles around the Equator and fixed continents assembled in a Pangea-like supercontinent placed above the “Atlantic” pile. At the start of the simulation, continents and piles are allowed to move freely. The relaxation to a new statistically steady state takes about 300 Myr. The first 300 Myr are thus not considered in the following analysis.

In contrast with model MF, plate kinematics are not imposed in this model. A plate-like behaviour is self-consistently produced using a pseudo-plastic rheology and a temperature-dependent viscosity

Regarding the objective of this work, the main advantages of model MC are the Earth-likeness of surface processes and its long time evolution. It captures the effect of realistic plate tectonics on a CMB heat flux that varies in space and time over nearly 1 Gyr. Model MC notably contains a complete cycle of breakup and assembly of a supercontinent, which is thought to modulate CMB heat flux

TPW is controlled by the change in the Earth's moment of inertia around its spin axis. The moment of inertia is obtained from the degree 2 components of the geoid, which we thus need to compute for our mantle flow models. The geoid is the equipotential surface of gravity measured or computed around a reference level, which is sea level for the Earth and the top of the model for simulations. It can be computed by integration of lateral density variations across the mantle model. Additional contributions arise from lateral mass heterogeneities produced by deflections of interfaces, in particular the surface and CMB. These deflections are not explicit in the models (which assume spherical boundaries) but can be computed from element

Model MF is forced at the surface by the plate model. To compute the geoid at a given time step, we solve the Stokes flow with self-gravitation at this time step with a free-slip condition at the surface as usually done in plate-driven models

Model MC is fully self-consistent: no forcing is imposed at the surface, where a stress-free boundary condition is applied. The total geoid computed in MC evolves smoothly. It is thus not necessary to remove the effect of lateral variations in the upper mantle to obtain a smoother geoid evolution as was the case in model MF, and only the total geoid is computed for this model. The geoid computation is done within a benchmarked module of the

TPW is governed by the conservation of Earth's angular momentum, yielding Liouville's equation

TPW is then implemented iteratively by rotating the mantle at each time step to ensure that the spin axis follows the position of the maximum inertia axis. The rotation direction is chosen so that the new north pole remains in the same hemisphere as the previous one, effectively limiting TPW amplitudes to a maximum of 90° per iteration. Since TPW is governed by the geoid, we have two different TPW paths for model MF. In the following, the TPW associated with the total geoid is called total TPW, while the TPW associated with the no-LVV geoid is called the no-LVV TPW.

Characteristics of the six cases analysed in this study.

This TPW implementation corresponds to a change in the reference frame in which data are represented. This new reference frame permanently wanders with respect to the initial simulation frame; we thus call it the wandering frame in the following. The simulation is not related to any forcing in model MC other than the initial conditions. The only preferential relation between the simulation frame and the wandering frame is thus through the initial conditions in model MC. In contrast, the simulation frame corresponds to the mantle reference frame of the plate reconstruction in model MF. If both the plate reconstruction and the resulting mantle convection simulation were perfectly tuned to the Earth, the wandering frame should merge with the simulation frame (to within a rotation in longitude) at the end of the simulation, i.e. for the present time. In past times, the two frames are expected to diverge because of the effect of TPW. In practice, a non-negligible shift exists between the simulation and the wandering frames at the end of the simulation in both the total TPW and no-LVV TPW cases.

An alternative case is considered for model MF by rotating the model results into the paleomagnetic reference frame. This case is more comparable to the work of

In total, six cases are considered in the subsequent analysis of CMB heat flux patterns. Their characteristics are summarized in Table

Spherical harmonic transforms and rotations are performed using the library SHTns

We use a principal component analysis (PCA) to obtain the dominant heat flux patterns at the bottom of the mantle in the different models. PCA is a data analysis tool that can be applied to a dataset comprising several observations, with each observation depending on several variables. It is used to express the dataset in a new orthonormal basis in order to limit the number of variables needed to explain the data. This is done by computing new variables (called principal components), which are combinations of the initial variables. A full mathematical description of PCA theory is given by

In the framework of this study, PCA is used to obtain the principal components corresponding to heat flux patterns that explain most of the heat flux signal at the CMB as a function of time. The dataset consists of the spherical harmonics coefficients of

Once PCA is performed, time-dependent CMB heat flux maps can be reconstructed as

Let us first present and discuss important characteristics of the two mantle simulations we exploit. The radial profiles of viscosity and temperature in model MF and MC are shown in Fig.

Viscosity and temperature profiles in model MF and MC. The profiles in model MF are the present-day (

Model MF. Maps of geoid undulations (first column panels: total geoid; second column panels: no-LVV geoid), surface topography (third column panels), and CMB heat flux (last column panels) at selected times. The maps are shown in a Mollweide projection. Black lines in the heat flux maps delineate the edges of basal chemical piles. The heat flux is low beneath chemical piles.

Model MC. Maps of geoid undulations (left panels), surface topography (centre panels), and CMB heat flux (right panels) at selected times. The maps are shown in a Mollweide projection. Black lines in the heat flux maps delineate the edges of basal chemical piles. Heat flux is low beneath chemical piles.

Both models reproduce the well-known bimodal topography of the Earth, reflecting the difference between continental and oceanic lithosphere. Oceanic ridges and trenches are well marked. Some plate boundaries can be easily recognized in line “0 Myr” of model MF in Fig.

CMB heat flux maps of both models show large-scale variations that strongly correlate with the presence of basal chemical piles (delineated by black lines). The heat flux is low beneath these piles, which act as thermal insulators. In both models, chemical piles are shaped and pushed by cold slabs that reach the CMB. The CMB heat flux is dominated by large-scale heterogeneities as shown by the spectra in Fig.

Time-averaged spherical harmonic spectra of the CMB heat flux in model MF and MC. The spectra are given relative to the power of the

Movies showing the time evolutions of the fields shown in Fig.

Our study requires computing the geoid in the models in order to deduce the resulting TPW. The geoid stems from a delicate balance between bulk density heterogeneities and flow-induced interface undulations. The geoid computed in model MC (first column in Fig.

The surface conditions of model MF are updated every 1 Myr. The total geoid being strongly affected by the positions of subducting slabs, the update of the surface conditions implies fast variations of the geoid from one snapshot to the next. We evaluate this effect by computing the no-LVV geoid, shown in the second column of Fig.

TPW paths in cases MF1, MF2, and MC1 in a Mollweide projection. Colour disks represent the successive positions of the maximum inertia axis in the simulation frames. The colour scale gives the time before the end of the simulation in millions of years (Myr). Black-circled magenta disk shows the position of the maximum inertia axis at the end of the simulation. The inertial interchange TPW event (IITPW) in case MF1 is highlighted by the black arrow.

TPW is applied by computing the successive positions of the maximum inertia axis and rotating the simulation frame to align this axis with the spin axis. The successive positions of the maximum inertia axis in the simulation frames of models MF and MC for cases MF1, MF2, and MC1 are shown in Fig.

Let us analyse the TPW paths for model MF, computed either from the total geoid (MF1) or the no-LVV geoid (MF2). The present-day position (black-circled magenta disk in Fig.

Turning to model MC, we recall that this model is not related to any plate reconstruction. Hence no particular relationship is expected between the simulation frame and the wandering frame. However, the two frames are linked due to the relationship between chemical piles and the geoid in the model. Chemical piles are introduced at the Equator at the beginning of the simulation and remain at low latitudes by spreading around the Equator. These piles are mostly associated with geoid lows throughout the simulation. They thus tend to move from low latitudes towards high latitudes due to the TPW correction. As a result, the maximum inertia axis tends to form a

In cases MF1 and MF2, the positions of the maximum inertia axes can be compared to the position of the pole in the paleomagnetic reference frame. By doing so, we can evaluate the consistency between the paleomagnetic constraints and the geoid produced by the mantle convection model rotated in the paleomagnetic reference frame as is done for case MF

Angular distance between the maximum inertia axis computed in MF1 and MF2 and the axis of the magnetic dipole (corresponding to the

The time evolution of TPW velocities (in ° Myr

TPW velocity in ° Myr

The average rotation rate in MF

Time evolution of degree 1 and 2 spherical harmonic coefficients of the CMB heat flux in cases MF0, MF1, MF2, and MF

Most studies exploring the effect of CMB heat flux heterogeneity on the geodynamo have focused on large-scale patterns of spherical harmonic degrees 1 and 2

Time evolution of degree 1 and 2 spherical harmonic coefficients of CMB heat flux in cases MC0 and MC1. Case MC1 is derived from case MC0 by correcting it for the TPW.

We have seen that CMB heat flux variations are strongly controlled by the distribution of basal chemical piles in our models. When piles are at high latitudes, for example at

True polar wander strongly impacts the behaviour of large-scale CMB heat flux patterns. We first note that TPW modifies the hierarchy of these patterns. For example, while the

More importantly, our study reveals that TPW changes large-scale patterns of CMB heat flux on timescales much shorter than typical mantle convection timescales. This is well illustrated by several sign reversals of the

As mentioned earlier, exploration of the effect of CMB heat flux heterogeneities on the geodynamo has previously mostly focused on degrees 1 and 2. An alternative is to explore the effect of heterogeneities inferred from seismic tomography of the lowermost mantle

PCA results consist of a set of components ranked by decreasing contribution to the total CMB heat flux variations. Each component is described by a heat flux pattern and an amplitude. The amount of variance

The variance explained by the first three PCs is shown in Fig.

Variance explained by the first three PCs normalized by the total variance in the different cases.

The patterns of the first three PCs are shown in Fig.

Patterns of the first three PCs of the CMB heat flux

Patterns of the first three PCs of the CMB heat flux

The associated amplitudes of the first three PCs are shown in Fig.

Time evolution of the amplitudes of the first three PCs of the CMB heat flux for cases MF0, MF1, MF2, and MF

Time evolution of the amplitudes of the first three PCs of the CMB heat flux for cases MC0 and MC1.

Recent models offer the possibility to study mantle convection related to plate tectonics on timescales of the order of 1 Gyr. These models reproduce plate tectonics either self-consistently

In the two mantle convection models studied here, we find that correcting for the TPW (or rotating the model in a paleomagnetic reference frame in case MF

The positions of chemical piles and subducted slabs in the lower mantle dominate the large-scale CMB heat flux signal of both MF and MC models. Piles keep the mantle material warm below them, while subducted slabs cool the surrounding mantle. Piles can also be correlated with large-scale geoid anomalies and be affected by TPW. The behaviour of the combined system formed by piles and slabs is thus of prime importance regarding the CMB heat flux and the way it is redistributed by TPW. It has been suggested by studying the locations of plume generation zones in the lower mantle that the chemical piles have stayed fixed over at least 300 Myr

The different PCAs we computed give the dominant heat flux patterns for each case. The PCs of the CMB heat flux largely reflect the positions of the piles. The insulating effect of the piles indeed dominates the large scales of the CMB heat flux. The dominant length scales of the heat flux patterns tend to decrease with the increase in the PC number, but the length scale separation between the first three components is rather weak. It is particularly true for the TPW-corrected cases. This is due to the addition of a source of complexity in these cases, which results in more different large-scale components required to explain the CMB heat flux time series. This additional source of complexity in the dataset also translates into a smaller amount of variance explained by the first PCs as shown in Fig.

The strong instabilities of the total geoid in model MF due to the imposed changes in surface conditions represent an important drawback of this geoid. These instabilities translate into fast motions of the poles. Though the rotation rate in case MF1 using this total geoid lies within expected values for the Earth, the pole wanders significantly faster than in the other cases that are not affected by this problem. The pole instabilities create high-frequency variations of the CMB heat flux throughout the whole simulation that are probably unrealistic. Our attempt to remove the surface contribution to the geoid by computing the no-LVV geoid is successful at cancelling the high-frequency variations in the geoid. The geoid pattern is, however, strongly impacted. The anomalies associated with subduction zones and chemical piles are notably affected, cancelling the first one and changing the sign of the second. This implies a widely different TPW path in case MF2 compared to case MF1. This path largely differs from the displacement of the magnetic dipole in both cases, which shows the inconsistency between the total geoid and the paleomagnetic reference frame and between the no-LVV geoid and the paleomagnetic reference frame in our plate-driven mantle convection model. A possibility would be to eliminate spurious fast variations of the geoid while securing agreement between the inertia axis of the mantle circulation model and the Earth's spin axis, as given by the magnetic dipole axis or (better) by paleogeographic constraints. Such constraints could be implemented in a data assimilation scheme

Plate-driven models introduce another complexity due to the choice of reference frame in which the plate reconstruction is considered. Depending on the reference frame, the surface can undergo net rotations, as is the case in the plate reconstruction of

From the point of view of core dynamics, rotations in longitude of a given CMB heat flux pattern are irrelevant. The PCA computed here would, however, see two identical patterns rotated in longitude as two distinct components. The analysis conducted in this study could thus be improved by gathering patterns that are similar by rotations in longitude into the same component.

The two main goals of this work are to investigate how TPW can affect the CMB heat flux in term of space and time behaviour and to provide heat flux maps representative of

This work also highlights the need to better constrain long-term mantle convection models using the moment of inertia of the mantle. Aiming for core–mantle coupling, self-consistent mantle convection models have to be repositioned in a frame that keeps the maximum inertia axis aligned with the rotation axis. This correction can also be used in plate-driven models. In this case, however, it is also possible to rotate the mantle convection model in a paleomagnetic reference frame if the relation between the paleomagnetic reference frame and the mantle reference frame of the model is known. This second option is not consistent with the model itself and does not ensure that the maximum inertia axis is aligned with the rotation axis. In this case, the potential differential rotation between the lithosphere and the deep mantle is poorly constrained. The next generation of models driven by reconstructed plate motions could consider including consistency with true polar wander as a constraint for data assimilation.

Full data files in HDF5 format are available for the model outputs and CMB heat flux PCA results, as are movies of the outputs and Python scripts, on Zenodo (

HCN and SL conceived the project. NC and NF provided data from mantle numerical simulations and computed additional outputs from the original models. TF wrote and ran all analysis scripts, prepared all figures, and led the writing of the paper. All authors discussed the results and contributed to the writing of the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The SHTns library

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-19-CE31-0019) and by the Australian Government's National Collaborative Research Infrastructure Strategy (NCRIS), with access to computational resources provided by the National Computational Infrastructure (NCI) through the National Computational Merit Allocation Scheme and through the University of Wollongong (UOW).

This paper was edited by Juliane Dannberg and reviewed by Bernhard Steinberger, Shijie Zhong, and one anonymous referee.