the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
On the impact of true polar wander on heat flux patterns at the coremantle boundary
Thomas Frasson
Stéphane Labrosse
HenriClaude Nataf
Nicolas Coltice
Nicolas Flament
Abstract. Heat flux across the coremantle boundary (CMB) is an important variable of Earth's thermal evolution and dynamics. Seismic tomography provides access to seismic heterogeneities in the lower mantle, which can be related to presentday 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 selfconsistent 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 use two recently published mantle convection models to explore the impact of TPW on the CMB heat flux over long timescales (~ 1 Gyr). One of the mantle convection models is driven by a plate reconstruction, while the other selfconsistently produces a platelike behavior. We compute the geoid in both models to correct for TPW. In the platedriven model, we compute a total geoid and a geoid in which lateral variations of viscosity and temperature are suppressed above 350 km depth. We show that TPW plays an important role in redistributing the CMB heat flux, notably at short time scales (≤ 10 Myr). Those rapid variations modify the latitudinal distribution of the CMB heat flux, which is known to affect the stability of the magnetic dipole in geodynamo simulations. A principal component analysis (PCA) is computed to obtain the dominant CMB heat flux pattern in the different cases. These heat flux patterns can be used as boundary conditions for geodynamo models as representative of the mantle convection cases studied here. We note that the geoids produced by the two models are widely different from each other and from the observed presentday geoid. Work thus still needs to be done to improve the computation of the geoid in mantle convection models related to platetectonics.
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Thomas Frasson et al.
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EC1: 'Comment on egusphere20231782', Juliane Dannberg, 21 Aug 2023
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Dear Thomas Frasson and coauthors,
For reasons of transparency, I wanted to disclose here that I currently work on a project with a similar focus, specifically the changes in patterns of coremantle boundary heat flux over time and their effect on the geodynamo (but not on the geoid/true polar wander). In particular, the new additions in this resubmission (models with a prescribed plate motion history throughout the last 1 billion years) are close to my own work (see for example https://doi.org/10.5194/egusphereegu239490 and https://agu.confex.com/agu/fm22/meetingapp.cgi/Paper/1073309).
I have discussed this with the Executive Editor, Susanne Buiter, and she encouraged me to act as topical editor for the resubmission (also because have already handled the initial submission). I will do my best to handle this submission well and objectively, but please let the editorial team know if you have any concerns.
Juliane Dannberg
Citation: https://doi.org/10.5194/egusphere20231782EC1 
AC3: 'Reply on EC1', Thomas Frasson, 28 Sep 2023
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Dear Juliane Dannberg,
We have chosen to submit and resubmit our article to Solid Earth, rather than to a higherimpact journal such as EPSL, because the topic we address concerns a wide community (mantle modeling, geological plate reconstructions, geodesy, geodynamo modeling, paleomagnetism), which we thought Solid Earth might better cover. Our paper addresses the fundamental consideration of the evolution of the spin axis on using CMB heat flow distribution for dynamo modeling, using as example two stateoftheart models of mantle convection published in highimpact journals.
You have chosen to send our new manuscript to Bernhard Steinberger again. This is fine and we understand this. However, considering the rather personal view he has on what should be published, and the rather narrow analysis he performed on our paper (see our reply to his review), it seems appropriate in our view to obtain a third set of comments.
Best regards,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC3

AC3: 'Reply on EC1', Thomas Frasson, 28 Sep 2023
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RC1: 'Comment on egusphere20231782', Anonymous Referee #1, 06 Sep 2023
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The manuscript by Frasson et al. takes a systematic approach on how true polar wander (TPW) affects the lateral variation of coremantle boundary (CMB) heat flux over a time scale of a billion years. The authors use output from two mantle convection models, integrated over this timescale, but forced in two different ways: one driven by the surface condition constrained by a plate reconstruction model and the other left to freely convect, but with rheological properties that reflect plate behavior. They track TPW throughout their simulation by tracking the longwavelength geoid. With these outputs, they perform a PC analysis in order to explore the dominant control on CMB heterogeneity.
While there are shortcomings in whether the mantle convection models reflect a realistic Earth given their large geoid misfit  which the authors both acknowledge, and the models themselves are not produced in this study  the authors have demonstrated that TPW should be considered when determining how CMB heat flux varies both spatially and temporally. Interestingly, TPW provides faster variation than would be expected from the convecting mantle alone and can thus potentially explain high frequency excursions in the paleomagnetic record.
Because of this, I believe it to be a novel contribution and have a few minor comments that only help to add clarity for the reader.
Specific comments:Page 2: lines 3545: perhaps discuss a short part about how we think that the chemical (and therefore negatively buoyant) heterogeneity may be confined to a small region at the base of the LLVPs. See Richards et al (2023; EPSL: "Geodynamic, geodetic, and seismic constraints favour deflated and densecored LLVPs").
Page 6: lines 160165: It would be good to elaborate on why there are two distinct ways to compute the geoid. This study will likely attract a varying audience (e.g., core dynamicists), so some background on this  even just 23 sentences  would be helpful. What does zeroing out the upper 350 km achieve?Why is the "No LVV" method not applied to the MC model? If this is because it was not calculated in the original study, is there a way to use the output you have access to to apply the same "No LVV" method. It would be better for comparison. Or maybe this is not applicable? If so, please explain why.Page 9: lines 250260. If the variations in the geoid are so large compared to today's actual geoid, what does this mean in terms of how "earth like" the CMB predictions would be? I realize now that you explain this later in the Discussion, so perhaps point the reader to it.Page 19: Section 4.2. Is it possible to give some idea of the timescales of the PCs? Perhaps even estimate the frequency content of the time series. Since these undulations time (figs 1011) reflect the mobility of the piles, can these be related to subducting slabs from above? Can you potentially derive some timescale for surface events to be translated to CMB events? I think this would be very interesting.Citation: https://doi.org/10.5194/egusphere20231782RC1 
AC1: 'Reply on RC1', Thomas Frasson, 28 Sep 2023
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Dear Referee,
We thank you for your prompt and accurate review of our manuscript. We are particularly encouraged by the fact that you have well captured both the novelty and the limitations of our study.
We will take into account your comments in our revised manuscript. We just give here short answers to your main questions:
 Page 2: lines 3545: perhaps discuss a short part about how we think that the chemical (and therefore negatively buoyant) heterogeneity may be confined to a small region at the base of the LLVPs. See Richards et al (2023; EPSL: "Geodynamic, geodetic, and seismic constraints favour deflated and densecored LLVPs").
Thank you for pointing out the recent study of Richards et al (2023), which we will refer to.
 Page 6: lines 160165: It would be good to elaborate on why there are two distinct ways to compute the geoid. This study will likely attract a varying audience (e.g., core dynamicists), so some background on this  even just 23 sentences  would be helpful. What does zeroing out the upper 350 km achieve?
The computation of the geoid is very sensitive to large lateral viscosity variations in the mantle (Cadek and Fleitout 2003, Flament 2019). The MF model is driven by a plate reconstruction model, updated every 1 Myr, which notably imposes the positions of viscous slabs. The update of the slab positions strongly affects the Total geoid, hence the scattered TPW path visible on figure 3 for the MF1 case. The No LVVs geoid in the MF model is much less affected by the update of the surface conditions, allowing for a smoother TPW closer to that given by MC1.
Radial viscosity profiles are moreover classically used to compute the geoid using geoid kernels (Richards and Hager 1984, Root et al. 2010, Steinberger et al. 2019). In our platelike models, the largest lateral variations of viscosity happen in the upper mantle. Removing the effects of these lateral variations in the upper mantle thus allows to compute a geoid that is closer to the one computed from radial geoid kernels.
 Why is the "No LVV" method not applied to the MC model? If this is because it was not calculated in the original study, is there a way to use the output you have access to to apply the same "No LVV" method. It would be better for comparison. Or maybe this is not applicable? If so, please explain why.
The MC model is fully selfconsistent. The Total geoid computed in MC is evolving smoothly, preventing the scattered TPW observed in MF1. It is thus not necessary to remove the effect of lateral variations in the upper mantle to obtain a smoother geoid, as it was the case in the MF model. However, we did try to compute a No LVVs geoid in the MC model in the hope that it would lower the amplitude of the geoid, without success.
 Page 9: lines 250260. If the variations in the geoid are so large compared to today's actual geoid, what does this mean in terms of how "earth like" the CMB predictions would be? I realize now that you explain this later in the Discussion, so perhaps point the reader to it.
This will be done.
 Page 19: Section 4.2. Is it possible to give some idea of the timescales of the PCs? Perhaps even estimate the frequency content of the time series. Since these undulations time (figs 1011) reflect the mobility of the piles, can these be related to subducting slabs from above? Can you potentially derive some timescale for surface events to be translated to CMB events? I think this would be very interesting.
We will see what can be said with some confidence.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC1

AC1: 'Reply on RC1', Thomas Frasson, 28 Sep 2023
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RC2: 'Comment on egusphere20231782', Bernhard Steinberger, 22 Sep 2023
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I am still not convinced of this paper. My main problem is that the geoid results shown are very different from the real Earth, and also very different from each other. It is not clear to me at all where these differences come from. It is all said in the end of the conclusion: "it would be of great interest to understand where these discrepancies come from". I agree, and I think it should be done in this paper and not in some future work  among other things, in order to reduce the chance that these discrepancies come from actual errors in the computations. I mean, if the paper in this form makes it in the literature, then it is being cited and it just confuses everyone and no one is helped or gains any insights.
More specifically, the methods would also have to be better described. It may be possible to extract these from the literature given, but at least some essentials need to be discussed: Particularly, what rheological model is used? Since the geoid strongly depends upon it, in particular on (average) radial viscosity structure. Is it the same in the MF and MC models, or different? If it is the same, why the geoids are so different? Also, the CMB heat flux is different for MF and MC models (line 245); which CMB temperature do they use? Is it the same?
The two cases MF1 and MF2 start more similar, but then evolve increasingly different. Does the density structure (below 350 km depth) also evolve differently in the two cases, or is it at each time the same density (below 350 km), only the geoid is computed differently? I think this would make more sense, i.e. you always insert slabs at each time step, but why would differences increase with time then?
Also, what boundary condition is used for geoid computation in MF? What I usually do is I use prescribed plate motions only for the flow and advection calculation, but freeslip for the subsequent geoid computation at each time step. Because prescribed surface motions are appropriate for flow computations, but may not give realistic surface radial stresses and topography, hence not realistic geoid. This would be important to know in order to understand the geoid in this case.
Regarding results, why the geoid in case MC has such high amplitude and is such strongly correlated with continents? In reality, continents are mostly isostatically compensated at shallow levels and are associated with a very weak signature, i.e. there is hardly any corrlation between geoid and the continentocean distribution. I think something is wrong here. On lines 286/287 you write that piles are mostly associated with geoid lows, but I don't see this; I see just the correlation with continents.
And why there is no such strong continent signal in MF? The difference in results between the three cases MC, MF1 and MF2 is really puzzling and some analysis should be given to understand the differences, e.g. by separating different contributions (topography, Moho, mantle density down to 350 km, mantle density below 350 km, CMB topography.Citation: https://doi.org/10.5194/egusphere20231782RC2 
AC2: 'Reply on RC2', Thomas Frasson, 28 Sep 2023
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Dear Bernhard,
We are very sorry to read that you are ‘still not convinced of this paper’, but we wonder what you are not convinced of. Is it about the impact of True Polar Wander (TPW) on heat flux patterns at the CoreMantle Boundary (CMB)? Is it about our message that the relevant frame for discussing CMB heat flux implications for the geodynamo must have the correct spin axis? Is it about our findings that TPW can be responsible for changes in CMB heat flux patterns on timescales much shorter than typical mantle convection timescales, even for their low degrees? We think that these messages, which are the focus of our paper, are novel and important and deserve to be published rapidly, particularly since several teams in the world are currently working on the implications of mantle convection on CMB heat flux maps and their implications for core dynamics.
There is nothing about these important findings in your review. You do not even acknowledge that we have done a lot of work to address your original comments by including a new set of models and presenting different calculations of the geoid. You simply declare that you don’t trust the geoids computed from these models. We agree with you that the geoids computed from the two mantle circulation models that we analyse differ from the geoid of the ‘real Earth’, to various degrees depending on the model, noting in passing that the only geological epoch for which the Earth’s geoid is known is the anthropocene… We have been explicit about this important limitation in the manuscript. For example, section 3.1 is devoted to a discussion of this discrepancy, where we note that ‘The geoid stems from a delicate balance between bulk density heterogeneities and flowinduced interface undulations’. Our revised study presents two different mantle circulation models, computed with stateoftheart codes (CitcomS and StagYY), each spanning one billion years. Geoids were computed using the intrinsic solver of these codes, implemented as in Zhong et al (2008) who benchmarked the results with available analytical solutions.
Our models follow two different strategies: model MF is forced by the motions of plates, following a stateoftheart geological reconstruction, while model MC is a selfconsistent realistic mantle circulation model in which plates form naturally. These are the two main strategies followed today. Model MC has the advantage of being selfconsistent and uses parameters that make its outcomes similar to observations for Earth in many respects , except for the geoid which was not investigated in the original publication of this model. Model MF is less selfconsistent, but it matches the geometry and velocities of plates, following a recent plate reconstruction spanning the past billion years. Concerning the computation of the geoid, it uses exactly the approach you advocate in your review: ‘What I usually do is I use prescribed plate motions only for the flow and advection calculation, but freeslip for the subsequent geoid computation at each time step. Because prescribed surface motions are appropriate for flow computations, but may not give realistic surface radial stresses and topography, hence not realistic geoid.’ This approach results in a model geoid that better matches Earth, however it is not physically selfconsistent. Using results from both types of models allows us to ascertain the robustness of our results about TPW and its implications for the timeevolution of CMB heat flux patterns.
We will follow your advice to give more details about the procedures used to compute the various geoids, but we refuse to further scrutinize their respective merits in the present article. This important question clearly falls outside of the scope of this paper. We are however very cautious about the geoid calculation, disclosing fully the difficulties with these results. There is therefore no danger of confusion to fear. We do not claim that our predicted CMB heat flux pattern evolutions represent those of the Earth during the past billion years. Although our study opens new possibilities, it is clearly not the end of the story…
However, we do believe that the impact of TPW on CMB heat flux patterns that we find and illustrate is robust and important, warranting rapid publication. In support of this view, we note that although no data are available to constrain Earth’s past geoid, there are constraints on TPW velocities from paleomagnetic observations. As illustrated in our Figure 4, the TPW velocities predicted from our two mantle circulation models fall within the observed range. In addition, our findings about the implication of TPW on the CMB heat flux are robust (in statistical terms) with respect to the choice of mantle convection model. As we mention in the section 4.3 of the manuscript, the effects of the TPW on the CMB heat flux we observe in our models would hold with more realistic geoids. We thus think it is important to share with the community how correcting for the TPW affects the CMB heat flux, and why it should be considered in future studies.
Best regards,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice, Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC2 
RC3: 'Reply on AC2', Bernhard Steinberger, 28 Sep 2023
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What I am not convinced of is that this paper is a useful contribution to the literature. This is perhaps because to me the impact of TPW on heat flux patterns at the CMB is obvious, and I wouldn't need a paper to appreciate that. But I acknowledge that many people even within the field may not be aware of this. I also acknowledge that you have done more work to address my original comments, but this has raised more questions, because now results not only strongly differ from presentday Earth but also from each other. For example, why in one computation there is this strong correlation with continents, and in the other there isn't? The last sentence in your conclusions indicates to me that even you don't understand this yourself, and as long as this is the case, I find the paper of limited use, because the suspicion remains that at least one of the geoid computations (especially the one with the strong correlation with continents, which is not observed, and also not predicted if continents are roughly in hydrostatic equilibrium) is wrong. I suggest that you address my specific comments. This will be less work than the previous round, because it won't require any new computations (unless my suspicion that at least one of the geoid computations is wrong turns out to be true), but probably requires a bit more analysis and explanation of the result. I don't want to stop you from publishing this; I just try to make suggestions to increase the potential impact and usefulness.
Citation: https://doi.org/10.5194/egusphere20231782RC3

RC3: 'Reply on AC2', Bernhard Steinberger, 28 Sep 2023
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AC2: 'Reply on RC2', Thomas Frasson, 28 Sep 2023
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Thomas Frasson et al.
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CMB heat flux PCA results Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice, Nicolas Flament https://doi.org/10.5281/zenodo.8205153
Thomas Frasson et al.
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