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
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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EC1: 'Comment on egusphere20231782', Juliane Dannberg, 21 Aug 2023
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
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 
AC4: 'Reply on AC3', Thomas Frasson, 26 Oct 2023
Dear Juliane Dannberg,
We have taken good note of the review by Shijie Zhong. We are preparing our responses to all three referees. This requires running several tests concerning the computation of the geoid of model MC. We understand from the guidelines of the peerreview process that we have four weeks after the end of the open discussion to reply to the comments we received. We would appreciate if you could leave open the possibility for us to upload our comments on line until the completion of the tests.
Best regards,
Thomas FrassonCitation: https://doi.org/10.5194/egusphere20231782AC4 
EC2: 'Reply on AC4', Juliane Dannberg, 30 Oct 2023
Dear Thomas Frasson (and coauthors),
Thank you for your comment, and I agree that this is a good way forward.
I have terminated the discussion phase (i.e. community members can not post comments any more), but you can still post author comments during that time.
Since two of the referees have expressed concerns about the geoid computations, it is important that in your revision/response you demonstrate the correctness/accuracy of the geoid computation, provide a more detailed description of the method you use, and clearly explain the differences in results, both between the different models, and between your models and Earth.
Best regards,
Juliane DannbergCitation: https://doi.org/10.5194/egusphere20231782EC2 
AC5: 'Reply on EC2', Thomas Frasson, 24 Nov 2023
Dear Juliane,
As mentioned in our previous comment, we started performing tests regarding the computation of the geoid in model MC.
First of all, the code itself has shown its efficiency for geoid calculation. We refer to Cammarano et al. (2011) and Guerri et al. (2016) for synthetic geoids for the Earth and Rolf et al. (2018) for Venus, in which Bernhard Steinberger is a coauthor. Because of computing infrastructure issues, Nicolas Coltice had to use the limited data he had at the time to reconstruct the chemical fields (continents and thermochemical piles). The problems have now been solved and computations using the full required data show that what the reviewers suspected was correct: the computed geoids in the manuscript were indeed flawed.We are currently recomputing the geoid for model MC. The first results suggest that this corrected geoid is much closer to the expectations put forward by the reviewers. For the snapshot at 250 Myr presented in the manuscript, we obtain RMS geoid amplitudes five times smaller after the correction. The positive geoid anomalies below continents that Bernhard Steinberger pointed out to be incorrect are no longer present. The slabs are associated with a negative geoid signal at small scales, in a similar way as in the total geoid of model MF. We show in the attached file the corrected geoid for this snapshot as well as several alternative geoids corresponding to various tests we made.
Computing the correct geoid will take several days, and we will have to modify the manuscript according to the new results. We will try to implement the revisions in a timely manner, and we hope to be able to send the revised manuscript before the end of the year.
We deeply apologise for this mistake and for having originally resisted to investigate the geoid in model MC. We are very grateful to the reviewers, and notably to Bernhard Steinberger and Shijie Zhong, for insisting on the need for investigation of this geoid.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
References
Cammarano, F., Tackley, P., & Boschi, L. (2011). Seismic, petrological and geodynamical constraints on thermal and compositional structure of the upper mantle: global thermochemical models. Geophysical Journal International, 187(3), 13011318.
Guerri, M., Cammarano, F., & Tackley, P. J. (2016). Modelling Earth’s surface topography: decomposition of the static and dynamic components. Physics of the Earth and Planetary Interiors, 261, 172186.
Rolf, T., Steinberger, B., Sruthi, U., & Werner, S. C. (2018). Inferences on the mantle viscosity structure and the postoverturn evolutionary state of Venus. Icarus, 313, 107123.
AC8: 'Reply on AC5', Thomas Frasson, 21 Dec 2023
Dear Juliane,
We have made substantial modifications to the article, notably following the referee comments and the correction of the geoid in model MC. Though we mostly converged on a final revised version, some adjustments still need to be made. We are actively working on these revisions, and it seems difficult for us to submit a satisfactory revised manuscript before December 24. We thus think that an extension would be necessary for us to submit the revised manuscript. We are going to ask for 15 days of extension, as that is the upper limit offered to us. Because of the Christmas break, we will not be able to work efficiently on the revision in the next few days. Thus, we would like to ask for a longer extension. If it were possible to add two more weeks to this extension, fixing the submission deadline on January 22, we would be able to submit a manuscript that would be more likely to meet the quality expectations. Would such an extension be possible?
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice, and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC8 
EC3: 'Reply on AC8', Juliane Dannberg, 21 Dec 2023
Dear Thomas (and coauthors),
It’s great to hear that the feedback from the reviewers has helped to correct a problem in the geoid computation and that the new results are closer to what is expected! I agree that it is much more important to make sure that everything is correct and that you can revise the manuscript according to these new results than to submit at a given deadline.
I got an email that said the new deadline is January 18, but if you need more time, feel free to reach out to the Copernicus staff about that (because I do not think that I can change the deadlines in the system myself).
I hope you have a good Christmas time and can enjoy the holidays!
Best regards,
JulianeCitation: https://doi.org/10.5194/egusphere20231782EC3

EC3: 'Reply on AC8', Juliane Dannberg, 21 Dec 2023

AC8: 'Reply on AC5', Thomas Frasson, 21 Dec 2023

AC5: 'Reply on EC2', Thomas Frasson, 24 Nov 2023

EC2: 'Reply on AC4', Juliane Dannberg, 30 Oct 2023

AC4: 'Reply on AC3', Thomas Frasson, 26 Oct 2023

AC3: 'Reply on EC1', Thomas Frasson, 28 Sep 2023

RC1: 'Comment on egusphere20231782', Anonymous Referee #1, 06 Sep 2023
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
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

RC2: 'Comment on egusphere20231782', Bernhard Steinberger, 22 Sep 2023
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
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
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 
AC7: 'Reply on RC3', Thomas Frasson, 24 Nov 2023
Dear Bernhard,
By performing new computations of the geoid in the MC model, we have realised that your suspicions regarding the correctness of the geoid were justified. We refer you to our comment to the editor for explanations of the mistake made in the previous computation. We are currently working on recomputing the geoid for the revised version of our manuscript. We thank you for highlighting this issue and apologise for not giving enough credit to your feedback, both on the first version of our paper and on this second version. Your persistence enabled us to avoid publishing flawed results and we expect it will significantly improve our manuscript.
As we are writing this reply, we only have access to a few snapshots of the geoid computed in the correct way. We notably computed some tests on the snapshot corresponding to the time 250 Myr in model MC as shown in the file attached to our comment to the editor. This corrected geoid significantly differs from the one shown in the manuscript, making it much closer to the expectations you share in your review. First, the RMS geoid amplitudes are reduced by a factor of five. This is much closer to what we obtained in model MF, although it is still larger than Earth's present geoid. Another important difference is the absence of strong correlations between the continents and the geoid. As you mentioned in your review, the continents are isostatically compensated and are thus not expected to leave an important signal in the geoid. Given these first results, we are confident that the corrected geoid will be of much greater scientific value.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC7

AC7: 'Reply on RC3', Thomas Frasson, 24 Nov 2023

RC3: 'Reply on AC2', Bernhard Steinberger, 28 Sep 2023

AC2: 'Reply on RC2', Thomas Frasson, 28 Sep 2023

RC4: 'Comment on egusphere20231782', Shijie Zhong, 24 Oct 2023
This manuscript presents a very interesting and novel idea that true polar wander may effectively alter the CMB heat flux pattern reference to the Earth's rotation axis, thus affecting the geodynamo if it is significantly influenced by the CMB heat flux pattern as suggested by previous studies. However, like Dr. Steinberger, I find the geoid results from the numerical models difficult to understand. Since the polar wander is controlled by the degree2 geoid and the geoid calculations are tricky in numerical models, it is important for the authors to explain more clearly how the geoid anomalies are computed.
The geoid modeling is tricky and sometimes even difficult in numerical models, because it requires accurate determination of dynamic topography. There are two contributions to the geoid: 1) mass anomalies associated with dynamic topography at the surface and CMB (and phase and compositional boundaries, if they are present) and 2) buoyancy from mantle thermochemical structure. The contributions from these two sources often have comparable amplitude but different signs, and the geoid anomalies represent the difference between these two terms. In general, dynamic topography is related to the pressure and derivative of flow velocity which is always oneoder less accurate than the primary variables like flow velocity. Therefore, a small error in dynamic topography tends to get amplified in the errors for the geoid. In finite element models like CitcomS, special techniques like consistent boundary flux method are often used to resolve this issue.
One class of convection models (MF) presented in this study used CitcomS. Given that CitcomS has been extensively benchmarked for the geoid problems, I would think that the geoid results from this class of models should be okay. However, the geoid results from this class of calculations also raise some concerns to me. For example, I do not quite understand why the geoid would be so different after the removal of shallow thermal structure (i.e., the top 350 km), because the longwavelength geoid (e.g., at degree2) often is insensitive to buoyancy at shallow depths where the geoid kernel goes to zero (the geoid kernel concept remains largely relevant even for models with 3D mantle viscosity).
We have done similar calculations of mantle convection driven by plate motion history with shorter time history of plate motion than that in this study (see Zhang et al., JGR, 2010; Mao and Zhong, JGR, 2021). Dynamic topography for Zhang et al. (2010) models was computed and discussed in Zhang et al., (2012, EPSL) in the context of continental uplift history. Mao and Zhong (JGR, 2021) computed the geoid (and dynamic topography) using plate motion history for the last 100 Ma and reproduced well the observed geoid between degrees 4 and 12 with certain mantle viscosity models. As discussed in Zhang et al. (2012) and especially in Mao and Zhong (2021), for convection models with prescribed plate motion as boundary conditions, it is important to recompute for dynamic topography by replacing the velocity boundary conditions with freeslip boundary conditions and appropriate lithospheric viscosity (i.e., to produce horizontal velocities that are similar to the prescribed plate motion). Note that in Mao and Zhong (2021), weak plate margins were introduced to achieve this purpose. Anyway, it is unclear from the manuscript how the geoid and dynamic topography were computed (e.g., were freeslip boundary conditions used together with some appropriate lithospheric viscosity?). The authors need to describe these issues in the revision. Prescribed surface velocity boundary conditions tend to produce spurious pressure field and hence dynamic topography, with high viscosity lithosphere.
The other class of convection models (MC) produced geoid anomalies of several kilometers that are only slightly smaller than that of surface dynamic topography (Fig. 2). These calculations are presumably for Rayleigh numbers that are comparable with that for the Earth's mantle and with that in the first class of convection models using CitcomS. In this type of situation, kilometers of geoid anomalies seem too large to me. Additionally, the geoid to topography ratio in most models in general should be around 0.10.2 (like admittance which is the ratio of gravity anomalies to topography, the geoid to topography ratio is only sensitive to viscosity structure, but significantly less sensitive to distribution of buoyancy). For reference, the observed long wavelength is ~ 100 meters, while the dynamic topography is ~1 km. Therefore, I recommend that the authors show some benchmark calculations for the geoid for their code by comparing with analytical solutions (see Zhong et al., G^3, 2008). Perhaps, such results are already available for their code, and then the authors can reference them (however, they still need to explain how + 3 km geoid anomalies can be generated from their models).
In summary, I think that the main scientific idea on polar wander and CMB heat flux pattern is very interesting and novel and deserves to be published, but I also think that the authors need to explain and justify their geoid results better. I hope that my review is useful.
Shijie Zhong
Citation: https://doi.org/10.5194/egusphere20231782RC4 
AC6: 'Reply on RC4', Thomas Frasson, 24 Nov 2023
Dear Shijie,
Thank you for your review and your enthusiasm for our study. As explained in our message to the editor, the tests we made in response to reviews by yourself and Bernhard Steinberger led us to realise that the geoid computations were erroneous in model MC. We are currently computing the geoid again in model MC, and we will modify the manuscript accordingly.
The first results we have obtained seem to answer your concerns about the amplitude of the geoid. We obtained RMS geoid anomalies that are five time smaller than previously. These amplitudes are still too large compared to the presentday Earth, but they are much closer to what is obtained in model MF. We conducted some tests on this snapshot, as shown in the file attached to our comment to the editor. We notably find that the topography and density components of the geoid are of opposite signs and largely compensate each other, as described in your review. We also show that choosing a radial viscosity profile significantly affects the results.
You also asked for precisions concerning the computation of the geoid. We will better describe the procedures in the revised version. Concerning the questions you raised, as you mention, model MF was computed using CitcomS, which has been benchmarked for this purpose. We compute two geoid outputs following Flament (2019). Both outputs are computed by restarting the model at a given time step with freeslip conditions applied at the surface. The first output, called Total geoid in the manuscript, was obtained without altering the density or viscosity distributions in the mantle. A second output called No LVVs was obtained by cancelling the density and viscosity lateral variations in the upper 350 km. This second output is significantly different from the Total geoid because of the removal of the lateral variations of viscosity. We also computed a third output, not shown in the manuscript, for which we only suppressed the lateral variations of density, keeping the viscosity distribution untouched. As you expected in your review, this third output is very similar to the Total geoid, leading to almost identical TPW paths. We thus decided not to include this case in our manuscript. It is clear from these tests that the differences between the Total geoid and the No LVVs geoid are due to the effect of viscosity variations rather than density variations.
The computation of the geoid in model MC uses the same approach as in model MF, based on Zhang and Christensen (1993). In this case, however, no changes in the surface conditions are necessary to compute the geoid as there is already a freeslip condition at the surface in the model. As in the Total geoid case of model MF, the density and viscosity fields are fully conserved for the geoid computation.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Flament, N. (2019). Presentday dynamic topography and lowermantle structure from palaeogeographically constrained mantle flow models. Geophysical Journal International, 216(3), 21582182.
Zhang, S., & Christensen, U. (1993). Some effects of lateral viscosity variations on geoid and surface velocities induced by density anomalies in the mantle. Geophysical Journal International, 114(3), 531547.Citation: https://doi.org/10.5194/egusphere20231782AC6

AC6: 'Reply on RC4', Thomas Frasson, 24 Nov 2023
Interactive discussion
Status: closed

EC1: 'Comment on egusphere20231782', Juliane Dannberg, 21 Aug 2023
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
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 
AC4: 'Reply on AC3', Thomas Frasson, 26 Oct 2023
Dear Juliane Dannberg,
We have taken good note of the review by Shijie Zhong. We are preparing our responses to all three referees. This requires running several tests concerning the computation of the geoid of model MC. We understand from the guidelines of the peerreview process that we have four weeks after the end of the open discussion to reply to the comments we received. We would appreciate if you could leave open the possibility for us to upload our comments on line until the completion of the tests.
Best regards,
Thomas FrassonCitation: https://doi.org/10.5194/egusphere20231782AC4 
EC2: 'Reply on AC4', Juliane Dannberg, 30 Oct 2023
Dear Thomas Frasson (and coauthors),
Thank you for your comment, and I agree that this is a good way forward.
I have terminated the discussion phase (i.e. community members can not post comments any more), but you can still post author comments during that time.
Since two of the referees have expressed concerns about the geoid computations, it is important that in your revision/response you demonstrate the correctness/accuracy of the geoid computation, provide a more detailed description of the method you use, and clearly explain the differences in results, both between the different models, and between your models and Earth.
Best regards,
Juliane DannbergCitation: https://doi.org/10.5194/egusphere20231782EC2 
AC5: 'Reply on EC2', Thomas Frasson, 24 Nov 2023
Dear Juliane,
As mentioned in our previous comment, we started performing tests regarding the computation of the geoid in model MC.
First of all, the code itself has shown its efficiency for geoid calculation. We refer to Cammarano et al. (2011) and Guerri et al. (2016) for synthetic geoids for the Earth and Rolf et al. (2018) for Venus, in which Bernhard Steinberger is a coauthor. Because of computing infrastructure issues, Nicolas Coltice had to use the limited data he had at the time to reconstruct the chemical fields (continents and thermochemical piles). The problems have now been solved and computations using the full required data show that what the reviewers suspected was correct: the computed geoids in the manuscript were indeed flawed.We are currently recomputing the geoid for model MC. The first results suggest that this corrected geoid is much closer to the expectations put forward by the reviewers. For the snapshot at 250 Myr presented in the manuscript, we obtain RMS geoid amplitudes five times smaller after the correction. The positive geoid anomalies below continents that Bernhard Steinberger pointed out to be incorrect are no longer present. The slabs are associated with a negative geoid signal at small scales, in a similar way as in the total geoid of model MF. We show in the attached file the corrected geoid for this snapshot as well as several alternative geoids corresponding to various tests we made.
Computing the correct geoid will take several days, and we will have to modify the manuscript according to the new results. We will try to implement the revisions in a timely manner, and we hope to be able to send the revised manuscript before the end of the year.
We deeply apologise for this mistake and for having originally resisted to investigate the geoid in model MC. We are very grateful to the reviewers, and notably to Bernhard Steinberger and Shijie Zhong, for insisting on the need for investigation of this geoid.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
References
Cammarano, F., Tackley, P., & Boschi, L. (2011). Seismic, petrological and geodynamical constraints on thermal and compositional structure of the upper mantle: global thermochemical models. Geophysical Journal International, 187(3), 13011318.
Guerri, M., Cammarano, F., & Tackley, P. J. (2016). Modelling Earth’s surface topography: decomposition of the static and dynamic components. Physics of the Earth and Planetary Interiors, 261, 172186.
Rolf, T., Steinberger, B., Sruthi, U., & Werner, S. C. (2018). Inferences on the mantle viscosity structure and the postoverturn evolutionary state of Venus. Icarus, 313, 107123.
AC8: 'Reply on AC5', Thomas Frasson, 21 Dec 2023
Dear Juliane,
We have made substantial modifications to the article, notably following the referee comments and the correction of the geoid in model MC. Though we mostly converged on a final revised version, some adjustments still need to be made. We are actively working on these revisions, and it seems difficult for us to submit a satisfactory revised manuscript before December 24. We thus think that an extension would be necessary for us to submit the revised manuscript. We are going to ask for 15 days of extension, as that is the upper limit offered to us. Because of the Christmas break, we will not be able to work efficiently on the revision in the next few days. Thus, we would like to ask for a longer extension. If it were possible to add two more weeks to this extension, fixing the submission deadline on January 22, we would be able to submit a manuscript that would be more likely to meet the quality expectations. Would such an extension be possible?
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice, and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC8 
EC3: 'Reply on AC8', Juliane Dannberg, 21 Dec 2023
Dear Thomas (and coauthors),
It’s great to hear that the feedback from the reviewers has helped to correct a problem in the geoid computation and that the new results are closer to what is expected! I agree that it is much more important to make sure that everything is correct and that you can revise the manuscript according to these new results than to submit at a given deadline.
I got an email that said the new deadline is January 18, but if you need more time, feel free to reach out to the Copernicus staff about that (because I do not think that I can change the deadlines in the system myself).
I hope you have a good Christmas time and can enjoy the holidays!
Best regards,
JulianeCitation: https://doi.org/10.5194/egusphere20231782EC3

EC3: 'Reply on AC8', Juliane Dannberg, 21 Dec 2023

AC8: 'Reply on AC5', Thomas Frasson, 21 Dec 2023

AC5: 'Reply on EC2', Thomas Frasson, 24 Nov 2023

EC2: 'Reply on AC4', Juliane Dannberg, 30 Oct 2023

AC4: 'Reply on AC3', Thomas Frasson, 26 Oct 2023

AC3: 'Reply on EC1', Thomas Frasson, 28 Sep 2023

RC1: 'Comment on egusphere20231782', Anonymous Referee #1, 06 Sep 2023
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
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

RC2: 'Comment on egusphere20231782', Bernhard Steinberger, 22 Sep 2023
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
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
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 
AC7: 'Reply on RC3', Thomas Frasson, 24 Nov 2023
Dear Bernhard,
By performing new computations of the geoid in the MC model, we have realised that your suspicions regarding the correctness of the geoid were justified. We refer you to our comment to the editor for explanations of the mistake made in the previous computation. We are currently working on recomputing the geoid for the revised version of our manuscript. We thank you for highlighting this issue and apologise for not giving enough credit to your feedback, both on the first version of our paper and on this second version. Your persistence enabled us to avoid publishing flawed results and we expect it will significantly improve our manuscript.
As we are writing this reply, we only have access to a few snapshots of the geoid computed in the correct way. We notably computed some tests on the snapshot corresponding to the time 250 Myr in model MC as shown in the file attached to our comment to the editor. This corrected geoid significantly differs from the one shown in the manuscript, making it much closer to the expectations you share in your review. First, the RMS geoid amplitudes are reduced by a factor of five. This is much closer to what we obtained in model MF, although it is still larger than Earth's present geoid. Another important difference is the absence of strong correlations between the continents and the geoid. As you mentioned in your review, the continents are isostatically compensated and are thus not expected to leave an important signal in the geoid. Given these first results, we are confident that the corrected geoid will be of much greater scientific value.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Citation: https://doi.org/10.5194/egusphere20231782AC7

AC7: 'Reply on RC3', Thomas Frasson, 24 Nov 2023

RC3: 'Reply on AC2', Bernhard Steinberger, 28 Sep 2023

AC2: 'Reply on RC2', Thomas Frasson, 28 Sep 2023

RC4: 'Comment on egusphere20231782', Shijie Zhong, 24 Oct 2023
This manuscript presents a very interesting and novel idea that true polar wander may effectively alter the CMB heat flux pattern reference to the Earth's rotation axis, thus affecting the geodynamo if it is significantly influenced by the CMB heat flux pattern as suggested by previous studies. However, like Dr. Steinberger, I find the geoid results from the numerical models difficult to understand. Since the polar wander is controlled by the degree2 geoid and the geoid calculations are tricky in numerical models, it is important for the authors to explain more clearly how the geoid anomalies are computed.
The geoid modeling is tricky and sometimes even difficult in numerical models, because it requires accurate determination of dynamic topography. There are two contributions to the geoid: 1) mass anomalies associated with dynamic topography at the surface and CMB (and phase and compositional boundaries, if they are present) and 2) buoyancy from mantle thermochemical structure. The contributions from these two sources often have comparable amplitude but different signs, and the geoid anomalies represent the difference between these two terms. In general, dynamic topography is related to the pressure and derivative of flow velocity which is always oneoder less accurate than the primary variables like flow velocity. Therefore, a small error in dynamic topography tends to get amplified in the errors for the geoid. In finite element models like CitcomS, special techniques like consistent boundary flux method are often used to resolve this issue.
One class of convection models (MF) presented in this study used CitcomS. Given that CitcomS has been extensively benchmarked for the geoid problems, I would think that the geoid results from this class of models should be okay. However, the geoid results from this class of calculations also raise some concerns to me. For example, I do not quite understand why the geoid would be so different after the removal of shallow thermal structure (i.e., the top 350 km), because the longwavelength geoid (e.g., at degree2) often is insensitive to buoyancy at shallow depths where the geoid kernel goes to zero (the geoid kernel concept remains largely relevant even for models with 3D mantle viscosity).
We have done similar calculations of mantle convection driven by plate motion history with shorter time history of plate motion than that in this study (see Zhang et al., JGR, 2010; Mao and Zhong, JGR, 2021). Dynamic topography for Zhang et al. (2010) models was computed and discussed in Zhang et al., (2012, EPSL) in the context of continental uplift history. Mao and Zhong (JGR, 2021) computed the geoid (and dynamic topography) using plate motion history for the last 100 Ma and reproduced well the observed geoid between degrees 4 and 12 with certain mantle viscosity models. As discussed in Zhang et al. (2012) and especially in Mao and Zhong (2021), for convection models with prescribed plate motion as boundary conditions, it is important to recompute for dynamic topography by replacing the velocity boundary conditions with freeslip boundary conditions and appropriate lithospheric viscosity (i.e., to produce horizontal velocities that are similar to the prescribed plate motion). Note that in Mao and Zhong (2021), weak plate margins were introduced to achieve this purpose. Anyway, it is unclear from the manuscript how the geoid and dynamic topography were computed (e.g., were freeslip boundary conditions used together with some appropriate lithospheric viscosity?). The authors need to describe these issues in the revision. Prescribed surface velocity boundary conditions tend to produce spurious pressure field and hence dynamic topography, with high viscosity lithosphere.
The other class of convection models (MC) produced geoid anomalies of several kilometers that are only slightly smaller than that of surface dynamic topography (Fig. 2). These calculations are presumably for Rayleigh numbers that are comparable with that for the Earth's mantle and with that in the first class of convection models using CitcomS. In this type of situation, kilometers of geoid anomalies seem too large to me. Additionally, the geoid to topography ratio in most models in general should be around 0.10.2 (like admittance which is the ratio of gravity anomalies to topography, the geoid to topography ratio is only sensitive to viscosity structure, but significantly less sensitive to distribution of buoyancy). For reference, the observed long wavelength is ~ 100 meters, while the dynamic topography is ~1 km. Therefore, I recommend that the authors show some benchmark calculations for the geoid for their code by comparing with analytical solutions (see Zhong et al., G^3, 2008). Perhaps, such results are already available for their code, and then the authors can reference them (however, they still need to explain how + 3 km geoid anomalies can be generated from their models).
In summary, I think that the main scientific idea on polar wander and CMB heat flux pattern is very interesting and novel and deserves to be published, but I also think that the authors need to explain and justify their geoid results better. I hope that my review is useful.
Shijie Zhong
Citation: https://doi.org/10.5194/egusphere20231782RC4 
AC6: 'Reply on RC4', Thomas Frasson, 24 Nov 2023
Dear Shijie,
Thank you for your review and your enthusiasm for our study. As explained in our message to the editor, the tests we made in response to reviews by yourself and Bernhard Steinberger led us to realise that the geoid computations were erroneous in model MC. We are currently computing the geoid again in model MC, and we will modify the manuscript accordingly.
The first results we have obtained seem to answer your concerns about the amplitude of the geoid. We obtained RMS geoid anomalies that are five time smaller than previously. These amplitudes are still too large compared to the presentday Earth, but they are much closer to what is obtained in model MF. We conducted some tests on this snapshot, as shown in the file attached to our comment to the editor. We notably find that the topography and density components of the geoid are of opposite signs and largely compensate each other, as described in your review. We also show that choosing a radial viscosity profile significantly affects the results.
You also asked for precisions concerning the computation of the geoid. We will better describe the procedures in the revised version. Concerning the questions you raised, as you mention, model MF was computed using CitcomS, which has been benchmarked for this purpose. We compute two geoid outputs following Flament (2019). Both outputs are computed by restarting the model at a given time step with freeslip conditions applied at the surface. The first output, called Total geoid in the manuscript, was obtained without altering the density or viscosity distributions in the mantle. A second output called No LVVs was obtained by cancelling the density and viscosity lateral variations in the upper 350 km. This second output is significantly different from the Total geoid because of the removal of the lateral variations of viscosity. We also computed a third output, not shown in the manuscript, for which we only suppressed the lateral variations of density, keeping the viscosity distribution untouched. As you expected in your review, this third output is very similar to the Total geoid, leading to almost identical TPW paths. We thus decided not to include this case in our manuscript. It is clear from these tests that the differences between the Total geoid and the No LVVs geoid are due to the effect of viscosity variations rather than density variations.
The computation of the geoid in model MC uses the same approach as in model MF, based on Zhang and Christensen (1993). In this case, however, no changes in the surface conditions are necessary to compute the geoid as there is already a freeslip condition at the surface in the model. As in the Total geoid case of model MF, the density and viscosity fields are fully conserved for the geoid computation.
Yours sincerely,
Thomas Frasson, Stéphane Labrosse, HenriClaude Nataf, Nicolas Coltice and Nicolas Flament
Flament, N. (2019). Presentday dynamic topography and lowermantle structure from palaeogeographically constrained mantle flow models. Geophysical Journal International, 216(3), 21582182.
Zhang, S., & Christensen, U. (1993). Some effects of lateral viscosity variations on geoid and surface velocities induced by density anomalies in the mantle. Geophysical Journal International, 114(3), 531547.Citation: https://doi.org/10.5194/egusphere20231782AC6

AC6: 'Reply on RC4', Thomas Frasson, 24 Nov 2023
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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
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Thomas Frasson
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