Dear Juliane Dannberg,

We have chosen to submit and re-submit our article to Solid Earth, rather than to a higher-impact 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 state-of-the-art models of mantle convection published in high-impact 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, Henri-Claude Nataf, Nicolas Coltice and Nicolas Flament

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 35-45: 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 dense-cored LLVPs").

Thank you for pointing out the recent study of Richards et al (2023), which we will refer to.

• Page 6: lines 160-165: 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 2-3 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 plate-like 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 self-consistent. 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 250-260. 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 10-11) 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, Henri-Claude Nataf, Nicolas Coltice and Nicolas Flament

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 Core-Mantle 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 flow-induced interface undulations’. Our revised study presents two different mantle circulation models, computed with state-of-the-art 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 state-of-the-art geological reconstruction, while model MC is a self-consistent realistic mantle circulation model in which plates form naturally. These are the two main strategies followed today. Model MC has the advantage of being self-consistent 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 self-consistent, 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 free-slip 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 self-consistent. Using results from both types of models allows us to ascertain the robustness of our results about TPW and its implications for the time-evolution 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, Henri-Claude Nataf, Nicolas Coltice, Nicolas Flament

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 present-day 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 free-slip 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 free-slip 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, Henri-Claude Nataf, Nicolas Coltice and Nicolas Flament

Flament, N. (2019). Present-day dynamic topography and lower-mantle structure from palaeogeographically constrained mantle flow models. Geophysical Journal International, 216(3), 2158-2182.

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), 531-547.