Inner core static tilt inferred from intradecadal oscillation in the Earth’s rotation

Domain: earth_science
Published: 2023-12-08
Editorial outcome: accept

Abstract
The presence of a static tilt between the inner core and mantle is an ongoing discussion encompassing the geodynamic state of the inner core. Here, we confirm an approximate 8.5 yr signal in polar motion is the inner core wobble (ICW), and find that the ICW is also contained in the length-of-day variations of the Earth’s rotation. Based on the determined amplitudes of the ICW and its good phase consistency in both polar motion and the length-of-day variations, we infer that there must be a static tilt angle θ between the inner core and the mantle of about 0.17 ± 0.03°, most likely towards ~90° W relative to the mantle, which is two orders of magnitude lower than the 10° assumed in certain geodynamic research. This tilt is consistent with the assumption that the average density in the northwestern hemisphere of the inner core should be greater than that in the other regions. Further, the observed ICW period (8.5 ± 0.2 yr) suggests a 0.52 ± 0.05 g/cm3 density jump at the inner core boundary.

Introduction
The oblate solid Earth consists mainly of a solid inner core, a liquid outer core, and a solid mantle with the same center of mass, which is reduced to a series of layered elliptical surfaces on which the density is constant in the classical model1,2. The current theories regarding the Earth’s rotation involve the consideration of the mantle’s elliptical surfaces of constant density, whose symmetry axes are aligned in the direction of rotation, and hydrostatic effects necessitate that the inner core’s figure axis Ωic (as defined in Appendix A of ref. 2) and rotation axis Ω′ic are aligned with the mantle’s figure or rotation axis Ωm (which are nearly identical due to centrifugal torque) in order to maintain equilibrium. The presence of random torques acting on the inner core results in a slight tilt and further excites a prograde rotation mode known as the inner core wobble (ICW), i.e., the inner core’s figure axis Ωic wobbles about its rotation axis Ω′ic 2–4 (also represents the direction of the lowest gravitational potential energy of the mantle-inner core system). The above ‘tilt’ between the Ωic and Ω′ic is a generally dynamic tilt, and in this case, the ICW theoretically appears only in the polar motion (PM) of the Earth’s rotation 2,3,5. In addition, the ICW period is very sensitive to the density jump ΔρICB at the inner core boundary (ICB; Δρ is still not well constrained6), and based on the PREM model1, theory predicts that its period falls in the range of 6.6–7.8 yr3,7–9. However, the elliptical surfaces of constant density within the heterogeneous mantle may exhibit random tilting around its rotation axis Ωm considering its solid properties, particularly with significant uncertainties at the core-mantle boundary (CMB). Consequently, the inner core’s rotation axis Ω′m, which signifies the direction of the static equilibrium of the inner core (and corresponds to the lowest gravitational potential energy of the mantle-inner core system10), was previously believed to deviate from alignment with the mantle’s axis Ωm and instead possess a tilt relative to it, and thereby the ‘tilt’ between the Ω′m and Ωm is called as static tilt. To explain the decadal oscillations in both the PM and the length-of-day variations (ΔLOD) as the possible ICW, the inner core’s rotation axis Ω′m was proposed to coincide with the dipole axis of the geomagnetic field (tilted 10° westwards from Ωm)11. Despite the absence of confirmed observations of the ICW12,13 and the lack of universal acceptance of the excessive static tilt of 10°, the possibility of a static-tilted inner core remains, and further investigation has been conducted to explore the impact of a static tilt on the period of the ICW12. Theoretically, such a static tilt must affect some modes that are sensitive to the inner core. However, no relevant eigenfrequency deviation has been clearly detected in the core-sensitive normal modes of Earth’s free oscillation14,15, which denotes that this static tilt is still uncertain, it may not exist or is quite small. Overall, a statically tilted inner core will be of great importance to some fundamental research about the Earth, such as the differential rotation of the inner core, the Earth’s surface gravity changes, the seismic tomography of the deep Earth, and the Geodynamo theory6,12,14,16–20.

A statically tilted inner core will induce changes in the rotational normal modes of the Earth, with the ICW being the most sensitive. In the presence of a statically tilted inner core, the ICW will manifest not only in the PM but also in the ΔLOD11. By identifying a similar periodic signal in the ΔLOD and establishing its correlation with the ICW identified in the PM, we can, in turn, ascertain the presence of a statically tilted inner core. Furthermore, the angle θ of static tilt can be determined by comparing the corresponding amplitudes of the two signals (in the PM and in the ΔLOD, respectively).

Results
In this study, we report the results from the ΔLOD and PM time series. The chosen ΔLOD timeseries is a yearly timeseries with a 1900–2020 timespan. For the PM timeseries, the 1900–2020 EOPC01 timeseries with one-year sampling is used (x and y components). The pretreatments of the ΔLOD and PM time series are shown in the Methods. Figure 1 shows the ΔLOD and PM records used. For the periodic signals present in the PM and ΔLOD, the consensus is that they are excited by the Earth’s internal or external sources through the conversion of angular momentum21. Hence, we need to rule out the influence of external excitation sources before determining that a target signal is from the Earth’s internal motion. There are three external excitation sources of the PM and ΔLOD changes, the atmospheric, oceanic, and hydrological effects. Of these, the first two effects are the two main external excitation sources21–23; although hydrological effects will also excite the Earth’s rotation changes, previous studies have proven that the hydrological effects have no significant contribution to the target 5.5–10 yr period band13,23 and different hydrological models have clear deviations13,24. Hence, similar to previous studies13,22, we only consider the atmospheric and oceanic effects. The PM and ΔLOD excited by the atmospheric angular momentum (AAM) and oceanic angular momentum (OAM) are also shown in Fig. 1. Supplementary Fig. 1 (in the Supplementary Information) also shows all datasets used before combining.

Different from the oceanic tidal signals that have both prograde and retrograde components in PM25–27, the ICW is a prograde motion (the same as the Chandler wobble, i.e., the mantle wobble); in the complex spectra of the PM, a prograde/retrograde wobble only has a positive/negative frequency. The identification of the well-known Chandler wobble is based on this feature28–30. Therefore, as a prograde motion, the ICW only appears on the positive frequency axis of the PM spectrum and this is a distinguishing feature for identifying it. Based on the 1960-2017 PM record without removing the AO (AAM+OAM) effects, a previous study13 used this feature to identify an ~8.7 yr signal for the ICW; here, we perform independent detection by using a longer record (1949–2020) and further consider the AO effects. Figure 2 shows the normalized AR-z spectra (see Methods) of the PM and ΔLOD records in the 1949–2020 timespan, in which the AO effects have been removed. Figure 2a shows four different harmonic signals (~5.9, ~7.3, ~8.5, and ~9–11 yr) in the positive frequency axis; only the ~8.5 yr signal has no corresponding spectral peak in the negative frequency axis (The AR-z method is meant for determining the presence of a signal and estimating its frequency, the amplitude of it contains no direct information about the actual complex amplitude of the detected signal). The corresponding Fourier spectra of the PMs (observed and AO excited) show similar findings (see Supplementary Fig. 2b). Among these harmonics, the ~5.9 yr signal has been suggested as the inner core oscillation coupled with torsional wave in the Earth’s core but still remains controversial31–33; the ~7.3 yr signal can be interpreted as the Magneto-Coriolis eigenmode in the Earth’s core based on a theoretical model34; the peak in the ~9–11 yr is possible from the ~11 yr Schwabe solar cycle. The adjacent ~13 yr signal (Fig. 2a) also has both positive and negative frequencies; similar period was also found in the geomagnetic dipole field35, but the underlying mechanism is still enigmatic; the ~18–23 yr spectral peak may be mainly caused by the 18.6 yr tidal signal and the ~22 yr Hale solar cycle or high-latitude MAC (magnetic-Archimedes-Coriolis forces) wave36 in the Earth’s core; those periods are too long to be the ICW3,7–9. Therefore, the 8.52 ± 0.19 yr signal is the only candidate for the ICW. Since no other mechanism has been proposed to account for such a prograde ~8.5 yr motion, and the AO effects have been removed, we can conclude that the 8.5 yr signal is the ICW. In addition, the uncertainties for the estimates in this study were based on a bootstrap procedure37.

Comparing Fig. 2a and b, a finding is that the six periodic/quasi-periodic signals in the positive frequency axis of the PM spectra are also present in the spectrum of the ΔLOD. This mainly benefits from the high-frequency resolution of the AR-z spectrum and its strong sensitivity to harmonic signals38; the Fourier spectra can only identify parts of those signals (see Supplementary Fig. 2). These consistencies deserve further attention, but we only focus on the 8.5 yr signal (the ICW signal). Figure 2b confirm that the ICW signal is also present in the ΔLOD (with an 8.47 ± 0.32 yr period); this finding preliminarily suggests that there should be a static tilt between the inner core and the mantle. Given that the AO effects have no significant contribution to the target signal, we use the 1900–2020 PM and ΔLOD records to extract the ~8.5 yr signal to obtain higher resolutions. For simplicity, we directly use a cosine least-square fitting process.

To further obtain the orientation of the static tilt angle θ and its magnitude, we need to determine the fluctuation characteristics of the axial torque Γz (∝dΔLOD/dt; see Methods) exerted on the mantle. Hence, we directly fit the ~8.5 yr signal from dΔLOD/dt; the fitted results from ΔLOD can be found in Fig. S3.

Figure 3 shows the fitted ICW from the dΔLOD/dt and the x and y components of the PM. Clearly, the ICW from the y-component is ahead of that from the x-component by ~π/2 (see green areas in Fig. 3); since the directions of x and y have a π/2 angle difference in the equatorial plane, these findings are acceptable. The most important point obtained from Fig. 3 is that, for the first time, we find that the ICW contained in the y component of the PM and dΔLOD/dt have almost synchronous phases; the extracted oscillations using a more complicated method (the normal time-frequency transform, NTFT39) show almost the same results (see Supplementary Fig. 3). This synchronicity is not a random phenomenon and at least demonstrates that the inner core tilts in a particular direction (see the possible scenario in Fig. 4). The axial torque Γz reaches its peak/trough only when the Γz is in the plane defined by the static tilted axis Ω′m and the rotation axis of the mantle Ωm; hence, we can deduce that the inner core tilts should be along the ~90° E–90° W direction.

Given the y component of the PM along the 90° W longitude, the phase synchronization in Fig. 3a, c indicates that the inner core is more likely tilted in the 90° W direction, which is also similar to that suggested by previous studies11,12,40,41. In terms of the long-term dynamic conservation of the Earth’s angular momentum, this static westwards tilt is consistent with the effect of the non-axisymmetric mass of the inner core, i.e., the western hemisphere (more specifically, the northwestern hemisphere) of the inner core should have greater averaged densities. To explain the asymmetry between the inner core’s eastern and western hemispheres in seismological observations42,43, a previous dynamical model considers the crystallization and melting at the surface of the inner core and has similar suggestions44, i.e., the western hemisphere of the inner core is denser than its eastern hemisphere. Interestingly, a seismological study has suggested that the western hemisphere of the inner core may be relatively denser45, and a thicker compacting layer at the top of the inner core’s western hemisphere was also suggested46; a more nuanced research found that the western zone is largely confined to the northern hemisphere47; those suggestions are generally consistent with the westwards statically tilted inner core that we found.

Here we can propose the following scenario as schematically depicted in Fig. 4: There is a static westward tilt θ between the inner core and the mantle, resulting in the inner core exhibiting a wobbling motion around the tilted axis Ω′m. This wobbling motion leads to the exchange of angular momentum between the mantle and inner core in both the equatorial and axial directions of the mantle, consequently giving rise to the appearance of the ICW in both the PM and ΔLOD. The inner core and mantle then obtain the maximum or minimum deviation at ~90° E−90° W (i.e., x ≈ 0) equatorial diameter; the torque on the inner core (equatorial plane of the inner core) thus has a maximum/minimum component on the axis of rotation when x ≈ 0. At the same time, the ΔLOD is also 0 due to a first derivative relationship with the exchange of the axial angular momentum (see Supplementary Fig. 3). Therefore, there will be a good phase consistency between the dΔLOD/dt and the y component of the PM for the ICW (confirmed in Fig. 3).

Next, we estimate the tilt θ. The heterogeneous mantle results in a tilted rotation axis of the inner core Ω′m. When the inner core’s figure axis Ωic is aligned with Ω′m, the mantle-inner core system has the lowest mantle-inner core gravitational (MICG) potential energy (the equilibrium state). Any deviation of the inner core’s figure axis Ωic from the equilibrium state caused by the ICW will result in a MICG restoring torque ΓMICG; the torque ΓMICG always bring the inner core back to the equilibrium state. Thus, the torque ΓMICG is always in the plane (the blue plane in Fig. 4) perpendicular to the inner core’s rotation axis Ω′m. The mean equatorial plane of the inner core is perpendicular to the axis Ω′m, so ΓMICG is always in the mean equatorial plane of the inner core. This MICG torque can also be decomposed into the equatorial and axial torques exerted on the mantle due to the tilt of the inner core. If there is no static tilt between the inner core and mantle, the axial torque exerted on the mantle will be 0, and the entire gravitational torque will provide the equatorial torque exerted on the mantle.

Under the MICG coupling, the equatorial torque exerted on the mantle can be written as the following (see Methods):

|ΓMICGeq| = (Cm − Am)Ω2|χICW(t)|

where χICW(t) is the excitation sequence of the ICW in the PM, and the observed ICW is almost the same as its excitation sequence due to the low frequency21,48 (see Methods). Since the ICW identified in the PM has an amplitude of 4.7 ± 0.4 mas, we can calculate |ΓMICGeq| to be (2.87 ± 0.24) × 1019 N·m. According to the theorem of angular momentum, the axial torque exerted on the mantle is written as the following (see Methods):

|ΓMICGz| = Ω0Cm d(ΔLODICW)/dt

Substituting the amplitude of 0.046 ± 0.005 ms/yr (corresponding to the amplitude of 0.061 ± 0.007 ms of ΔLODICW) and the period of 8.5 yr for the observed ICW signal in the ΔLOD into Eq. (2), the axial torque |ΓMICGz| is calculated to be (8.61 ± 0.95) × 1016 N·m; this is only a small component of the equatorial torque of the inner core caused by the ICW due the inner core static tilt. Therefore, the static tilt angle θ, or the angle between the axis about which the inner core wobbles and the rotation axis of the mantle, is calculated as arctan(|ΓMICGz|/|ΓMICGeq|) (see Fig. 4) and equal to 0.17 ± 0.03°; this is much smaller than previous assumptions.

Our observed ICW period is slightly larger than the theoretical values (6.6–7.8 yr)3,7–9, but considering that even the generally accepted Chandler wobble observation period of prograde ~430 days is ~30 days longer than its theoretical periods28–30, free core nutation observation period of retrograde ~430 days is ~20 days shorter than its theoretical periods49,50, and that the density jump ΔρICB at the ICB was also poorly determined6,51, this deviation is accepted. Considering this newly determined period of the ICW, we can also invert the density jump ΔρICB. Taking the density profiles of the PREM model as a reference, we finally obtained ΔρICB = 0.52 ± 0.05 g/cm3 (see Methods), which is smaller than that of the PREM model (0.598 g/cm3).

In summary, based on the Earth’s rotation observations (PM and ΔLOD), we experimentally confirmed for the first time that the 8.5 yr signal is the ICW. The evidence indicates that the inner core is tilted to the mantle along ~90° W, and the inverted tilt angle is 0.17 ± 0.03°; this static tilt angle means that the average density in the northwest hemisphere of the inner core should be greater. The larger observed period may also indicate that the eastwards differential rotation rate of the inner core should be much less than 1° per year12,16. Besides, the density jump of 0.52 ± 0.05 g/cm3 at the ICB is also inverted based on the observed ICW period. Undeniably, it is difficult for seismological observations to detect such inner core static tilt directly, but interestingly, the results from seismological studies showed that the western/northwestern hemisphere (or at least its top layer) of the inner core may be relatively denser42–47. These suggestions, although they have some uncertainties, are qualitatively consistent with our finding of a westwards-tilted inner core, and we suggest such consistency should be helpful to the inner core oscillation or differential rotation.

Methods
Conservation of angular momentum of the mantle and inner core
Considering the mantle alone, the law of angular momentum can be rewritten as48:

dH/dt + Ω × Hm = Γm

where Earth’s angular velocity Ω = Ω0 [m1, m2, 1+m3]T; the mantle angular momentum is:

Hm = Im Ω

The asymmetric part of the mantle mentioned above is insignificant relative to its axisymmetric part, and the mantle can still be approximately as axisymmetric in the calculation of torque for simplicity; Im is the mantle moment of inertia tensor initially expressed in the principal axes:

Im = diag(Am, Am, Cm)

Combining the above equations with the eigenfrequency of the free Euler wobble replaced by the Chandler wobble σCW, the equatorial torque exerted on the mantle can be obtained by the observed PM

Γeq / (σCW (Cm − Am)Ω0) + m = χ(t)

where Am and Cm are the equatorial and axial moments of inertia of the mantle, respectively; m = m1 + im2 = x − iy is the observed PM; χ(t) = χ1 + iχ2 is its excitation function. The relation between an excitation function of complex frequency σ and the motion of the observed pole is χ(t) = (1 − σ/σCW)m (in which σ = ω + iγ, ω = 2π·0.843 cpy and γ = ω / 2Q; Q ≈ 30–150)52,53. Thus, there is little difference between excitation and observation in the low-frequency band.

Similarly, the axial torque exerted on the mantle can be obtained by the following:

Ω0 Cm dm3/dt = Γz

where m3 = −ΔLOD/LOD; LOD = 86400 s. Combining with Eqs. (1), (2), (6) and (7), we can directly infer the static tilted angle of the inner core from the ICW signal in the ΔLOD and PM, which is impossible in related previous studies11,12.

Stabilized AR-z spectrum
A real discrete time series with the length of N equally spaced samples, which contains M harmonics, is written as (which satisfies the AR relation54):

x(n) = Σj=1M Aj exp(inσj) + A*j exp(-inσ*j), n = 1, 2, 3, ..., N

where Aj = Aj exp(iϕj)/2 is the complex amplitude (Aj and ϕj are the amplitude and initial phase) and σj = ωj + iαj is the complex frequency of a given harmonic (ωj and αj are the angular frequency and decay rate). By using a frequency-domain AR method, the complex frequency σj can be estimated38. A Lorentzien power spectrum in the complex z plane can be formed as follows38:

P(σ) = 1 / |exp(σ~i) - exp(iσi)|2, i = 1, 2, ..., N

where σ~i and σi are the estimated and referred complex frequencies, respectively. For the specific execution of the stabilized AR-z spectrum, please see the Supplementary Text (in the Supplementary Information).

Constraint for the density jump at the inner core boundary
The frequency of the ICW can be written as follows (in cpsd: cycle per solar day)3:

σICW = [αg(1 + αg)(e + Sg34 + Sp34)] / (1 + KICB)

where the elastic compliances Sg34 = –1.812 × 10–6, Sp34 = –2.686 × 10–4 and KICB is a dimensionless coupling constant55 and Real(KICB) = 1.11 × 10–3; αg and αs have the following forms:

αg = 1 - (A0e0 / Ases)αs
αs = 3G / (a5 Ω2) (5ρf + 1) (A0e0 / Ases - 1)

where ρf is the fluid density just outside the ICB, ρs is the mean density of the inner core, A0 and e0 are the equatorial moment of inertia and the dynamical ellipticity of the inner core, respectively, and A’ and e’ have similar definitions but for a body of the inner core radius with the constant mass of that of the fluid core at the ICB2.

In which ɛ(r) is the geometrical ellipticity of the Earth and a is the inner core radius. Taking the PREM model as a reference because it is the generally accepted model, underlying the conservation of the whole Earth’s mass and angular momentum, we can modify the density of the outer core (based on the related expression given in PREM) and hence change the inner core density profile to obtain the observed ICW period. When the observed 8.5 yr period is obtained, the corresponding ΔρICB is the one we recommend using.

Datasets and preprocessing
The PM observations were obtained from the EOPC01 dataset (1861/01-1889/12 with 0.1 yr sampling and 1900/01–2019/12 with 0.05 yr sampling); the ΔLOD record was combined with a long-term dataset56 (1623/06-2008/06 with 1 yr sampling from IERS; EOPC01) and the EOPC04 ΔLOD record57 (1962/01–2019/12 with 1-day sampling); the AAM (1949/01–2019/12, sampling at 6 h) record was from the Special Bureau for the Atmosphere58–60. The AAM was calculated from NCEP/NCAR reanalyses archived on pressure surfaces, and the inverted barometer (IB) pressure term was chosen as the mass term. The OAM record was obtained from the Special Bureau for the Oceans’ datasets: ECCO_50yr61 (1949/01–2003/01, sampling at 10 days) and ECCO_kf080i62 (1993/01–2020/3, sampling at 1 day). Those datasets are shown in Supplementary Fig. 1. To standardize the sampling intervals of the records, we down-sampled all records to 1 yr, and to avoid aliasing effects in this down-sampling process, a low-pass filter (with a cut-off frequency fc = 0.5 cpy) was used prior to down-sampling.

Note that although the theoretical amplitudes of the 8.85 yr and 9.3 yr zonal tides are quite small (only ~2 μs for the 9.3 yr tide and < 1 μs for the 8.85 yr tide) and far less than the background noise level of the ΔLOD time series, they were removed from this ΔLOD record based on a given model63 to avoid the effects of some well-known signals on the target ~8 yr period band. The dΔLOD/dt was obtained by a classical discrete numerical derivation algorithm, i.e., dΔLOD(t)/dt = [ΔLOD(ti+1) - ΔLOD(ti)] / Δt.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "A review of the paper:\nInner core static tilt: Evidence from intradecadal oscillation in the Earth's rotation\nby An et al.\nThis is a paper on the possibility that the inner core has a static tilt with the Earth's rotation axis.\nConsider the very first sentence of your manuscript.\n“The Earth consists mainly of a solid inner core, a liquid outer core, and a solid inner core with the same\ncenter of mass.”\nI don’t know what the above sentence means, but the sloppiness, unfortunately, persists throughout the\nentire manuscript. Although geodynamics is not my area of expertise, and this paper needs to be\nassessed by someone with good knowledge of the current research on the tilt to judge the novelty you\nclaim, you did not convince me that you can observe the tilt of the Earth’s inner core, let alone prove it.\nThere are two main reasons: firstly, in the abstract and elsewhere, you say that the tilt you infer is 0.17\ndegrees away from the Earth’s spin axis. However, you do not present the uncertainty of that estimate,\nand given that it is probably quite large, it could be said that you obtained a 0.0-degree tilt or no tilt at\nall. Secondly, a correlation is not causation. You will need more robust evidence that the inner core has\na tilt apart from showing the two spectra and deducing that the signal you observe in the spectrum is\nthe only candidate for the ICW. Take, as an example, a recent claim by seismologists that the rotation of\nthe inner core has some correlation with the LOD change – no one in the seismological and the broad\ngeophysical community can take it as proof that the two are related or caused by the same\nphenomenon due to a significant uncertainty of those estimates and a lack of physical mechanism\nexplaining the correlation. This lack of uncertainty analysis and solid proof that you are observing a tilt\nof the inner core makes your paper highly speculative. Furthermore, you use some seismological papers\nthat showed complexities in the inner core to reinforce that the inner core might have a non-\naxisymmetric mass, but in my opinion, a more rigorous understanding of what these papers are all\nabout is lacking from the manuscript.\nSo, a tilt and a differential rotation of the inner core might be real as they emerge intuitively and\nnumerically from geodynamical modeling (as shown in multiple papers), but you need more in this\npaper to convince me that what you have achieved is a conceptual advance.\nLet me comment further on your reference to the seismological papers.\nYou misunderstood the goal of most inner-core seismological papers you cite. The goal was not to\nestablish a tilt of the inner core but to study anisotropy. Then, by making a strong assumption that\nanisotropy could be approximated by a cylindrical model, one can perform a grid search to see if the\ndata one has would point to the main axis of anisotropy oriented in a slightly different direction than\nthe spin axis of the Earth. You can then obtain some mean, which cannot be oriented exactly along the\nEarth's rotation axis because of the noise in the traveltime data. What’s worse than that is that we don’t\nhave an even volumetric sampling of the ray paths sensitive to the inner core. You cite the paper by Su\net al. (1996) and their estimate of the tilt within the range of 5–10 degrees, but you are perhaps\nunaware that this tilt was shown to be meaningless by Souriau et al. (1997) because of the bias in the\ndata. By the way, they didn’t find any wobble in time, even with that biased estimate. The bottom line is\nthat these early estimates of the anisotropy axis were shown to be unreliable and erroneous. For that\nreason, they were abandoned, and the differential rotation of the inner core estimates were conducted\nby other methods. Therefore, your underlying thesis that your estimate of the tilt is two orders of\nmagnitude lower than what is obtained by seismology is wrong and misleading. Seismology has not yet\ndetermined the tilt of the inner core or anything like that.\nAlso, let me comment on your invoking seismology as a method that found that the mass distribution in\nthe Earth’s inner core is not axisymmetric. While heterogeneities are documented to exist near the inner\ncore boundary, everything beyond that is much more speculative. If we accept that one hemisphere of\nthe inner core is faster than the other, would that necessarily indicate a denser material and more\nmass? Why would anisotropic fabric be a denser fabric? The opposite can be true. From seismology,\nanisotropy appears weaker in the outer part of the inner core and stronger towards its center. It does\nnot necessarily have anything to do with cylindrical anisotropy's fast and slow axis but with the\npronounced slow axis in the innermost part of the inner core. For instance, deeper into the inner core,\ntwo recent studies that you cited show an offset of the anisotropic (as hypothesized) volume from the\ncenter at two different locations of the inner core. In all honesty, there is no consensus as to which one\nis correct, so your inference that a tilt is to the western side needs more seismology support. But if you\nhonestly present seismological work as it is, you will find it more difficult to substantiate your claims.\nThere is no such thing as “super-rotation”. (Line 34) We talk about the differential rotation of the inner\ncore relative to the mantle. Besides, the paper would benefit from a discussion how the inner core tilt\nwould impact the differential rotation studies.\nThe quasi-eastern and quasi-western anisotropy is not that established. The robust differences are\ndocumented in absolute velocity between the eastern and the western hemisphere, but only at the top\nof the inner core. Given the sparse coverage of the inner core on the western side, whether the same\nholds for anisotropy is hard to tell. As you go deeper toward the center, the hemispherical division\nloses its meaning, and there is no consensus that any hemispherical pattern exists.", "reviewer_2": "First of all, I would like to state for the record that I believe that the authors have clearly and\nconvincingly demonstrated that the results they present in the article are true. Moreover, I would like\nto add that in my view they are truly important and sound. Applying a methodology that has already\nshown its power in other applications to geodetic problems and using precise earth rotation data, they\nhave proved that the axis of the earth's solid inner core is tilted with respect to the polar axis of the\nwhole earth, and at the same time they have determined for the first time empirically the value of the\nso-called ICW period. This motion can be thought of as analogous to the precession motion of the earth\nunder lunisolar attraction, but in this case it is the inner core under the attraction of the earth's mantle\nsurrounding it. This free mode of the Earth's rotation has so far only been known from theoretical\ndevelopments and the value of its period inferred from geophysical models, as well as more indirectly\nfrom its effects on the Earth's rotation, more precisely on nutation, since the introduction in 1991 of a\nbasic earth model made of 3 ellipsoidal, concentric and aligned layers, which was later refined and fitted\nto VLBI data, becoming the official IAU2000 nutation model.\nAlthough the authors only highlight the importance of the experimental determination of the inner core\ninclination from earth rotation data, more precisely from pole motion (PM) and length-of-day (LOD),\nI consider that the determination of the value of this ICW period is not less important than the previous\nfinding and I could even dare to say that for me it could be even more important in that the accuracy of\nthe method they use to identify weak harmonic signals (stabilized AR-z spectrum) is very precise in\ndetermining the value of the periods, but does not allow determining the amplitudes with the same\nrelative accuracy as far as I know.\nIn my opinion, there is no doubt about the veracity of the results, since they clearly identify a harmonic\nconstituent of the pole motion with a period close to 8.5 years, formed only by the prograde\ncomponent, and also an oscillation of smaller amplitude in LOD with the same period; current\ntheoretical knowledge can only explain this coincidence if the origin of the signal is precisely the\naforementioned free oscillation ICW (which we can imagine as the precession of a small planet located\ninside the Earth's mantle following the analogy made in one of the references cited by the authors).\nTherefore, my main suggestion is to make some changes in the wording of the article to show that not\nonly the static inclination of the inner core axis has been determined, but also the value of the period of\nits free ICW motion has been determined for the first time experimentally, based on Earth rotation data.\nI consider that the empirical determination of this value should be highlighted since in the last 30 years\nseveral values of the period, inferred from theoretical considerations, have been published (as cited in\nthe references) and even in some recent work it has been obtained that the motion should occur in the\nretrograde direction and not in the prograde direction as most of the previous authors concluded (\nChao, B. F. (2017), Dynamics of the inner core wobble under mantle-inner core gravitational interactions,\nJ. Geophys. Res. Solid Earth, 122, doi:10.1002/ 2017JB014405). This shows how difficult is deriving\naccurate values of the ICW period from theory and geophysical models of the earth’s interior. On the\nother hand, I would like to remind that the observed value of the Chandler period has not yet been fully\njustified, and that the most complete theoretical deductions often differ from the observed value by a\nfew tens of days.\nI think the authors could emphasize their experimental determination of the period without making\nmajor changes to the text or the abstract, although perhaps the title should be modified so as not to\nemphasize only the inclination. Regarding the magnitude of the difference with former estimations of\nthe period, based on quite different methodologies because none uses accurately observed earth\nrotation data, I’d like to recall that reliable error bars are not available for any of the published\ndeterminations, given the difficulty of the implied calculations. That fact must be taken into account in\nany controversy that may be risen.\nOn the other hand, while the wording poses no problems for those with an intermediate knowledge of\nthe earth's rotation, I think from my experience that a large number of potential readers interested in\nthe subject do not reach that level. They are possibly well acquainted with the Chandler period arising\nfrom the existence of the liquid core, but rather less familiar with the two additional normal modes of\nrotation that appear when the solid earth model also contains an inner core. I therefore suggest that the\nauthors consider adding a short paragraph describing the phenomenon.\nBelow, I suggest that the authors please consider making some minor editorial changes, but this\nsuggestion does not mean that they should make them. They are (by lines):\nL 21. Change \"Earth\"->\"solid Earth\".\nL 26 Specify which theory is meant if the sentence refers to a specific one or make a more generic\nreference such as \"most published theories...\"..\nL 34-35 It would be useful to include some reference, either among those already cited or in addition to\nthose already cited.\nL 39 I think it would be useful to indicate that this fact will be demonstrated later, or to give a reference,\nfor the convenience of readers.\nL 54 It is actually in the caption of Figure 1, not in the figure itself.\nL 55 The wording is not completely accurate, it is valid for the so-called Earth Rotation Parameters (ERP)\nwhich are the ones used in the derivation of the results, but not for the remaining two, namely\nprecession-nutation ones.\nL 79-80 Perhaps it would be useful to reinforce the reasoning made to reach the conclusion. Actually, it\nis not only possible to conclude, but it must be concluded because current theoretical knowledge offers\nno other possibility to explain the origin of this 8.5-year signal with the above-mentioned\nproperties.\nL 107-108 Although the sentence is true, it is perhaps worth expanding the explanation slightly.\nL 191 Including a reference for the conventional value of Omega, among the many available sources\nL 194 Rather than observation, one should say the motion of the observed pole (as a response).\nL 208 Not necessarily in this line, but somewhere in the text or supplementary material, I think more\ndetails should be given on how the method has been applied, e.g., the size of the bins, whether the Q\nfactor that appears as one of the input variables in the referenced routine (in L 228) has been given an\ninfinite or finite value, etc., so as to ensure that the calculations are transparent and repeatable.\nL 306-307 Correct the reference (Fong Chao, B…)", "reviewer_3": "Major points\n1) Discussion of seismology\nThe discussion of the previous seismic work is not entirely reflective of what those studies found. I do\nnot think it is fair to say that the studies are collectively arguing for a large static tilt of the inner core.\nCollectively, the cited studies (and others) have investigated the anisotropic and heterogeneous\nstructure of the inner core, and whether there is differential axial rotation between the inner core and\nmantle. Some of the differential rotation studies (e.g. Wang & Vidale 2022, ref 10 of this work; and the\nrecent article by Yang & Song 2023, Nature Geoscience) measure differential rotation by\ntracking/estimating seismic variations in the shallow inner core; these lateral variations provide no\ninformation on a potential tilt of the inner core. Studies of inner core anisotropy (e.g. refs 7 and 9 of\nthis work) do lead to estimates of anisotropy axis that is tilted relative to the net rotation axis of the\nEarth. However, the physical mechanism responsible for that seismic anisotropy and, in particular, the\nphysical meaning of the orientation of that fast axis are debated (even within individual observational\nstudies), as noted in L118-120. The alignment of the fast axis from seismic anisotropy certainly need\nnot be attributed to a tilt of the inner core. Furthermore, the review by Deuss (2014, ref 12 of this work) notes\n“The hemispherical variations removed the need for a tilted anisotropy axis and provide a better fit to\nthe body wave data”.\nThe inner core structure outlined by the recent study of Brett et al (2022, ref 29) and discussed on L130\nhave the western anisotropic zone confined to the northern hemisphere; I do not agree with the\nsubsequent statement on L131 that this is indicative of a tilted figure axis, which should produce a\ncomparable anomaly in the antipodal eastern + southern region of the inner core. Overall, I think the\nstatements made in this work that a 5-10 degree tilt of the inner core figure axis are suggested by\nseismology need to be removed as being unreflective of the overall message put forward by those\nstudies.\n2)Is 8.5 years the ICW and, if so, why?\nAs in their previous work (ref 16 of this work), the authors have found a number of peaks in polar\nmotion. The 8.5 yr peak is argued to be the ICW as it shows in the prograde spectrum. The other peaks\nare of similar amplitude to the 8.5 yr signal and also show inconsistency between positive and negative\nPM spectra (e.g., the 7.3 yr signal is much larger in the negative half of figure 2a, the 5.9 year peak is\nconsiderably stronger in the positive half). Without understanding what those other peaks represent\nphysically and why their amplitudes vary, it is not obvious that the 8.5 yr peak should be picked out as\nthe ICW.\nLet us accept that the 8.5 yr peak is the ICW. This is not the period predicted from previous theoretical\nmodels that account for Earth structure and match other observed wobbles and nutations (e.g., refs 1-\n6). Why does the period not match those models and predictions? This must tell us something about the\nstructure of the inner core.\nI grant that these are not the questions that the authors have focused on, but I think the existence of\nthe other peaks and the mismatch between predicted and “observed” ICW periods are both points that\nare too significant to be left unexplored and unexplained.\nModerate points\n•L99: How is the time series for the rate of change of ΔLOD constructed? I don’t think it is provided\ndirectly by the data service; so, the authors have presumably calculated it. If so, what method was used\nand was the differencing done before/after filtering of ΔLOD? Or did the authors calculate the derivative\nof their fit to the ΔLOD data? Such details would be useful for anyone trying to reproduce the work.\n•Related to the above, it would be useful to have a specific data section in the Methods where the\nsources and full processing flows of each time series were clearly laid out rather than needing to pick\nthe details out of the caption to supplementary figure 1.\n•L108-110: I do not follow the argument here. What is “it” referring to in the phrase “…only when it is in the same plane…”? What do you mean by being in the same plane as the tilt angle? The axial torque vector Gamma_z is always in the same plane as the vector defining the inner core figure axis, since any two (non-colinear) vectors from the origin define a plane.\n•The Methods section shows how the equatorial and axial torques on the mantle are related to the\nchanges in mantle rotation vector (observed polar motion and LOD signals); however, there is no\nexplicit demonstration that the ratio of those torques equates to the tilt angle of the inner core. Given\nthat the complete description of couple rotations and torques for the mantle-outer core-inner core\nsystem is far from trivial (particularly if we allow that the solid regions of the Earth may be somewhat\ntriaxial), I think the authors should clearly articulate the reasoning here.\nMinor points\n•L21: Repetition of “solid inner core” one of which should be “solid mantle”.\n•L35: “and etc.” is unnecessary.\n•L67: “that has a significantly different feature” Different than what?\n•L106: “using a more complicated method” is too vague. You don’t have to explain the method here but\nshould at least be specific on what it was.\n•L108- I would use “trough” rather than “valley”.\n•L192- Am and Cm should have units of kg m^2."}, "author_response": "Responses to Reviewers\nReviewer #1 (Remarks to the Author):\nA review of the paper: Inner core static tilt: Evidence from intradecadal oscillation in the Earth's\nrotation\nby An et al.\nThis is a paper on the possibility that the inner core has a static tilt with the Earth's rotation axis.\nConsider the very first sentence of your manuscript. “The Earth consists mainly of a solid inner core,\na liquid outer core, and a solid inner core with the same center of mass.”\nI don’t know what the above sentence means, but the sloppiness, unfortunately, persists throughout the\nentire manuscript.\nResponse: Thanks for your so careful review, which help us to further improve our MS. First, we\napologize for our carelessness, we did have some spelling mistakes (one of the ‘solid inner core’ should\nbe changed to ‘solid mantle’; we have modified it, please see line 21). This time, we carefully re-\nchecked the full text and made some adjustments. Besides, we have used a professional editing service\n(American Journal Experts) to polish the manuscript and asked a native speaker to help us to further\nimprove the English. We hope the new MS will be much better.\n-------------------------------------------------------------------------------------------------------\nAlthough geodynamics is not my area of expertise, and this paper needs to be assessed by someone\nwith good knowledge of the current research on the tilt to judge the novelty you claim, you did not\nconvince me that you can observe the tilt of the Earth’s inner core, let alone prove it. There are two\nmain reasons: firstly, in the abstract and elsewhere, you say that the tilt you infer is 0.17 degrees away\nfrom the Earth’s spin axis. However, you do not present the uncertainty of that estimate, and given that\nit is probably quite large, it could be said that you obtained a 0.0-degree tilt or no tilt at all.\nResponse: Thanks for your constructive comments. According to your comments, we added the\nestimated uncertainties and explained how to estimate them. Here, the bootstrap procedure (Efron &\nTibshirani, 1986) was used to evaluate the uncertainties, and a short explanation is given here:\n1) We first chose the frequency-domain AR method to estimate the amplitude and period of the target\n~8.5 yr spectral peak from the PM (1900-2020) and ΔLOD (1900-2020) records; the obtained\nestimated period/amplitude for the PM and ΔLOD are 8.52yr/4.7mas and 8.47yr/0.061ms,\nrespectively.\n2) We chose the 0.09-0.105cpy and 0.12-0.15cpy as the background noise frequency bands, and the\nmean power of those two bands was chosen as the surrounding noise level R.\n3) Taking the estimated 8.52yr signal with the amplitude of 4.7 mas as an example; we then synthesize\na noise-free time series f (t) containing a 1/8.52cpy frequency signal with an estimated complex\n0\namplitude of 4.7 mas. We also synthesize 300 white noise time series f(t) (i=1,2….300) with the\ni\nsame noise level R (300 is sufficient statistically).\n4) After putting f (t) into the 300 noise time series f(t), 300 new time series containing the target signal\n0 i\nwere obtained. Consequently, 300 new period/amplitude estimations for the target ~8.5yr signal\nwere obtained. Finally, the standard deviation of those 300 estimations is taken as the period (and\namplitude) uncertainty.\nSuch process has been widely used for uncertainty estimations in different geophysical research\n(such as Earth’s free oscillations (e.g., Widmer, et al., 1992; Häfner and Widmer, 2013; Ding and Chao,\n2015) and the trend estimations from space geodetic observations (e.g., Chen et al., 2013; Han et al.,\n2017)).\nThe final estimated uncertainties are 0.19yr/0.4mas for the estimates from PM, and\n0.32yr/0.007ms for the estimates from ΔLOD. The final estimate for the period can be obtained by\nweighting those period estimates, i.e., 8.5±0.2yr. Combing with the amplitude estimate 0.061±0.007\nms (0.046±0.005 ms/yr) in ΔLOD, we obtained that |Γ |= (2.87±0.24)×1019 Nm, and |Γ |=\neq z\n(8.61±0.95)×1016 Nm, and according to the law of error propagation, we obtained that the static tilt\nangle θ= 0.17±0.03°. Given that the frequency-domain AR method has high precision for estimating\nthe amplitude/period of a harmonic signal, we can see that our estimates are quite accurate, hence the\nuncertainty for the tilt is not so large. Hence, we tend to believe this static tilt should be reliable.\nAccording to your comments, we added the corresponding uncertainties and expressed that we\nused a bootstrap procedure, and considering the paper-length limitation, we didn’t explain the details.\nPlease see lines 91 and 92 (In addition, the uncertainties for the estimates in this study were based on\na bootstrap procedure29.) and lines 89, 98, and 172, 176, 178, 181.\n___________________\nReference:\nEfron, B., & Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and\nother measures of statistical accuracy. Statistical Science, 54-75.\nHäfner, R., & Widmer-Schnidrig, R. (2013). Signature of 3-D density structure in spectra of the\nspheroidal free oscillation S . Geophysical Journal International, 192(1), 285-294.\n0 2\nWidmer, R., Masters, G., & Gilbert, F. (1992). Observably split multiplets—data analysis and\ninterpretation in terms of large-scale aspherical structure. Geophysical Journal International,\n111(3), 559-576.\nDing, H., & Chao, B. F. (2015). Detecting harmonic signals in a noisy time-series: the z-domain\nAutoregressive (AR-z) spectrum. Geophysical Journal International, 201(3), 1287-1296.\nChen, J. L., C. R. Wilson, and B. D. Tapley (2013), Contribution of ice sheet and mountain glacier\nmelt to recent sea level rise, Nat. Geosci., 6, 549–552, doi:10.1038/ngeo1829.\nHan, J., N. Tangdamrongsub, C. Hwang,and H. Z. Abidin (2017), Intensified water storage loss by\nbiomass burningin Kalimantan: Detection by GRACE, J. Geophys. Res. Solid Earth,122,2409–\n2430, doi:10.1002/2017JB01412\n-------------------------------------------------------------------------------------------------------\nSecondly, a correlation is not causation. You will need more robust evidence that the inner core has a\ntilt apart from showing the two spectra and deducing that the signal you observe in the spectrum is the\nonly candidate for the ICW. Take, as an example, a recent claim by seismologists that the rotation of\nthe inner core has some correlation with the LOD change – no one in the seismological and the broad\ngeophysical community can take it as proof that the two are related or caused by the same phenomenon\ndue to a significant uncertainty of those estimates and a lack of physical mechanism explaining the\ncorrelation. This lack of uncertainty analysis and solid proof that you are observing a tilt of the inner\ncore makes your paper highly speculative.\nResponse: Thanks for your constructive comments and suggestions. We fully agree with you that we\ncannot only use the related correlation to claim that the inner core must be static tilted. Until now, the\nonly theoretical explanation that the ICW can also appear in ΔLOD is that there is a static tilt angle\nbetween the inner core’s rotation axis and the mantle’s rotation axis (see details at below). Without this\nstatic tilt angle, the ICW can only appear in the PM. At present, there is no other theoretical mechanism\nto explain the presence of the ICW in ΔLOD. Therefore, we only need to confirm that the ICW also\nappears in ΔLOD to verify that the inner core has a static tilt.\nWe have also noticed some recent seismological results about the rotation of the inner core, and\nour corresponding author, Hao Ding, has participated in the discussion of some of the papers before\ntheir publication (such as Wang & Vidale, 2022 SA; Yang & Song, 2023, NG). Similar periods or\nsimilar waveforms are weaker constraints, for example, although the ~70yr signal and the ~64yr signal\nhave similar waveforms in the time domain, they are very likely two different signals (the 18.6yr signal\nand the ~22.5yr signals in the ΔLOD have the same situation). But the same period and good phase\nconsistency are different cases, coincidence is possible, but it will be quite small. Even though, we still\nneed to rule out that small probability.\nWe eliminate this small probability in our data processing. The Earth’s rotation changes have two\nexcited sources, one from the Earth’s interior motions and another from the Earth’s external mass\nchanges. The latter includes three excitation sources, atmospheric press and non-tidal ocean and\nhydrological loading effects. Only when the external sources were ruled out can we be confident that\nthe signal is caused by the internal sources. Taking the ΔLOD as an example, we have shown that\natmospheric and oceanic effects have no significant contribution to the target frequency band (see Fig.\nS2a in the SI), here we further show the hydrological effects. Fig. R1 show the Fourier spectrum of the\nhydrological effects (HAM) of the ΔLOD. We can see that the amplitudes in the target frequency band\nare only ~0.005 ms, which is far less than the noise level of the observed ΔLOD in the same frequency\nband. Hence, we can also rule out the hydrological effects as the excited sources of the ~8.5yr signal.\nFigure R1. The Fourier spectrum of the hydrological effects (HAM) of the ΔLOD. The vertical dashed\nline denotes the 8.5yr.\nIn short, our recommendation was based on the following:\n1) we ruled out the possibility of external excitation sources, and hence the 8.5yr must come from\nthe Earth’s interior motion;\n2) the theoretical period of ICW was suggested in the range of 6.6-7.8yr (based on the PREM\nmodel, and different coupling effects were considered). This period is mainly dependent on the density\njump Δρ at the inner core boundary, although the Δρ of the PREM was not particularly accurate,\nICB ICB\nit will not deviate too much; if too much, the frequencies of other normal modes of the free oscillations\nwhich are sensitive to the core structures will have larger deviations with the theoretical values based\non the PREM, but no such results were found. Hence, we can be sure that the period of the ICW should\nbe close to 6.6-7.8 yr.\n3) the ICW is a prograde motion that same as the well-known Chandler wobble; in the complex\nspectrum of the PM (with the complex from x−i∙y), a prograde wobble has a positive frequency and a\nretrograde wobble has a negative frequency; the Chandler wobble was just identified by that it is only\npresent in the positive frequency axis of the PM’s spectrum, the observed period of the CW is also ~30\ndays large than its theoretical value (Dumberry, 2009; Crossley & Rochester, 2014)). There are no\nother physical explanations for the signal based on current theoretical knowledge except the ICW.\nHence, we determine that the 8.5yr signal must be the ICW.\n4) the 8.5yr oscillation extracted from the dΔLOD/dt is synchronized with the 8.5yr oscillation\nextracted from the y component of the PM, and has a fixed π/2 phase difference with that from the x\ncomponent of the PM. This is not simply correlation; they have the same periods (considering estimate\nerrors) and good phase consistency, and they are different components of the Earth’s rotation.\nBased on those, we can carefully suggest the 8.5yr oscillation in ΔLOD is also the ICW signal. As\nexplained above, the only possibility is that there is a static tilt angle between the inner core’s rotation\naxis and the mantle’s rotation axis. As explained above, the only possibility is that there is a static tilt\nangle between the inner core’s rotation axis and the mantle’s rotation axis. Thus, the coupling torque Γ\nchanges on the equatorial plane of the inner core and will produce a component in the rotation direction\nof the mantle, which will cause the corresponding change in the ΔLOD. Therefore, the ICW revealed in\nboth the ΔLOD and the PM can be used to demonstrate the tilt of the inner core relative to the mantle.\nBesides, a tilted inner core will cause the ICW shown in the ΔLOD was also theoretically suggested by\nprevious studies (Greiner-Mai et al. 2000).\nAccording to your comments, we added some relevant expressions, please see lines 55-58: For\nthe periodic signals present in the PM and ΔLOD, the consensus is that they were excited by the Earth's\ninternal or external sources through the conversion of angular momentum17. Hence, we need to rule\nout the influence of external excitation sources before determining that a target signal is coming from\nthe Earth's internal motion; and see lines 72-77: Different from the oceanic tidal signals that have both\nprograde and retrograde components in PM22-24, the ICW is a prograde motion (the same as the\nChandler wobble, i.e., the mantle wobble); in the complex spectra of the PM, a prograde/retrograde\nwobble only has a positive/negative frequency. The identification of the well-known Chandler wobble\nis based on this feature25-28. Therefore, as a prograde motion, the ICW only appears on the positive\nfrequency axis of the PM spectrum and this is a distinguishing feature for identifying it.\n___________________\nReference:\nCrossley, D. J., & Rochester, M. G. (2014). A new description of Earth's wobble modes using Clairaut\ncoordinates 2: Results and inferences on the core mode spectrum. Geophysical Journal\nInternational, 198(3), 1890-1905.\nDumberry, M. (2009). Influence of elastic deformations on the inner core wobble. Geophysical Journal\nInternational, 178(1), 57-64.\nGreiner-Mai, H., Jochmann, H., & Barthelmes, F. (2000). Influence of possible inner-core motions on\nthe polar motion and the gravity field. Physics of the Earth and Planetary Interiors, 117(1-4), 81-\n93.\nWang, W., & Vidale, J. E. (2022). Seismological observation of Earth’s oscillating inner core.\nScience Advances, 8(23), eabm9916.\nYang, Y., & Song, X. (2023). Multidecadal variation of the Earth’s inner-core rotation. Nature\nGeoscience, 16(2), 182-187.\n-------------------------------------------------------------------------------------------------------\nFurthermore, you use some seismological papers that showed complexities in the inner core to\nreinforce that the inner core might have a non-axisymmetric mass, but in my opinion, a more rigorous\nunderstanding of what these papers are all about is lacking from the manuscript.\nSo, a tilt and a differential rotation of the inner core might be real as they emerge intuitively and\nnumerically from geodynamical modeling (as shown in multiple papers), but you need more in this\npaper to convince me that what you have achieved is a conceptual advance.\nResponse: Thanks for your very useful comments, we do have problems with accurate understanding\nand citation of these papers. Actually, as we have written in lines 118-120 in the old MS:’ However,\nthere is no evidence for the interpretation of the seismic anisotropy axis as the figure or rotation axis\nof the inner core and there is no consensus on what the anisotropy axis represents even in the mineral\nphysics community’. We found that some papers expressed that the seismic anisotropy axis was not the\nfigure axis of the inner core, but we didn’t express them well in the old MS. Our initial idea was to\nstate that such a perspective exists, without concern as to whether it has been falsified; because our\nunderstanding was that there were certain uncertainties about the inner core results from seismic\nobservations, and are often controversial, so we cannot judge which one is right. Based on this\ncomment and the following comments, we have changed the related expressions (more details will be\ngiven in the next response); we tend to emphasize that our results should be available as a constraint\non the inner core structure retrieved from seismic observations, that is, the relevant seismological\nresults should conform to our observations. Of course, such conformity can only be qualitative;\nalthough we are not experts in seismology, we believe that such a small tilt should be difficult to\nidentify directly from seismic observations.\nThe related seismological expressions in lines 115-130 in the old MS were mainly used to support\nthe idea that the inner core might be tilted to the west. In fact, we don't need to refer to the anisotropic\naxis results from seismology (of course, as you commented, they are different things). Ever since the\npossibility of a static tilt of the inner core was proposed for explaining the decadal variations in the\nPM might be the ICW, it has been assumed that it should be in line with the dipole axis of the\ngeomagnetic field, i.e., 10° away from the rotation axis of the mantle (e.g., Schmutzer, 1978; Szeto\nand Smylie, 1984, 1989; Greiner-Mai, 1997; Greiner-Mai et al. 2000). Guo & Ning (2002) followed\nthis suggestion, they simply considered that the inner core and mantle are rigid and axially symmetrical,\nand the mantle is rotating uniformly in the space, then they found that a 10° tilt of the inner core will\nincrease the ICW period by about 70 days. Although the decadal variations in the PM previously\nproposed have been confirmed cannot be the ICW, the possibility of a westward tilted inner core was\nnot denied. Besides, the phase synchronization of the ~8.5yr oscillations in the dΔLOD/dt and in the y\ncomponent of the PM can directly infer that the inner core should be tilted to the west, because the\ny component of the PM along the 90°W longitude. Hence, combing with your comments, we changed\nthe related expressions as: “To explain the decadal oscillations in both the PM and ΔLOD as the\npossible ICW, the rotation axis Ω′ of the inner core was thought to no longer be the common axis Ω\nm m\nbut have a static tilt relative to the mantle, i.e., the inner core rotation axis coincides with the dipole\naxis of the geomagnetic field (tilted 10° westwards from Ω )8. Although there is still no confirmed\nm\nobservation of the ICW9,10, a static tilted inner core remains possible, and the effect of a 10° static tilt\non the ICW period was further investigated9.” (please see lines 30-35); and “Given the y component\nof the PM along the 90°W longitude, the phase synchronization in Figures 3a and 3c indicates that the\ninner core is more likely tilted in the 90°W direction, which is also similar to that suggested by previous\nstudies8,9,31,32.” (please see lines 128-130).\nExcept for the above-mentioned reference, few other studies have specifically considered a static\ntilted inner core (e.g., Mathews et al., 1991; 2002; Rochester & Crossley, 2009; Rochester et al., 2014;\nDumberry, 2009), the ‘tilt’ they called is just the ‘dynamic tilt’ that we explained in lines 26-28.\nDifferent from previous studies, we further considered an ellipsoidal inner core, and under the core-\nmantle coupling mechanism, we derived for the first time the variations of ΔLOD caused by the ICW\nwhen the inner core has a static tilt. In addition, none of Greiner-Mai et al. and Guo et al.’s formulas\ncan directly infer the static tilted angle of the inner core from the ICW signal in the ΔLOD and PM.\nTherefore, in contrast to previous studies, we believe that we are still taking a certain conceptual\nadvance -- of course, we do not want to emphasize a new concept from scratch, we just want to confirm\nan existing but doubtful concept. From the literature available so far, we tend to think that we have\nindeed found strong evidence for the existence of this static tilt for the first time. According to your\ncomments, we added a new note in lines 223-225: “Combing with Eqs. (1), (2), (6) and (7), we can\ndirectly infer the static tilted angle of the inner core from the ICW signal in the ΔLOD and PM,\nwhich is impossible in related previous studies8,9.”.\n___________________\nReference:\nDumberry, M. (2009). Influence of elastic deformations on the inner core wobble. Geophysical Journal\nInternational, 178(1), 57-64.\nGreiner‐Mai, H. (1997). Possible relations between the rotational axis of the earth's inner core and the\nmagnetic dipole axis. Astronomische Nachrichten, 318(1), 63-71.\nGreiner-Mai, H., Jochmann, H., & Barthelmes, F. (2000). Influence of possible inner-core motions on\nthe polar motion and the gravity field. Physics of the Earth and Planetary Interiors, 117(1-4), 81-\n93.\nGuo, J. Y., & Ning, J. S. (2002). Influence of inner core rotation and obliquity on the inner core wobble\nand the free inner core nutation. Geophysical Research Letters, 29(8), 45-1.\nMathews, P. M., Buffett, B. A., Herring, T. A., & Shapiro, I. I. (1991). Forced nutations of the Earth:\nInfluence of inner core dynamics: 1. Theory. Journal of Geophysical Research: Solid Earth,\n96(B5), 8219-8242.\nMathews, P. M., Herring, T. A., & Buffett, B. A. (2002). Modeling of nutation and precession: New\nnutation series for nonrigid Earth and insights into the Earth's interior. Journal of Geophysical\nResearch: Solid Earth, 107(B4), ETG-3.\nRochester, M. G., & Crossley, D. J. (2009). Earth's long-period wobbles: a Lagrangean description of\nthe Liouville equations. Geophysical Journal International, 176(1), 40-62.\nRochester, M. G., Crossley, D. J., & Zhang, Y. L. (2014). A new description of Earth's wobble modes\nusing Clairaut coordinates: 1. Theory. Geophysical Journal International, 198(3), 1848-1877.\nSchmutzer, E. (1978). Investigation on the influence of the global magnetic field of the Earth on the\nmotion of the solid core.(Declination, westward drift, northward drift etc.). Gerlands Beitraege\nzur Geophysik, 87(6), 455-468.\nSzeto, A. M. K., & Smylie, D. E. (1984). Coupled motions of the inner core and possible geomagnetic\nimplications. Physics of the Earth and Planetary Interiors, 36(1), 27-42.\nSzeto, A. M. K., & Smylie, D. E. (1989). Motions of the inner core and mantle coupled via mutual\ngravitation: regular precessional modes. Physics of the Earth and Planetary Interiors, 54(1-2),\n38-49.\n-------------------------------------------------------------------------------------------------------\nLet me comment further on your reference to the seismological papers.\nYou misunderstood the goal of most inner-core seismological papers you cite. The goal was not\nto establish a tilt of the inner core but to study anisotropy. Then, by making a strong assumption that\nanisotropy could be approximated by a cylindrical model, one can perform a grid search to see if the\ndata one has would point to the main axis of anisotropy oriented in a slightly different direction than\nthe spin axis of the Earth. You can then obtain some mean, which cannot be oriented exactly along the\nEarth's rotation axis because of the noise in the traveltime data. What’s worse than that is that we don’t\nhave an even volumetric sampling of the ray paths sensitive to the inner core. You cite the paper by Su\net al. (1996) and their estimate of the tilt within the range of 5–10 degrees, but you are perhaps\nunaware that this tilt was shown to be meaningless by Souriau et al. (1997) because of the bias in the\ndata. By the way, they didn’t find any wobble in time, even with that biased estimate. The bottom line is\nthat these early estimates of the anisotropy axis were shown to be unreliable and erroneous. For that\nreason, they were abandoned, and the differential rotation of the inner core estimates were conducted\nby other methods. Therefore, your underlying thesis that your estimate of the tilt is two orders of\nmagnitude lower than what is obtained by seismology is wrong and misleading. Seismology has not yet\ndetermined the tilt of the inner core or anything like that.\nResponse: Thanks a lot for your detailed explanation, and we have benefited a lot. As we responded\nabove, we were indeed aware that Su et al. (1996) 's results were not considered to be the inner core\nstatic tilted; as you said, Su et al. (1996) 's results have been proven wrong by Souriau et al. (1997).\nWe also don't think that seismic observations can find the ICW signals directly.\nIn the old MS, we referred to an 8° tilted inner core from seismological results, this was based\nWang & Vidale (2022, EPSL). Applying a novel back-projection method, Wang & Vidale (2022, EPSL)\nlocate the inner-core regions that scatter the energy within the PKiKP coda based on its slowness and\nthe lapse time. By relaxing the constraint that the inner core rotation axis is the same as the Earth's\nrotation axis, Wang & Vidale (2022 EPSL) grid-search the tilt of the rotation axis, the azimuth of the\ntilt, and the rotation rate to find the best match to the time shifts of the high-energy points. The best-\nfitting result indicates a rotation axis that is 8° off the geographic (mantle) north pole. However, Wang\n& Vidale (2022 EPSL) denotes that this tilted axis is only a marginally better fit to the observed time\nshifts, and the seismic waves lack resolution for the tilt of the inner core axis. After discussing with\ntheir first author (W. Wang), and combining with your comments, we decided not to show this again;\nand as we responded above, we directly refer to the theoretical suggestions given by geodynamical\nstudies.\nThe differential rotation of the inner core and the ICW are two different concepts, the former\nmainly refers to the change of the inner core rotation rate, while the latter refers to the change of the\ninner core’s pole, which are similar to the Earth's ΔLOD and PM (they are also two different concepts).\nAccording to your suggestions, in order to avoid misleading readers, we have deleted those related\nexpressions and made corresponding modifications (please see lines 30-35 and 130-142).\n___________________\nReference:\nSouriau, A., Roudil, P., & Moynot, B. (1997). Inner core differential rotation: facts and artefacts.\nGeophysical Research Letters, 24(16), 2103-2106.\nSu, W. J., Dziewonski, A. M., & Jeanloz, R. (1996). Planet within a planet: rotation of the inner core\nof Earth. Science, 274(5294), 1883-1887.\nWang, W., & Vidale, J. E. (2022). Earth's inner core rotation, 1971 to 1974, illuminated by inner-core\nscattered waves. Earth and Planetary Science Letters, 577, 117214.\n-------------------------------------------------------------------------------------------------------\nAlso, let me comment on your invoking seismology as a method that found that the mass\ndistribution in the Earth’s inner core is not axisymmetric. While heterogeneities are documented to\nexist near the inner core boundary, everything beyond that is much more speculative. If we accept that\none hemisphere of the inner core is faster than the other, would that necessarily indicate a denser\nmaterial and more mass? Why would anisotropic fabric be a denser fabric? The opposite can be true.\nFrom seismology, anisotropy appears weaker in the outer part of the inner core and stronger towards\nits center. It does not necessarily have anything to do with cylindrical anisotropy's fast and slow axis\nbut with the pronounced slow axis in the innermost part of the inner core. For instance, deeper into the\ninner core, two recent studies that you cited show an offset of the anisotropic (as hypothesized) volume\nfrom the center at two different locations of the inner core. In all honesty, there is no consensus as to\nwhich one is correct, so your inference that a tilt is to the western side needs more seismology support.\nBut if you honestly present seismological work as it is, you will find it more difficult to substantiate\nyour claims.\nResponse: Thanks for your advice and we learn a lot. Indeed, as you say, if only one hemisphere is\nfaster than the other, it is difficult to confirm whether the density is denser or not, which is similar to\nthe controversy about the Large Low Shear Velocity Provinces (LLSVPs).\nIn the old MS, we weren't focused enough, so there was some ambiguity. We have deleted some\nunnecessary discussion about some literature this time, and mainly focus on three relevant papers\n(Alboussière et al. (2010 Nature), Monnereau et al. (2010 Science) and Zhang et al. (2019 JGR-SE)).\nIn the caption of Figure 3 of Monnereau et al. (2010 Science) (also Figure R3), they noted that:\n“In a superadiabatic regime, any thermal heterogeneity of harmonic degree-one shifts the center of\nmass of the inner core toward its colder and denser hemisphere (left). The equilibrium position at the\ncenter of mass of Earth (o) is restored by a translation of the inner core (from dashed to solid positions).\nThis induces a topography h that is not in equilibrium with the pressure and temperature conditions\nwithin the outer core. Crystallization on the denser hemisphere (left) and melting on the opposite side\nact to remove the topography, but in return amplify the density heterogeneity and maintain the center\nof mass shifted toward the crystallizing side….”. Although they wanted to explain a possible inner\ntranslational mechanism, they did discuss the possibility that the inner core was denser in the western\nhemisphere. This study is related to Alboussière et al. (2010, Nature), which discussed a uniform flow\ncaused by the crystallization and melting at the surface of the inner core, the associated changes in\ndensity and gravitational potential lead to the inner core has an eastward translation along the x-axis.\nIn Alboussière et al. (2010)’s dynamical model, the western hemisphere of the inner core is denser than\nits eastern hemisphere (see their Fig. 4). In addition, Zhang et al. (2019 JGR-SE) concluded that\n“Earth's western hemisphere has a thicker compacting layer (F′ layer) than the eastern hemisphere on\naverage” based on the significant difference in seismic velocity between the eastern and western\nhemispheres, this may also mean denser material in the western hemisphere.\nAs we have responded above, we don't really need seismological results to tell us that the inner\ncore's static tilt is westward. However, it is possible that these seismological results are at least\nqualitatively consistent with ours (denser material in the northwest hemisphere, even may only be in\nthe top layer of the inner core, may cause the inner core’s figure axis tilts to the west).\nConsidering your comments, we modified the related expressions, please see lines 133-142:\n“Considering the crystallization and melting at the surface of the inner core, a previous dynamical\nmodel has similar suggestions33, i.e., the western hemisphere of the inner core is denser than its eastern\nhemisphere. Interestingly, although it has some uncertainties34, a seismological study has suggested\nthat the western hemisphere of the inner core may be relatively denser35 due to its lopsided growth\nbased on different seismic anisotropy between its eastern and western hemispheres36,37, and a thicker\ncompacting layer at the top of the inner core's western hemisphere was also suggested38; the more\nnuanced researches also revealed the differences in latitude distributions of those anisotropy regional\nstructures39 and found that the western anisotropic zone is largely confined to the northern\nhemisphere40; those suggestions are generally consistent with the westwards statically tilted inner core\nthat we found.”. In addition, we further modified the related discussion parts, we emphasized\nqualitative analysis of the possible correlation between the westward tilted inner core we found and\nthe seismological results, rather than using the seismological results as a prerequisite (because it is not\nnecessary); please see lines 196-202: “Undeniably, it is difficult for seismological observations to\ndetect such inner core static tilt directly, but interestingly, the results from seismological studies\nshowed that the western/northwestern hemisphere (or at least its top layer) of the inner core may be\nrelatively denser34,35,38,40. These suggestions, although they have some uncertainties, are qualitatively\nconsistent with our finding of a westwards-tilted inner core. Therefore, we also suggest that the static\ntilted inner core can be used as a qualitative constraint on the inner core structure retrieved from\nseismic observations.”.\n[Redacted]\nFigure R3. From Monnereau et al. (2010; Science).\n[Redacted]\nFigure R4. From Zhang et al. (2019 JGR-SE)\n___________________\nReference:\nAlboussière, T., Deguen, R., & Melzani, M. (2010). Melting-induced stratification above the Earth’s\ninner core due to convective translation. Nature, 466(7307), 744-747.\nMonnereau, M., Calvet, M., Margerin, L., & Souriau, A. (2010). Lopsided growth of Earth's inner core.\nScience, 328(5981), 1014-1017.\nZhang, Y., Nelson, P., Dygert, N., & Lin, J. F. (2019). Fe alloy slurry and a compacting cumulate pile\nacross Earth's inner‐core boundary. Journal of Geophysical Research: Solid Earth, 124(11),\n10954-10967.\n-------------------------------------------------------------------------------------------------------\nThere is no such thing as “super-rotation”. (Line 34) We talk about the differential rotation of the\ninner core relative to the mantle. Besides, the paper would benefit from a discussion how the inner core\ntilt would impact the differential rotation studies.\nResponse: Thanks for your comments. We also prefer to believe the inner core should be differential\nrotation but not ‘super-rotation’, in particular, recent results suggest that the inner core is more likely\nto oscillate. The ‘super-rotation’ also has appeared in many previous literature, even on Prof. Song’s\nhomepage (https://xiaodongsong.web.illinois.edu/news/), in which, they highlighted that “2005\nConfirmation of the inner core super-rotation”.\nAccording to your suggestion, we changed it to ‘differential rotation’ (please see line 41). As for\n“how the inner core tilt would impact the differential rotation studies”, quantitative interpretation is\nnot an easy task, and it is difficult for us to adequately and accurately state it in this study. Qualitatively,\nhowever, we know that the eastward differential rotation rate of the inner core will shorten the period\nof the ICW. Hence, we only added a note: “The larger observed period may also indicate that the\neastwards differential rotation rate of the inner core should be much less than 1° per year9,14.” (please\nsee lines 193-195).\n-------------------------------------------------------------------------------------------------------\nThe quasi-eastern and quasi-western anisotropy is not that established. The robust differences are\ndocumented in absolute velocity between the eastern and the western hemisphere, but only at the top\nof the inner core. Given the sparse coverage of the inner core on the western side, whether the same\nholds for anisotropy is hard to tell. As you go deeper toward the center, the hemispherical division\nloses its meaning, and there is no consensus that any hemispherical pattern exists.\nResponse: Thanks for your useful comments and kind reminder. As we responded above, we only\nwant to qualitatively state that some of the seismological results are likely to be consistent with the\nwestward tilt we have obtained, and in the new manuscript, we do not want to use the seismological\nresults as a prerequisite for determining the specific mechanism of our results. In addition, it is not\nnecessary to go deep into the inner core; the outermost uneven density has the greatest effect on the\nfigure axis of the inner core because the moment of inertia (defining the figure axis) is the integral of\nthe density and the square of the distance over the body. Therefore, assuming that the anomalies are\nonly at the top of the core, they should be considered consistent with our findings as long as they are\nlikely to indicate that the average density in the eastern and western hemispheres of the inner core is\nuneven. In the new manuscript, we have changed to a qualitative description of this possible correlation\n(please see lines 133-142 and 196-202).\nFinally, we sincerely thank you for your constructive suggestions and comments. In the old MS,\nwe over-emphasized the results of the tilt determined by seismology, which led to some confusion in\nthe discussion. In the new MS, we made a logical adjustment; we emphasize that the tilt is determined\nby the Earth's rotation, and it is most likely caused by the western hemisphere of the inner core having\na greater average density, which may be consistent with some seismological results (but still waiting\nfor future seismological verification). We hope you will find our reply satisfactory. Thank you!\n-------------------------------------------------------------------------------------------------------\n###########################################################################\nReviewer #2 (Remarks to the Author):\nFirst of all, I would like to state for the record that I believe that the authors have clearly and\nconvincingly demonstrated that the results they present in the article are true. Moreover, I would like\nto add that in my view they are truly important and sound. Applying a methodology that has already\nshown its power in other applications to geodetic problems and using precise earth rotation data, they\nhave proved that the axis of the earth's solid inner core is tilted with respect to the polar axis of the\nwhole earth, and at the same time they have determined for the first time empirically the value of the\nso-called ICW period. This motion can be thought of as analogous to the precession motion of the earth\nunder lunisolar attraction, but in this case it is the inner core under the attraction of the earth's mantle\nsurrounding it. This free mode of the Earth's rotation has so far only been known from\ntheoretical developments and the value of its period inferred from geophysical models, as well as more\nindirectly from its effects on the Earth's rotation, more precisely on nutation, since the introduction in\n1991 of a basic earth model made of 3 ellipsoidal, concentric and aligned layers, which was later\nrefined and fitted to VLBI data, becoming the official IAU2000 nutation model.\nResponse: Dear Prof. Ferrandiz,\nWe appreciate your recognition and encouragement of our research, which gives us the confidence and\nmotivation to further revise and improve our MS. We carefully addressed each of your comments and\nupdated the manuscript accordingly. In the following, we listed detailed responses and made\ncorrections carefully based on your comments and suggestions.\n-------------------------------------------------------------------------------------------------------\nAlthough the authors only highlight the importance of the experimental determination of the inner\ncore inclination from earth rotation data, more precisely from pole motion (PM) and length-of-day\n(LOD), I consider that the determination of the value of this ICW period is not less important than the\nprevious finding and I could even dare to say that for me it could be even more important in that the\naccuracy of the method they use to identify weak harmonic signals (stabilized AR-z spectrum) is very\nprecise in determining the value of the periods, but does not allow determining the amplitudes with\nthe same relative accuracy as far as I know.\nResponse: Thank you very much for the thoughtful review and encouragement to help improve this\nstudy. We further modified the related parts according to your insightful comments and suggestions,\nand to emphasize the importance of the observed ICW, we also further apply this observed period to\nconstrain density jumps at the ICB, which is also a very important parameter but still controversial\n(please see lines 183-189, 195 and 196, and 239-255).\n-------------------------------------------------------------------------------------------------------\nIn my opinion, there is no doubt about the veracity of the results, since they clearly identify a\nharmonic constituent of the pole motion with a period close to 8.5 years, formed only by the prograde\ncomponent, and also an oscillation of smaller amplitude in LOD with the same period; current\ntheoretical knowledge can only explain this coincidence if the origin of the signal is precisely the\naforementioned free oscillation ICW (which we can imagine as the precession of a small planet located\ninside the Earth's mantle following the analogy made in one of the references cited by the authors).\nTherefore, my main suggestion is to make some changes in the wording of the article to show that not\nonly the static inclination of the inner core axis has been determined, but also the value of the period\nof its free ICW motion has been determined for the first time experimentally, based on Earth rotation\ndata.\nResponse: Thanks for your very useful comments! “which we can imagine as the precession of a small\nplanet located inside the Earth's mantle” is a very revealing idea, the phenomenon we want to explain\nis indeed similar to this. According to your suggestion, we further enhanced the importance of the\nobserved 8.5yr period. Please see lines 190 and 191: “based on the Earth's rotation observations (PM\nand ΔLOD), we experimentally confirmed for the first time that the 8.5yr signal is the ICW.”\n-------------------------------------------------------------------------------------------------------\nI consider that the empirical determination of this value should be highlighted since in the last 30\nyears several values of the period, inferred from theoretical considerations, have been published (as\ncited in the references) and even in some recent work it has been obtained that the motion should occur\nin the retrograde direction and not in the prograde direction as most of the previous authors concluded\n(Chao, B. F. (2017), Dynamics of the inner core wobble under mantle-inner core gravitational\ninteractions, J. Geophys. Res. Solid Earth, 122, doi:10.1002/ 2017JB014405). This shows how difficult\nis deriving accurate values of the ICW period from theory and geophysical models of the earth’s\ninterior. On the other hand, I would like to remind that the observed value of the Chandler period has\nnot yet been fully justified, and that the most complete theoretical deductions often differ from the\nobserved value by a few tens of days.\nResponse: Thanks for your very useful comments. As you noted, even the theoretical period of the\nICW is quite difficult to determine, a small oversight may lead to a wrong result. I (H. Ding)\nparticipated in the discussion of Chao (2017)'s draft, but didn’t find that the used magnitude of q M\n20\nhas a mistake; after it was published, Rochester and Crossley found this and co-authored with Ben\nChao, they published a Correction: Rochester, M. G., Crossley, D. J., & Chao, B. F. (2018). On the\nphysics of the inner core wobble; corrections to “Dynamics of the inner-core wobble under mantle-\ninner-core gravitational interactions” by B. F. Chao. Journal of Geophysical Research: Solid Earth,\n123, 9998–10,002.”. In which, they confirmed that the theoretical period of the ICW based on the\nPREM model is still a ~+7yr prograde motion, not a 15yr retrograde motion.\nYour comments about the Chandler wobble will help the readers to further understand the ICW,\nso we added some new discussions about it. Please see lines 183-186: “Our observed ICW period is\nslightly larger than the theoretical values (6.6-7.8yr)1,5-7, but considering that even the generally\naccepted Chandler wobble observation period of ~430 days is ~30 days longer than its theoretical\nperiods and that the density jump Δρ at the ICB was also poorly determined, this deviation is\nICB\naccepted.”.\n-------------------------------------------------------------------------------------------------------\nI think the authors could emphasize their experimental determination of the period without making\nmajor changes to the text or the abstract, although perhaps the title should be modified so as not to\nemphasize only the inclination. Regarding the magnitude of the difference with former estimations of\nthe period, based on quite different methodologies because none uses accurately observed earth\nrotation data, I’d like to recall that reliable error bars are not available for any of the published\ndeterminations, given the difficulty of the implied calculations. That fact must be taken into account\nin any controversy that may be risen.\nResponse: Thank you very much for your understanding that we did not give the uncertainty estimates\nin the old MS. As the other reviewers have pointed out that we didn’t give the uncertainties, hence, in\nthis version, based on the used PM and ΔLOD records, we further estimated the uncertainties of the\nperiod and amplitudes of this ~8.5yr signal.\n-------------------------------------------------------------------------------------------------------\nOn the other hand, while the wording poses no problems for those with an intermediate knowledge\nof the earth's rotation, I think from my experience that a large number of potential readers interested\nin the subject do not reach that level. They are possibly well acquainted with the Chandler period\narising from the existence of the liquid core, but rather less familiar with the two additional normal\nmodes of rotation that appear when the solid earth model also contains an inner core. I therefore suggest\nthat the authors consider adding a short paragraph describing the phenomenon.\nResponse: Thanks for your useful suggestions. In the new MS, we added some expressions, please see\nlines 26-35, and 72-77.\n-------------------------------------------------------------------------------------------------------\nBelow, I suggest that the authors please consider making some minor editorial changes, but this\nsuggestion does not mean that they should make them. They are (by lines):\nResponse: Thanks for your careful review. We modified them as you suggested. Please see below.\n-------------------------------------------------------------------------------------------------------\nL 21. Change \"Earth\"->\"solid Earth\".\nResponse: Done. Please see line 21.\n-------------------------------------------------------------------------------------------------------\nL 26 Specify which theory is meant if the sentence refers to a specific one or make a more generic\nreference such as \"most published theories...\"..\nResponse: Thanks for your suggestion! To more clearly explain the “static tilt”, we have rewritten this\nsentence as “The above ‘tilt’ is a generally dynamic tilt, and in this case, the ICW theoretically appears\nonly in the polar motion (PM) of the Earth's rotation2,4.”. Please see lines 26-28.\n-------------------------------------------------------------------------------------------------------\nL 34-35 It would be useful to include some reference, either among those already cited or in addition\nto those already cited.\nResponse: Done as you suggested. Please see line 41.\n-------------------------------------------------------------------------------------------------------\nL 39 I think it would be useful to indicate that this fact will be demonstrated later, or to give a reference,\nfor the convenience of readers.\nResponse: Thanks for your kind reminder! We added some references for this. Please see line 44.\n-------------------------------------------------------------------------------------------------------\nL 54 It is actually in the caption of Figure 1, not in the figure itself.\nResponse: Thanks for your kind reminder! Combing with the suggestions given by reviewer #3,\nwe moved the pretreatments into the Method parts. Please see lines 54, 257-274.\n-------------------------------------------------------------------------------------------------------\nL 55 The wording is not completely accurate, it is valid for the so-called Earth Rotation Parameters\n(ERP) which are the ones used in the derivation of the results, but not for the remaining two, namely\nprecession-nutation ones.\nResponse: Thanks for your suggestions! We changed it to “the PM and ΔLOD changes”. Please see\nlines 58 and 59.\n-------------------------------------------------------------------------------------------------------\nL 79-80 Perhaps it would be useful to reinforce the reasoning made to reach the conclusion. Actually,\nit is not only possible to conclude, but it must be concluded because current theoretical knowledge\noffers no other possibility to explain the origin of this 8.5-year signal with the above-mentioned\nproperties.\nResponse: This is a very useful suggestion! According to this, we further modified this part, please see\nlines 89 and 90: “Since there is no other mechanism has been proposed to account for such a prograde\n~8.5yr motion, and…”.\n-------------------------------------------------------------------------------------------------------\nL 107-108 Although the sentence is true, it is perhaps worth expanding the explanation slightly.\nResponse: Thanks for your reminder! Here we added a note to suggest the reader see Figure 4 (please\nsee lines 120 and 121).\n-------------------------------------------------------------------------------------------------------\nL 191 Including a reference for the conventional value of Omega, among the many available sources\nResponse: Done as your suggestion! Please see line 215.\n-------------------------------------------------------------------------------------------------------\nL 194 Rather than observation, one should say the motion of the observed pole (as a response).\nResponse: Done as you suggested. Please see lines 218 and 219.\n-------------------------------------------------------------------------------------------------------\nL 208 Not necessarily in this line, but somewhere in the text or supplementary material, I think more\ndetails should be given on how the method has been applied, e.g., the size of the bins, whether the Q\nfactor that appears as one of the input variables in the referenced routine (in L 228) has been given an\ninfinite or finite value, etc., so as to ensure that the calculations are transparent and repeatable.\nResponse: Thanks for your suggestion. We modified this part and added some notes in the SI. Please\nsee lines 233-237, and the SI.\n-------------------------------------------------------------------------------------------------------\nL 306-307 Correct the reference (Fong Chao, B…)\nResponse: Corrected. Please see line 379.\n-------------------------------------------------------------------------------------------------------\nSincerely,\nJose M Ferrandiz\n###########################################################################\nReviewer #3 (Remarks to the Author):\nMajor points\n1) Discussion of seismology\nThe discussion of the previous seismic work is not entirely reflective of what those studies found. I do\nnot think it is fair to say that the studies are collectively arguing for a large static tilt of the inner core.\nCollectively, the cited studies (and others) have investigated the anisotropic and heterogeneous\nstructure of the inner core, and whether there is differential axial rotation between the inner core and\nmantle. Some of the differential rotation studies (e.g. Wang & Vidale 2022, ref 10 of this work; and\nthe recent article by Yang & Song 2023, Nature Geoscience) measure differential rotation by\ntracking/estimating seismic variations in the shallow inner core; these lateral variations provide no\ninformation on a potential tilt of the inner core. Studies of inner core anisotropy (e.g. refs 7 and 9 of\nthis work) do lead to estimates of anisotropy axis that is tilted relative to the net rotation axis of the\nEarth. However, the physical mechanism responsible for that seismic anisotropy and, in particular, the\nphysical meaning of the orientation of that fast axis are debated (even within individual observational\nstudies), as noted in L118-120. The alignment of the fast axis from seismic anisotropy certainly need\nnot be attributed to a tilt of the inner core. Furthermore, the review by Deuss (2014, ref 12 of this work)\nnotes “The hemispherical variations removed the need for a tilted anisotropy axis and provide a better\nfit to the body wave data”.\nThe inner core structure outlined by the recent study of Brett et al (2022, ref 29) and discussed on\nL130 have the western anisotropic zone confined to the northern hemisphere; I do not agree with the\nsubsequent statement on L131 that this is indicative of a tilted figure axis, which should produce a\ncomparable anomaly in the antipodal eastern + southern region of the inner core. Overall, I think the\nstatements made in this work that a 5-10 degree tilt of the inner core figure axis are suggested by\nseismology need to be removed as being unreflective of the overall message put forward by those\nstudies.\nResponse: Thank you for your very useful and constructive suggestions and comments. As you\ncommented, our discussion about the anisotropic axis being the inner core's static tilt axis was not\nappropriate. In the new MS, we have revised this, and we no longer emphasized the seismological\nresults about the 5-10º anisotropic axis (namely, we have deleted those parts as your suggestion);\nthe existing geodynamic literature has directly indicated that the inner core may have a westward tilt of\n10°. Hence, according to your comments, we have modified some related expressions, such as line 14\nand 15: “which is two orders of magnitude lower than that 10° assumed in geodynamics.”, and lines\n30-35: “To explain the decadal oscillations in both the PM and ΔLOD as the possible ICW, the rotation\naxis Ω′ of the inner core was thought to no longer be the common axis Ω but have a static tilt relative\nm m\nto the mantle, i.e., the inner core rotation axis coincides with the dipole axis of the geomagnetic field\n(tilted 10° westwards from Ω )8. Although there is still no confirmed observation of the ICW9,10, a\nm\nstatic tilted inner core remains possible, and the effect of a 10° static tilt on the ICW period was further\ninvestigated9.” (please see lines 30-35).\nAs for the related expressions in lines 115-132 in the old MS, our primary intention was to further\nillustrate from the seismological results that the 90ºE-90ºW tilt of the inner core should be more\nspecifically westward. But as you commented, this should not be appropriate. In fact, we don't need to\nrefer to the anisotropic axis results from seismology. Since the possibility of a static tilt of the inner\ncore was proposed for explaining the decadal variations in the PM might be the ICW, it has been\nassumed that it should be in line with the dipole axis of the geomagnetic field, i.e., westward 10° away\nfrom the rotation axis of the mantle (Greiner-Mai et al. 2000). Although the decadal variations in the\nPM have been confirmed cannot be the ICW (Guo et al., 2005), the possibility of a westward tilted\ninner core was not denied. Besides, the phase synchronization of the ~8.5yr oscillations in the\ndΔLOD/dt and in the y component of the PM can directly infer that the inner core should be tilted to\nthe west, because the y component of the PM along the 90°W longitude. Thus, considering your\nsuggestion, we find that the logic in which we discussed seismological results needs to be changed,\nnot to treat them as prerequisites, but as a possible correlation. Hence, we changed the related\nexpressions (lines 115-132 in the old MS) as: “Given the y component of the PM along the 90°W\nlongitude, the phase synchronization in Figures 3a and 3c indicates that the inner core is more likely\ntilted in the 90°W direction, which is also similar to that suggested by previous studies8,9,31,32. In terms\nof the long-term dynamic conservation of the Earth's angular momentum, this static westwards tilt is\nmost likely due to the effect of the non-axisymmetric mass of the inner core, i.e., the western\nhemisphere (more specifically, the northwestern hemisphere) of the inner core should have greater\naverage densities. Considering the crystallization and melting at the surface of the inner core, a\nprevious dynamical model has similar suggestions33, i.e., the western hemisphere of the inner core is\ndenser than its eastern hemisphere. Interestingly, although it has some uncertainties34, a seismological\nstudy has suggested that the western hemisphere of the inner core may be relatively denser35 due to its\nlopsided growth based on different seismic anisotropy between its eastern and western hemispheres36,37,\nand a thicker compacting layer at the top of the inner core's western hemisphere was also suggested38;\nthe more nuanced researches also revealed the differences in latitude distributions of those anisotropy\nregional structures39 and found that the western anisotropic zone is largely confined to the northern\nhemisphere40; those suggestions are generally consistent with the westwards statically tilted inner core\nthat we found.” (please see lines 128-142). As you can see, the expression of the seismological results\nhas been changed to qualitative possible correlations. \nAs for your comment: “which should produce a comparable anomaly in the antipodal eastern +\nsouthern region of the inner core”, we fully agree with you, this is indeed a possible scenario. This\ntime, we only express a simpler scenario, i.e., mass anomaly only in the northwest hemisphere of the\ninner core also causes a similar tilt westward, and we show this result in Figure R5a. Focusing on the\nfigure axis (defined by the direction of maximum moment of inertia), the slightly flattened inner core\ncan be viewed as the sum of a standard large sphere and two small spheres at both poles (north and\nsouth). The large sphere can be removed due to the axisymmetric mass, and only the two small spheres\nare shown in Figure R5a. We can see that the C-axis figure axis points in a north-south direction. With\nan identical anomaly in the western + northern hemisphere (i.e., add an identical small sphere to the\nwestern + northern hemisphere), as shown in Figure R5b, we can clearly see that the center of mass\nshifts slightly north to the west and the figure axis of the core to shift westward. Thus, only anomaly\nin the western + northern hemisphere of the inner core also causes a tilt westward. In fact, the anomaly\nis small, the inner core tilt and the northward shift of the center of mass are small, and it is mainly a\nslow translational motion eastward of the inner core caused by the lopsided growth of the inner core\nin the western hemisphere and eastern hemisphere under the influence of gravity (Monnereau et al.,\n2010).\nFigure R5. Simplified oblate sphere model, the axisymmetric sphere has been removed. (a) A standard\noblate sphere is simplified to two small spheres at the north and south poles; the center of mass O is in\nthe center, and the figure C-axis points in a north-south direction. (b) Add the same sphere (an identical\nanomaly) to the western + northern hemisphere, and the center of mass shifts slightly north to the west\nand the figure axis of the inner core shifts westward. \nIn addition, we tend to believe that the inner core static tilt cannot be detected directly by\ntracking/estimating seismic variations in the shallow inner core (as you have pointed out); because\neven the wobble angle of the ICW itself is hard to be detected in this way. Instead, the inner core\noscillation may be detected in this way; the inner core oscillation is quite different from the ICW, the\nformer is related to the inner core rotation speed while the latter is related to the pole’s motion of the\ninner core.\n___________________\nReference:\nGreiner-Mai, H., Jochmann, H., & Barthelmes, F. (2000). Influence of possible inner-core motions on\nthe polar motion and the gravity field. Physics of the Earth and Planetary Interiors, 117(1-4), 81-\n93.\nGuo, J. Y., & Ning, J. S. (2002). Influence of inner core rotation and obliquity on the inner core wobble\nand the free inner core nutation. Geophysical Research Letters, 29(8), 45-1.\nMonnereau, M., Calvet, M., Margerin, L., & Souriau, A. (2010). Lopsided growth of Earth's inner core.\nScience, 328(5981), 1014-1017.\n-------------------------------------------------------------------------------------------------------\n2)Is 8.5 years the ICW and, if so, why?\nAs in their previous work (ref 16 of this work), the authors have found a number of peaks in polar\nmotion. The 8.5 yr peak is argued to be the ICW as it shows in the prograde spectrum. The other peaks\nare of similar amplitude to the 8.5 yr signal and also show inconsistency between positive and negative\nPM spectra (e.g., the 7.3 yr signal is much larger in the negative half of figure 2a, the 5.9 year peak is\nconsiderably stronger in the positive half). Without understanding what those other peaks represent\nphysically and why their amplitudes vary, it is not obvious that the 8.5 yr peak should be picked out as\nthe ICW.\nResponse: Thanks for your useful comments. It was an oversight on our part not to emphasize the\ncharacteristics of the AR-z spectrum. The AR-z spectrum is a detection method, which only is meant\nfor signal detection, its amplitudes have only a relative meaning, not an absolute meaning; namely, if\nwe find a spectral peak over the background noise level, it can be identified as the quasi-harmonic\nsignal, while the absolute amplitude of the AR-z spectral peak contains no direct information about the\nactual complex amplitude of the detected signal.\nHere we first give an example of the normal mode S of the free oscillation, as shown in Figure\n0 2\nR6. We can see that the real excited amplitudes for the m= ±1 singlets are less than those of m= ±2\nsinglets from the Fourier spectrum, while in the AR-z spectrum, the ‘amplitudes’ of the m= ±1 singlets\nare even larger than those of m= ±2 singlets; and the SNRs and frequency resolution are highly\nimproved than the Fourier spectrum.\nFigure R6. The normalized Fourier and AR-z spectra of S from the Moxa SG station after the 2004\n0 2\nSumatra earthquake.\nAs for the AR-z spectrum for the PM (as shown in Figure R7), the red area denotes the background\nnoise level (the mean amplitudes which were calculated after removing some strong signals), we can\nsee that the signals indicated in the main text are all over this background level, and can be considered\nas signals. According to the above, the absolute ‘amplitudes’ of those spectral peaks are not indications\nof the absolute signals’ power. This result can indicate that the 7.3yr and 5.9yr signals have both\npositive and negative components. Even if we don’t consider the positive component of the 7.3yr signal,\nit still cannot be the ICW. As a prograde motion, the ICW can only be present in the positive part,\nnamely, it has no negative component, which is similar to the well-known Chandler wobble.\nFigure R7. The AR-z spectrum of the PM.\nBased on your comments, we added some notes, please see lines 72-77: “Different from the\noceanic tidal signals that have both prograde and retrograde components in PM22-24, the ICW is a\nprograde motion (the same as the Chandler wobble, i.e., the mantle wobble); in the complex spectra of\nthe PM, a prograde/retrograde wobble only has a positive/negative frequency. The identification of the\nwell-known Chandler wobble is based on this feature25-28. Therefore, as a prograde motion, the ICW\nonly appears on the positive frequency axis of the PM spectrum and this is a distinguishing feature for\nidentifying it.”, and lines 83-85: “(The AR-z method is meant for signal detection, the amplitude of it\ncontains no direct information about the actual complex amplitude of the detected signal)”.\n-------------------------------------------------------------------------------------------------------\nLet us accept that the 8.5 yr peak is the ICW. This is not the period predicted from previous theoretical\nmodels that account for Earth structure and match other observed wobbles and nutations (e.g., refs 1-\n6). Why does the period not match those models and predictions? This must tell us something about\nthe structure of the inner core.\nI grant that these are not the questions that the authors have focused on, but I think the existence of\nthe other peaks and the mismatch between predicted and “observed” ICW periods are both points that\nare too significant to be left unexplored and unexplained.\nResponse: Thanks for your constructive comments. The 8.5yr signal does not match the predicted\nperiods from previous theoretical models; this is similar to the well-known Chandler wobble, which\nhas a ~430 days observed period but a ~400 days theoretical period. As we know, the density jump\nΔρ at the ICB is still not well constrained, different Earth model has different Δρ (such as the\nICB ICB\n1066A, PREM, AK135, CORE11 models), while the period of the ICW is directly dependent on the\nΔρ . Therefore, we believe our result (8.5yr period) is mainly because of that the Δρ of the real\nICB ICB\nEarth is different from those of the models, and hence can be used to constrain the Δρ at the ICB.\nICB\nAccording to your comments, we added the related parts to explain what can we learn from the\n‘observed’ period of the ICW. Please see lines 183-189: “Our observed ICW period is slightly larger\nthan the theoretical values (6.6-7.8yr)1,5-7, but considering that even the generally accepted Chandler\nwobble observation period of ~430 days is ~30 days longer than its theoretical periods and that the\ndensity jump Δρ at the ICB was also poorly determined, this deviation is accepted. Considering this\nICB\nnewly determined period of the ICW, we can also invert the density jump Δρ . Taking the density\nICB\nprofiles of the PREM model as a reference, we finally obtained Δρ = 0.520±0.05 g/cm3 (see\nICB\nMethods), which is smaller than that of the PREM model (0.598 g/cm3).”; and lines 195-196: “Besides,\nthe density jump of 0.520±0.05 g/cm3 at the ICB is also inverted based on the observed ICW period.”;\nand the ‘Method’ parts: “Constraint for the density jump at ICB. …” (please see lines 239-255).\n-------------------------------------------------------------------------------------------------------\nModerate points\n•L99: How is the time series for the rate of change of ΔLOD constructed? I don’t think it is provided\ndirectly by the data service; so, the authors have presumably calculated it. If so, what method was used\nand was the differencing done before/after filtering of ΔLOD? Or did the authors calculate the derivative\nof their fit to the ΔLOD data? Such details would be useful for anyone trying to reproduce the work.\nResponse: Thanks for your comments. As for the dΔLOD/dt (here we refer it as to g(t)), because the\namplitudes of the signals contained in the ΔLOD may be time-varying, so it’s hard to select suitable\nfunctions to fit the ΔLOD data, and such fitting process may cause some deviations; hence, we directly\nuse a classical discrete numerical derivation algorithm, i.e., g(t\ni\n)= [ΔLOD (t\ni+1\n)−ΔLOD (t\ni\n)]/Δt.\nGiven that the final used ΔLOD has a time span of 1900-2020, which was reconstructed by two\ndifferent records with different sample intervals. The first ΔLOD record is a long-term dataset\n(1623/06-2008/06) with 1-year sampling from IERS; while the second record is the EOPC04 ΔLOD\n(1962/01-2019/12) with 1-day sampling. To merge those two records, we first need to resample them\nto the same interval, so we used a low-pass filter for the second record. If no filter was used, the high-\nfrequency signals will be leaked to the low-frequency parts, i.e., the so-called aliasing signals. Here\nwe show a comparison of this.\nFrom Figure R8a, we can see that the down-sampled ΔLOD with and without a low-pass filter\nhave clear differences, such differences are caused by the high-frequency signals’ leak. Figure R8b can\nmore clearly show this, we can see that only with a low-pass filter, the obtained down-sampled ΔLOD\nhas the same low-frequency parts as the original record. If we didn’t use a filter before the down-\nsampling process, the final obtained result for the fitted ~8.5yr certainly will have some difference\nfrom our used recommended result (as Figure R9 shows). Figure R9 shows the results from those two\ndifferent records have some amplitude differences (dΔLOD/dt have similar results, we will not further\nshow them). Thus, we recommend the result obtained from the ΔLOD with a low-pass filter, as filtering\nbefore a down-sampling process is the standard process in preprocessing.\nAccording to this comment and your following comments and suggestions, we added a new part\nin the ‘Methods’ to explain those related problems. Please see the following response.\nFigure R8. (a) the original ΔLOD and the down-sampled ΔLOD (with and without using a low-pass\nfilter). (b) the corresponding Fourier amplitude spectra of the records in (a).\nFigure R9. The fitted 8.5yr oscillation from the ΔLOD with and without using a low-pass filter.\n-------------------------------------------------------------------------------------------------------\n•Related to the above, it would be useful to have a specific data section in the Methods where the\nsources and full processing flows of each time series were clearly laid out rather than needing to pick\nthe details out of the caption to supplementary figure 1.\nResponse: Thanks for your useful suggestions. We once want to explain those in the main text, but\nconsidering the length limitation, we removed them to the SI the last time. This time, according to your\nsuggestions, we added a new part “Datasets and Preprocessing” in the Methods. Please see lines 257-\n274: “The PM observations were obtained from the EOPCO1 dataset (1861/01-1889/12 with 0.1yr\nsampling and 1900/01-2019/12 with 0.05yr sampling); the ΔLOD record was combined with a long-\nterm dataset45 (1623/06-2008/06 with 1yr sampling from IERS; EOPC01) and the EOPC04 ΔLOD\nrecord46 (1962/01-2019/12 with 1-day sampling); the AAM (1949/01-2019/12, sampling at 6 hours)\nrecord was from the Special Bureau for the Atmosphere47-49. The AAM was calculated from\nNCEP/NCAR reanalyses archived on pressure surfaces, and the inverted barometer (IB) pressure term\nwas chosen as the mass term. The OAM record was obtained from the Special Bureau for the Oceans'\ndatasets: ECCO_50yr50 (1949/01-2003/01, sampling at 10 days) and ECCO_kf080i51 (1993/01-2020/3,\nsampling at 1 day). To standardize the sampling intervals of the records, we down-sampled all records\nto 1yr, and to avoid aliasing effects in this down-sampling process, a low-pass filter (with a cut-off\nfrequency f = 0.5 cpy) was used prior to down-sampling.\nc\nNote that although the theoretical amplitudes of the 8.85yr and 9.3yr zonal tides are quite small\n(only ~2 μs for the 9.3yr tide and less than 1 μs for the 8.85yr tide) and far less than the background\nnoise level of the ΔLOD time series, they were removed from this ΔLOD record based on a given\nmodel52 to avoid the effects of some well-known signals on the target ~8yr period band. The dΔLOD/dt\nwas obtained by a classical discrete numerical derivation algorithm, i.e., dΔLOD(t)/dt=\ni\n[ΔLOD(t )−ΔLOD(t)]/Δt.”.\ni+1 i\n-------------------------------------------------------------------------------------------------------\n•L108-110: I do not follow the argument here. What is “it” referring to in the phrase “…only when it\nis in the same plane…”? What do you mean by being in the same plane as the tilt angle? The axial\ntorque vector Gamma_z is always in the same plane as the vector defining the inner core figure axis,\nsince any two (non-colinear) vectors from the origin define a plane.\nResponse: We are sorry for our lack of clarity here and thank you for catching this oversight. As you\nsaid, the two vectors the static tilted axis Ω′ and the rotation axis of the mantle Ω will define a plane,\nm m\nand the axial torque Г will reach its peak/valley when Г is in the plane. So, we restate it as: “The\nz z\naxial torque Г reaches its peak/trough only when the Г is in the plane defined by the static tilted axis\nz z\nΩ′ and the rotation axis of the mantle Ω ;” . Please see lines 121-122.\nm m\n-------------------------------------------------------------------------------------------------------\n•The Methods section shows how the equatorial and axial torques on the mantle are related to the\nchanges in mantle rotation vector (observed polar motion and LOD signals); however, there is no\nexplicit demonstration that the ratio of those torques equates to the tilt angle of the inner core. Given\nthat the complete description of couple rotations and torques for the mantle-outer core-inner core\nsystem is far from trivial (particularly if we allow that the solid regions of the Earth may be somewhat\ntriaxial), I think the authors should clearly articulate the reasoning here.\nResponse: Thanks for your valuable reminder. When the inner core wobbles around a tilted axis\nrelative to the mantle, the torque of mantle-inner core gravitational coupling Γ in the equatorial\nMICG\nplane of the inner core causes the angular momentum exchange between the mantle and the inner core.\nThis torque Γ produces a component Г in the equatorial plane of the mantle, which changes the\nMICG eq\nEarth's polar motion, and produces a component Г in the direction of Earth's rotation, which is\nz\nperpendicular to the equatorial plane of the mantle, and Г causes the ΔLOD (length-of-day variation).\nz\nTherefore, according to the vector addition, the angle between the inner core's equatorial plane and the\nmantle's equatorial plane is arctan(|Γ |/|Γ |)= 0.17°, which is also the tilted angle of the inner core. To\nz eq\nfurther illustrate this problem, we draw a simple schematic in Figure R9. We can see that the angle θ\nbetween the equatorial plane of the inner core and the equatorial plane of the mantle is arctan(|Γ |/|Γ |).\nz eq\nFigure R9. The red circle represents the equatorial plane of the inner core, and the gray circle\nrepresents the equatorial plane of the mantle. The torque of mantle-inner core gravitational coupling\nΓ moves on the equatorial plane of the inner core because of the inner core wobble; the Γ can\nMICG MICG\nbe decomposed into the vector sum of the torque Г on the equatorial plane of the mantle and the\neq\ntorque Γ perpendicular to the equatorial plane of the mantle.\nz\nAccording to your comments, we added Figure R9 in the new Figure 4 of the main text and added\na note in line 153.\n-------------------------------------------------------------------------------------------------------\nMinor points\n•L21: Repetition of “solid inner core” one of which should be “solid mantle”.\nResponse: Done as your reminder, thanks!\n------------------------------------------------------------------------------------------------------\n•L35: “and etc.” is unnecessary.\nResponse: Deleted it as your suggestion.\n-------------------------------------------------------------------------------------------------------\n•L67: “that has a significantly different feature” Different than what?\nResponse: Thanks for your reminder! In the new MS, we modified this sentence as: “Different from\nthe oceanic tidal signals that have both prograde and retrograde components in PM22-24, the ICW is a\nprograde motion (the same as the Chandler wobble, i.e., the mantle wobble); in the complex spectra of\nthe PM, a prograde/retrograde wobble only has a positive/negative frequency. The identification of the\nwell-known Chandler wobble is based on this feature25-28. Therefore, as a prograde motion, the ICW\nonly appears on the positive frequency axis of the PM spectrum and this is a distinguishing feature for\nidentifying it.”. Please see lines 72-77.\n-------------------------------------------------------------------------------------------------------\n•L106: “using a more complicated method” is too vague. You don’t have to explain the method here\nbut should at least be specific on what it was.\nResponse: Thanks for your suggestion. We modified it as: “the extracted oscillations using a more\ncomplicated method (the normal time-frequency transform, NTFT) show almost the same results (see\nFigure S3).”. Please see line 118 and 119.\n-------------------------------------------------------------------------------------------------------\n•L108- I would use “trough” rather than “valley”.\nResponse: Done as your suggestion. Please see line 121.\n-------------------------------------------------------------------------------------------------------\n•L192- Am and Cm should have units of kg m^2.\nResponse: Thanks for your kind reminder! We added the unit. Please see line 216.\n-------------------------------------------------------------------------------------------------------\nREVIEWER COMMENTS\nReviewer #1 (Remarks to the Author):\nThanks for the detailed and comprehensive responses, which are highly appreciated.\nThe manuscript has significantly improved relative to the original version, at least in its part of making\nconnections with seismological findings. Another reviewer had similar concerns, which helped improve\nthe links with the seismological work and corresponding discussions.\nI now have minor concerns remaining that I hope can be addressed. However, they are optional, and it is\nbetween the Editor and you to consider if you want to implement them.\n- I find the new abstract version better than the previous one, but still a bit unpolished. As far as I know,\nthe use of wording such as “for the first time, we newly find…” or “Our novel finding…” is discouraged.\nI suggest removing it.\n- I also suggest removing the statement “(which is also a pending problem)” at the end of the abstract.\nLike many other parameters, such as anisotropy, lateral heterogeneity, and differential rotation, the\ninner-outer core density contrast is a subject of seismological studies and comes with uncertainty. Your\ninferred value of 0.52 g.cm^3 (by the way, you should remove the last zero from the value reported in\nthe abstract because your uncertainty estimate is 0.05) is a valuable contribution and stands on its own\nwithout explicitly stating that it is “a pending problem.” A relatively recent review of the inner-outer\ncore density contract and the differential rotation, from a seismological point of view, is given in the\ninner core book. You can consider adding this reference to the seismological references made at the\nend of the first paragraph of the introduction: https://www.cambridge.org/core/books/earths-inner-\ncore/3994E0608D352380FAF75165A4125DDD\n- Line 135-136: I suggest removing “although it has some uncertainties,” as from your referencing, it\nlooks as though the first study contradicts the second, although they were published parallelly\nsupporting the same model.\n- Line 138: there is a problem here as you cite two studies (#36 and #37) that found hemispherical\ndifferences in a) velocity and b) attenuation. Note that this is not the same as a hemispherical difference\nin anisotropy. In fact, such a dynamical model (proposed by #34 and #35) would erase any anisotropic\nsignature, which is why some contest it. Following this, I have difficulties understanding how\nreferencing #39 can follow from this discussion. If the proposed geodynamical model that would create\na hemispherical difference (#34 and #35) is indeed in operation, it will contradict anisotropy at the top\nof the inner core. I think you are safer excluding anisotropy completely from your discussions where\nyou try to establish a link with seismological observations. Many readers will not understand why\nanisotropy is brought into these discussions in the first place.\n- I don’t understand the last sentence of the main part of the manuscript (before the Methods). You are\nsaying that the static tilt should be used as a constraint on the inner core structure retrieved from\nseismology, but it is unclear in what sense. For example, if it is meant that it should be used in the\nsense that a tilt would also require a density jump of 0.52 g/cm^3 at the ICB or that it would require a\ndifferential rotation much less than 1.0 degrees per year, then OK. But if it is meant in the sense that it\nwould require some sort of a travel time correction, then this wouldn’t make much sense because 0.17\ndegrees is less than 20 km on the Earth’s surface and negligible at the ICB in comparison with the\nwavelengths the seismic probes are sensitive to. At the Earth’s surface, this approaches the error in the\nhypocenter location, and it is within the uncertainty we deal with in seismology. It would be good to\nclarify what you meant.\nReviewer #2 (Remarks to the Author):\nThe authors have answered properly my concerns and met all my suggestions. Besides, I think the\nmanuscript has been improved with the comments of the other reviewers. Therefore, I am satisfied with\nthe revision and can recommend it for publication.\nThere are some minor changes I’d like to suggest. They are:\nL 15 “This tilt implies that…” sounds to a proven causality relation. I’d suggest changing it to “this tilt is\nconsistent with the assumption that…”. Or rewording the sentence with the same idea to make it\nmore compact\nL 21 “The solid Earth consists mainly of…”. As the result is based on Earth rotation data, and quite a\nlot people might not be familiar with the current theory of earth rotation, I’d suggest modifying the\nfirst sentence so that it recalls somehow the results are based on earth rotation theory. For instance,\nyou might start this way: “Current theories of the Earth rotation consider a basic model of the solid\nEarth that consists mainly of…”\nL 22-23 ff. The paper has many references to the “axes of figure” of earth components. However, I’m\nafraid that most readers won’t be familiar with the concept of axis of figure used in earth rotation,\neven though many of them might work in that field. My guess is because the exact definition of this\nconcept is not simple at all but takes up more than 4 pages (with 34 Eqs.!) of Appendix A in the reference\n2, i.e., the 1991 paper by Mathews et al - in which the so-called three-layer earth model was introduced\nas well as the associated four normal rotational modes (2 known + 2 new ones due to the SIC). Of\ncourse, it will be out of place providing details of such intricate definitions, but it may be appropriate\nto recall that the concept of axis of figure used in a work relying on earth rotation data is (and must be)\nthe one used in the corresponding earth rotation theory and resulting equations needed to obtain those\ndata. I think this can be made straightforward, e.g., by including a footnote (or a comment) at the first\nappearance of the word “axis”, clarifying that figure axis is ever understood as defined in appendix A\nof the current ref. 2.\nL 25 Please remove “geometric”, it may be confusing\nL 55 …they were… -> … they are…\nL 84 I think the explanation might be slightly more detailed. For instance: “for signal detection” -> “for\nthe accurate detection of the frequencies of harmonic components of a signal”\nL 119 NTFT needs a reference\nL 169 - Eq (1) - The notation for the equatorial torque exerted for the mantle on the SIC used in the left-hand\nside of the equation is the same as that used in the right-hand side of Eq (6) in L 214, where it\nrefers to the total torque. Furthermore, chi and m in (6) refer to the observed (total) excitation and\npolar motion, whereas the chi/m used in L 170-172 correspond only to the 8.5-year component, as\nimplicit in the calculations described in L 171-172. This may raise some confusion that could even\nlead to a serious misunderstanding. Therefore, I suggest either changing the notation or clarifying the\nsituation by explaining that the same notation is being used for ease of visualization but that the torques,\ncomplex PM and excitation functions in Eqs 1 & 2 correspond only to the determined 8.5 yr signal as\nopposed to the same symbols when they appear in Eqs 6 & 7. Perhaps the second option is clearer;\nyou could also combine both clarifications.\nL 183-186. Concerning the values of the periods of rotational normal modes derived either from\ngeophysical earth models or from earth rotation data, FCN (free core nutation) has been observed over\n30 years and its period is also a good example of how much those values can differ. A good sample of\nvalues is available in table 3 of (doi: 10.1093/gji/ggab079); the rule there seems to be that ‘theoretic’\nand ‘observed’ periods never match each other. Please consider whether you wish to include a\ncomment on FCN too.\nL 210-211 I think you should explain in the text, here or earlier, that the mantle inertia ellipsoid is\nassumed to have axial symmetry. I know this is the usual assumption in most of Earth rotation related\nwork, but NatCom readers are a much wider audience.\nL 219-220 Some readers may not be very familiar with the various estimates of the CW complex period,\nparticularly those of its Q factor, which exhibit greater variability. I suggest including a sample of some\nreference values/intervals and the corresponding bibliography. You could include, for example, a\nreference to a table from a recent and easy to find article. That way it will be evident that the complex\nfactor that appear in this line is very close to 1.\nL 223 Combing -> Combining\nL 258 EOPCO1 -> EOPC01\nReviewer #3 (Remarks to the Author):\nI appreciate the efforts made by the authors in response to the previous round of reviews. The work is\nimproved; however, I still have some concerns.\nMajor points\nThe translation from torques to tilt\nIt remains unclear to me why the torques deduced for the equatorial and axial components resolve the\ninner core tilt. Certainly, the two vectors can be combined in this way, but I see no clear demonstration\nthat the torques on the mantle that cause polar motion and LOD variations should lie in the equator of\nthe inner core aligned with the longitude of the tilt. LOD changes arising from inner core-mantle\ngravitational coupling have often been attributed to a spherical harmonic degree 2, order 2 component\nof the inner core shape (e.g. Dumberry and Mandea, 2021; Buffett, 1996), in which case the axial\ntorque has no relation to the tilted figure axis. That mechanism would imply a triaxial inner core, but\neven if the authors wish to consider the case of an oblate spheroidal inner core, then it seems to me\nthat a gravitational attraction between a tilted inner core and the mantle would result in an equatorial\ntorque aligned with the intersection of the inner core and mantle equatorial planes. Perhaps I am\nthinking about the dynamics incorrectly, but since this calculation is central to the argument for the tilt,\nI think that the theory needs to be more fully explained.\nThe framing of previous geodynamic work suggesting a large inner-core tilt\nWhile it is true that Greiner-Mai et al (2000, ref. 8 of the manuscript) propose alignment of the inner\ncore spin axis with the geomagnetic dipole axis as a hypothesis from which to start their analysis, I do\nnot think that this is a well-founded and widely accepted idea. Greiner-Mai & Barthelmes (2001) argue\nfrom polar motions data for an inner-core figure axis tilted by about 1 degree relative to the mantle\nrotation axis. Dumberry and Mandea (2021) argue from gravity data for constraints on the inner-core\nfigure axis tilt on the order of 0.014 - 0.07 degrees. Just as I do not think that the weight of previous\nseismic studies supports a large inner core tilt, I do not think that the overall view of previous\ngeodynamic studies favours a large tilt.\nModerate points\n• L25-26, L31-32, L121-123 and figure 4: From the outset I think that the potential offset between the\ninner core rotation axis and the mantle rotation axis should be mentioned. L25-26 imply that they\nshould be the same, but the main point of the paper is to argue that they are not. Therefore, the first\nmention of the inner core and mantle rotation axes should make that distinction and use the Omega_m\nand Omega^prime_m notation used elsewhere in the manuscript.\n• I had not appreciated that the amplitude information from the AR-z process was not particularly\nmeaningful, thank you for that clarification. I think that the background noise level indication in the\nfigure included in the response should also be used in the paper (figure 2a) to illustrate whether the\nvarious peaks rise above the detection threshold. I remain concerned that the other peaks detected in\nthe PM and LOD time series have no apparent physical explanation, but the reasoning for choosing\nthat one peak as indicative of the ICW is now clearer.\nMinor points\n• L14: “than that” --> “than the”\n• L41: a citation to Triana et al (2022) might be included as another recent review relevant to the\nquestion of the importance of the topic.\n• L89-90: “Since there is no other mechanism has been proposed…” --> “Since no other mechanism\nhas been proposed…”\n• L95: “super frequency resolution”, the use of “super” here feels awkward to me.\n• L117: “ICWs” --> “ICW signals” or similar\n• L167: “inner-mantle” is missing “core”\nReferences\nBuffett, B. A. Gravitational oscillations in the length of day. Geophys Res Lett 23, 2279–2282 (1996).\nDumberry, M. & Mandea, M. Gravity Variations and Ground Deformations Resulting from Core\nDynamics. Surv Geophys 1–35 (2021) doi:10.1007/s10712-021-09656-2.\nGreiner-mai, H. & Barthelmes, F. Relative wobble of the Earth’s inner core derived from polar motion\nand associated gravity variations. Geophys J Int 144, 27–36 (2001).\nTriana, S. A. et al. Core Eigenmodes and their Impact on the Earth’s Rotation. Surv Geophys 43, 107–148\n(2022).\nResponse to Reviewers\n###########################################################################\n###########################################################################\nReviewer #1 (Remarks to the Author):\nThanks for the detailed and comprehensive responses, which are highly appreciated. The manuscript\nhas significantly improved relative to the original version, at least in its part of making connections\nwith seismological findings. Another reviewer had similar concerns, which helped improve the links\nwith the seismological work and corresponding discussions.\nResponse: We appreciate this positive assessment and would like to thank you for the thoughtful\nreview and the insightful comments to help improve this study. We have carefully considered your\nsuggestion and revised this MS according to the comments, and hope that the revised version is more\nsuitable.\n-------------------------------------------------------------------------------------------------------\nI now have minor concerns remaining that I hope can be addressed. However, they are optional, and it\nis between the Editor and you to consider if you want to implement them.\n- I find the new abstract version better than the previous one, but still a bit unpolished. As far as I know,\nthe use of wording such as “for the first time, we newly find…” or “Our novel finding…” is\ndiscouraged. I suggest removing it.\nResponse: Thank you very much for your kind reminder, we have changed the related expressions in\nthe new MS, please see lines 11 and 13.\n-------------------------------------------------------------------------------------------------------\n- I also suggest removing the statement “(which is also a pending problem)” at the end of the abstract.\nLike many other parameters, such as anisotropy, lateral heterogeneity, and differential rotation, the\ninner-outer core density contrast is a subject of seismological studies and comes with uncertainty. Your\ninferred value of 0.52 g.cm^3 (by the way, you should remove the last zero from the value reported in\nthe abstract because your uncertainty estimate is 0.05) is a valuable contribution and stands on its own\nwithout explicitly stating that it is “a pending problem.” A relatively recent review of the inner-outer\ncore density contract and the differential rotation, from a seismological point of view, is given in the\ninner core book. You can consider adding this reference to the seismological references made at the\nend of the first paragraph of the introduction: https://www.cambridge.org/core/books/earths-inner-\ncore/3994E0608D352380FAF75165A4125DDD\nResponse: Thank you for your useful suggestions and for bringing a useful textbook to our attention,\nwhich greatly helps me to track the progress of seismological observations of the inner core and deepen\nmy understanding of the structure of the inner core. We fully agree with your comments about the\ninner-outer core density contrast, and we have removed the latest sentence at the end of the abstract\n(please see line 17) and deleted the last zero in “0.520 g/cm3” (please see lines 17, 209, and 216), and\nwe also added a reference as you suggested (please see lines 34, 53, and 207).\n-------------------------------------------------------------------------------------------------------\n- Line 135-136: I suggest removing “although it has some uncertainties,” as from your referencing, it\nlooks as though the first study contradicts the second, although they were published parallelly\nsupporting the same model.\nResponse: Thank you for your kind reminder, we have removed it in the new MS, please see line 155.\n-------------------------------------------------------------------------------------------------------\n- Line 138: there is a problem here as you cite two studies (#36 and #37) that found hemispherical\ndifferences in a) velocity and b) attenuation. Note that this is not the same as a hemispherical difference\nin anisotropy. In fact, such a dynamical model (proposed by #34 and #35) would erase any anisotropic\nsignature, which is why some contest it. Following this, I have difficulties understanding how\nreferencing #39 can follow from this discussion. If the proposed geodynamical model that would create\na hemispherical difference (#34 and #35) is indeed in operation, it will contradict anisotropy at the top\nof the inner core. I think you are safer excluding anisotropy completely from your discussions where\nyou try to establish a link with seismological observations. Many readers will not understand why\nanisotropy is brought into these discussions in the first place.\nResponse: Thank you for your suggestion, we found some discussions and statements in the old MS\nare really confusing, so we have deleted and revised those statements. We prefer to retain some\npresentations of the anisotropy, as the results of the studies may be relevant to the further conclusions\nwe suggest for the northwestern hemisphere. The specific modification is expressed as follows: “To\nexplain the asymmetry between the inner core's eastern and western hemispheres in seismological\nobservations42,43, a previous dynamical model considers the crystallization and melting at the surface\nof the inner core and has similar suggestions44, i.e., the western hemisphere of the inner core is denser\nthan its eastern hemisphere. Interestingly, a seismological study has suggested that the western\nhemisphere of the inner core may be relatively denser45, and a thicker compacting layer at the top of\nthe inner core's western hemisphere was also suggested46; a more nuanced research found that the\nwestern zone is largely confined to the northern hemisphere47”, please see lines 151-158.\n-------------------------------------------------------------------------------------------------------\n- I don’t understand the last sentence of the main part of the manuscript (before the Methods). You are\nsaying that the static tilt should be used as a constraint on the inner core structure retrieved from\nseismology, but it is unclear in what sense. For example, if it is meant that it should be used in the\nsense that a tilt would also require a density jump of 0.52 g/cm^3 at the ICB or that it would require a\ndifferential rotation much less than 1.0 degrees per year, then OK. But if it is meant in the sense that it\nwould require some sort of a travel time correction, then this wouldn’t make much sense because 0.17\ndegrees is less than 20 km on the Earth’s surface and negligible at the ICB in comparison with the\nwavelengths the seismic probes are sensitive to. At the Earth’s surface, this approaches the error in the\nhypocenter location, and it is within the uncertainty we deal with in seismology. It would be good to\nclarify what you meant.\nResponse: Thanks for your constructive suggestions! This sentence is really vague. We originally\nintended to say that this static tilt could qualitatively constrain some core structures or motions from\nseismic observations, such as the density differences between the eastern and western hemispheres,\nthe inner core differential rotation, or inner core oscillations. As you commented, it is currently very\ndifficult to quantitatively constrain the inner structures obtained from seismic waves. We have\ntherefore revised this sentence to: and we suggest such consistency should be helpful to the inner core\noscillation or differential rotation (please see lines 220-222).\n-------------------------------------------------------------------------------------------------------\n###########################################################################\n###########################################################################\nReviewer #2 (Remarks to the Author):\nThe authors have answered properly my concerns and met all my suggestions. Besides, I think the\nmanuscript has been improved with the comments of the other reviewers. Therefore, I am satisfied\nwith the revision and can recommend it for publication.\nResponse: Thank you very much for your thorough read of our MS and insightful comments. We\ncarefully addressed each of your comments and updated the manuscript accordingly. Please see our\ndetailed responses below with tracking for the exact edits.\n-------------------------------------------------------------------------------------------------------\nThere are some minor changes I’d like to suggest. They are:\nL 15 “This tilt implies that…” sounds to a proven causality relation. I’d suggest changing it to “this\ntilt is consistent with the assumption that…”. Or rewording the sentence with the same idea to make it\nmore compact\nResponse: Thank you for the suggestions to make it more readable, we changed it as you suggested,\nplease see line 15.\n-------------------------------------------------------------------------------------------------------\nL 21 “The solid Earth consists mainly of…”. As the result is based on Earth rotation data, and quite a\nlot people might not be familiar with the current theory of earth rotation, I’d suggest modifying the\nfirst sentence so that it recalls somehow the results are based on earth rotation theory. For instance,\nyou might start this way: “Current theories of the Earth rotation consider a basic model of the solid\nEarth that consists mainly of…”\nResponse: Thanks very much for your suggestions. According to your and Reviewer #3's suggestions,\nwe have made detailed revisions here, adding the background on the theories of the Earth's rotation.\nPlease see lines 21-27.\n-------------------------------------------------------------------------------------------------------\nL 22-23 ff. The paper has many references to the “axes of figure” of earth components. However, I’m\nafraid that most readers won’t be familiar with the concept of axis of figure used in earth rotation,\neven though many of them might work in that field. My guess is because the exact definition of this\nconcept is not simple at all but takes up more than 4 pages (with 34 Eqs.!) of Appendix A in the reference\n2, i.e., the 1991 paper by Mathews et al - in which the so-called three-layer earth model was introduced\nas well as the associated four normal rotational modes (2 known + 2 new ones due to the SIC). Of\ncourse, it will be out of place providing details of such intricate definitions, but it may be appropriate\nto recall that the concept of axis of figure used in a work relying on earth rotation data is (and must be)\nthe one used in the corresponding earth rotation theory and resulting equations needed to obtain those\ndata. I think this can be made straightforward, e.g., by including a footnote (or a comment) at the first\nappearance of the word “axis”, clarifying that figure axis is ever understood as defined in appendix A\nof the current ref. 2.\nResponse: Thanks a lot for your very useful suggestions, we have added a note about this according\nto your suggestions, please see line 26.\n-------------------------------------------------------------------------------------------------------\nL 25 Please remove “geometric”, it may be confusing\nResponse: Done as you suggested. Please see line 25.\n-------------------------------------------------------------------------------------------------------\nL 55 …they were… -> … they are…\nResponse: Done as you suggested. Please see line 67.\n-------------------------------------------------------------------------------------------------------\nL 84 I think the explanation might be slightly more detailed. For instance: “for signal detection” ->\n“for the accurate detection of the frequencies of harmonic components of a signal”\nResponse: Done as you suggested. Please see line 96.\n-------------------------------------------------------------------------------------------------------\nL 119 NTFT needs a reference\nResponse: Done as you suggested. Please see line 137.\n-------------------------------------------------------------------------------------------------------\nL 169 - Eq (1) - The notation for the equatorial torque exerted for the mantle on the SIC used in the\nleft-hand side of the equation is the same as that used in the right-hand side of Eq (6) in L 214, where\nit refers to the total torque. Furthermore, chi and m in (6) refer to the observed (total) excitation and\npolar motion, whereas the chi/m used in L 170-172 correspond only to the 8.5-year component, as\nimplicit in the calculations described in L 171-172. This may raise some confusion that could even\nlead to a serious misunderstanding. Therefore, I suggest either changing the notation or clarifying the\nsituation by explaining that the same notation is being used for ease of visualization but that the torques,\ncomplex PM and excitation functions in Eqs 1 & 2 correspond only to the determined 8.5 yr signal as\nopposed to the same symbols when they appear in Eqs 6 & 7. Perhaps the second option is clearer;\nyou could also combine both clarifications.\nResponse: Thank you for the suggestion to make it more readable accessible. We have changed it as\nyour suggestions, please see lines 176 and 187-202.\n-------------------------------------------------------------------------------------------------------\nL 183-186. Concerning the values of the periods of rotational normal modes derived either from\ngeophysical earth models or from earth rotation data, FCN (free core nutation) has been observed over\n30 years and its period is also a good example of how much those values can differ. A good sample of\nvalues is available in table 3 of (doi: 10.1093/gji/ggab079); the rule there seems to be that ‘theoretic’\nand ‘observed’ periods never match each other. Please consider whether you wish to include a\ncomment on FCN too.\nResponse: This is a good idea, we have added a comment on FCN “free core nutation observation\nperiod of retrograde ~430 days is ~20 days shorter than its theoretical periods49,50” as you suggested,\nplease see lines 205 and 206.\n-------------------------------------------------------------------------------------------------------\nL 210-211 I think you should explain in the text, here or earlier, that the mantle inertia ellipsoid is\nassumed to have axial symmetry. I know this is the usual assumption in most of Earth rotation related\nwork, but NatCom readers are a much wider audience.\nResponse: This is a good idea, we have made this change as your suggestions, please see lines 230\nand 231.\n-------------------------------------------------------------------------------------------------------\nL 219-220 Some readers may not be very familiar with the various estimates of the CW complex period,\nparticularly those of its Q factor, which exhibit greater variability. I suggest including a sample of some\nreference values/intervals and the corresponding bibliography. You could include, for example, a\nreference to a table from a recent and easy to find article. That way it will be evident that the complex\nfactor that appear in this line is very close to 1.\nResponse: Thank you for your insightful comments and suggestions, we have made corresponding\nchanges and added relevant references as you suggested. Please see lines 241 and 242.\n-------------------------------------------------------------------------------------------------------\nL 223 Combing -> Combining\nResponse: Our apologies for this oversight. We modified it according to your comment. Please see\nline 246.\n-------------------------------------------------------------------------------------------------------\nL 258 EOPCO1 -> EOPC01\nResponse: Thanks for your kind reminder! We modified it as your comment. Please see line 281.\n-------------------------------------------------------------------------------------------------------\nSincerely\nJose M Ferrandiz\n###########################################################################\nReviewer #3 (Remarks to the Author):\nI appreciate the efforts made by the authors in response to the previous round of reviews. The work is\nimproved; however, I still have some concerns.\nResponse: Thank you very much for some recognition of our work in the last round and thoughtful\nreview. The remaining concerns and the helpful comments drive us to further improve this MS. We\ncarefully addressed each of your comments and modified this MS based on your helpful feedback.\nPlease see our detailed responses below with tracking for the exact edits.\n-------------------------------------------------------------------------------------------------------\nMajor points\nThe translation from torques to tilt\nIt remains unclear to me why the torques deduced for the equatorial and axial components resolve the\ninner core tilt. Certainly, the two vectors can be combined in this way, but I see no clear demonstration\nthat the torques on the mantle that cause polar motion and LOD variations should lie in the equator of\nthe inner core aligned with the longitude of the tilt. LOD changes arising from inner core-mantle\ngravitational coupling have often been attributed to a spherical harmonic degree 2, order 2 component\nof the inner core shape (e.g. Dumberry and Mandea, 2021; Buffett, 1996), in which case the axial\ntorque has no relation to the tilted figure axis. That mechanism would imply a triaxial inner core, but\neven if the authors wish to consider the case of an oblate spheroidal inner core, then it seems to me\nthat a gravitational attraction between a tilted inner core and the mantle would result in an equatorial\ntorque aligned with the intersection of the inner core and mantle equatorial planes. Perhaps I am\nthinking about the dynamics incorrectly, but since this calculation is central to the argument for the tilt,\nI think that the theory needs to be more fully explained.\nResponse: Thank you for your insightful comments, in an ideal Earth model that the symmetry axes\nof mantle's elliptical surfaces of constant density are aligned along the axis of rotation (see Figure 10a),\nsome cases are indeed as you commented. Here, we consider a more realistic and complex Earth model\nthat the symmetry axes of elliptical surfaces of constant density are no longer aligned for homogeneous\nmantle (see Figure 10b), but this was not specifically stated in the previous drafts. Next, we will explain\nin detail why the torques deduced for the equatorial and axial components resolve the inner core tilt.\nWe first approach the problem of the mantle-inner core gravitational (MICG) torque. The torque that\nchanges the polar motion (PM) is always on the equatorial plane of the mantle, and the torque that\nchanges the ΔLOD is always aligned with the rotation axis of the mantle. In other words, any torque\nexerted on the equatorial plane of the mantle can change the PM, and any torque exerted on the rotation\naxis of the mantle can change the ΔLOD. The ~8.5yr ICW signal is detected from the PM,\ndemonstrating that there is a torque Γ always in the equatorial plane of the mantle (the gray plane in\neq\nFigure 4) which changes its direction with the period of ~8.5yr. However, the ~8.5yr ICW signal is\nalso detected in the ΔLOD, which demonstrates that there should be a torque Γ in the direction of the\nz\nmantle's rotation which changes its magnitude with the period of ~8.5yr. Given that the interaction\nbetween the mantle and the inner core is mainly through the MICG coupling (as suggested by Buffett\n(1996) and Dumberry (2009)), the MICG torque can be targeted immediately and the two torques\nshould be originated from the MICG torque (and hence can be further re-written as ΓMICG and\neq\nΓMICG). The only way these two torques ΓMICG and ΓMICG can be combined is by vector addition,\nz eq z\ncreating the torque Γ which changes direction continuously in the tilted plane (blue plane in\nMICG\nFigure 4) relative to the mantle. Next, we explain why the MICG torque Γ caused by the ICW is\nMICG\nin the tilted plane and it is the mean equatorial plane of the inner core. Before that, we introduce the\nbackground of the Earth's rotation and the Earth model used. The mantle's rotation axis and its figure\naxis are very close due to the centrifugal force, and they can be seen as the same axis Ω .\nSimilarly,\nm\nthe figure axis Ω of the inner core is very close to its rotation axis Ω′ (but much larger than the\nic m\nmantle) which locates the direction of the lowest MICG potential energy (equilibrium state; similar\nto the lowest point of a simple pendulum model). When the inner core's figure axis Ω deviates from\nic\nits rotation axis Ω′ (the equilibrium state), it will excite a rotation mode that the inner core's figure\nm\naxis Ω wobbles around its rotation axis Ω′ under the influence of spin, i.e., the ICW. Current theories\nic m\nof the Earth's rotation are based on the classical 2-D Earth model; in this model, the flat solid Earth\nconsists mainly of a solid inner core, a liquid outer core, and a solid mantle with the same center of\nmass; this model is reduced to a series of layered elliptical surfaces on which the density is constant,\nand the mantle's elliptical surfaces of constant density is aligned in the rotation's axis Ω (see Figure.\nm\nR10a). Thus, the direction of the lowest MICG potential energy is the mantle's rotation axis Ω ,\nand\nm\nthe inner core's rotation axis Ω′ is aligned with of the axis Ω . However, for the real Earth, the\nm m\nsurfaces of constant density of the highly heterogeneous mantle are no longer the ideal ellipsoids, and\ntheir symmetry axes may be slightly tilted randomly relative to the mantle's rotation axis Ω , especially\nm\nwith large uncertainties at the core-mantle boundary (CMB), as shown in Figure. R10b. The direction\nof the mantle's figure axis is only sensitive to the heterogeneity or ellipticity of the upper mantle\naccording to its definition (the direction corresponding to the maximum moment of inertia). However,\nthe closer matters have the stronger gravitational force, and therefore the lowest MICG potential\nenergy state (the equilibrium state) is that the inner core's figure axis Ω is aligned with the mean axis\nic\nof the lower mantle's elliptical surfaces of constant density; the mean axis is usually tilted relative to\nthe mantle's rotation axis Ω and it coincides with the inner core's rotation axis Ω′ (which represents\nm m\nthe direction of the lowest MICG potential energy). When the inner core's figure axis Ω deviates from\nic\nthis equilibrium state, it excites the ICW that the figure axis Ω wobbles around its rotation axis Ω′ .\nic m\nIn the process, the MICG torque Γ always brings the inner core back to the equilibrium state,\nMICG\nthat is, the torque Γ always perpendicular to Ω′ , and therefore torque Γ always in the\nm\nMICG MICG\nplane perpendicular to Ω′ , which is the mean equatorial plane of the inner core (blue plane in Figure\nm\n4). The decomposition of the MICG torque at a certain moment is shown in Figure 4.\nPrivately, we have communicated with Prof. Benjamin F. Chao about this part for the\nheterogeneous mantle and have received positive supports (we are planning to work together to further\ndevelop the theory of Earth rotation in response to this problem). We also sent the MS to Prof. Chao,\nand he strongly suggested that we should include a description of the heterogeneous mantle and the\nalignment of the inner core with the mean axis of the lower mantle at the static equilibrium due to the\nstronger attraction of nearby objects in this MS, which may improve the theory of the Earth's rotation.\nTo clarify, we added the background of the Earth's rotation ‘The flat solid Earth consists mainly of a\nsolid inner core, a liquid outer core, and a solid mantle with the same center of mass, which is reduced\nto a series of layered elliptical surfaces on which the density is constant in the classical model1,2. The\ncurrent theories regarding the Earth's rotation involve the consideration of the mantle's elliptical\nsurfaces of constant density, whose symmetry axes are aligned in the direction of rotation, and\nhydrostatic effects necessitate that the inner core's figure axis Ω (as defined in Appendix A of ref. 2)\nic\nand rotation axis Ω′ are aligned with the mantle's figure or rotation axis Ω (which are nearly identical\nm m\ndue to centrifugal torque) in order to maintain equilibrium. The presence of random torques acting on\nthe inner core results in a slight tilt and further excites a prograde rotation mode known as the inner\ncore wobble (ICW), i.e., the inner core's figure axis Ω wobbles about its rotation axis Ω′ 2-4 (also\nic m\nrepresents the direction of the lowest gravitational potential energy of the mantle-inner core system).\nThe above ‘tilt’ between the Ω and Ω′ is a generally dynamic tilt’ (please see lines 21-32), the\nic m\ndescription of a possible highly heterogeneous mantle for the real Earth ‘However, the elliptical\nsurfaces of constant density within the heterogeneous mantle may exhibit random tilting around its\nrotation axis Ω considering its solid properties, particularly with significant uncertainties at the core-\nm\nmantle boundary (CMB). Consequently, the inner core's rotation axis Ω′ , which signifies the direction\nm\nof the static equilibrium of the inner core (and corresponds to the lowest gravitational potential energy\nof the mantle-inner core system10), was previously believed to deviate from alignment with the mantle's\naxis Ω and instead possess a tilt relative to it, and thereby the ‘tilt’ between the Ω′ and Ω is called\nm m m\nas static tilt.’ (please see lines 35-42), and explanation of the torque Γ always in the mean\nMICG\nequatorial plane of the inner core ‘Next, we estimate the tilt θ. The heterogeneous mantle forms a tilted\nrotation axis of the inner core Ω′ which has the lowest mantle-inner core gravitational (MICG)\nm\npotential energy (the axis at the equilibrium state); the deviation of the inner core's figure axis Ω\nic\nfrom the equilibrium state caused by the ICW will result in a MICG restoring torque Γ which always\nMICG\nbrings the inner core back to the equilibrium state, that is, the torque Γ always in the plane (blue\nMICG\nplane in Figure 4) perpendicular to the inner core's rotation axis Ω′ ; the plane is also the mean\nm\nequatorial plane of the inner core.’ (please see lines 177-183). The mean equatorial plane of the inner\ncore perpendicular to the inner core's rotation axis Ω′ , so we have also further modified the Figure. 4\nm\nfor easier understanding, please see line 171.\nFigure R10. Schematic diagram of the mantle's elliptical surfaces of constant density. (a) The\nmantle in the classical 2-D Earth model which is used in current theories of the Earth's rotation; the all\nsymmetry axes of the mantle's elliptical surfaces of constant density are aligned in the rotation's\ndirection. (b) The more realistic model for the highly heterogeneous mantle; the symmetry axes of the\nmantle's elliptical surfaces of constant density are slightly tilted randomly relative to the mantle's\nrotation axis Ω (the tilt angle is exaggerated to show).\nm\nSecondly, we fully agree with you that the inner core (gravitational) oscillation mode can change\nthe LOD and the axial torque caused by it has no relation to the tilted figure axis. Based on the\nrotationally symmetric (biaxiality) three-layered Earth, the classical Earth rotation theories predicted\ntwo rotation modes related to the inner core, the free inner core nutation (FICN; irrelevant to this study),\ninner core wobble (ICW; Mathews et al., 1991, 2002; Dumberry, 2009; Rochester et al., 2014).\nConsidering the triaxiality of Earth, Buffett (1996) predicted a new inner core's rotation mode, the\ninner core (gravitational) oscillation, to explained the intradecadal fluctuation in the ΔLOD. The ICW\nand inner core oscillation are two completely different modes, and they have different theoretical\nperiods. The ICW refers to that the inner core's figure axis wobbles counterclockwise (as seen from\nabove the North Pole) around the mantle's axis in a circular motion, taking one cycle to complete;\nwhile inner core oscillation refers to that the inner core's rotation rate changes periodically relative to\nthe mantle's rotation rate. The geophysical mechanisms of the ICW and inner oscillation are different;\nthe ICW is caused by the Earth's rotation effect and the equatorial torque Γ which is caused by the\neq\nmisalignment of the degree-2 order-0 component of the inner core and mantle's shape; while the inner\ncore oscillation is caused by the axial torque (Γ) which is from the misalignment of the degree-2 order-\nz\n2 component of the inner core and mantle's shape. The two modes are decoupled from each other, they\ncause different results and appear in different observations due to different orders. Normally, the ICW\nmode only changes the polar motion with its period, and the inner core oscillation mode can cause the\nΔLOD. Unless the inner core and mantle have the static tilt described above, the ICW does not appear\nin the ΔLOD.\n___________________\nReference:\nBuffett, B. A. (1996). Gravitational oscillations in the length of day. Geophysical Research Letters,\n23(17), 2279-2282.\nDumberry, M. (2009). Influence of elastic deformations on the inner core wobble. Geophysical Journal\nInternational, 178(1), 57-64.\nMathews, P. M., Buffett, B. A., Herring, T. A., & Shapiro, I. I. (1991). Forced nutations of the Earth:\nInfluence of inner core dynamics: 1. Theory. Journal of Geophysical Research: Solid Earth,\n96(B5), 8219-8242.\nMathews, P. M., Herring, T. A., & Buffett, B. A. (2002). Modeling of nutation and precession: New\nnutation series for nonrigid Earth and insights into the Earth's interior. Journal of Geophysical\nResearch: Solid Earth, 107(B4), ETG-3.\nRochester, M. G., Crossley, D. J., & Zhang, Y. L. (2014). A new description of Earth's wobble modes\nusing Clairaut coordinates: 1. Theory. Geophysical Journal International, 198(3), 1848-1877.\n-------------------------------------------------------------------------------------------------------\nThe framing of previous geodynamic work suggesting a large inner-core tilt\nWhile it is true that Greiner-Mai et al (2000, ref. 8 of the manuscript) propose alignment of the inner\ncore spin axis with the geomagnetic dipole axis as a hypothesis from which to start their analysis, I do\nnot think that this is a well-founded and widely accepted idea. Greiner-Mai & Barthelmes (2001) argue\nfrom polar motions data for an inner-core figure axis tilted by about 1 degree relative to the mantle\nrotation axis. Dumberry and Mandea (2021) argue from gravity data for constraints on the inner-core\nfigure axis tilt on the order of 0.014 - 0.07 degrees. Just as I do not think that the weight of previous\nseismic studies supports a large inner core tilt, I do not think that the overall view of previous\ngeodynamic studies favours a large tilt.\nResponse: Thanks for your useful comments. Greiner-Mai et al. (2000) assumed alignment of the inner\ncore's rotation axis with the geomagnetic dipole axis and proposed a tilt of 10° of the inner core relative\nto the mantle, this ‘tilt’ actually refers to static tilt (deviation of Ω and Ω′ ). Subsequently, Greiner-\nm m\nMai and Barthelmes (2001) calculated a tilt of 1° from the excitation sequence of the polar motion,\nthis ‘tilt’ actually refers to dynamic tilt (deviation of Ω and Ω′ ). These two concepts are different,\nic m\nbased on your comments, and we distinguish them (please see lines 31-32 and 41-42) and explain the\nreason why they are produced (please see lines 23-31 and 35-41).\nIn fact, the inverted the dynamic tilt in Greiner-Mai and Barthelmes (2001) is from the entire\nfrequency band of the excitation sequence of the polar motion, which contains not only the ICW signal,\nbut also what they found ~70yr, ~30yr, ~20yr and trend term with large amplitudes, so the result is\nrelatively large. At present, a very small inner core's dynamic tilt (no more than 0.1°) has been agreed\nin the geodynamics (Dehant, et al., 1993; Dumberry and Bloxham, 2002; Guo et al., 2005; Dumberry,\n2008; Ding et al., 2019), which also includes the dynamic tilt of 0.014°-0.07° calculated by Dumberry\nand Mandea (2021). In this study, the dynamic tilt angle of the inner core can be easily estimated from\nthe amplitude and period of the ICW observed in the polar motion, i.e., we can infer the true amplitude\nof the ICW from the amplitude observed in the polar motion m= 3.685 m= 3.685×4.7 mas= ~17.3\ns\nmas (Dumberry, 2010), and hence the dynamic tilt is n= m/σ = ~53673 mas= ~0.015° (Mathews\ns s ICW\net al., 1992; Dumberry, 2010), where the σ = 1/(8.5×365) is in unit of cycle per days (cpd);\nICW\nthis dynamic tilt is consistent with the results of Dumberry and Mandea (2021) and previous studies\n(Dehant, et al., 1993; Dumberry and Bloxham, 2002; Guo et al., 2005; Dumberry, 2008; Ding et al.,\n2019). The static tilt is not included in the current Earth models, and the current theories of Earth's\nrotation (Mathews et al., 1991, 2002; Dumberry, 2009; Rochester et al., 2014) are based on the\nexisting Earth model, so the static tilt is not considered in theories of Earth's rotation. Although the\nstatic tilt postulated by Greiner-Mai et al. (2000) was large and unproven, their proposal represents a\npossibility. This is why we explain it that way in old MS. According to your comments, we have\nadjusted the relevant statements: ‘which is two orders of magnitude lower than the 10° assumed in\ncertain geodynamic researches.’ (please see lines 14 and 15) and ‘Despite the absence of confirmed\nobservations of the ICW12,13 and the lack of universal acceptance of the excessive static tilt of 10°,\nthe possibility of a static-tilted inner core remains, and further investigation has been conducted to\nexplore the impact of a static tilt on the period of the ICW12.’ (please see lines 44-47).\n___________________\nReference:\nDehant, V., Hinderer, J., Legros, H., & Lefftz, M. (1993). Analytical approach to the computation of\nthe Earth, the outer core and the inner core rotational motions. Physics of the Earth and Planetary\nInteriors, 76(3-4), 259-282.\nDing, H., Pan, Y., Xu, X. Y., Shen, W., & Li, M. (2019). Application of the AR‐z spectrum to polar\nmotion: A possible first detection of the inner core wobble and its implications for the density of\nEarth's core. Geophysical Research Letters, 46(23), 13765-13774.\nDumberry, M., & Bloxham, J. (2002). Inner core tilt and polar motion. Geophysical Journal\nInternational, 151(2), 377-392.\nDumberry, M. (2008). Decadal variations in gravity caused by a tilt of the inner core. Geophysical\nJournal International, 172(3), 921-933.\nDumberry, M. (2009). Influence of elastic deformations on the inner core wobble. Geophysical Journal\nInternational, 178(1), 57-64.\nDumberry, M., & Mandea, M. (2022). Gravity variations and ground deformations resulting from core\ndynamics. Surveys in Geophysics, 1-35.\nGreiner-Mai, H., Jochmann, H., & Barthelmes, F. (2000). Influence of possible inner-core motions on\nthe polar motion and the gravity field. Physics of the Earth and Planetary Interiors, 117(1-4), 81-\n93.\nGreiner-Mai, H., & Barthelmes, F. (2001). Relative wobble of the Earth's inner core derived from polar\nmotion and associated gravity variations. Geophysical Journal International, 144(1), 27-36.\nGuo, J. Y., Greiner‐Mai, H., & Ballani, L. (2005). A spectral search for the inner core wobble in Earth's\npolar motion. Journal of Geophysical Research: Solid Earth, 110(B10).\nMathews, P. M., Buffett, B. A., Herring, T. A., & Shapiro, I. I. (1991). Forced nutations of the Earth:\nInfluence of inner core dynamics: 1. Theory. Journal of Geophysical Research: Solid Earth,\n96(B5), 8219-8242.\nMathews, P. M., Herring, T. A., & Buffett, B. A. (2002). Modeling of nutation and precession: New\nnutation series for nonrigid Earth and insights into the Earth's interior. Journal of Geophysical\nResearch: Solid Earth, 107(B4), ETG-3.\nRochester, M. G., Crossley, D. J., & Zhang, Y. L. (2014). A new description of Earth's wobble modes\nusing Clairaut coordinates: 1. Theory. Geophysical Journal International, 198(3), 1848-1877.\n-------------------------------------------------------------------------------------------------------\nModerate points\n? L25-26, L31-32, L121-123 and figure 4: From the outset I think that the potential offset between the\ninner core rotation axis and the mantle rotation axis should be mentioned. L25-26 imply that they\nshould be the same, but the main point of the paper is to argue that they are not. Therefore, the first\nmention of the inner core and mantle rotation axes should make that distinction and use the Omega_m\nand Omega^prime_m notation used elsewhere in the manuscript.\nResponse: Thanks for your reminder! We have modified the relevant parts as you suggested, please\nsee lines 26, 27, 30, 31, 36, 38, and 41.\n-------------------------------------------------------------------------------------------------------\n? I had not appreciated that the amplitude information from the AR-z process was not particularly\nmeaningful, thank you for that clarification. I think that the background noise level indication in the\nfigure included in the response should also be used in the paper (figure 2a) to illustrate whether the\nvarious peaks rise above the detection threshold. I remain concerned that the other peaks detected in\nthe PM and LOD time series have no apparent physical explanation, but the reasoning for choosing\nthat one peak as indicative of the ICW is now clearer.\nResponse: Thank you, it's a good idea to add the background noise level in Figure. 2, we have made\nthese changes (please see line 121). As for the other signals, the ~5.9yr signal has been suggested as\nthe inner core oscillation coupled with torsional wave in the Earth's core but still remains controversial\n(Buffett, 1996; Buffett and Mound, 2005; Gillet et al., 2010); Gillet et al. (2022) identified and modeled\na non-axisymmetric ~7.3yr Magneto-Coriolis wave in the Earth's outer core that can explain the ~7.3yr\nsignal in the Earth’s rotation; the peak in the ~9-11yr is possible from the ~11yr Schwabe solar cycle;\nthe period of ~13yr was also found in the geomagnetic dipole field (Jin and Thomas, 1977), but relevant\nphysical mechanism has been not proposed; the ~18-23yr spectral peak may be mainly caused by the\n18.6yr tidal signal and the ~22yr Hale solar cycle or high-latitude MAC (magnetic-Archimedes-\nCoriolis forces) wave (Nicolas and Buffett, 2023) in the Earth's core. Thus, we have made\ncorresponding changes and added relevant physical explanations as your suggestions. Please see lines\n99-106.\n___________________\nReference:\nBuffett, B. A. (1996). Gravitational oscillations in the length of day. Geophysical Research Letters,\n23(17), 2279-2282.\nBuffett, B. A., & Mound, J. E. (2005). A Green's function for the excitation of torsional oscillations in\nthe Earth's core. Journal of Geophysical Research: Solid Earth, 110(B8).\nGillet, N., Jault, D., Canet, E., & Fournier, A. (2010). Fast torsional waves and strong magnetic field\nwithin the Earth’s core. Nature, 465(7294), 74-77.\nGillet, N., Gerick, F., Jault, D., Schwaiger, T., Aubert, J., & Istas, M. (2022). Satellite magnetic data\nreveal interannual waves in Earth’s core. Proceedings of the National Academy of Sciences,\n119(13), e2115258119.\nJin, R. S., & Thomas, D. M. (1977). Spectral line similarity in the geomagnetic dipole field variations\nand length of day fluctuations. Journal of Geophysical Research, 82(5), 828-834.\nNicolas, Q., & Buffett, B. (2023). Excitation of high-latitude MAC waves in Earth’s core. Geophysical\nJournal International, 233(3), 1961-1973.\n-------------------------------------------------------------------------------------------------------\nMinor points\n? L14: “than that” --> “than the”\nResponse: Done as you suggested. Please see line 14.\n-------------------------------------------------------------------------------------------------------\n? L41: a citation to Triana et al (2022) might be included as another recent review relevant to the\nquestion of the importance of the topic.\nResponse: Thanks for your suggestion and sharing this paper with us, it is an insightful review paper\nworth reading. We have added this reference as your suggestion. Please see line 53.\n-------------------------------------------------------------------------------------------------------\n? L89-90: “Since there is no other mechanism has been proposed…” --> “Since no other mechanism\nhas been proposed…”\nResponse: Done as you suggested. Please see line 108.\n-------------------------------------------------------------------------------------------------------\n? L95: “super frequency resolution”, the use of “super” here feels awkward to me.\nResponse: Our apologies for using this inappropriate word, we have modified it as “high-frequency\nresolution”, please see line 113.\n-------------------------------------------------------------------------------------------------------\n? L117: “ICWs” --> “ICW signals” or similar\nResponse: Done as you suggested. Please see line 135.\n-------------------------------------------------------------------------------------------------------\n? L167: “inner-mantle” is missing “core”\nResponse: Our apologies for this oversight. Given that the word “the mantle-inner core gravitational”\nappears many times, we chose to abbreviate it as “MICG” the first time it appears (please see line 178),\nso we use the abbreviation directly here (please see line 187).\n-------------------------------------------------------------------------------------------------------\nReferences\nBuffett, B. A. Gravitational oscillations in the length of day. Geophys Res Lett 23, 2279–2282 (1996).\nDumberry, M. & Mandea, M. Gravity Variations and Ground Deformations Resulting from Core\nDynamics. Surv Geophys 1–35 (2021) doi:10.1007/s10712-021-09656-2.\nGreiner-mai, H. & Barthelmes, F. Relative wobble of the Earth’s inner core derived from polar motion\nand associated gravity variations. Geophys J Int 144, 27–36 (2001).\nTriana, S. A. et al. Core Eigenmodes and their Impact on the Earth’s Rotation. Surv Geophys 43, 107–\n148 (2022).\nREVIEWERS' COMMENTS\nReviewer #2 (Remarks to the Author):\nThe authors have answered properly my concerns and met my suggestions. I’ve also seen the changes\nmade as a result of the other reviews. In general, I am satisfied with the revision and think that the\nobservation of the ICW free mode of the earth rotation is very relevant and that the current version is\nsuitable for publication.\nI have still a few minor suggestions and I’d like to be sure that I’ve understood some points properly.\nThey are:\n1. The paper says nothing about the observed ICW oscillation being steady and showing no decay.\nThis must be so because the ICW frequency must be real according to theory, but I’m afraid the\ncontact of most people with free modes limits to the two ones known so far, CW and FCN, and\nboth experience dissipation. I also think the property I said is not quite popular, since I have\nseen it published only in the proceedings of an IAU Colloquium. The series of papers coauthored\nby Mathews in 1991 and 2002 do not contain any theoretical approach to the topic of ICW\ndissipation nor give any estimation of its Q factor (they never even suggest a 0 value). I think it is\na further evidence supporting you found the ICW signal, please consider it. The paper I referred\nto is coauthored by me, and I think it’s not fair to point at my own papers in a review. Please\ndon’t feel you have to comment this point at all. It can be accessed at\nA Escapa, J Getino, JM Ferrandiz (2000) Free frequencies for a three layered Earth model International\nAstronomical Union Colloquium 178, 481-485. https://www.cambridge.org/core/services/aop-\ncambridge-\ncore/content/view/7BA4FC87472F405EDF0C8A848A1E9AE7/S0252921100061637a.pdf/free_frequencie\ns_for_a_three_layered_earth_model.pdf\n2. I’m not sure if I understand properly the meaning of figure 4b. I wonder if it is intended to\ndisplay the location of the torque in a generic case, of a special position of it, namely an\nextreme. My reasoning for asking this is the following: the torque equatorial projection must\nrotate keeping perpendicular to the equatorial angular velocity that describes a circumference\nunder the ICW oscillation, so that it cannot be fixed in the equator. However, if the SIC axis lies\nin the meridian plane containing 90W, 90E, when PM_ICW is in the perpendicular (containing\nthe 0 meridian) y-PM vanishes and dLOD/dt also because you report its observed variation at\nthe ICW period is in phase with y-PM. Therefore, when PM_ICW is in the 90W-90E plane\ndLOD/dt will be maximum/minimum and the z-torque too because of the pi/2 phase shift. In\nsuch a case, the bars applied to the torques should mean “extreme”, but I haven’t seen it in the\ntext. Please, explain me if this is true or I missed or misunderstood something.\nWhat I have difficulties to fully understand is why the 3D-torque must lie in the SIC equator and not\nin another plane passing by the same line of nodes, please clarify it further to me.\nI take this opportunity to say that I liked the related explanation of the R1 version (lines 159-166)\nbetter than the new R2 one (177-186).\nReviewer #3 (Remarks to the Author):\nThank you for your detailed and lengthy response, and for making the changes to the manuscript. Your\nreasoning and notation are now much clearer. The determination of the IC static tilt and ICB density\njump will be of interest to the deep Earth research community.\nI have only a few minor comments on the latest version.\nline 21: \"flat solid Earth\", the word flat here is confusing. Do you mean \"laterally homogeneous\"?\nline 43: \"is proposed to coincide\" -> \"was proposed to coincide\" as this is reporting on a previous study.\nFigure 2: Thank you for adding the grey bar to the figure to indicate the level required to rise above\nbackground noise. I think that the meaning of this region should be included in the caption.\nLine 177: \"forms a tilted\" might be better as \"results in a tilted\"\nLine 178: \"which has\" might be better as \"having\"\nLine 181: \"always in the plane\" -> \"is always in the plane\"\nLine 177-183: This is a single lengthy sentence. It might read better if it was broken into two or three.\nResponses to Reviewers\n###########################################################################\nReviewer #2 (Remarks to the Author):\nThe authors have answered properly my concerns and met my suggestions. I’ve also seen the changes\nmade as a result of the other reviews. In general, I am satisfied with the revision and think that the\nobservation of the ICW free mode of the earth rotation is very relevant and that the current version is\nsuitable for publication.\nI have still a few minor suggestions and I’d like to be sure that I’ve understood some points properly.\nThey are:\n1. The paper says nothing about the observed ICW oscillation being steady and showing no decay.\nThis must be so because the ICW frequency must be real according to theory, but I’m afraid the\ncontact of most people with free modes limits to the two ones known so far, CW and FCN, and\nboth experience dissipation. I also think the property I said is not quite popular, since I have\nseen it published only in the proceedings of an IAU Colloquium. The series of papers coauthored\nby Mathews in 1991 and 2002 do not contain any theoretical approach to the topic of ICW\ndissipation nor give any estimation of its Q factor (they never even suggest a 0 value). I think it is\na further evidence supporting you found the ICW signal, please consider it. The paper I referred\nto is coauthored by me, and I think it’s not fair to point at my own papers in a review. Please\ndon’t feel you have to comment this point at all. It can be accessed at\nA Escapa, J Getino, JM Ferrandiz (2000) Free frequencies for a three layered Earth model International\nAstronomical Union Colloquium 178, 481-\n485.https://www.cambridge.org/core/services/aop-cambridge-\ncore/content/view/7BA4FC87472F405EDF0C8A848A1E9AE7/S0252921100061637a.pdf/free_freq\nuencies_for_a_three_layered_earth_model.pdf\nResponse: Thanks for your support for our study and your kind reminder about the Q value of ICW.\nAlthough you think that we do not have to reply, we are very happy to see your reminder. As you\nmentioned, there is currently little research about the Q value of ICW, and the calculation of\n(theoretical and observational) Q value of ICW is another important work and needs further study. The\nliterature you provided is of great reference significance, and we will refer to it when we do relevant\nresearch about the Q of ICW in the future.\n-------------------------------------------------------------------------------------------------------\n2. I’m not sure if I understand properly the meaning of figure 4b. I wonder if it is intended to display\nthe location of the torque in a generic case, of a special position of it, namely an extreme. My reasoning\nfor asking this is the following: the torque equatorial projection must rotate keeping perpendicular to\nthe equatorial angular velocity that describes a circumference under the ICW oscillation, so that it\ncannot be fixed in the equator. However, if the SIC axis lies in the meridian plane containing 90W,\n90E, when PM_ICW is in the perpendicular (containing the 0 meridian) y-PM vanishes and dLOD/dt\nalso because you report its observed variation at the ICW period is in phase with y-PM. Therefore,\nwhen PM_ICW is in the 90W-90E plane dLOD/dt will be maximum/minimum and the z-torque too\nbecause of the pi/2 phase shift. In such a case, the bars applied to the torques should mean “extreme”,\nbut I haven’t seen it in the text. Please, explain me if this is true or I missed or misunderstood something.\nWhat I have difficulties to fully understand is why the 3D-torque must lie in the SIC equator and not\nin another plane passing by the same line of nodes, please clarify it further to me.\nI take this opportunity to say that I liked the related explanation of the R1 version (lines 159-166)\nbetter than the new R2 one (177-186).\nJose M Ferrandiz\nResponse: Thanks for your thoughtful review and insightful comments. You have got the message\nproperly. Here, we will further explain that the 3D-torque must lie in the SIC equator and not in another\nplane passing by the same line of nodes. Grey shaded plane represents the mantle's equatorial plane,\nand it is perpendicular to the figure (or rotation) axis of the mantle. Blue shaded plane represents the\ninner core's equatorial plane, and it is perpendicular to the rotation axis (or mean figure axis) of the\ninner core. The mean figure axis of the inner core should be aligned with the internal axis of the mantle\n(or the mean figure axis of the lower mantle). This is because the inner core and lower mantle are\ncloser and therefore they have a stronger gravitational pull. Because of the density imbalance or\nasymmetry in the Earth's interior, the mantle results in a tilted internal axis relative to its rotation axis,\nand the mantle-inner core system will have the lowest gravitational potential energy when the mean\nfigure axis of the inner core is aligned with the tilted internal axis of the mantle. When the inner core\nwobbles and deviates from this tilted axis, the MICG torque forces the inner core back to this tilted\naxis. Thus, MICG torque is in a plane perpendicular to this tilted axis, i.e., MICG torque is in the blue-\nshaded plane rather than the grey shaded plane. The internal axis of the heterogeneous mantle is\ninnovative, and we have developed a detailed mathematical formulation after discussions with\nBenjamin F. Chao. Considering your comments and the suggestions of Reviewer# 3, we have made\nsimple changes, please see lines 158-165.\n-------------------------------------------------------------------------------------------------------\n###########################################################################\nReviewer #3 (Remarks to the Author):\nThank you for your detailed and lengthy response, and for making the changes to the manuscript. Your\nreasoning and notation are now much clearer. The determination of the IC static tilt and ICB density\njump will be of interest to the deep Earth research community.\nResponse: Thank you very much for your recognition of our work and positive assessment. We have\ncarefully considered your suggestion and revised this MS according to the comments, and hope that\nthe revised version is more suitable.\n-------------------------------------------------------------------------------------------------------\nI have only a few minor comments on the latest version.\nline 21: \"flat solid Earth\", the word flat here is confusing. Do you mean \"laterally homogeneous\"?\nResponse: Thank you very much for your kind reminder, we think that “oblate” is a more appropriate\nexpression, please see line 21.\n-------------------------------------------------------------------------------------------------------\nline 43: \"is proposed to coincide\" -> \"was proposed to coincide\" as this is reporting on a previous study.\nResponse: Thank you for your kind reminder, we have changed it in the new MS, please see line 43.\n-------------------------------------------------------------------------------------------------------\nFigure 2: Thank you for adding the grey bar to the figure to indicate the level required to rise above\nbackground noise. I think that the meaning of this region should be included in the caption.\nResponse: Thank you for your kind reminder, we have added a note about this according to your\nsuggestions, please see line 464.\n-------------------------------------------------------------------------------------------------------\nLine 177: \"forms a tilted\" might be better as \"results in a tilted\"\nResponse: Done as you suggested. Please see line 158.\n-------------------------------------------------------------------------------------------------------\nLine 178: \"which has\" might be better as \"having\"\nResponse: Thanks for your suggestions, we have changed this long sentence into several clauses\naccording to your last suggestion. Please see line 159.\n-------------------------------------------------------------------------------------------------------\nLine 181: \"always in the plane\" -> \"is always in the plane\"\nResponse: Done as you suggested. Please see line 163.\n-------------------------------------------------------------------------------------------------------\nLine 177-183: This is a single lengthy sentence. It might read better if it was broken into two or three.\nResponse: Thank you for the suggestions to make it more readable, we changed it as you suggested,\nplease see lines 158-165."}]

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