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′m 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′m2,3,4 (also represents the direction of the lowest gravitational potential energy of the mantle-inner core system). The above ‘tilt’ between the Ωic and Ω′m is a generally dynamic tilt, and in this case, the ICW theoretically appears only in the polar motion (PM) of the Earth’s rotation2,3,5. In addition, the ICW period is very sensitive to the density jump ΔρICB at the inner core boundary (ICB; ΔρICB 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 yr3,7,8,9. 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 theory6,12,14,16,17,18,19,20.

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 and discussion

Inner core wobble in the polar motion and the length-of-day variations

In this study, we report the results from the ΔLOD and PM time series. The chosen ΔLOD time series is a yearly time series with a 1900–2020 time span. For the PM time series, the 1900–2020 EOPC01 time series 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 sources21,22,23; 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.

Fig. 1: The used polar motion (PM) and length-of-day variations (ΔLOD) records.
figure 1

The x component (a) and y component (b) of the observed PM records from 1900 to 2020 (gray curves) and the PM excited by the atmospheric and oceanic angular momentum (AAM + OAM, abbreviated as AO: black curve) from 1949 to 2020; c the observed ΔLOD record from 1900 to 2020 (gray curves) and the ΔLOD excited by the AAM + OAM (AO: black curve) from 1949 to 2020.

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Different from the oceanic tidal signals that have both prograde and retrograde components in PM25,26,27, 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 feature28,29,30. 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 time span, 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 controversial31,32,33; 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 ICW3,7,8,9. 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.

Fig. 2: Normalized AR-z spectra of the used polar motion (PM) and length-of-day variations (ΔLOD) records (1949–2020).
figure 2

The AR-z spectra of the PM (a) and ΔLOD (b) records in which the atmospheric and oceanic angular momentum effects were removed. Different colored rectangles indicate different periodic signals; the gray area represents 95% confidence intervals (C.I.).

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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 confirms 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.

Static tilt between the inner core and mantle

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 Гz (∝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 signals 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 Гz reaches its peak/trough only when the Гz 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.

Fig. 3: The least-square fitted ~8.5 yr signals from different records.
figure 3

From (a) the first-order time derivative of length-of-day variations (dΔLOD/dt); (b) the x component of the polar motion (PM), and (c) the y component of the PM. The time span used is 1900–2020, and the sampling interval is one year.

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Fig. 4: Schematic depiction of the tilted inner core wobble (ICW).
figure 4

The figure axis of the inner core wobbles about its rotation axis tilted to the mantle in a small circular orbit (see blue shaded area). Ωm is the rotation axis of the mantle, Ω′m and Ωic are respectively the rotation axis and dynamic figure axis of the inner core, and θ is the static tilt angle between Ωm and Ω′m. The upper right subgraph shows how to get θ from the mantle-inner core gravitational torque in the mantle’s axial and equatorial directions \(|{{{{{{\boldsymbol{\Gamma }}}}}}}_{z}^{{{{{{\rm{MICG}}}}}}}|\) and \(|{{{{{{\boldsymbol{\Gamma }}}}}}}_{{{{{{\rm{eq}}}}}}}^{{{{{{\rm{MICG}}}}}}}|\) (see also in Methods).

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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 average 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 brings 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):

$$\left|{{{{{{\boldsymbol{\Gamma }}}}}}}_{{{{{{\rm{eq}}}}}}}^{{{{{{\rm{MICG}}}}}}}\right|=({C}_{m}-{A}_{m}){\varOmega }_{0}^{2}|{{{{{{\boldsymbol{\chi }}}}}}}_{{{{{{\rm{ICW}}}}}}}(t)|$$
(1)

where \({{{{{{\boldsymbol{\chi }}}}}}}_{{{{{{\rm{ICW}}}}}}}(t)\) 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 \(|{{{{{{\boldsymbol{\Gamma }}}}}}}_{{{{{{\rm{eq}}}}}}}^{{{{{{\rm{MICG}}}}}}}|\) 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):

$$\left|{{{{{{\boldsymbol{\Gamma }}}}}}}_{z}^{{{{{{\rm{MICG}}}}}}}\right|=\frac{{\varOmega }_{0}{C}_{m}}{{{{{{\rm{LOD}}}}}}}\frac{d(\varDelta {{{{{{\rm{LOD}}}}}}}_{{{{{{\rm{ICW}}}}}}})}{dt}$$
(2)

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 \(|{{{{{{\boldsymbol{\Gamma }}}}}}}_{z}^{{{{{{\rm{MICG}}}}}}}|\) 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(\(|{{{{{{\boldsymbol{\Gamma }}}}}}}_{z}^{{{{{{\rm{MICG}}}}}}}|/|{{{{{{\boldsymbol{\Gamma }}}}}}}_{{{{{{\rm{eq}}}}}}}^{{{{{{\rm{MICG}}}}}}}|\)) (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)3,7,8,9, but considering that even the generally accepted Chandler wobble observation period of prograde ~430 days is ~30 days longer than its theoretical periods28,29,30, 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 denser42,43,44,45,46,47. 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:

$$\frac{d}{dt}{{{{{{\bf{H}}}}}}}_{m}+{{{{{\boldsymbol{\Omega }}}}}}\times {{{{{{\bf{H}}}}}}}_{m}={{{{{{\boldsymbol{\Gamma }}}}}}}_{m}$$
(3)

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

$${{{{{{\bf{H}}}}}}}_{m}={{{{{{\bf{I}}}}}}}_{m}{{{{{\boldsymbol{\Omega }}}}}}$$
(4)

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:

$${{{{{{\bf{I}}}}}}}_{m}=\left[\begin{array}{ccc}{A}_{m} & 0 & 0\\ 0 & {A}_{m} & 0\\ 0 & 0 & {C}_{m}\end{array}\right]$$
(5)

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

$$\frac{i}{{\sigma }_{{{{{{\rm{CW}}}}}}}}\frac{d{{{{{\bf{m}}}}}}}{dt}+{{{{{\bf{m}}}}}}={{{{{\boldsymbol{\chi }}}}}}(t)=\frac{{{{{{{\boldsymbol{\Gamma }}}}}}}_{{{{{{\rm{eq}}}}}}}}{i({C}_{m}-{A}_{m}){{\varOmega }_{0}}^{2}}$$
(6)

where Γeq = Γ1 + iΓ2 is the equatorial torque exerted on the mantle; Ω0 = 7.29212 × 10−5 s−1 is the mean (sidereal) rotation rate9; Am = 7.0999 × 1037 kg·m2 and Cm = 7.1236 × 1037 kg·m2 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 σCW = ωCW + iγCW, ωCW = 2π·0.843 cpy and γCW = ωCW/2QCW; QCW ≈ 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:

$${\varOmega }_{0}{C}_{m}\frac{d{m}_{3}}{dt}={{{{{{\boldsymbol{\Gamma }}}}}}}_{z}$$
(7)

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):

$$x(n)=\mathop{\sum }\limits_{j=1}^{M}\left[{{{{{{\bf{A}}}}}}}_{j}\exp (in{\sigma }_{j})+{{{{{{\bf{A}}}}}}}_{j}^{\ast }\exp (-in{\sigma }_{j}^{\ast })\right],n=1,2,3,\ldots,N$$
(8)

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:

$$P({\sigma }_{i})=\frac{1}{{|\exp ({\tilde{\sigma }}_{i})-\exp (i{\sigma }_{i})|}^{2}},\,i=1,2,\ldots,N$$
(9)

where \({\tilde{\sigma }}_{i}\) 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:

$${\sigma }_{{{{{{\rm{ICW}}}}}}}=[{\alpha }_{3}(1+{\alpha }_{g})({e}_{s}+{S}_{34}^{g}+{S}_{34}^{p})]/(1+{K}^{{{{{{\rm{ICB}}}}}}})$$
(10)

where the elastic compliances \({S}_{34}^{g}\)= –1.812 × 10–6, \({S}_{34}^{p}\)= –2.686×10–4 and KICB is a dimensionless coupling constant55 and Real(KICB) = 1.11 × 10–3; α3 and αg have the following forms:

$$\left\{\begin{array}{c}{\alpha }_{3}=1-(A^{\prime} e^{\prime} /{A}_{s}{e}_{s}){\alpha }_{g}\\ {\alpha }_{g}=\frac{3G}{{a}_{s}^{5}{\varOmega }^{2}}\left(\left[\frac{5\bar{\rho }}{3{\rho }_{f}}+1\right]A^{\prime} e^{\prime} -{A}_{s}{e}_{s}\right)-1\end{array}\right.$$
(11)

where ρf is the fluid density just outside the ICB, \(\bar{\rho }\) is the mean density of the inner core, As and es 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.

$$\left\{\begin{array}{c}{A}_{s}=\frac{8\pi }{3}{\int }_{0}^{{a}_{s}}\rho (r)\{{r}^{4}-\frac{1}{15}\frac{d[\varepsilon (r)\cdot {r}^{5}]}{dr}\}dr\\ A^{\prime} e^{\prime}=\frac{8\pi }{15}{\rho }_{f}{a}_{s}^{5}{\varepsilon }_{s}\\ A^{\prime} (1+\frac{e^{\prime} }{3})=\frac{8\pi }{15}{\rho }_{f}{a}_{s}^{5}\end{array}\right.$$
(12)

in which ɛ(r) is the geometrical ellipticity of the Earth and as 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 Atmosphere58,59,60. 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_50 yr61 (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(ti)/dt = [ΔLOD(ti+1) − ΔLOD(ti)]/Δt.