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

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 flat 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 model 1,2 .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′m [2][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 rotation 2,3,5 .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 system 10 ), 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 variation (ΔLOD) as the possible ICW, the inner core's rotation axis Ω′m is 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 ICW 12,13 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 ICW 12 .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 oscillation 14,15 , 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 theory 6,12,14,[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 ΔLOD 11 .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).

The ~8.5yr ICW signal appearing in the PM and ΔLOD
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 momentum 21 .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 sources [21][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 5.5-10yr period band 13,23 and different hydrological models have clear deviations 13,24 .Hence, similar to previous studies 13,22 , 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 Figure 1.Different from the oceanic tidal signals that have both prograde and retrograde components in PM [25][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 feature [28][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 study 13 used this feature to identify an ~8.7yr 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 (~5.9yr, ~7.3yr, ~8.5yr, and ~9-11yr) in the positive frequency axis; only the ~8.5yr 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 Figure .S2b in the Supplementary Information).Among these harmonics, the ~5.9yr signal has been suggested as the inner core oscillation coupled with torsional wave in the Earth's core but still remains controversial 31- ; the ~7.3yr signal can be interpreted as the Magneto-Coriolis eigenmode in the Earth's core based on a theoretical model 34 ; the peak in the ~9-11yr is possible from the ~11yr Schwabe solar cycle.The adjacent ~13yr signal (Figure 2a) also has both positive and negative frequencies; similar period was also found in the geomagnetic dipole field 35 , but the underlying mechanism is still enigmatic; the ~18-23yr spectral peak may be mainly caused by the 18.6yr tidal signal and the ~22yr Hale solar cycle or high-latitude MAC (magnetic-Archimedes-Coriolis forces) wave 36 in the Earth's core; those periods are too long to be the ICW 3,[7][8][9] .Therefore, the 8.52±0.19yrsignal is the only candidate for the ICW.
Since no other mechanism has been proposed to account for such a prograde ~8.5yr motion, and the AO effects have been removed, we can conclude that the 8.5yr signal is the ICW.In addition, the uncertainties for the estimates in this study were based on a bootstrap procedure 37 .
Comparing Figures 2a and 2b, 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 signals 38 ; the Fourier spectra can only identify parts of those signals (see Figure S2).These consistencies deserve further attention, but we only focus on the 8.5yr signal (the ICW signal).Figure 2b confirms that the ICW signal is also present in the ΔLOD (with an 8.47±0.32yrperiod); 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 ~8.5yr 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 ~8.5yr signal from dΔLOD/dt; the fitted results from ΔLOD can be found in Figure 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 Figure 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 Figure 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 timefrequency transform, NTFT 39 ) show almost the same results (see Figure S3).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 Figure 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.Given the y component of the PM along the 90°W longitude, the phase synchronization in Figures 3a and 3c indicates that the inner core is more likely tilted in the 90°W direction, which is also similar to that suggested by previous studies 11,12,40,41 .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 nonaxisymmetric 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 observations 42,43 , a previous dynamical model considers the crystallization and melting at the surface of the inner core and has similar suggestions 44 , 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 denser 45 , and a thicker compacting layer at the top of the inner core's western hemisphere was also suggested 46 ; a more nuanced research found that the western zone is largely confined to the northern hemisphere 47 ; 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 Figure 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 Figure S3).Therefore, there will be a good phase consistency between the dΔLOD/dt and the y component of the PM for the ICW (confirmed in Figure 3).Next, we estimate the tilt θ.The heterogeneous mantle forms a tilted rotation axis of the inner core Ω′m which has the lowest mantle-inner core gravitational (MICG) potential energy (the axis at the equilibrium state); the 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 which always brings the inner core back to the equilibrium state, that is, the torque ΓMICG always in the plane (blue plane in Figure 4) perpendicular to the inner core's rotation axis Ω′m; the plane is also 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): where ICW () t χ 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 frequency 21,48  According to the theorem of angular momentum, the axial torque exerted on the mantle is written as the following (see Methods): Substituting the amplitude of 0.046±0.005ms/yr (corresponding to the amplitude of 0.061±0.007ms of ΔLODICW) and the period of 8.5yr for the observed ICW signal in the ΔLOD into Equation ( 2), the axial torque MICG z Γ is calculated to be (8.61±0.95)×10 16N•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( MICG MICG eq / z Γ Γ ) (see Figure 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 (6.6-7.8yr) 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 periods [28][29][30] , free core nutation observation period of retrograde ~430 days is ~20 days shorter than its theoretical periods 49,50 , and that the density jump ΔρICB at the ICB was also poorly determined 6,51 , 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±0.05g/cm 3 (see Methods), which is smaller than that of the PREM model (0.598 g/cm 3 ).
In summary, based on the Earth's rotation observations (PM and ΔLOD), we experimentally confirmed for the first time that the 8.5yr 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 year 12,16 .Besides, the density jump of 0.52±0.05g/cm 3 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 denser [42][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.

Conservation of angular momentum of the mantle and inner core
Considering the mantle alone, the law of angular momentum can be rewritten as 48 : where Earth's angular velocity Ω= Ω0[m1, m2, 1+m3] T ; the mantle angular momentum is: 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: () () where Γeq= Γ1+iΓ2 is the equatorial torque exerted on the mantle; Ω0 = 7.29212×10 −5 s −1 is the mean (sidereal) rotation rate 9 ; Am= 7.0999×10 37 kg•m 2 and Cm= 7.1236×10 37 kg•m 2 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·0.843cpy 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: where m3= −ΔLOD/LOD; LOD =86400s.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 studies 11,12 .

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 relation 54 ): ( ) ( ) 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 estimated 38 .A Lorentzien power spectrum in the complex z plane can be formed as follows 38 : , exp exp where 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 Information.

Constraint for the density jump at the ICB
The frequency of the ICW can be written as follows (in cpsd: cycle per solar day) 3 : ( ) where the elastic compliances 34 g S = -1.812×10-6 , 34 p S = -2.686×10-4 and K ICB is a dimensionless coupling constant 55 and Real(K ICB )= 1.11×10 -3 ; α3 and αg have the following forms: where ρf is the fluid density just outside the ICB,  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 ICB 2 .
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 8.5yr period is obtained, the corresponding ΔρICB is the one we recommend using.
Note that although the theoretical amplitudes of the 8.85yr and 9.3yr zonal tides are quite small (only ~2 μs for the 9.3yr tide and less than 1 μs for the 8.85yr 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 model 63 to avoid the effects of some well-known signals on the target ~8yr 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.

Figure 1 .
Figure 1.The used PM and ΔLOD records.The x component (a) and y component (b) of the

Figure 2 .
Figure 2. Normalized AR-z spectra of the PM (a) and ΔLOD (b) records from the 1949-2020 time

Figure 3 .
Figure 3.The least-square fitted ~8.5 yr signals from different records.From (a) the dΔLOD/dt;

Figure 4 .
Figure 4. Schematic depiction of the tilted ICW.The figure axis of the inner core wobbles about its 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 eq 2 CW 0 (see Methods).Since the ICW identified in the PM has an amplitude of 4.7±0.4mas,we can calculateMICG eqΓto be (2.87±0.24)×10 19N•m.