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

The study addresses whether Earth’s solid inner core is statically tilted relative to the mantle and, if so, by how much and in what direction. In classical Earth rotation theory, figure and rotation axes of mantle and inner core align to maintain hydrostatic equilibrium. Random torques cause a dynamic tilt that excites the inner core wobble (ICW), expected to appear in polar motion (PM) with a period sensitive to the density jump at the inner core boundary, theoretically 6.6–7.8 years. Heterogeneity in the solid mantle could introduce a static tilt between the inner core and mantle axes, which would make the ICW manifest in both PM and axial length-of-day variations (ALOD). Previous suggestions of a ~10° tilt aligned with the geomagnetic dipole lacked observational confirmation of the ICW and are not widely accepted. The authors propose that detecting a common ~8.5-year ICW signal in both PM and ALOD, and examining its phase and amplitude relations, can diagnose the presence, direction, and magnitude of a static tilt, which has broad implications for inner core dynamics, gravity changes, deep Earth seismology, and geodynamo theory.

Theoretical work predicts the ICW period depends sensitively on the density jump at the inner core boundary (ICB), with predictions of ~6.6–7.8 years based on PREM. Prior efforts proposed a large (~10°) static tilt to explain decadal oscillations in PM and ALOD and potential alignment with the geomagnetic dipole axis. However, confirmed observational evidence for ICW has been lacking, and core-sensitive normal mode eigenfrequencies show no clear deviations expected from a large static tilt, suggesting any static tilt is small. Other decadal signals in Earth rotation have been attributed to atmospheric and oceanic excitations, torsional oscillations, MAC waves, and solar cycles. Earlier spectral analyses identified candidate ~-8.7-year prograde signals in PM, but comprehensive treatment of external excitations and corroboration in ALOD remained necessary.

- Data: Annual ALOD series spanning 1900–2020; PM (x, y) from EOPC01 with annual sampling 1900–2020. For spectral identification, 1949–2020 records were also used. External excitations obtained from atmospheric angular momentum (AAM; NCEP/NCAR reanalysis) and oceanic angular momentum (OAM; ECCO solutions). Hydrological excitations were not modeled due to their negligible contribution in the 5.5–10-year band and model discrepancies. - Preprocessing: All datasets were down-sampled to 1-year intervals after applying a low-pass filter with a 0.5 cpy cut-off to prevent aliasing. Small long-period zonal tides at ~8.85 and 9.3 years, though theoretically only microsecond-level in ALOD, were removed based on a model to avoid interference in the ~8-year band. dALOD/dt was computed via finite differences. - Spectral analysis: Applied the stabilized AR-z spectrum to PM and ALOD (1949–2020) after removing AO (AAM+OAM) effects to identify harmonic content. Prograde/retrograde discrimination used the sign of spectral frequency in complex spectra; ICW, as a prograde wobble, should appear only at positive frequencies. - Signal extraction: For higher resolution, the ~8.5-year component was extracted from the 1900–2020 PM and ALOD using cosine least-squares fitting. Additionally, the normal time-frequency transform (NTFT) was used to verify phase relations. - Phase analysis: Compared the phases of the y-component of PM and dALOD/dt for the ~8.5-year signal to infer tilt direction in the equatorial plane. - Torque and tilt estimation: Using mantle–inner core gravitational (MICG) coupling theory, the equatorial torque on the mantle was related to the ICW excitation in PM via τ_MICG = (C_m − A_m) X_ICW(t). The axial torque was related to ALOD via τ^z_MICG = C_m Ω_m d(ALOD_ICW)/dt. From observed amplitudes (PM ICW amplitude 4.7 ± 0.4 mas; ALOD_ICW amplitude 0.061 ± 0.007 ms with derivative amplitude 0.046 ± 0.005 ms/yr and period 8.5 yr), equatorial and axial torques were computed, and the static tilt angle θ was obtained from arctan(τ^z/τ^x). - Uncertainty: Period and amplitude uncertainties were derived via bootstrap procedures. - Density jump inversion: Using an ICW frequency formulation with elastic compliances and coupling constants (based on PREM as reference) and conserving Earth’s mass and angular momentum, the outer and inner core density profiles were adjusted to match the observed 8.5-year ICW period, yielding an estimate for Δρ_ICB.

- A robust ~8.5-year prograde signal is detected in PM after removing atmospheric and oceanic excitations and appears only at positive frequency in the AR-z spectra (period 8.52 ± 0.19 years), uniquely identifying it as the inner core wobble (ICW). - The same ~8.5-year signal is detected in ALOD (8.47 ± 0.32 years), indicating that the ICW manifests in both PM and ALOD, which implies a static tilt between the inner core and mantle. - Phase synchrony: The y-component of PM and dALOD/dt exhibit nearly synchronous phases for the ICW, indicating the tilt direction is approximately along −90°W. - Amplitudes and torques: PM ICW amplitude is 4.7 ± 0.4 mas, yielding an equatorial MICG torque τ_MICG = (2.87 ± 0.24) × 10^19 Nm. ALOD_ICW amplitude 0.061 ± 0.007 ms (derivative amplitude 0.046 ± 0.005 ms/yr) with 8.5-year period gives axial torque τ^z_MICG = (8.61 ± 0.95) × 10^16 Nm. - Static tilt magnitude: θ = arctan(τ^z/τ^x) = 0.17 ± 0.03°, most likely toward −90°W relative to the mantle, two orders of magnitude smaller than the previously assumed 10°. - Density structure: The observed ICW period implies a density jump at the ICB of Δρ_ICB = 0.52 ± 0.05 g/cm³ (vs. PREM’s 0.598 g/cm³). - Geodynamic implication: The results are consistent with higher average density in the northwestern hemisphere of the inner core and suggest the eastward differential rotation rate of the inner core is likely much less than 1°/yr.

Identifying the ICW unequivocally in PM and ALOD and establishing their phase relationship resolves a key question regarding the existence of a static tilt between the inner core and mantle. The prograde ~8.5-year signal’s presence in both observables, after removing major external atmospheric and oceanic excitations, directly supports a statically tilted inner core. The near-synchronous phase between the PM y-component and dALOD/dt constrains the tilt direction to approximately −90°W. Quantitative torque estimates from the observed amplitudes yield a small tilt angle (0.17°), addressing prior debates that posited a much larger (~10°) tilt. The inferred tilt aligns with independent seismological indications of a denser western/northwestern inner core hemisphere, providing cross-disciplinary consistency. The ICW period slightly exceeds theoretical predictions, but given uncertainties in the ICB density jump and analogies to other free rotational modes (e.g., Chandler wobble and free core nutation discrepancies), this is plausible. Moreover, the derived Δρ_ICB refines interior density constraints. Overall, the findings substantiate a small but nonzero static tilt that modifies both equatorial and axial angular momentum exchanges between mantle and inner core, with implications for models of inner core dynamics, gravity changes, and the geodynamo.

The study provides the first experimental confirmation that the ~8.5-year signal in Earth’s rotation is the inner core wobble and demonstrates that it appears in both polar motion and length-of-day variations, requiring a static tilt of the inner core relative to the mantle. The tilt is small (0.17 ± 0.03°) and directed toward −90°W, consistent with a denser northwestern inner core hemisphere. The observed ICW period leads to an estimated ICB density jump of 0.52 ± 0.05 g/cm³. These results refine our understanding of inner core–mantle coupling and place new constraints on core density structure and possible differential rotation. Future work could further constrain Δρ_ICB with improved datasets and seek independent seismological or normal-mode signatures of such a small static tilt.

- Static tilt magnitude is inferred indirectly from rotational data and torque models; direct seismological detection of such a small tilt is currently difficult. - Hydrological excitations were not modeled due to uncertainties, though prior work suggests negligible impact in the 5.5–10-year band. - The spectral identification relies on the stabilized AR-z method and preprocessing choices (low-pass filtering, annual down-sampling), which may influence resolution and noise characteristics. - Small long-period tidal contributions in ALOD were removed based on models despite being below noise levels; residual model errors could persist. - The observed ICW period slightly exceeds theoretical predictions; interpretation depends on uncertain parameters, notably the ICB density jump and coupling constants within the PREM-based framework. - No clear deviations in core-sensitive normal mode eigenfrequencies have been observed to corroborate the static tilt, implying it is small or its modal signatures are below detection thresholds.