Earth's inner core is widely held to rotate at a slightly different rate than the mantle ('super-rotation'), and because the inner core's figure axis is slightly tilted, this rotation should make the external gravity field at any fixed surface point vary with time. Chen, Shen and Han (2008) computed this signal: for a steady super-rotation of 0.27–0.53°/yr, the global annual gravity change is only of order 10−310^{-3} μGal — below all current gravimetry — while the local signal at Beijing and Wuhan varies like a sine with a maximum of roughly 0.3 μGal per cycle, leading them to suggest that continuous gravity measurements over one or two decades might detect the inner core's differential motion. Their specific question has not been taken up: the seismological premise the paper relied on has since been transformed — the rotation now appears time-variable and possibly reversing, with concurrent inner-core shape change. Later gravimetric and geodetic work has advanced neighboring targets — satellite-gravity analyses have placed upper bounds on oscillatory inner-core rotation and quantified the noise that masks deep-Earth gravity signals, and Earth-rotation data have been used to infer the inner core's static tilt — but no study has recomputed the super-rotation gravity signature under the new rotation history, assessed its detectability in surface or satellite gravimetry against realistic noise, or built the combined-observable inversion. The open problem is therefore to determine, using a forward model that includes the inner core's other motions and its coupling to the mantle and outer core, whether the gravity signature of inner-core differential rotation is separable from environmental and instrumental noise in existing and foreseeable gravimetry, and to develop and validate an inversion that recovers inner-core motion from combined geophysical observables.

Background

Earth's solid inner core, a sphere of iron alloy about 1220 km in radius discovered seismologically in 1936, floats within the convecting, electrically conducting liquid outer core that generates the geomagnetic field. Self-consistent geodynamo simulations — computer models of the magnetic-field-generating flow in the liquid core (Glatzmaier and Roberts, 1996) — predicted that electromagnetic torques spin up the inner core so that it rotates slightly faster than the mantle — a differential rotation, commonly called 'super-rotation'. Song and Richards (1996) sought this effect seismologically: waves that graze the bottom of the outer core (seismic phase PKP(BC)) serve as a reference against waves that pass through the inner core (PKP(DF)), so that differential travel times between earthquake pairs recorded at the same station isolate changes in inner-core structure. Their finding of systematic travel-time changes implied an eastward super-rotation of order 1°/yr, and subsequent analyses of repeated earthquake 'doublets' (nearly identical earthquake pairs recorded at the same station) revised the estimate to a few tenths of a degree per year (Zhang et al., 2005). Because super-rotation is an angular-momentum exchange between the core and the mantle, it also leaves geodetic fingerprints: geodynamo-based comparisons of geodetic and seismic rotation estimates (Buffett and Creager, 1999), decade-scale fluctuations of the length of day (LOD, the small time-variations of Earth's rotation period), and polar motion (the wander of the rotation axis over the surface) have all been used to constrain inner-core dynamics.

A further, less explored consequence is a time-variable gravity field. The inner core is a stratified, slightly flattened ellipsoid whose density structure is usually taken from PREM (the Preliminary Reference Earth Model, the standard seismological one-dimensional Earth model). Its figure axis (the symmetry axis of its slightly ellipsoidal mass distribution) is tilted slightly with respect to the mantle-fixed frame. If the inner core super-rotates, this tilted mass asymmetry is carried around relative to the mantle, so the external gravitational acceleration at a fixed surface point varies quasi-periodically with the rotation period. The gravity unit used here is the μGal (10−8 m s−210^{-8}\,\mathrm{m\,s^{-2}}, about one billionth of normal gravity); the most precise continuous instruments are superconducting gravimeters (SGs), cryogenic relative gravimeters whose precision reaches nanogal levels at periods of minutes to hours but whose useful long-period accuracy is limited by instrumental drift and by unmodeled environmental loading (groundwater, atmosphere, oceans), which reaches μGal levels at annual-to-decadal periods.

Chen, Shen and Han (2008) improved earlier forward models of this effect (their group had already proposed testing inner-core super-rotation with gravity variations in Shen and Ning, 2006). Assuming a steady super-rotation of 0.27–0.53°/yr, a fixed small tilt, and a PREM-fitted density ellipsoid, they found that the global temporal variation over the year 2007 has a maximum of about 0.79–1.54×10−3\times 10^{-3} μGal — explicitly beyond the accuracy of contemporary gravimetry and even SG data — while at Beijing and Wuhan the gravity varies like a sine with maxima of about 0.33 and 0.29 μGal per cycle, prompting their suggestion that continuous measurements over one or two decades might detect the inner core's differential motion. They themselves listed the unresolved issues: the practical measurability of the signal; the neglected inner-core wobble (a free oscillation in which the inner core's figure axis moves relative to the mantle's rotation axis), nutation (small periodic deviations of Earth's rotation axis), oscillation and time-varying tilt, and the mechanical coupling of the inner core to the mantle and outer core; the sensitivity of the prediction to the observationally contested rotation rate and tilt; and the absence of an inversion strategy that could recover inner-core motion from combined observables (gravity, polar motion, LOD, seismic doublets).

Since 2008, the seismological premise of a steady super-rotation has itself been upended. Hemispherical asymmetry of the inner core was reconciled with a much slower rotation (Waszek, Irving and Deuss, 2011); the rotation interpretation itself was contested, with temporal changes of the inner-core surface proposed as an alternative (Yao et al., 2019); waveform doublets revealed multidecadal variation including a pause around 2010 (Yang and Song, 2023) and then backtracking — sub-rotation followed by reversal — on a sub-decadal timescale (Wang et al., 2024); and the most recent analysis concludes that the inner core changes shape as well as rotation rate, since seismic recordings differ even when the core has returned to the same orientation (Vidale et al., 2025). In parallel, adjacent work has quantified gravity variations from inner-core tilt (Dumberry, 2008), searched geodetic records for the inner-core wobble (Greiner-Mai and Barthelmes, 2001; Guo et al., 2005; Rosat et al., 2016), and pursued superconducting-gravimeter detections of the Slichter triplet — the three translational oscillation modes of the inner core (Shen and Ding, 2013) — while the theory of mantle–inner-core gravitational coupling and locking continues to be revised (Dumberry and Mound, 2010; Chao, 2017; Dumberry, 2025). Closest to the question itself, satellite gravimetry has recently been turned on inner-core motion: Lecomte, Rosat, Mandea and Dumberry (2023a) reconstructed the time-variable degree-2, order-2 Stokes coefficient (a coefficient of the spherical-harmonic expansion of Earth's gravity field) from Satellite Laser Ranging and the GRACE and GRACE Follow-On missions, and used the gravity variation produced when inner-core rotation carries its degree-2 topography out of gravitational alignment with the mantle to place upper bounds on an oscillating rotation angle at periods of about 4, 6 and 12 years — bounds they themselves note are overestimates because the reconstructed signal remains polluted by hydrology — while a companion study quantified how much uncertainty the surface-process corrections add to any core-origin gravity signal (Lecomte et al., 2023b). The source paper's own group, using Earth-rotation observations, confirmed an about 8.5-year inner-core wobble in polar motion and length-of-day data and inferred a static inner-core tilt of about 0.17° (An et al., 2023), and a 2024 preprint by that group reports a satellite-gravity detection of an about six-year inner-core oscillation correlated with the length-of-day signal (An et al., 2024, arXiv preprint). None of this, however, has addressed the specific question left open by the 2008 study: whether the gravity signature of the secular differential rotation itself — the tilted-ellipsoid signal computed by Chen, Shen and Han — can be detected, attributed, and inverted.

Problem Statement

Determine whether the time-variable gravity field produced by the differential rotation of the tilted inner core can be detected and unambiguously attributed to that rotation in continuous gravimetric records, and whether inner-core motion can be recovered from combined geophysical observables. Concretely, this requires: (1) a forward model that extends the Chen–Shen–Han (2008) computation to a time-variable (possibly reversing) inner-core rotation rate with evolving tilt, and that includes the inner core's other motions — wobble, nutation, oscillation — together with its gravitational, electromagnetic and viscous coupling to the mantle and outer core; (2) a quantitative detectability analysis comparing the predicted signal (from about 10−310^{-3} μGal annual global changes up to roughly 0.3 μGal site-level cycle amplitudes under the 2008 assumptions) against realistic long-period gravimetric noise — hydrological, atmospheric and ocean-loading effects and instrumental drift — for current superconducting-gravimeter networks, absolute gravimeters, satellite gravimetry (the GRACE and GRACE Follow-On missions and satellite laser ranging), and foreseeable instruments, stating which observing strategies could separate the secular super-rotation signature from that noise and from the oscillatory inner-core rotation signals at roughly 4-to-12-year periods already bounded in satellite gravity (Lecomte et al., 2023a); and (3) a validated inversion strategy that recovers inner-core motion parameters from combined observables (gravity, polar motion, length-of-day, and seismic doublet constraints), with the sensitivity of the recovered motion to the observationally contested rotation-rate and tilt values quantified. The problem is answered either by a demonstrated detection with attribution, or by a definitive detectability bound together with the conditions (instrument class, record length, station distribution, model completeness) under which the signature becomes accessible. The forward model, the detectability analysis and the validated inversion are jointly posed: separating the differential-rotation signature from the long-period noise to attribute it, and recovering the inner-core motion from the combined observables, are established through all three parts together.

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