Earth's inner core can undergo degree-one translational oscillations (Slichter modes) inside the liquid outer core, split by rotation into one polar and two equatorial modes, but the modes have never been unambiguously detected. Personnettaz, Cébron, Schaeffer, Deguen and Mandea (2026) used magnetohydrodynamic simulations to show that viscous damping in the outer core is negligible and that Ohmic (Joule) damping in the conducting liquid yields decay times of roughly 3–16 years. They explicitly leave open three coupled issues that together prevent a closed damping budget: (i) the Ohmic lifetime is only indicative because the radial magnetic-field strength and electrical conductivity at the inner-core boundary (ICB) are poorly constrained; (ii) anelastic dissipation in the solid inner core was not modeled, and if the ~420-day anelastic decay time estimated by Crossley, Hinderer and Legros (1991) is correct, anelasticity — not Ohmic dissipation — would dominate; and (iii) stable stratification of the liquid layer (Earth's outer core; Mercury's stratified layer or shallow subsurface oceans) was neglected although it can modify both periods and damping. Their model additionally imposes an axial magnetic field and thereby excludes any coupling of the oscillations to dynamo action and self-consistent core dynamics, but the paper states this as a scope limitation of its simulations, not as an open issue it poses. The open problem is to complete this dissipation budget: determine the dominant damping mechanism, its decay time with quantified uncertainty, and the associated period shifts for Earth's Slichter triplet and for translational modes in stratified planetary cores.

Background

The Slichter modes are the gravest free oscillations of the Earth's inner core: the solid inner core, being slightly denser than the liquid outer core surrounding it, can oscillate as a rigid body along a line through the center (a degree-one spheroidal motion). Coriolis forces from Earth's rotation split the oscillation into a triplet — one polar mode (displacement along the rotation axis) and two equatorial modes (prograde and retrograde motion in the equatorial plane) — with periods of several hours governed mainly by the density contrast between the inner and outer core. The modes were first proposed by L. B. Slichter after the 1960 Chile earthquake, but despite decades of searches with superconducting gravimeters (the most sensitive continuously recording gravity sensors, based on levitating a superconducting sphere) they have never been unambiguously detected. A central question is therefore whether they are hard to excite or rapidly damped. Damping is conventionally expressed through the quality factor QQ (the number of oscillation cycles after which the energy has decayed by a factor e2πe^{2\pi}), related to the e-folding decay time τ=2Q/ω\tau = 2Q/\omega for angular frequency ω\omega. Three dissipation mechanisms have been proposed: anelasticity (internal friction) of the solid inner core and mantle, viscous shear in the liquid outer core, and Ohmic dissipation — Joule heating from electric currents induced when the conducting liquid moves through the core's magnetic field. Crossley, Hinderer and Legros (1991) estimated the anelastic contribution by extrapolating the quality factors of the Preliminary Reference Earth Model (PREM), the standard radially averaged seismic model of Earth's interior, from seismic frequencies down to core-mode periods, obtaining Q≈5.2×103Q\approx 5.2\times 10^{3}, i.e. a decay time of about 420 days with the inner core as the main dissipation site; whether such an extrapolation across many decades of period is valid is itself doubtful, as Lau and Faul (2019) showed for the related problem of extrapolating mantle anelasticity from seismic to tidal periods. Buffett and Goertz (1995) built the first quantitative model of magnetic damping of the translational oscillations, an analytical boundary-layer theory for the polar mode in an unbounded fluid neglecting viscosity and rotation, predicting decay times spanning 150 days to 2200 years depending on the assumed radial field at the ICB. Personnettaz et al. (2026) finally computed both viscous and Ohmic dissipation with numerical simulations in a rotating spherical shell with an imposed axial magnetic field, deriving scaling laws for QQ in three magnetohydrodynamic regimes (diffusive, where magnetic diffusion smooths the induced field over the whole liquid; skin-layer, where the field is confined to a thin layer of thickness set by the oscillation frequency and the magnetic diffusivity; and Alfvén-wave emission — dissipation via magnetic waves radiated from the oscillating sphere), and their companion asymptotic theory (Cébron and Personnettaz 2026) unified the classical drag theories. For Earth they find viscous damping negligible (centuries-long decay) and Ohmic decay times of about 16 years (polar) and 3–4 years (equatorial), implying the modes could persist for years and that their non-detection points to weak excitation. However, these Ohmic values inherit large parameter uncertainties — the radial magnetic field strength and electrical conductivity at the ICB are not directly observable — and the study deliberately excludes inner-core rheology, stable stratification of the liquid layer, and any interaction with the geodynamo (the convective process that generates the core field; its convective turnover time of about 130 years is separated from the hours-long oscillation period by five orders of magnitude). Translational oscillations are also expected in other bodies: for Mercury, period estimates exist (e.g., Coyette et al. 2012), and Mercury's fluid core may host an outer stably stratified layer (Seuren et al. 2023), a configuration — like shallow subsurface oceans on icy moons — where confinement plus stratification could strongly enhance dissipation relative to the unbounded models used so far.

Problem Statement

Determine the damping of the inner-core translational (Slichter) oscillations — and, where coupled to damping, their periods — well enough to identify the dominant dissipation mechanism and its decay time with quantified uncertainty, for Earth and for planetary cores with stratified liquid layers. Concretely, relative to the imposed-field, unstratified, purely fluid-dissipation model of Personnettaz et al. (2026), the following linked gaps must be closed: (1) sharpen the Ohmic decay times beyond the current indicative range (about 16 years for the polar mode, 3–4 years for the equatorial modes) by accounting for the poorly constrained radial magnetic-field strength and electrical conductivity at the inner-core boundary; (2) determine the anelastic (rheological) damping of the solid inner core at core-mode periods, deciding whether the extrapolated ~420-day decay time of Crossley et al. (1991) survives and hence whether anelasticity, rather than Ohmic dissipation, is the leading damping mechanism; and (3) quantify how a stably stratified layer in the liquid outer core — and, for bodies such as Mercury or icy moons with shallow oceans or stratified cores, the combined effect of stratification and confinement — modifies both the periods and the quality factors of the translational modes. These three components jointly complete one scientific objective: a closed, predictive damping budget for inner-core translational oscillations that settles whether the modes survive for years and what a future detection would constrain.

References

  1. Personnettaz, P., Cébron, D., Schaeffer, N., Deguen, R., Mandea, M. (2026). Ohmic and viscous damping of inner core translational oscillations. Earth and Planetary Science Letters 690, 120156. DOI: 10.1016/j.epsl.2026.120156. https://doi.org/10.1016/j.epsl.2026.120156
  2. Personnettaz, P., Cébron, D., Schaeffer, N., Deguen, R., Mandea, M. (2026). Ohmic and viscous damping of inner core translational oscillations. arXiv:2604.21417. https://arxiv.org/abs/2604.21417
  3. Cébron, D., Personnettaz, P. (2026). Magnetohydrodynamic drag on an oscillating sphere in a rotating spherical cavity. arXiv:2604.10594. https://arxiv.org/abs/2604.10594
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