The Slichter modes are the degree-one translational oscillations of Earth's solid inner core within the fluid outer core, with classical (impermeable-boundary) theory predicting eigenperiods of several hours. Grinfeld and Wisdom (2010) showed that if phase transformations (melting/solidification) at the inner-core boundary (ICB) proceed with instantaneous kinetics — the interface pressure pinned to the phase-coexistence pressure — the outer-core added-mass effect (the inertia of outer-core fluid entrained by the oscillating inner core) vanishes and the density contrast enters the frequency formula in the denominator, yielding a period of about 7.5 minutes, an order of magnitude below Busse's classical 5.42-hour estimate. The paper itself left the physically decisive question open: how finite-rate transformation kinetics alter this prediction. No later work has derived the translational-mode eigenfrequency with a finite-rate kinetic law at the ICB; the most recent treatment (Personnettaz et al. 2026), extending Wu and Rochester (1994), argues by latent-heat energy balance that even an instantaneous thermodynamic response produces only nanometre-scale phase change per 10 cm displacement, so the effect should be negligible. The open problem is a self-consistent eigenmode treatment coupling a finite-rate phase-change law (and its energy budget) to the translational oscillation, recovering both the instantaneous-kinetics and impermeable limits, and determining which regime — if either — applies to Earth.

Background

The Slichter modes are the three translational normal modes of the Earth: the solid inner core, immersed in the liquid outer core, can oscillate as a rigid body about its equilibrium position, with Earth's rotation splitting the oscillation into one polar (axial) mode and two equatorial modes, the 'Slichter triplet'. They are named after L. B. Slichter, who proposed searching for them after the 1960 Chile earthquake; because a displaced inner core perturbs the gravity field, the modes are searched for with superconducting gravimeters (instruments measuring sub-nanogal gravity variations), though no definitive detection exists. In the classical treatment initiated by Busse (1974), a non-rotating three-layer model — rigid inner core of radius R1R_1 and density ρ1\rho_1, inviscid incompressible outer core of density ρ2\rho_2 out to radius R2R_2, stationary mantle — the boundary is impermeable (no mass crosses the inner-core boundary, ICB), so the oscillating inner core must entrain outer-core fluid (the added-mass effect), and the eigenperiod is of order hours (about 5.4 hours for canonical Earth parameters). Comprehensive models including rotation, elasticity, and stratification (Rieutord 2002) confirm the hour-scale triplet.

Grinfeld and Wisdom (2010) asked what changes if phase transformations of the first kind (solidification/melting) are allowed at the ICB. Under the idealization of instantaneous kinetics — the pressure at the moving interface equals the phase-coexistence pressure P0P_0 at every instant, so thermodynamic equilibrium is maintained continuously — they derived the closed-form squared frequency of the degree-one mode,

ω2=4πGρ23  3ρ2R23+(ρ1−ρ2)(2R13+R23)(ρ1−ρ2)(2R13+R23),\omega^{2}=\frac{4\pi G \rho_{2}}{3}\;\frac{3\rho_{2}R_{2}^{3}+(\rho_{1}-\rho_{2})(2R_{1}^{3}+R_{2}^{3})}{(\rho_{1}-\rho_{2})(2R_{1}^{3}+R_{2}^{3})},

with GG Newton's gravitational constant. The density contrast ρ1−ρ2\rho_{1}-\rho_{2} now appears in the denominator: because material converts phase at the interface, the outer-core fluid need not be entrained (added mass is eliminated) and the effective displacement driving the gravitational restoring force is amplified, so smaller contrasts paradoxically give higher frequencies. With canonical parameters the period is about 7.5 minutes versus Busse's 5.42 hours. The same inversion appears in a planar-interface wave problem (Grinfel'd 1988), where allowing phase change converts the dispersion ω2=gK(ρ1−ρ2)/(ρ1+ρ2)\omega^{2}=gK(\rho_{1}-\rho_{2})/(\rho_{1}+\rho_{2}) into ω2=gK(ρ1+ρ2)/(ρ1−ρ2)\omega^{2}=gK(\rho_{1}+\rho_{2})/(\rho_{1}-\rho_{2}). The treatment is an idealization: real transformations relax pressure/temperature disequilibria at a finite rate, and the assumed mechanically-driven transformation (rather than transformation limited by latent-heat transport) has not been validated for Earth.

Two earlier threads frame the issue. Wu and Rochester (1994), in a comprehensive rotating-Earth treatment, neglected melting/solidification because the oscillation amplitude is small, estimating undercoolings of order 2×10−52\times10^{-5} K per metre of displacement. Peng (1997) found that a kilometre-thick mushy transition zone (a two-phase solid–liquid layer) at the ICB would modify the Slichter period by only about 0.5%. Most recently, Personnettaz et al. (2026), in a numerical study of viscous and Ohmic (magnetic) damping of the translational modes, revisited the phase-change question: assuming an instantaneous thermodynamic response as a conservative upper bound, an energy balance between the sensible heat of the undercooled displaced material (governed by the difference between the outer-core adiabatic gradient and the solidification-gradient set by the Clapeyron slope, i.e. the slope of the melting curve) and the latent heat shows that only a thickness of order one nanometre can transform per 10 cm displacement — concluding that phase-change effects are negligible and that the Grinfeld–Wisdom reduction rests on a simplified model that omits the latent-heat budget. Thus the literature contains two unreconciled positions: an idealized mechanical-equilibrium model predicting minute-scale periods, and thermodynamic energy arguments predicting a negligible effect — while the finite-rate eigenproblem itself has never been solved.

Problem Statement

Derive and quantify how finite-rate phase-transformation kinetics at the inner-core boundary alter the eigenfrequency of the Earth's Slichter (degree-one translational) mode. Concretely: extend the three-layer, self-gravitating, linear translational-mode eigenproblem of Grinfeld and Wisdom (2010) — rigid inner core (R1R_1, ρ1\rho_1), inviscid incompressible outer core (ρ2\rho_2, R2R_2), stationary mantle — by replacing the instantaneous-kinetics condition (interface pressure equal to the coexistence pressure P0P_0 at all times) with a finite-rate transformation law relating interfacial disequilibrium to melting/solidification rate, characterized by a kinetic-rate (or relaxation-time) parameter κ\kappa. The solution should (i) yield the dispersion relation ω(κ)\omega(\kappa) as a function of that rate, (ii) recover the instantaneous-kinetics limit (the Grinfeld–Wisdom formula, ~7.5 minutes for canonical Earth parameters) and the impermeable-interface limit (Busse's formula, ~5.42 hours) as opposite endpoints, and (iii) determine which regime — if either — applies to Earth for plausible values of κ\kappa, adjudicating between the two unreconciled positions in the literature.

References

  1. P. Grinfeld and J. Wisdom, 'The effect of phase transformations at the inner core boundary on the Slichter modes', Physics of the Earth and Planetary Interiors 178(3–4), 183–188 (2010). doi:10.1016/j.pepi.2009.10.006, https://doi.org/10.1016/j.pepi.2009.10.006
  2. F. H. Busse, 'On the free oscillation of the Earth's inner core', Journal of Geophysical Research 79(5), 753–757 (1974). doi:10.1029/jb079i005p00753, https://doi.org/10.1029/jb079i005p00753
  3. L. B. Slichter, 'The fundamental free mode of the Earth's inner core', Proceedings of the National Academy of Sciences 47(2), 186–190 (1961). doi:10.1073/pnas.47.2.186, https://doi.org/10.1073/pnas.47.2.186
  4. M. Grinfel'd, 'Continuum methods in the theory of phase transitions in solids', Physics of the Earth and Planetary Interiors 50(1), 99–109 (1988). doi:10.1016/0031-9201(88)90099-4, https://doi.org/10.1016/0031-9201(88)90099-4
  5. P. Grinfeld and J. Wisdom, 'Motion of the mantle in the translational modes of the Earth and Mercury', Physics of the Earth and Planetary Interiors 151(1–2), 77–87 (2005). doi:10.1016/j.pepi.2005.01.003, https://doi.org/10.1016/j.pepi.2005.01.003
  6. Wen-Jing Wu and M. G. Rochester, 'Gravity and Slichter modes of the rotating Earth', Physics of the Earth and Planetary Interiors 87(1–2), 137–154 (1994). doi:10.1016/0031-9201(94)90027-2, https://doi.org/10.1016/0031-9201(94)90027-2
  7. Z. R. Peng, 'Effects of a mushy transition zone at the inner core boundary on the Slichter modes', Geophysical Journal International 131(3), 607–617 (1997). doi:10.1111/j.1365-246x.1997.tb06602.x, https://doi.org/10.1111/j.1365-246x.1997.tb06602.x
  8. M. Rieutord, 'Slichter modes of the Earth revisited', Physics of the Earth and Planetary Interiors 131(3–4), 269–278 (2002). doi:10.1016/s0031-9201(02)00039-0, https://doi.org/10.1016/s0031-9201(02)00039-0
  9. R. Deguen, T. Alboussière, and P. Cardin, 'Thermal convection in Earth's inner core with phase change at its boundary', Geophysical Journal International 194(3), 1310–1334 (2013). doi:10.1093/gji/ggt202, https://doi.org/10.1093/gji/ggt202
  10. A. Coyette and T. Van Hoolst, 'Slichter modes of large icy satellites', Icarus 231, 287–299 (2014). doi:10.1016/j.icarus.2013.11.024, https://doi.org/10.1016/j.icarus.2013.11.024
  11. Hao Ding and Benjamin F. Chao, 'The Slichter mode of the Earth: Revisit with optimal stacking and autoregressive methods on full superconducting gravimeter data set', Journal of Geophysical Research: Solid Earth 120(10), 7261–7272 (2015). doi:10.1002/2015JB012203, https://doi.org/10.1002/2015JB012203
  12. P. Personnettaz, D. Cébron, N. Schaeffer, R. Deguen, and M. Mandea, 'Ohmic and viscous damping of inner core translational oscillations', Earth and Planetary Science Letters 690, 120156 (2026). doi:10.1016/j.epsl.2026.120156, https://doi.org/10.1016/j.epsl.2026.120156; preprint arXiv:2604.21417, https://arxiv.org/abs/2604.21417