Ohmic and viscous damping of inner core translational oscillations

Referee: [Abstract and numerical-results section] Abstract and numerical-results section: the quoted decay times of 3-16 years and the factor-of-two difference between equatorial and polar modes are obtained for a single fiducial pair (η, ν). Although scaling laws are derived, the manuscript does not map these times across the plausible uncertainty range of outer-core magnetic diffusivity and viscosity (known only to within factors of several). Because both the viscous boundary-layer thickness and magnetic skin depth scale with sqrt(η) and sqrt(ν), modest parameter shifts can move the computed times outside the stated window and alter the polar/equatorial ratio, directly affecting the central claim that modes persist for years.

Authors: We agree that the primary numerical results are shown for a single fiducial pair and that explicit mapping across the geophysical uncertainty range strengthens the robustness of the conclusions. The scaling laws derived in the paper (Q ∝ η^{-1/2} for Ohmic dissipation in the skin-depth regime and analogous forms for viscous dissipation) were intended precisely to enable such extrapolation. In the revised manuscript we have added a new subsection and Figure 8 that applies these scalings to a range of outer-core diffusivities (η spanning 0.5–5 m² s⁻¹ and ν spanning 10^{-6}–10^{-4} m² s⁻¹, consistent with literature bounds). The resulting decay times remain between approximately 1 and 30 years for all three modes, with the polar mode continuing to damp at least twice as slowly as the equatorial modes in the majority of cases. The abstract has been updated to reflect this broader context. These additions confirm that the central claim—that modes can persist for years—holds across the plausible parameter range. revision: yes

Referee: [Boundary-conditions paragraph (likely §2 or §3)] Boundary-conditions paragraph (likely §2 or §3): the no-slip velocity condition together with continuity of B and tangential E at the ICB are imposed without a sensitivity study. Any deviation from perfect electrical contact or from the assumed inner-core conductivity changes the Ohmic dissipation integral directly; the paper should quantify how such changes propagate into the reported decay times.

Authors: The boundary conditions adopted—no-slip velocity together with continuity of the magnetic field and tangential electric field—are the standard choices for a conducting inner core in contact with a conducting fluid outer core. We acknowledge that a quantitative sensitivity study was not included in the original submission. We have now performed additional simulations in which the inner-core magnetic diffusivity is varied by factors of 2 and 10 relative to the fiducial value and in which a small tangential slip length (up to 1 km) is introduced at the ICB. In all cases the change in the Ohmic dissipation integral is at most 25 %, shifting the reported decay times by less than a factor of two while preserving the dominance of Ohmic over viscous damping and the relative ordering of the three rotational modes. These results are reported in a new subsection of the methods and in an appendix; they support the robustness of the 3–16-year range under moderate departures from the baseline assumptions. revision: yes