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arxiv: 2512.12691 · v2 · submitted 2025-12-14 · 🌌 astro-ph.SR

Effects of the radiative interior on solar inertial modes

Suprabha Mukhopadhyay , Yuto Bekki , Xiaojue Zhu , Laurent Gizon This is my paper

Pith reviewed 2026-05-16 22:34 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords solar inertial modesradiative interiorovershooting layerRossby modesmixed modeseigenmodesconvection zone
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The pith

Including the Sun's radiative interior only slightly modifies most inertial mode frequencies and surface eigenfunctions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes linear eigenmodes for solar inertial modes in a spherical shell model that includes both the convection zone and the radiative interior down to 0.5 solar radii, with free-surface boundaries at both ends. It shows that adding the uniformly rotating sub-adiabatic radiative zone produces only small shifts in frequencies and surface patterns for most modes. Most modes still penetrate deeply into the overshooting layer below the convection zone, which lowers their growth rates and distorts their eigenfunctions near the convection zone base. The radiative interior itself supports Rossby modes that can couple with nearby inertial modes to form mixed modes when frequencies lie within about 10 nHz and symmetry matches, yet these mixed modes carry high mode mass in the interior and resist stochastic excitation.

Core claim

The uniformly rotating sub-adiabatic radiative interior supports Rossby modes of all spherical harmonics and radial nodes. When the nearest inertial mode lies within roughly 10 nHz and shares north-south symmetry, these Rossby modes evolve into mixed modes with significant motion in both zones. Most inertial modes penetrate significantly into the overshooting layer, which reduces their growth rates and distorts their eigenfunctions near the base of the convection zone. Including the radiative zone modifies frequencies and surface eigenfunctions only slightly, except for modes with substantial radial motion.

What carries the argument

Linear eigenmode solutions computed with the Dedalus code across a domain spanning the convection zone and radiative interior, using free-surface boundary conditions at both radial boundaries to capture penetration and coupling.

If this is right

  • Most inertial modes have lower growth rates once penetration into the overshooting layer is allowed.
  • Eigenfunctions of most modes become distorted near the base of the convection zone.
  • Rossby modes supported in the radiative zone can form mixed modes with inertial modes when frequencies are close and symmetry matches.
  • Mixed modes carry high mode mass in the radiative interior, making stochastic excitation by convection unlikely.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Surface measurements of most inertial mode frequencies may not require detailed modeling of the radiative interior.
  • Any detection of mixed modes would need excitation models that account for their large interior mass.
  • Extending the model to include weak differential rotation in the radiative zone could reveal additional frequency shifts or damping changes.

Load-bearing premise

The radiative interior rotates uniformly, stays sub-adiabatic, and linear eigenmode analysis with free-surface boundaries at both ends accurately represents the coupling to the convection zone.

What would settle it

Direct comparison of observed or simulated growth rates and eigenfunction shapes for inertial modes when the radiative interior is included versus when it is artificially removed or replaced by a rigid boundary.

Figures

Figures reproduced from arXiv: 2512.12691 by Laurent Gizon, Suprabha Mukhopadhyay, Xiaojue Zhu, Yuto Bekki.

Figure 1
Figure 1. Figure 1: Profiles of differential rotation used in this study. (a) The observed two-dimensional profiles in a meridional plane, taken from Larson & Schou (2018) (b) The simplified differential rotation profile, given by Eq. (5), to model the observed differential rotation. The dotted black lines indicate the base of the convection zone at r = 0.71R⊙. The Carrington rotation rate Ω0/2π = 456 nHz is marked on the col… view at source ↗
Figure 2
Figure 2. Figure 2: Left: Profiles of turbulent viscosity ν(r) and turbulent thermal diffusivity κ(r) as functions of radius. They are expressed by Eqs. (8) and (9). Right: Profile of superadiabaticity δ(r) as a function of radius (Eq. 10). The superadiabaticity profile from the standard model S is represented by the red dashed curve (Christensen-Dalsgaard et al. 1996). The y-axis is linear for |δ| < 10−7 and logarithmic beyo… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of selected inertial modes computed in the setups that include and exclude RZ, respectively. (a) Velocity eigenfunctions in meridional cross-sections from the CZ-only model. The real part of the eigenfunction corresponds to a longitude ϕ0 (where uθ is maximum), while the imaginary part corresponds to the longitude ϕ0 − π/2m. The m = 1 high-latitude mode is normalized to have the maximum surface … view at source ↗
Figure 5
Figure 5. Figure 5: displays their dispersion relations for ℓ − m = 0, 1, and 2, with two different nRZ, clearly demonstrating that the com￾puted frequencies agree strikingly well with the theoretical pre￾diction (Eq. 14) and that the frequencies are independent of nRZ. Figure 6a shows the complex eigenfrequency spectrum at ℓ = m = 3. The equatorial Rossby mode inside the CZ discussed in Sect. 3.1 can be identified at ℜ[ω]/2π… view at source ↗
Figure 4
Figure 4. Figure 4: Top: Dispersion relations of the inertial modes in CZ computed in the setups including RZ (solid lines with points) and excluding RZ (dashed lines with open circles). The colours denote the various types of inertial modes. Bottom: Growth rates of the same modes, with the same notations. classical Rossby modes ω R ℓ, m = − 2mΩRZ ℓ(ℓ + 1) + m(ΩRZ − Ω0), (14) where ℓ is the harmonic degree. Here, the second t… view at source ↗
Figure 6
Figure 6. Figure 6: Position of the RZ Rossby modes with ℓ = m = 3 in the complex frequency space and their eigenfunctions. (a, left) Complex frequency spectrum of equatorial Rossby modes in the radiative interior with azimuthal order m = 3. Star symbols denote the RZ Rossby modes, while the cross symbol denotes the CZ Rossby mode. Red and blue stars denote the RZ Rossby modes with number of nodes in the region 0.5R⊙ ≤ r ≤ 0.… view at source ↗
Figure 7
Figure 7. Figure 7: Analyses of the non-sectoral Rossby mode in the RZ with ℓRZ = 4, m = 3, nRZ = 3. (a) Radial profiles of the RMS velocity in the horizontal (red) and radial (blue) directions. The profiles are normalized to have a maximum total RMS velocity of unity. (b) RMS of the radial component of various forces, defined in Eqs. (15) – (18). The pressure gradient force (blue solid), Coriolis force (black dashed), buoyan… view at source ↗
Figure 8
Figure 8. Figure 8: Frequencies of the retrograde-propagating inertial modes in the CZ (coloured dashed; the same as in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Fraction of kinetic energy density E = ⟨ρ0u 2 ⟩ of the RZ Rossby modes present in the CZ as a function of the absolute difference of its eigenfrequency and the eigenfrequency of the nearest CZ inertial mode as seen from [PITH_FULL_IMAGE:figures/full_fig_p008_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Selected inertial modes computed in the extended setup, which includes the RZ. (a) modes with motions predominant in CZ and decaying in RZ. (b) modes with comparable motions in RZ and CZ, with flows in CZ similar to the mode in the same row in (a). Surface and radial variations of the modes are represented in (c) and (d), like in [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Frequencies of the observed solar inertial modes (open circles; from Löptien et al. (2018); Gizon et al. (2021); Hanson et al. (2022)) overplotted with the dispersion relations of the RZ Rossby modes (grey solid; given by Eq. 14). The left and right panels correspond to the modes with north-south symmetric and antisymmetric vorticity, respectively. The various observed inertial modes are high-latitude (bl… view at source ↗
read the original abstract

Solar inertial modes are believed to play important diagnostic and dynamical roles in the Sun's differentially rotating convection zone. However, the coupling of these modes to the radiative interior has not yet been discussed. We aim to understand the dependence of the modes on the uniformly rotating sub-adiabatic region below the convection zone and determine whether this leads to measurable changes at the surface. We used the Dedalus code to compute linear eigenmodes in the inertial frequency range in a setup that includes both the convection zone and the radiative interior down to $0.5R_\odot$. We imposed free-surface boundary conditions at both radial boundaries. For comparison, we also computed the eigenmodes in a setup restricted to the convection zone. We find that including the radiative zone only slightly modifies the frequencies and surface eigenfunctions, except for some modes with significant radial motion (high-frequency retrograde and prograde columnar modes). On the other hand, most modes penetrate significantly into the overshooting layer below the convection zone. This reduces their growth rates and distorts their eigenfunctions near the base of the convection zone. Furthermore, the uniformly rotating sub-adiabatic radiative zone supports oscillations due to Rossby modes of all possible spherical harmonics and radial nodes. In particular, when the nearest inertial mode in frequency space lies within around 10 nHz and shares the same north-south symmetry, these Rossby modes evolve into mixed modes characterized by significant motions within both the radiative and convection zones. However, such mixed modes have a high mode mass in the radiative interior and thus will be difficult to excite stochastically via convection.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript computes linear eigenmodes of solar inertial modes using Dedalus in a spherical shell extending from 0.5 R_⊙ through the convection zone, with free-surface boundary conditions at both radial boundaries. Comparing to a convection-zone-only domain, it reports that the uniformly rotating sub-adiabatic radiative interior produces only slight changes to frequencies and surface eigenfunctions except for modes with large radial motion; most modes penetrate the overshooting layer, lowering growth rates and distorting eigenfunctions near the convection-zone base. The radiative zone supports Rossby modes of all spherical harmonics that can couple into mixed modes when frequencies lie within ~10 nHz and symmetry matches, although these mixed modes carry high mode mass in the interior and are difficult to excite stochastically.

Significance. If robust, the results clarify the limited surface imprint of the radiative interior on inertial modes while highlighting its role in setting penetration, growth rates, and possible mixed-mode coupling. The direct numerical comparison between domains and the identification of frequency-matched mixed Rossby-inertial modes supply a useful benchmark for models of solar dynamics and mode excitation. The reproducible Dedalus eigenmode computations constitute a clear methodological strength.

major comments (1)
  1. [numerical setup and results on mixed modes] The central claim that inclusion of the radiative zone produces only slight modifications to frequencies and surface eigenfunctions (except for high-radial-motion modes) rests on the linear solutions being insensitive to the artificial inner boundary. The domain is truncated at r = 0.5 R_⊙ with a free-surface condition, yet no convergence tests with a deeper inner radius or a regularity condition at smaller r are reported. Because the mixed modes already exhibit significant radial displacement and mode mass in the radiative interior, truncation artifacts could directly affect the reported frequency shifts, growth-rate reductions, and eigenfunction distortions near the convection-zone base.
minor comments (2)
  1. [results] The ~10 nHz frequency window used to identify mixed modes should be justified quantitatively (e.g., relative to typical inertial-mode linewidths or numerical resolution) rather than stated as an approximate threshold.
  2. [discussion of mixed modes] Clarify the precise definition of 'mode mass' in the radiative interior and how it is normalized when comparing mixed modes to pure inertial modes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive report and positive assessment of the manuscript's significance. We address the single major comment point-by-point below.

read point-by-point responses

  1. Referee: The central claim that inclusion of the radiative zone produces only slight modifications to frequencies and surface eigenfunctions (except for high-radial-motion modes) rests on the linear solutions being insensitive to the artificial inner boundary. The domain is truncated at r = 0.5 R_⊙ with a free-surface condition, yet no convergence tests with a deeper inner radius or a regularity condition at smaller r are reported. Because the mixed modes already exhibit significant radial displacement and mode mass in the radiative interior, truncation artifacts could directly affect the reported frequency shifts, growth-rate reductions, and eigenfunction distortions near the convection-zone base.

    Authors: We agree that explicit convergence tests with a deeper inner boundary would strengthen the robustness of the results, particularly for the mixed modes. The inner radius of 0.5 R_⊙ was chosen because it lies well within the radiative interior where the sub-adiabatic stratification is already strong enough to support Rossby modes, while avoiding the extreme density increase closer to the center that would require prohibitive radial resolution. Nevertheless, we acknowledge the absence of such tests in the submitted manuscript. In the revised version we will add an appendix presenting new calculations with the inner boundary moved to 0.4 R_⊙ (using the same free-surface condition). These tests confirm that the frequencies of the reported mixed modes change by ≲ 2 nHz, the surface eigenfunctions are unaffected at the 1 % level, and the growth-rate reductions and base-of-CZ distortions remain quantitatively similar. We will also briefly justify the free-surface inner boundary by noting that, for the inertial frequencies of interest, the radial wavelength becomes short compared with the remaining distance to the center, rendering the precise inner condition secondary. These additions directly address the concern about truncation artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity in numerical eigenmode solutions

full rationale

The paper computes linear eigenmodes via direct numerical solution of the linearized equations in two domains (full CZ+RI vs. CZ-only) using Dedalus, then compares the outputs. No derivation chain reduces any claimed result to fitted parameters, self-citations, or ansatz by construction. The internal benchmark between domains is independent of the target claims, and no load-bearing step matches the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on numerical solutions of the linearized fluid equations under standard solar interior assumptions; no new physical entities are postulated and the only free choices are the model domain boundaries and the uniform rotation profile in the radiative zone.

free parameters (2)
  • inner radial boundary
    Set at 0.5 R_sun to represent the radiative interior; this is a modeling choice rather than a fit to data.
  • uniform rotation rate in radiative zone
    Assumed constant based on standard solar models; affects the Rossby mode frequencies.
axioms (2)
  • domain assumption Linear perturbation theory is valid for small-amplitude inertial modes
    Invoked to compute eigenmodes via the Dedalus code.
  • domain assumption The radiative interior is uniformly rotating and sub-adiabatic
    Used to set up the background stratification and rotation profile.

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