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arxiv: 2605.19534 · v1 · pith:UZ2QOB3Znew · submitted 2026-05-19 · 🌌 astro-ph.SR

Towards inertial-mode helioseismology: Direct sensing of solar rotation at 75 deg latitude and 0.8 Rsun

Prithwitosh Dey , Laurent Gizon , Yuto Bekki This is my paper

Pith reviewed 2026-05-20 02:40 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords inertial modeshelioseismologysolar rotationhigh latitudesconvection zonedifferential rotationHMI observations
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The pith

The m=1 inertial mode frequency implies solar rotation at 75 deg latitude and 0.8 solar radii is 365.3 nHz, exceeding p-mode estimates by 8.1 nHz.

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

The paper establishes that the frequency of the m=1 high-latitude inertial mode can be inverted to measure solar rotation in a region poorly constrained by acoustic modes. The authors calculate a sensitivity kernel showing peak response at 75 degrees latitude and 0.8 solar radii, then use the observed retrograde frequency of -87.9 nHz to derive a local rotation rate. A sympathetic reader would care because this supplies a direct probe into the deep convection zone that could clarify the Sun's internal differential rotation and its role in the dynamo. The analysis covers averaged data from 2010-2024 and proposes an additional smooth adjustment to high-latitude rotation beyond linear perturbations.

Core claim

Using a validated eigenvalue solver, the linear sensitivity kernel of the m=1 high-latitude inertial mode is shown to peak at 75 deg latitude and 0.8 R_sun. From the observed frequency in the Carrington frame of -87.9 ± 1.9 nHz, the solar rotation rate at that location is inferred to be 365.3 ± 2.0 nHz, which exceeds the reference p-mode estimate by 8.1 nHz. This constitutes the first example of spatially resolved inertial-mode helioseismology for the bulk of the convection zone.

What carries the argument

The linear sensitivity kernel of the m=1 high-latitude inertial mode, computed from perturbations to the reference p-mode rotation profile and peaking at 75 deg latitude and 0.8 R_sun with widths of 7 deg and 0.13 R_sun.

If this is right

  • Individual inertial modes supply direct constraints on rotation throughout the bulk of the solar convection zone.
  • A latitudinally smooth, radially independent modification to the high-latitude rotation rate accounts for the observations outside the small-perturbation regime.
  • Inertial-mode helioseismology enables spatially resolved diagnostics of solar differential rotation at high latitudes.

Where Pith is reading between the lines

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

  • The method could be applied to other inertial modes to construct a fuller map of rotation across different depths and latitudes.
  • The discrepancy with p-mode estimates may motivate inclusion of magnetic effects when modeling mode frequencies at high latitudes.
  • Repeated measurements of this mode over multiple solar cycles could test whether the inferred rotation varies with time.

Load-bearing premise

The frequency perturbation of the m=1 mode is accurately captured by a linear sensitivity kernel from the reference p-mode rotation profile, without significant contributions from magnetic fields, nonlinear effects, or other unmodeled physics.

What would settle it

An independent measurement of the rotation rate at 75 deg latitude and 0.8 R_sun, for instance from additional helioseismic techniques or long-term mode-frequency monitoring, that matches the p-mode reference instead of the 8.1 nHz higher value would falsify the central inference.

Figures

Figures reproduced from arXiv: 2605.19534 by Laurent Gizon, Prithwitosh Dey, Yuto Bekki.

Figure 2
Figure 2. Figure 2: Spatial sensitivity of the frequency of the HL1 mode to localised changes in the background rotation rate. The kernel K(r, θ) is symmetric across the equator. Gridsizes dr = 2.3 Mm and dθ = 2 ◦ have been used for this calculation. For reference, a patch of area 10dr × 2rdθ = 20dΣ is outlined with a solid black contour. The dashed black contour corresponds to the value of the kernel at half maximum. mode is… view at source ↗
Figure 3
Figure 3. Figure 3: Smooth high-latitude modification to the background rotation rate, ∆Ω(θ; A), and its effect on the HL1 mode frequency. (a) Functional form of ∆Ω(θ; A) as defined by Eq. (6), symmetric across the equa￾tor. (b) Variation of the real part of the mode frequency with A, using Solver 1 (green curve). By construction ω = ωref at A = 0. Agreement with the observed eigenfrequency (gray shaded region) is obtained fo… view at source ↗
Figure 4
Figure 4. Figure 4: Constraints on high-latitude solar rotation rate at r = 0.8 R⊙. The green cross shows the constraint from the HL1 mode frequency (2010–2024), according to Eq. (5), with the horizontal bar indicating the width of the kernel. The green curve shows the rotation rate Ωref(r, θ) + ∆Ω(θ; A) for A/2π = 13.6 nHz and the green shaded region indicates the associated uncertainty (see [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
read the original abstract

Solar internal rotation at high latitudes is poorly constrained by acoustic-mode helioseismology. Global inertial modes observed on the Sun are highly sensitive to solar differential rotation and may provide new diagnostics of rotation in these regions. We aim to constrain solar rotation with the measured frequency of the $m=1$ high-latitude inertial mode, starting from the HMI/SDO reference rotation profile given by p-mode helioseismology for 2010-2024. Using a validated and accurate eigenvalue solver, we compute the perturbation to the mode frequency resulting from localised changes in the differential rotation rate throughout the solar interior. We find that the linear sensitivity kernel of the $m=1$ high-latitude mode peaks at latitude 75 deg and radius $0.8 R_\odot$, with full widths of 7 deg and $0.13 R_\odot$. From the observed mode frequency in the Carrington frame, $-87.9 \pm 1.9$ nHz (retrograde, averaged over 2010-2024), we infer that the solar rotation rate near this location is $365.3\pm 2.0$ nHz, which exceeds the reference p-mode estimate by $8.1$ nHz. Additionally, we propose a latitudinally smooth, radially independent modification to the rotation rate at high latitudes beyond the linear (small-perturbation) regime. This work demonstrates that individual inertial modes can provide direct constraints on rotation in the bulk of the solar convection zone, well below the surface, representing the first example of spatially resolved inertial-mode helioseismology.

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

2 major / 2 minor

Summary. The paper develops inertial-mode helioseismology to constrain solar differential rotation at high latitudes. Using a reference rotation profile from p-mode helioseismology (2010-2024), the authors compute a linear sensitivity kernel for the m=1 high-latitude inertial mode via an eigenvalue solver. The kernel peaks at 75° latitude and 0.8 R_⊙. From the observed retrograde frequency of −87.9 ± 1.9 nHz in the Carrington frame, they infer a local rotation rate of 365.3 ± 2.0 nHz (8.1 nHz above the p-mode reference) and additionally propose a latitudinally smooth, radially independent modification to the rotation profile beyond the linear regime.

Significance. If the linear kernel and its error budget are validated, the work would provide the first spatially resolved inertial-mode constraint on rotation deep in the convection zone at high latitudes, where acoustic-mode sensitivity is weak. The explicit location of the kernel peak (75° , 0.8 R_⊙) and the quantitative frequency-to-rotation mapping constitute concrete, falsifiable outputs that could be tested with future observations or forward modeling.

major comments (2)
  1. [Abstract / Methods] The central inference of a 365.3 ± 2.0 nHz rotation rate (8.1 nHz excess) rests on the linear sensitivity kernel computed from the reference p-mode profile. The manuscript provides no quantitative validation metrics, synthetic-data recovery tests, or error budget for this kernel (abstract and associated methods description). Without these, it is unclear whether the reported offset can be attributed solely to a localized rotation perturbation rather than unmodeled contributions.
  2. [Abstract / Results] The linear-kernel assumption—that the entire observed frequency shift arises from a small, localized change in the differential-rotation profile—requires explicit justification at 0.8 R_⊙ and 75°. The abstract notes a separate latitudinally smooth modification proposed beyond the linear regime, yet the quoted numerical result and uncertainty are derived from the linear kernel alone. A concrete test (e.g., forward modeling with added magnetic or nonlinear terms) is needed to bound possible systematic contributions.
minor comments (2)
  1. [Abstract] Clarify the precise definition of the Carrington-frame frequency and how the ±1.9 nHz uncertainty is propagated into the final rotation-rate uncertainty.
  2. [Abstract] The phrase 'validated and accurate eigenvalue solver' should be accompanied by at least one quantitative accuracy metric or comparison to an independent code.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address the two major comments point by point below, indicating the changes we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Methods] The central inference of a 365.3 ± 2.0 nHz rotation rate (8.1 nHz excess) rests on the linear sensitivity kernel computed from the reference p-mode profile. The manuscript provides no quantitative validation metrics, synthetic-data recovery tests, or error budget for this kernel (abstract and associated methods description). Without these, it is unclear whether the reported offset can be attributed solely to a localized rotation perturbation rather than unmodeled contributions.

    Authors: We agree that the current version lacks explicit synthetic recovery tests and a fuller error-budget discussion for the kernel. The eigenvalue solver itself was validated in our prior work on inertial modes against analytic limits and independent codes, but we did not repeat those tests for the specific high-latitude m=1 kernel here. In the revised manuscript we will add a dedicated Methods subsection that (i) summarizes the solver validation, (ii) presents synthetic recovery experiments in which known localized rotation perturbations are injected into the reference profile and recovered via the kernel, and (iii) details the linear propagation of the observed frequency uncertainty (1.9 nHz) through the kernel to obtain the quoted ±2.0 nHz. We will also note the dominant remaining systematic (uncertainty in the reference p-mode profile itself) and quantify its contribution at the kernel peak. revision: yes

  2. Referee: [Abstract / Results] The linear-kernel assumption—that the entire observed frequency shift arises from a small, localized change in the differential-rotation profile—requires explicit justification at 0.8 R_⊙ and 75°. The abstract notes a separate latitudinally smooth modification proposed beyond the linear regime, yet the quoted numerical result and uncertainty are derived from the linear kernel alone. A concrete test (e.g., forward modeling with added magnetic or nonlinear terms) is needed to bound possible systematic contributions.

    Authors: The 8.1 nHz perturbation is only ~2 % of the local rotation rate, which lies well inside the regime where the linear eigenvalue perturbation is expected to be accurate. We will insert a short paragraph in the Results section that justifies this scale and cites the fractional perturbation size. The latitudinally smooth, radially independent modification is presented as an exploratory extension, not as the primary result; the quoted 365.3 ± 2.0 nHz value remains the linear-kernel inference. To provide the requested concrete test, the revised manuscript will include a forward-modeling check: we solve the full eigenvalue problem with the proposed smooth high-latitude modification added to the reference profile and confirm that the resulting mode frequency lies within the linear prediction plus the observational uncertainty. This exercise bounds the size of nonlinear corrections for the observed shift. revision: yes

Circularity Check

0 steps flagged

Derivation uses independent inertial-mode datum and external p-mode reference without reduction by construction

full rationale

The paper computes a linear sensitivity kernel for the m=1 inertial mode by applying an eigenvalue solver to localized perturbations around the external HMI/SDO p-mode reference rotation profile. It then maps the independently measured observed frequency (−87.9 ± 1.9 nHz) through this kernel to infer a local rotation rate of 365.3 ± 2.0 nHz at the kernel peak (75° latitude, 0.8 R⊙). This inference is not equivalent to the reference profile by construction; the observed frequency supplies new information, and the reported 8.1 nHz excess is the direct result of matching that datum under the linear approximation. No self-definitional steps, fitted inputs relabeled as predictions, or load-bearing self-citations appear in the provided derivation chain. The central claim remains self-contained against the external reference and the new observational input.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of the linear perturbation kernel derived from an external reference rotation profile and on the assumption that the observed frequency shift is caused solely by rotation changes at the kernel peak.

free parameters (1)
  • observed inertial-mode frequency
    Measured value −87.9 ± 1.9 nHz used directly to back out the rotation rate.
axioms (1)
  • domain assumption Frequency perturbation is linear in differential-rotation changes
    Invoked to compute the sensitivity kernel from the reference p-mode profile.

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