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Investigating Variations in Solar Differential Rotation by Helioseismology

T0 review · 5 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims the dynamo-wave signature in solar torsional oscillations is genuine and persistent, appearing in both zonal flow and zonal acceleration across GONG, MDI, and HMI data, with low-latitude branches taking about 5–6 years…

desk verdict A genuinely useful time-radius inversion and a mostly confirmatory multi-instrument dynamo-wave study that overclaims three-instrument confirmation and needs a null test. read the letter →

arxiv 2505.10756 v1 pith:DB7ECN6I submitted 2025-05-15 astro-ph.SR

Krishnendu Mandal , Alexander G. Kosovichev , Valery V. Pipin , Sylvain G. Korzennik This is my paper · ORCID
classification astro-ph.SR
keywords solardifferentialrotationtorsionaloscillationszonalflowshelioseismologydynamowavesregularizedleastsquaresinversionnear-surfaceshearlayercycle
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to establish that the recently discovered dynamo-wave signature in the Sun's torsional oscillations is real, not a quirk of one instrument or one analysis pipeline. Using p-mode frequency-splitting data from GONG, MDI, and HMI, with time series from 72 days to eight times that length, the authors report the same tilted wave pattern in both the zonal flow and its time derivative, the zonal acceleration, throughout the convection zone: low-latitude branches take about 5–6 years to rise from the base of the convection zone to the surface, while high-latitude changes appear almost simultaneously at all depths. To achieve this, they introduce a time-dependent inversion that smooths the solution in radius and time jointly, eliminating the need for separate post-processing smoothing. The paper also characterizes the dimensionless radial gradient of rotation in the near-surface shear layer, finding values near −1 at the surface that increase with depth and show a torsional-oscillation-like variation, with high-latitude results left inconclusive because of inter-instrument systematics.

What carries the argument

The load-bearing tool is a time-dependent Regularized Least Squares inversion: the misfit function (Eq. 6) adds second-derivative smoothing in radius and first-derivative smoothing in time to a fit of the frequency-splitting coefficients, with rotation expanded in cubic B-splines along both acoustic depth and time. Smoothing in time is thus internal to the inversion, controlled by a regularization weight rather than applied afterward as a Gaussian filter, and it can be made depth-dependent through the radial weighting function f(r). This device turns the 72-day time series into a continuous time–radius solution for the zonal flow coefficients ws(r,t), from which both the flow and its acceleration are reconstructed and the dynamo-wave tilt is measured.

What would settle it

Invert a synthetic dataset built from a time-independent rotation profile plus noise matching the observed uncertainties using the same time-dependent inversion; if tilted patterns resembling 5–6-year propagation appear in the recovered zonal flow or acceleration, the claimed dynamo-wave pattern is an artifact of the regularization. Alternatively, if independent local-helioseismology measurements of zonal flows at 0.8–0.98 R⊙ disagree with the global-inversion propagation speeds by more than the combined uncertainties, the pattern does not survive.

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Extended reading notes

Core claim

The central claim is that the dynamo-wave pattern first reported by Kosovichev & Pipin (2019) appears unambiguously in every dataset analyzed here: GONG, MDI, and HMI frequency splittings, binned in 72-day, 4×72-day, 5×72-day, and 8×72-day segments, and processed both by the Korzennik (2023) pipeline and by the JSOC pipeline. The pattern is visible in the zonal flow itself and in its acceleration, and the two are phase-shifted; the tachocline onset at high latitudes correlates in sign and timing with the approach of the next solar cycle. For the near-surface shear layer, the paper claims the logarithmic radial gradient of rotation is close to −1 at the surface, rises toward zero with depth, stays nearly constant from equator to mid-latitudes in the top 13–35 Mm, and exhibits a torsional-oscillation-like variation whose equatorward branch matches the magnetic butterfly diagram.

Load-bearing premise

The argument stands on the assumption that the measured frequency-splitting coefficients, after pipeline-specific cuts such as limiting MDI to harmonic degrees below 120 and GONG below 150, represent solar rotation with systematic errors smaller than the roughly 0.5 m/s zonal-flow signals being interpreted.

Editorial extensions

If this is right

  • Dynamo models must reproduce a low-latitude branch that takes about 5–6 years to travel from the base of the convection zone to the surface and a nearly instantaneous high-latitude branch, because the same phase pattern appears in zonal flow and zonal acceleration.
  • The time-dependent inversion removes post-processing temporal smoothing and produces consistent features from 72-day to 8×72-day datasets, so it can be reused for other time-varying helioseismic inversions.
  • The dimensionless radial rotation gradient near −1 at the surface, increasing with depth and nearly constant from equator to mid-latitudes in the top 13–35 Mm, gives a quantitative target for models of the near-surface shear layer and its role in the dynamo.
  • The torsional-oscillation-like variation of the radial gradient, with an equatorward branch matching the butterfly diagram, ties the near-surface shear layer's cycle variations to the same dynamo-wave process.
  • The phase difference between zonal flow and acceleration near the tachocline may provide timing and strength information for the upcoming solar cycle, though the paper notes that the correlation with magnetic field must first be established.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the authors leave implicit: comparing the measured travel-time and phase-lag pattern against flux-transport versus distributed dynamo models could discriminate between dynamo families, because the two predict different relations between deep magnetic torque and surface flow.
  • The high-latitude inter-instrument discrepancies suggest the systematics may be depth-dependent; resolving them with high-degree modes (ℓ > 300) or local helioseismology would test whether the polar branch of the dynamo wave is real.
  • The bump in the rotation gradient near 0.98 R⊙, attributed to the He II ionization transition, is testable with independent ring-diagram measurements; a matching bump in their gradient profiles would strengthen the interpretation.
  • Extending the same joint time–radius inversion to meridional flows could reveal whether the dynamo-wave coupling inferred for zonal flows also organizes the meridional circulation on similar timescales.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The paper introduces a time-dependent regularized least-squares inversion that carries out the inversion jointly in radius and time using B-spline basis functions, thereby avoiding separate post-hoc temporal smoothing. The method is applied to p-mode frequency-splitting coefficients from GONG, MDI, and HMI, using both the Korzennik (2023) sets and JSOC pipeline products at several time-window lengths (1x72-day through 8x72-day). The authors report dynamo-wave-like signatures in both zonal flow and zonal acceleration throughout the convection zone, with a low-latitude branch that takes approximately 5-6 years to rise from the base of the convection zone to the surface, and a nearly vertical pattern at high latitudes. They also analyze the dimensionless radial gradient of rotation in the near-surface shear layer, finding values near -1 that increase with depth and show torsional-oscillation-like temporal variations, while acknowledging high-latitude results as inconclusive. The inversion is validated with forward-modeled synthetic splittings from the Pipin & Kosovichev (2020) dynamo model.

Significance. If the central detection holds, the paper provides a useful methodological contribution with the first simultaneous time-radius RLS inversion for global helioseismology, and it places a concrete constraint on dynamo-wave propagation: a 5-6 year rise time at low latitudes. The use of multiple independent instruments and window lengths strengthens the earlier detection by Kosovichev & Pipin (2019). The paper also honestly reports high-latitude inter-instrument discrepancies and inconclusive high-latitude gradient results. However, the strong claims in the abstract and conclusions are not fully supported by the displayed evidence: the main pattern figures come from GONG only, no null test is shown for the temporal regularization, and the central figures do not display error bars. The forward-model validation uses the authors' own dynamo model for both the input and the interpretation, which limits its ability to rule out inversion artifacts.

major comments (5)
  1. [Section 2.2, Eq. (6)] The regularization weights λ_r and λ_t are selected without a documented procedure, and the only validation shown assumes f(r)=g(r)=1. Because the central claim is the inclined 5-6 year pattern in Figures 3-5, please provide an L-curve or grid scan for λ_r and λ_t and, crucially, a null test in which synthetic splittings from a time-independent rotation profile with realistic noise are inverted and the recovered time-dependent residual is shown to be consistent with zero. Without such a test, the tilted branches could be an artifact of temporal regularization.
  2. [Section 3.1, Figures 3-7] The maps that directly display the dynamo-wave pattern are produced from GONG data only, with Figure 2 combining MDI and HMI. The abstract and Section 4 claim confirmation across all three instruments, but no same-epoch cross-instrument zonal-flow map is shown. Please add an apples-to-apples comparison over overlapping MDI/HMI/GONG epochs with identical inversion settings to demonstrate that the tilted branches are not an artifact of a particular dataset or pipeline.
  3. [Section 3.2, Figure 9] The manuscript states that high-latitude (>60°) differences between instruments 'exceed significantly' and 'indicate potential systematics' in the frequency measurements. Since the same mode sets are used for the lower-latitude dynamo-wave detection, the paper needs to show that these systematics are confined to high latitudes. For example, present inter-instrument residuals in the latitude range 0-60° and depths 0.75-0.98 R⊙, or repeat the central inversions after applying a more conservative mode selection.
  4. [Section 2.2, Figures 3-7 and 11] The Monte Carlo error estimation is described but the resulting uncertainties are not displayed on the figures that support the central claim. Because the zonal-flow signals are of order 0.5 m/s, the reader needs error bars or shaded uncertainty regions (or an explicit statement of the typical 1σ error amplitude) to judge whether the reported branches are significant.
  5. [Section 3.1, MDI harmonic-degree cut] The choice to exclude MDI modes with ℓ>120 is justified as a consistency requirement with other instruments, which is a post hoc selection. The same applies to the GONG ℓ<150 limit. Please repeat the time-dependent inversion for several degree cutoffs (e.g., MDI with ℓ_max = 120, 150, 180) and confirm that the 5-6 year rise time and the branch directions remain unchanged, to rule out that the consistency cut removes modes that would otherwise alter the inferred wave pattern.
minor comments (6)
  1. [Appendix A] The word 'Equatuon' should read 'Equation'.
  2. [Section 2.2] The sentence 'and a third temporal grid point as a knot in the time direction' is ambiguous; please clarify whether the time knots are equally spaced and how many knots are used.
  3. [Figures 3 and 4] The vertical axes are labeled 'dv/dt' and 'v' without units; please add m/s year⁻¹ and m/s, respectively.
  4. [Section 3.2] The statement that error bars are 'very small' is not quantitative; please report the actual 1σ uncertainty in ∇rΩ.
  5. [Figure 5 caption] The caption mentions an online video; please specify the URL or repository where it can be accessed.
  6. [Section 4] The phrase 'unambiguously confirming the original detection' is stronger than the evidence presented given the high-latitude systematics acknowledged in Section 3.2; consider tempering this wording and the corresponding abstract phrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dynamo-wave detection is an independent inversion of observed splittings; model comparisons serve as validation rather than fitted inputs.

full rationale

The central claim, that zonal flow and zonal acceleration inferred from GONG, MDI, and HMI frequency splittings show a dynamo-wave-like pattern with a 5-6 yr low-latitude rise time, is derived from observational frequency-splitting coefficients via Eq. (5) and the RLS inversion in Eq. (6). No dynamo-model parameter is fitted to the observed data. The forward-modeling test using the Pipin & Kosovichev (2020) model is a methodological check that a known rotation profile can be recovered after inversion, and the later qualitative similarity between Figure 8 and Figure 11 is a model-data comparison, not a prediction forced by the inversion. The MDI l<120 cut is a documented systematic cut motivated by prior literature, not a parameter fitted to the target dynamo-wave pattern. Self-citations to Kosovichev & Pipin (2019), Mandal et al. (2024), and Pipin & Kosovichev (2020) provide the original detection, previous methodology, and comparison model, but the present detection does not reduce to those citations: it uses independent mode-splitting datasets and a new time-dependent inversion. The NSSL gradient results are also compared with external results such as Barekat et al. (2014), Antia & Basu (2022), and Komm (2023). The absence of a null test for the regularization choices is a robustness or evidence limitation, not circularity.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard helioseismic forward and inversion assumptions plus a handful of hand-chosen regularization parameters and harmonic-degree cuts. No new physical entities are introduced. The main non-standard input is the authors' own dynamo model, used both to validate the inversion and to interpret the observed pattern; that self-citation is noted under circularity.

free parameters (6)
  • lambda_r (radial smoothing weight)
    Controls second-derivative smoothing in radius in Eq. (6); chosen by hand with no value or objective criterion such as L-curve or GCV reported.
  • lambda_t (temporal smoothing weight)
    Controls first-derivative smoothing in time in Eq. (6); chosen by hand, and no null test shows that it does not reshape the wave patterns.
  • Smoothing functions f(r) and g(r)
    Chosen as 1, r^-1, or r^-2; Appendix C shows the choice changes recovery of the s=3 component near the tachocline.
  • B-spline knot grid = 50 radial knots and a third temporal grid point as a knot
    The knot spacing sets the resolution of the joint time-radius solution and is chosen without a resolved sensitivity test.
  • MDI harmonic degree cutoff = ell < 120
    Modes above this cutoff are excluded to force consistency among instruments, which is a post hoc data-selection choice.
  • GONG harmonic degree cutoff = ell <= 150
    GONG data are limited to ell up to 150, which the paper states may cause discrepancies near the surface.
assumptions (5)
  • domain assumption Odd-order a-coefficients in the frequency-splitting expansion represent rotation without significant contamination from even-order asphericities or magnetic effects.
    Standard helioseismic assumption introduced through Eq. (3); the paper relies on it when inverting a-coefficients for zonal flows.
  • domain assumption The sensitivity kernels computed from generalized spherical harmonics are accurate for the forward and inverse mapping between splitting coefficients and rotation.
    Used in Eq. (5) and the Appendix; this is standard methodology but is not independently verified in this paper.
  • domain assumption Regularized least squares with second-derivative radial smoothing and first-derivative temporal smoothing is an appropriate constraint for solar zonal flows.
    The inversion is ill-posed, so smoothness is imposed through Eq. (6); the form of the regularizer is a modeling choice that affects the recovered wave pattern.
  • ad hoc to paper The Pipin & Kosovichev (2020) dynamo model produces a representative zonal-flow pattern for validating the inversion and for interpreting the observed pattern.
    The model is from the same research group and is used both to generate synthetic splittings and as the reference pattern for the observed dynamo-wave interpretation (Figures 8 and 11).
  • domain assumption Mode coverage with ell up to 150 and 120 resolves the near-surface shear layer at the quoted depths of 13 to 35 Mm.
    The paper acknowledges near-surface limitations for GONG and uses the assumption when presenting the radial gradient as a function of depth.

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Pith. "Pith review of Investigating Variations in Solar Differential Rotation by Helioseismology." pith.science (2026). https://pith.science/paper/DB7ECN6I

@misc{pith2026250510756,
  author       = {Pith},
  title        = {Pith review of: Investigating Variations in Solar Differential Rotation by Helioseismology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DB7ECN6I}},
  note         = {Machine review of arXiv:2505.10756}
}
read the original abstract

Helioseismic signatures of dynamo waves have recently been discovered in variations of the solar differential rotation, offering valuable insights into the type of dynamo mechanism operating in the solar convection zone. To characterize these variations, we analyze p-mode frequency-splitting data estimated using time intervals of various lengths to enhance the signal-to-noise ratio in inversions of zonal flows. We introduce a novel time-dependent inversion method that inherently smooths the solution over time, eliminating the need for separate post-processing smoothing. By applying this approach to observational data from the SOHO Michelson Doppler Imager, SDO Helioseismic Magnetic Imager, and Global Oscillation Network Group, we identify similar dynamo wave patterns in both the zonal acceleration and the zonal flow throughout the entire convection zone. Our analysis shows that while using longer time series smooths out temporal variations, the fundamental features observed in the short time series (i.e. 72-day long) persist when inverting datasets covering different time periods. These findings reinforce earlier detections and offer further validation of solar dynamo models. We additionally investigate the dimensionless radial gradient of rotation. Its value is close to -1 and increases in the deeper layers, remaining nearly constant from the equator to mid-latitudes within the depth range of 13 to 35 Mm below the surface; the results at high latitudes remain somewhat inconclusive. The variation of this quantity displays a torsional oscillation-like pattern, albeit with certain differences.

Figures

Figures reproduced from arXiv: 2505.10756 by the authors.

Figure 1
Figure 1. The mean differential rotation profile is shown from left to right using observations from GONG, MDI, and HMI, respectively. For all instruments, we use 5 × 72-day data sets based on the data products of Korzennik (2023). We have used entire time periods of each instrument. High latitudes (75◦ ) and deeper regions in the radiative interior are subject to large uncertainties due to poorly localized averaging kernels.… view at source ↗
Figure 2
Figure 2. Evolution of solar zonal flow in the solar convection zone from the analysis of MDI and HMI combined data sets. We use 5 × 72-days datasets from Korzennik (2023) for this analysis. We plot the torsional oscillation velocity (measured in m/s) for different time (mentioned as well in each panel) to illustrate the evolution of zonal flow. The total time span is chosen to extend beyond the duration of Solar Cycle 23, en… view at source ↗
Figure 3
Figure 3. Zonal flow acceleration, derived using the GONG observations, plotted as a function of radius and time for various latitudes (15◦ , 30◦ , 45◦ , and 60◦ ). The panels on the left present results derived from 8 × 72-day datasets, while the panels on the right correspond to the analysis using 4 × 72-day datasets, both from Korzennik (2023), derived from GONG observation. high-latitude and near-equatorial features. In e… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: For comparison, the zonal flow is shown using 8 ×72-day (left) and 4 ×72-day (right) datasets, both from Korzennik (2023). 0.1 0.4 0.7 1.0 r/ R 1997-11 1999-02 2000-09 2001-11 0.1 0.4 0.7 1.0 r/R 0.1 0.4 0.7 1.0 r/ R 2004-03 0.1 0.4 0.7 1.0 r/R 2006-12 0.1 0.4 0.7 1.0 …
Figure 5
Figure 5. Figure 5: The acceleration of the zonal flow is shown as a function of radius and latitude at various time instances, as indicated in the legend of each panel. Note the difference between [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: We compare the acceleration of zonal flow, V˙ , at latitudes 15◦ (left panel), 30◦ (middle panel) and 50◦ (right panel) across multiple depths (as indicated in the title of each panel) using two different GONG dataset lengths: 5 × 72-day, and 8 × 72-day (black and red …
Figure 7
Figure 7. Figure 7: We compare evolution of zonal flow V (r, θ, t), at latitudes 15◦ (left panel), 30◦ (middle panel) and 50◦ (right panel) across multiple depths (as indicated in the title of each panel) using two different GONG dataset lengths: 5 × 72-day, and 8 × 72-day (black and red …
Figure 8
Figure 8. Figure 8: The left panels show the variation in the dimensionless radial gradient, ∇r(Ω), as obtained from the dynamo model of Pipin & Kosovichev (2020). The right panels display the corresponding results derived from the inversion, using the dynamo model profile as input. Resul…
Figure 9
Figure 9. Figure 9: The mean radial gradient of rotation, ∇r(Ω) is shown as a function of latitude at three different depths—0.95, 0.97, and 0.99 R⊙ —represented by blue, red, and black curves, respectively. The data are taken from GONG, MDI, and HMI observations, indicated by dashed, das…
Figure 10
Figure 10. Figure 10: We plot ∇r(Ω) as a function of radius at several different latitudes (indicated in the title of each panel). The left panel shows the results using the data series from Korzennik (2023), while the right panel shows the results using the data series from the JSOC pipel…
Figure 11
Figure 11. Figure 11: The variation of angular velocity gradient, ∇r(Ω) as a function of time and latitude for three different depths as indicated at the top of each panel. Sunspot locations are shown as gray dots overlaid on the plot to facilitate comparison with the magnetic butterfly di…
Figure 12
Figure 12. Figure 12: Zonal flow from the dynamo model undergoes spherical transformation according to Equation 4 to obtain ws. We use these values to perform forward modeling and inversion. The inverted profile is then compared with the original profile in this Figure for harmonic degrees…
Figure 13
Figure 13. Figure 13: Inversion results using different functional forms of f(r) in Equation 6 are shown: the top panel uses f(r) = r −1 , while the bottom panel uses f(r) = r −2 for smoothing. We compare the inverted profiles with the original ones for harmonic degrees s = 1 and s = 3. Ho…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.