Pith. sign in

REVIEW 4 major objections 4 minor 46 references

Properties of solar Rossby waves from normal mode coupling and characterizing its systematics

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Twenty years of normal-mode coupling data from two solar observatories yield frequencies, line widths, and amplitudes for sectoral Rossby modes, with surface root-mean-square velocities near 0.5 m/s.

desk verdict A solid incremental extension of the authors' Rossby-wave mode-coupling work to 20 years of data, with a useful leakage kernel, but the abstract's 0.5 m/s amplitude needs reconciling with the body's ~4 m/s, and leakage+noise are never tested together. read the letter →

arxiv 1908.05890 v2 pith:XBVMMDCV submitted 2019-08-16 astro-ph.SR

Krishnendu Mandal , Shravan Hanasoge This is my paper · ORCID
classification astro-ph.SR
keywords solarRossbywavesnormalmodecouplinghelioseismologyconvectionzoneMDIHMIleakagecorrectionr-modes
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

Solar Rossby waves are large-scale, retrograde waves in the convection zone whose restoring force is the Coriolis force; in rotating fluids they carry angular momentum, so measuring their properties tests the physics of solar and stellar interiors. This paper tries to measure their frequencies, line widths, and amplitudes by coupling acoustic normal modes over 20 years of space-based Doppler observations: 12 years from MDI and 8 years from HMI. Because we observe only the visible hemisphere, spatial windowing mixes neighboring spherical-harmonic modes, so the authors build a leakage-correction scheme and validate it with synthetic tests before applying it to the data. They report that the measured sectoral modes follow the classical dispersion relation $\sigma_s = 2\Omega/(s+1)$ closely enough to identify each ridge with a Rossby mode, and that the root-mean-square velocity at the surface is of order 0.5 m/s. A sympathetic reader would care because this turns a recent detection into a 20-year, two-instrument catalog of mode parameters that can be compared with theory and with independent helioseismic measurements.

What carries the argument

The measurement rests on normal-mode coupling: cross-correlating spherical-harmonic coefficients of line-of-sight Doppler velocity, $\langle \varphi^*_{\ell m}(\omega)\varphi_{\ell,m+t}(\omega+\sigma)\rangle$, and condensing the correlations into $B$-coefficients through a weighted least-squares fit. The central object is the leakage matrix $L^{\ell' m'}_{\ell m}$, which describes how the spatial window of the visible disk mixes a mode $(\ell,m)$ into other modes; from it the paper constructs the new diagonal sensitivity kernel $\Theta^s_s(n,\ell,\sigma,r)$ of Eq. (19), which connects the observed $B$-coefficients directly to the toroidal Rossby velocity profile $w_{s,-s}(r)$. A Wigner $3j$ symbol enforces the angular-momentum selection rules in the coupling. This kernel, combined with optimally localized averaging and regularized least-squares inversions, is what lets the authors convert the measured correlations into depth-dependent amplitudes and assess the leakage systematics.

What would settle it

Build a synthetic data set from a known sectoral Rossby depth profile plus a second non-sectoral mode ($s\neq|t|$) just below the current detection threshold, run the paper's diagonal-only inversion, and check whether the recovered frequency and amplitude shift by more than the quoted uncertainties; any such shift would falsify the sectoral-only assumption. A cheaper check: compare MDI and HMI measurements over their overlapping 2010–2011 data after cross-calibrating the two instruments, since instrument systematics would not produce the same parameter differences as genuine solar-cycle variation.

Watch

Extended reading notes

Core claim

The paper claims that normal-mode coupling of 12 years of SOHO/MDI and 8 years of SDO/HMI Doppler observations yields a consistent set of solar Rossby-mode parameters for sectoral modes of odd harmonic degree $s=1,3,5,\ldots,15$ (with $s=13$ not fitted for HMI). The frequencies in the co-rotating frame are close to the sectoral dispersion relation $\sigma_s=2\Omega/(s+1)$, for example $233\pm3$ nHz for $s=3$ from HMI and $249\pm0.4$ nHz from MDI; line widths are typically 5–70 nHz, amplitudes decline with $s$, and the surface root-mean-square velocity of the modes is about 0.5 m/s. The load-bearing methodological claim is that leakage from observing only part of the Sun can be handled by a new diagonal sensitivity kernel $\Theta^s_s$ built from the leakage matrix, and that synthetic inversions using this kernel recover input depth profiles better than the earlier no-leakage approximation, which may bias amplitudes by up to a factor of a few. The authors deliberately do not combine the two data sets because MDI and HMI are not cross-calibrated for this measurement, and they stop short of declaring the $s=1$ mode a detection because its frequency coincides with the tracking rate.

Load-bearing premise

The analysis assumes that only sectoral Rossby waves exist in the low-frequency band and that leakage from neighboring harmonic degrees is negligible there, so the inversion can keep only the diagonal kernel; if non-sectoral waves or that off-diagonal leakage are significant, the inferred frequencies and amplitudes will be biased.

Editorial extensions

If this is right

  • If the measurements are right, the tabulated frequencies give a direct test of the sectoral dispersion relation $\sigma_s=2\Omega/(s+1)$ for the Sun's interior Rossby waves.
  • The measured line widths and amplitudes provide the first long-baseline reference set for modeling wave damping and excitation in the convection zone.
  • A surface root-mean-square velocity of about 0.5 m/s places a concrete scale on the angular-momentum transport that Rossby waves could mediate in the Sun and other stars.
  • The diagonal leakage kernel offers a reusable correction for future normal-mode-coupling measurements of weakly excited sectoral flows.
  • The MDI–HMI parameter differences, if not instrument systematics, would encode solar-cycle dependence, but the paper does not claim that without a cross-calibration.

Reading between the lines

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

  • A reader could push further: inject a non-sectoral mode ($s\neq|t|$) just below the detection threshold into the paper's synthetic pipeline; if the diagonal-only inversion shifts the recovered sectoral frequency or amplitude beyond the quoted errors, the sectoral-only ansatz would need to be relaxed.
  • Cross-calibrating MDI and HMI normal-mode coupling over their overlapping years (2010–2011) would turn the two-instrument parameter differences into a clean solar-cycle test rather than an instrument ambiguity.
  • Coupling acoustic modes of different harmonic degree with $\delta\ell=1,3,\ldots$ should bring the even-harmonic-degree Rossby modes into view, which the identical-degree coupling used here cannot see.
  • The leakage pattern that moves a mode at $\sigma_s+2\Omega$ suggests a template for detecting other large-scale sectoral flows: any rotating sectoral perturbation will reappear at shifted temporal frequencies.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper applies normal-mode coupling to 20 years of MDI and HMI Doppler observations (1999–2018) to measure frequencies, linewidths, and amplitudes of solar Rossby modes. It extends an earlier analysis by Hanasoge & Mandal (2019), using a longer baseline and two instruments, and introduces a leakage-correction scheme based on a diagonal sensitivity kernel (Eq. 19). The method is validated with synthetic inversions: one noiseless test that includes leakage and one noise test that ignores leakage. The authors report frequencies consistent with the classical Rossby dispersion relation for several low-degree modes, compare them with the independent studies of Löptien et al. (2018) and Liang et al. (2018), and quote a surface rms velocity of about 0.5 m/s. They also investigate whether the s=1 mode could be a tracking-rate artifact and conclude it is likely real while remaining cautious. The two datasets are analyzed separately and not combined.

Significance. If the result holds, this paper provides the first long-baseline, two-instrument characterization of solar Rossby-mode frequencies, linewidths, and amplitudes, and it demonstrates a practical way to handle spatial leakage in normal-mode coupling. The strengths are the explicit comparison with the analytic dispersion relation, the use of two independent datasets, the careful discussion of the s=1 systematics, and the synthetic tests that at least partially validate the inversion pipeline. The main caveat is that the validation is split: leakage and noise are never tested together, and the inversion forward model and leakage matrices come from the authors' own earlier papers, so the synthetic tests do not independently validate the forward model. The frequency measurements, however, are checked against external results, which supports the central detection claim.

major comments (4)
  1. [Section 4.2, Figures 3 and 4] The validation of the leakage-correction scheme is split. Figure 3 tests the diagonal-kernel inversion (Eq. 19) on noiseless synthetic data that include leakage, while Figure 4 tests inversion with realistic noise but assumes Eq. (7), i.e., no leakage. No test includes both leakage and noise, even though both are present in the actual observations. The paper itself concedes in Section 4.2 that ignoring leakage biases amplitudes by at most a factor of a few, so the statistical error bars in Tables 1 and 2 (e.g., √A = 70 ± 24 cm/s for s=3 HMI) do not include the leakage systematic. The claim that the leakage-correction scheme makes the measurements reliable requires either a combined leakage-plus-noise synthetic test or an explicit propagation of the factor-of-few amplitude uncertainty into the tabulated parameters.
  2. [Section 3, Eq. (8), and Section 4.2] The analysis assumes that only sectoral Rossby modes are present, so b^σ_st ≈ δ_{s,-t}, and that in the 0–0.5 µHz band the off-diagonal leakage terms Θ_{s'}^s with s' ≠ s are negligible, allowing the inversion to use only the diagonal kernel (Eq. 19). The synthetic tests are built from the same sectoral-mode assumption and the same leakage model, so they do not test whether non-sectoral power or non-negligible off-diagonal leakage would bias the recovered frequencies and amplitudes. I ask for a sensitivity test that injects non-sectoral or off-diagonal-leakage components at a few percent of the sectoral amplitude and quantifies the resulting bias.
  3. [Section 3, before Eq. (7)] The asymptotic kernel used in Eq. (7) is stated to be valid only when s ≪ ℓ or s ≪ ℓ′. With ℓ ∈ [10,180] and s ≤ 20, there are contributing pairs with s ≈ ℓ (for example ℓ = 11, s = 11), for which Hanasoge (2018) finds the asymptotic kernel to be less accurate. The paper justifies the asymptotic form only by noting s ≤ 20, which is insufficient for the lowest harmonic degrees. The authors should either restrict the inversion to ℓ ≫ s, or quantify the bias from low-ℓ modes using exact kernels.
  4. [Tables 1 and 2] Several fitted frequencies deviate strongly from the classical dispersion relation and from the earlier measurements the paper cites. For example, HMI s=11 gives 54 ± 2 nHz versus 75.5 nHz theoretical and 75 ± 7 nHz in Löptien et al., and HMI s=15 gives 18 ± 1 nHz versus 56.6 nHz theoretical. Since the identification of these features as Rossby modes rests partly on agreement with the dispersion relation, these outliers need discussion (e.g., misidentification, line blending, or systematic frequency shifts) before the tabulated frequencies can be taken as a reliable characterization of Rossby-mode properties.
minor comments (4)
  1. [Equation (20)] Equation (20) uses τ/2 in the Lorentzian denominator, but the text defines Γ as the full width at half maximum. The relationship between τ and Γ is never stated; please use one symbol consistently.
  2. [Section 4.3] The sentence "Fitted spectrum for HMI and MDI are shown in Figure (6) and (7) respectively" has a subject-verb mismatch. Also, "line-widths" in the abstract and elsewhere should be written consistently as "linewidths" or "line widths".
  3. [Figure 3 caption] The notation Θ^{s,-s}_{s,-s} in the right panel is confusing and does not clearly match the kernels defined in Eqs. (17)–(19). Please align the notation between the equations and the figure caption.
  4. [Section 4.4, Figure 8] The right panel of Figure 8 shows the s=1, t=1 power spectrum without error bars or a quantitative upper limit. A noise level or significance threshold would make the argument that there is no spurious power at 453 nHz more compelling.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the measured Rossby-mode parameters are data-driven and cross-checked against analytic dispersion and independent prior detections.

full rationale

The derivation chain is not circular. The B-coefficients are defined directly from observed MDI/HMI Doppler time series (Eq. 1), and the inversion in Section 3 (Eqs. 12-19) maps them to the Rossby flow profile w via sensitivity kernels. The reported frequencies, linewidths, and amplitudes are obtained by Lorentzian fits to the resulting power spectra in Section 4.3 (Eq. 20). No fitted parameter is relabeled as a prediction: the theoretical dispersion relation 2Ω/(s+1) is used only as a comparison curve (Fig. 5), and the frequencies are checked against the independent measurements of Löptien et al. and Liang et al. The leakage treatment is tested with synthetic B-coefficients generated from the same forward model, but that is a self-consistency test of the inversion, not a derivation of the observed quantities from the model; the observed quantities remain empirically determined. Self-citations to Hanasoge (2018) and Hanasoge & Mandal (2019) provide the mode-coupling and leakage formalism, but no load-bearing uniqueness theorem or ansatz is imported by those citations: the sectoral-mode and diagonal-leakage assumptions are stated and examined, and any weakness there is a validation concern, not circularity. The paper even attaches caveats to the s=1 mode in Section 4.4, showing that the measured signal is not forced by the method. No step reduces by definition to its own inputs, so the central claim is not circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central measurements rely on the toroidal-flow description of Rossby waves, the sectoral-mode dominance, and the Gauss-Markov error model. The tracking rate, regularization parameters, and B-spline discretization are free analysis choices. No new physical entities are introduced.

free parameters (3)
  • Tracking rate Ω = 453 nHz
    Chosen as the equatorial rotation rate; not fitted, but affects the co-rotating frame frequencies and the location of leakage peaks.
  • Regularization parameter λ (OLA/RLS inversions) = not specified
    Regularization parameters for inversions in Sections 3.1 and 3.2 are chosen but values are not reported; they affect the inferred depth profiles and amplitudes.
  • B-spline knot count = 50 knots
    The RLS inversion expands the flow profile in 50 cubic B-spline knots up to depth 0.1 R⊙; this is a discretization choice that can influence the resolved depth structure.
assumptions (5)
  • domain assumption Rossby waves are accurately described as a toroidal flow with velocity u = w(r) r̂ × ∇_h Y_st (Equation 4).
    This restricts the measurement to toroidal components and ignores poloidal or compressible contributions to the wave field.
  • domain assumption Only sectoral modes (s=|t|) have significant power; non-sectoral modes are absent or below detection (b^σ_st ≈ δ_{s,-t}).
    Based on prior studies (Löptien et al. 2018, Hanasoge and Mandal 2019); permits simplification of the general coupling equation to Equation (8).
  • domain assumption Errors in the mode-coupling measurements are uncorrelated with equal variance and zero mean (Gauss-Markov theorem).
    Invoked in Section 2 to justify that the B-coefficient estimator is unbiased; real solar noise may not satisfy these conditions exactly.
  • domain assumption The asymptotic kernel of Vorontsov (2011) is accurate for s << ℓ; here s ≤ 20 and ℓ ≥ 10.
    Hanasoge (2018) showed the asymptotic kernel is accurate when s < ℓ, with deviations when s ≈ ℓ; the paper justifies this range.
  • domain assumption The Sun rotates as a solid body at 453 nHz for the purpose of tracking features.
    The co-rotating frame uses a single equatorial rotation rate, neglecting differential rotation, which can shift apparent mode frequencies.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Properties of solar Rossby waves from normal mode coupling and characterizing its systematics." pith.science (2026). https://pith.science/paper/XBVMMDCV

@misc{pith2026190805890,
  author       = {Pith},
  title        = {Pith review of: Properties of solar Rossby waves from normal mode coupling and characterizing its systematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XBVMMDCV}},
  note         = {Machine review of arXiv:1908.05890}
}
read the original abstract

Rossby waves play an important role in mediating the angular momentum of rotating spherical fluids, creating weather on Earth and tuning exoplanet orbits in distant stellar systems (Ogilvie 2014). Their recent discovery in the solar convection zone provides an exciting opportunity to appreciate the detailed astrophysics of Rossby waves. Large-scale Rossby waves create subtle drifts in acoustic oscillations in the convection zone, which we measure using helioseismology to image properties of Rossby waves in the interior. We analyze 20 years of space-based observations, from 1999 to 2018, to measure Rossby-mode frequencies, line-widths, and amplitudes. Spatial leakage affects the measurements of normal model coupling and complicates the analysis of separating out specific harmonic degree and the azimuthal number of features on the Sun. Here we demonstrate a novel approach to overcome this difficulty and test it by performing synthetic tests. We find that the root-mean-square velocity of the modes is of the order of 0.5 m/s at the surface.

Figures

Figures reproduced from arXiv: 1908.05890 by the authors.

Figure 1
Figure 1. Rossby modes of harmonic degree s and frequency σs leak into degree s + 2 with frequency σs + 2Ω. Red dashed lines in the lower and upper parts of the figure show the classical Rossby-wave dispersion relation (Equation 16) and leakage of those modes into higher frequencies. frequency bin in the range [0.0, 0.5] µHz and since the contribution to our desired frequency bins from neighbouring modes is negligible in that… view at source ↗
Figure 2
Figure 2. Left panel displays leakage of modes when tracking is not applied. Leakage then occurs at the same frequency. The right panel displays leakage of modes when tracking is considered and bears a strong resemblance to observations, i.e. Figure (1). In both cases, modes with odd harmonic degrees leak into neighbouring odd harmonic degrees.       ⊙            r ⊙ [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 3
Figure 3. Left panel shows inversion results without noise. The input profile is plotted using a solid line and the inverted profile with and without leakage are marked by dashed and dot-dashed lines respectively. In the right panel we compare kernels, f0,sKnℓ(r) (red dashed line) with Θs,−s s,−s (black solid line) for s = 7. It can be seen that two kernels are of the same shape but slightly differing in magnitude from each o… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Plot shows the inversion result with noise. In this case, we ignore leakage. The black solid line is the original profile we put in. The red dashed line is the inferred profile and corresponding error ( ±1σ around the mean) in the inferred profile is shown by orange sh…
Figure 5
Figure 5. Figure 5: The left panel shows the normalized average power spectrum of Rossby waves at depth 0.98R⊙ by analyzing 12 years of MDI data divided into three four-year chunks. The right panel shows the same as the left panel but with 8 years of HMI data divided into two four-year ch…
Figure 6
Figure 6. Figure 6: Averaged power spectrum from the analysis of SDO/HMI data (blue solid line with cycle). We fit a Lorentzian profile with a constant background to the power spectrum as described in section (4.3). The fitted parameter values are tabulated in [PITH_FULL_IMAGE:figures/fu…
Figure 7
Figure 7. Figure 7: Same as in Figure (6). Averaged power spectrum (blue solid line with cycle) and corresponding fit (red solid line) from the MDI analysis. The fitted parameter values are listed in Table (2). We have extended the work by Hanasoge & Mandal (2019) by analyzing 8 years of …
Figure 8
Figure 8. Figure 8: Left panel:Frequencies of the mode s = 1 (circles) for different tracking rates and corresponding theoretical values (triangle) are plotted. Right panel: we plot the power for harmonic degree, s = 1 and azimuthal number, t = 1. In the spectra, we do not find any extra …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 38 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month note number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.co...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION format.doi doi empty "" "doi:" doi * if FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix ":" * if eprint field.or.null * if FUNCTION format.pid eprint empty format.doi format.eprint if FUNCTION n.dashify 't := "" t...

  3. [3]

    hY @d좖I ,f]:y?Dj D NZ G D iXXm #! \ tIUj PXͯ!bu)Ȥ 'ST+VU 4H AmˍָkQ<F1Hݐ-@لV xu*JE vD32 )b鐙 AYhj6,Th qxjAx 'ʔ1? I Wc nL4GP 2÷ji0,aֆ `sg(V

    thebibliography [1] 20pt to REFERENCES 6pt =0pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command Each re...

    work page arXiv 2017

  4. [4]

    R., Duvall , Jr., T

    Anderson , E. R., Duvall , Jr., T. L., & Jefferies , S. M. 1990, , 364, 699

    work page 1990

  5. [5]

    E., & Gilbert, J

    Backus, G. E., & Gilbert, J. 1967, Geophysical Journal International, 13, 247

    work page 1967

  6. [6]

    J., & Tolhoek , H

    Brussaard , P. J., & Tolhoek , H. A. 1957, Physica, 23, 955

    work page 1957

  7. [7]

    B., & Schlax , M

    Chelton , D. B., & Schlax , M. G. 1996, Science, 272, 234

    work page 1996

  8. [8]

    A., & Tromp , J

    Dahlen , F. A., & Tromp , J. 1998, Theoretical Global Seismology ( Princeton University Press )

    work page 1998

Show all 46 references
  1. [9]

    A., & McIntosh , S

    Dikpati , M., Belucz , B., Gilman , P. A., & McIntosh , S. W. 2018 a , , 862, 159

    2018

  2. [10]

    W., Bothun , G., et al

    Dikpati , M., McIntosh , S. W., Bothun , G., et al. 2018 b , , 853, 144

    2018

  3. [11]

    L., Jefferies , S

    Duvall , Jr., T. L., Jefferies , S. M., Harvey , J. W., Osaki , Y., & Pomerantz , M. A. 1993 a , , 410, 829

    1993

  4. [12]

    L., Jefferies , S

    Duvall , Jr., T. L., Jefferies , S. M., Harvey , J. W., & Pomerantz , M. A. 1993 b , , 362, 430

    1993

  5. [13]

    2018, , 861, 46

    Hanasoge , S. 2018, , 861, 46

    2018

  6. [14]

    2019, arXiv e-prints, arXiv:1901.06479

    Hanasoge , S., & Mandal , K. 2019, arXiv e-prints, arXiv:1901.06479

    2019 arXiv

  7. [15]

    Hanasoge , S. M. 2017, , 470, 2780

    2017

  8. [16]

    M., Woodard , M., Antia , H

    Hanasoge , S. M., Woodard , M., Antia , H. M., Gizon , L., & Sreenivasan , K. R. 2017, , 470, 1404

    2017

  9. [17]

    1988, , 333, 996

    Hill , F. 1988, , 333, 996

    1988

  10. [18]

    R., Armstrong , J

    Kuhn , J. R., Armstrong , J. D., Bush , R. I., & Scherrer , P. 2000, , 405, 544

    2000

  11. [19]

    M., & Ritzwoller , M

    Lavely , E. M., & Ritzwoller , M. H. 1992, Philosophical Transactions of the Royal Society of London Series A, 339, 431

    1992

  12. [20]

    C., Duvall , Jr., T

    Liang , Z.-C., Birch , A. C., Duvall , Jr., T. L., Gizon , L., & Schou , J. 2017, , 601, A46

    2017

  13. [21]

    C., & Duvall , Jr, T

    Liang , Z.-C., Gizon , L., Birch , A. C., & Duvall , Jr, T. L. 2018, arXiv e-prints, arXiv:1812.07413

    2018 arXiv

  14. [22]

    C., et al

    L \"o ptien , B., Gizon , L., Birch , A. C., et al. 2018, Nature Astronomy, arXiv:1805.07244

    2018 arXiv

  15. [23]

    W., Cramer , W

    McIntosh , S. W., Cramer , W. J., Pichardo Marcano , M., & Leamon , R. J. 2017, Nature Astronomy, 1, 0086

    2017

  16. [24]

    Ogilvie , G. I. 2014, , 52, 171

    2014

  17. [25]

    Papaloizou , J., & Pringle , J. E. 1978, , 182, 423

    1978

  18. [26]

    1981, , 94, 126

    Provost , J., Berthomieu , G., & Rocca , A. 1981, , 94, 126

    1981

  19. [27]

    Rossby , C. 1939, J. Marine Res. 2, 38

    1939

  20. [28]

    2003, , 405, 779

    Roth , M., & Stix , M. 2003, , 405, 779

    2003

  21. [29]

    2008, , 251, 77

    ---. 2008, , 251, 77

    2008

  22. [30]

    1982, , 256, 717

    Saio , H. 1982, , 256, 717

    1982

  23. [31]

    2011, , 734, 97

    Schad , A., Timmer , J., & Roth , M. 2011, , 734, 97

    2011

  24. [32]

    2013, , 778, L38

    ---. 2013, , 778, L38

    2013

  25. [33]

    H., Bush , R

    Schou , J., Scherrer , P. H., Bush , R. I., et al. 2012, , 275, 229

    2012

  26. [34]

    1981, , 78, 483

    Smeyers , P., Craeynest , D., & Martens , L. 1981, , 78, 483

    1981

  27. [35]

    A., Bush , R., Gough , D

    Sturrock , P. A., Bush , R., Gough , D. O., & Scargle , J. D. 2015, , 804, 47

    2015

  28. [36]

    Ulrich , R. K. 2001, , 560, 466

    2001

  29. [37]

    1989, Nonradial oscillations of stars , 2nd edn

    Unno , W., Osaki , Y., Ando , H., Saio , H., & Shibahashi , H. 1989, Nonradial oscillations of stars , 2nd edn. (University of Tokyo Press)

    1989

  30. [38]

    Vorontsov, S. V. 2011, Monthly Notices of the Royal Astronomical Society, 418, 1146. + http://dx.doi.org/10.1111/j.1365-2966.2011.19564.x

    2011

  31. [39]

    2014, , 289, 1085

    Woodard , M. 2014, , 289, 1085

    2014

  32. [40]

    C., & Larson , T

    Woodard , M., Schou , J., Birch , A. C., & Larson , T. P. 2013, , 287, 129

    2013

  33. [41]

    Woodard , M. F. 1989, , 347, 1176

    1989

  34. [42]

    2006, , 649, 1140

    ---. 2006, , 649, 1140

    2006

  35. [43]

    2016, , 460, 3292

    ---. 2016, , 460, 3292

    2016

  36. [44]

    V., Carbonell , M., Oliver , R., & Ballester , J

    Zaqarashvili , T. V., Carbonell , M., Oliver , R., & Ballester , J. L. 2010 a , , 709, 749

    2010

  37. [45]

    2010 b , , 724, L95

    ---. 2010 b , , 724, L95

    2010

  38. [46]

    V., Oliver , R., Hanslmeier , A., et al

    Zaqarashvili , T. V., Oliver , R., Hanslmeier , A., et al. 2015, , 805, L14

    2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.