Effects of density stratification on Rossby waves in deep atmospheres

Bradley W. Hindman; Catherine C. Blume

arxiv: 2511.03832 · v2 · submitted 2025-11-05 · 🌌 astro-ph.SR

Effects of density stratification on Rossby waves in deep atmospheres

Catherine C. Blume , Bradley W. Hindman This is my paper

Pith reviewed 2026-05-18 00:37 UTC · model grok-4.3

classification 🌌 astro-ph.SR

keywords Rossby wavessolar interiordensity stratificationradial eigenfunctionsbeta-plane approximationLagrangian pressure fluctuationwave cavitiesModel S

0 comments

The pith

Rossby waves trapped in the Sun's radiative interior maintain nearly constant vorticity through the entire convection zone.

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

The paper derives a vertical structure equation for Rossby waves in a generally stratified atmosphere using the beta-plane approximation and the Lagrangian pressure fluctuation as the primary variable. It applies this to Model S and finds two distinct wave cavities, one in the radiative interior and one in the convection zone. For modes localized in the radiative cavity the associated radial vorticity eigenfunctions turn out to be almost flat across the convection zone. A sympathetic reader would care because Rossby waves have already been detected at the solar surface; a constant interior vorticity profile would make those surface signals a direct window into the radiative interior.

Core claim

In a deep, generally stratified solar model the vertical wave equation for Rossby waves possesses two propagation cavities. The radial vorticity eigenfunctions belonging to the radiative-interior cavity remain nearly constant with radius throughout the overlying convection zone.

What carries the argument

The vertical structure equation expressed in the Lagrangian pressure fluctuation δP, which yields singularity-free eigenfunctions and exposes the two-cavity structure under the β-plane approximation.

If this is right

Rossby-wave propagation occurs in separate radial cavities rather than throughout the whole interior.
Modes trapped in the radiative interior can still produce observable surface vorticity signals because their eigenfunctions do not decay in the convection zone.
The Lagrangian-pressure formulation removes internal singularities that appeared in earlier formulations.
General density stratification produces wave behavior that differs from both purely radiative and purely convective models.

Where Pith is reading between the lines

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

Surface observations of Rossby waves could be used to infer properties of the radiative interior without needing to resolve the convection zone directly.
The same vertical-equation approach might be applied to other stars that possess both a radiative core and a convective envelope.
If the near-constant vorticity holds, it suggests a dynamical decoupling between the radiative interior and the convection zone for these particular waves.

Load-bearing premise

The beta-plane approximation remains valid over the full radial extent of the solar interior when the vertical structure equation is derived for a generally stratified fluid.

What would settle it

A direct measurement, from helioseismic or surface-flow data, showing that the radial vorticity profile of a low-frequency Rossby wave varies significantly with depth inside the convection zone.

Figures

Figures reproduced from arXiv: 2511.03832 by Bradley W. Hindman, Catherine C. Blume.

**Figure 1.** Figure 1: Propagation diagram for Model S—The orange (purple) region denotes propagation for Rossby waves in the radiative interior (convection zone). The blue (yellow) lines mark the possible values of the separation constant in each region respectively. The inset shows the very top of the convection zone, with y-axis values ranging from −5×10−10 to 0. The thick black curve corresponds to the vertical profile of ω… view at source ↗

**Figure 2.** Figure 2: Radial eigenfunctions for Rossby waves—Radial eigenfunctions for the radiative interior modes (top) and convection zone modes (bottom) in Lagrangian pressure fluctuation δP (left column), reduced pressure δP/ρ0 (middle column) and radial vorticity ζz(right column). The radiative interior mode eigenfunctions are distributed across the region and are roughly constant in the convection zone. The convection zo… view at source ↗

**Figure 3.** Figure 3: Fractional frequency shift—The fractional frequency shift (ωn − ω2D)/ω2D with respect to the two-dimensional dispersion relation for modes of the (a) radiative interior and (b) convection zone. The figure was generated for a low-latitude β-plane located at θ = 10 degrees. The black dashed line represents the asymptotic behavior for large m. The frequency shift for the radiative interior modes is so small a… view at source ↗

**Figure 4.** Figure 4: Artificial spectra—Artificial spectra calculated for the ℓ = m = 3 mode at about (a) 0.03 nHz and (b) 3 nHz. Each panel displays the first 11 modes of increasing radial order for both the radiative interior and convection zone families given an arbitrary linewidth of 0.1 nHz and amplitude falling off like 1/n n. The zeroth order mode for the radiative interior (convection zone) family is marked with an ora… view at source ↗

read the original abstract

Though Rossby waves have been observed on the Sun, their radial eigenfunctions remain a mystery. The prior theoretical work either considers quasi-2D systems, which do not apply to the solar interior, or only considers fully radiative or fully convective atmospheres. This project calculates the radial eigenfunctions for Rossby waves in a deep atmosphere for a general stratification. Here, we use the $\beta$-plane approximation to derive a vertical equation in terms of the Lagrangian pressure fluctuation $\delta P$, and we then calculate radial eigenfunctions for Rossby waves in a standard solar model, Model S. We find that working in the Lagrangian pressure fluctuation results in cleaner wave equations that lack internal singularities that have been encountered in prior work. The resulting radial wave equation makes it abundantly clear that there are two wave cavities in the solar interior, one in the radiative interior and another in the convection zone. Surprisingly, our calculated radial vorticity eigenfunctions for the radiative interior modes are nearly constant throughout the convection zone, raising the possibility that they may be observable at the solar surface.

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.

Desk Editor's Note private letter to a colleague

The paper finds two Rossby-wave cavities in a stratified solar model and reports nearly constant vorticity for radiative-interior modes across the convection zone.

read the letter

The main point is that Rossby waves in a deep, stratified atmosphere split into two separate cavities, one in the radiative interior and one in the convection zone, and that the radial vorticity eigenfunctions for the radiative modes stay nearly flat through the convection zone. That flatness is what raises the possibility of surface signals from interior modes. The calculation uses the beta-plane equations on Model S and works with the Lagrangian pressure fluctuation to get the vertical structure equation. This approach produces cleaner equations without the internal singularities that appeared in earlier treatments. The explicit radial eigenfunctions for general stratification are new compared with the quasi-2D or uniform-layer cases cited in the abstract. The paper does a straightforward job showing how the two cavities appear once realistic density variation is included and how the constant-vorticity behavior follows from the numerics for the radiative modes. The soft spot is the beta-plane approximation itself when it is stretched over the full radial domain. The stress-test note correctly flags that curvature, changes in the local vertical, and strong stratification could make the local Cartesian frame and linear beta less reliable from the radiative interior out to the surface. Without seeing detailed checks or error estimates on that approximation, the eigenfunctions and the surface-observability suggestion rest on an assumption that may need more defense. The derivation starts from standard equations and a standard solar model with no obvious fitted parameters, so the circularity burden looks low. This paper is for helioseismologists and solar-dynamics modelers who need concrete radial structures rather than uniform-layer idealizations. A reader already working on Rossby-wave observations or interior wave cavities will get direct value from the eigenfunctions and the cavity picture. It deserves a serious referee because it supplies a missing calculation on a realistic stratification even though the beta-plane range is open to question. I would send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript derives a vertical structure equation for Rossby waves in a generally stratified deep atmosphere by adopting the β-plane approximation and working with the Lagrangian pressure fluctuation δP. Applied to the standard solar model (Model S), the resulting eigenproblem reveals two distinct wave cavities (radiative interior and convection zone). The central result is that radial vorticity eigenfunctions for modes trapped in the radiative interior remain nearly constant throughout the convection zone, raising the prospect of surface observability.

Significance. If the derivation and eigenfunctions hold, the work supplies the first explicit radial structure for Rossby waves across a realistic solar stratification, moving beyond quasi-2D or uniform-atmosphere models. The Lagrangian-pressure formulation is credited for producing singularity-free equations, and the reported near-constant vorticity in the convection zone constitutes a falsifiable prediction that could link interior modes to surface observations.

major comments (2)

[derivation of the vertical equation (abstract and §2)] The central claim that radiative-interior modes have nearly constant vorticity eigenfunctions across the convection zone rests on the vertical structure equation derived under the β-plane approximation extended over the full radial domain (radiative interior through convection zone). This approximation replaces spherical geometry with a local Cartesian frame and assumes linear latitudinal variation of the Coriolis parameter; the manuscript does not quantify the size of neglected curvature or radial-vertical reorientation terms in a strongly stratified medium, which directly affects whether the reported eigenfunction constancy is robust.
[vertical wave equation for δP] The statement that the Lagrangian-pressure formulation removes internal singularities is load-bearing for the cleanliness of the wave equation and the subsequent eigenfunction calculation. Without an explicit side-by-side comparison (e.g., the singular terms that appear in the Eulerian formulation versus their absence here) or a demonstration that the new equation remains regular at all turning points in Model S, the improvement cannot be verified from the given derivation.

minor comments (2)

[abstract and §3] The reference to 'Model S' should include the full citation (Christensen-Dalsgaard et al.) at first mention for reproducibility.
[figures] Figure captions for the eigenfunction plots should explicitly state the normalization and the radial range shown (e.g., 0.2–1.0 R⊙) to allow direct comparison with the text claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We address the two major comments below, clarifying the scope of the β-plane approximation and the advantages of the Lagrangian formulation while agreeing to strengthen the manuscript with additional discussion and comparisons where feasible.

read point-by-point responses

Referee: [derivation of the vertical equation (abstract and §2)] The central claim that radiative-interior modes have nearly constant vorticity eigenfunctions across the convection zone rests on the vertical structure equation derived under the β-plane approximation extended over the full radial domain (radiative interior through convection zone). This approximation replaces spherical geometry with a local Cartesian frame and assumes linear latitudinal variation of the Coriolis parameter; the manuscript does not quantify the size of neglected curvature or radial-vertical reorientation terms in a strongly stratified medium, which directly affects whether the reported eigenfunction constancy is robust.

Authors: We agree that a quantitative assessment of the neglected curvature and radial-vertical terms would strengthen the justification for extending the β-plane over the full domain. Our choice follows standard practice in deep-atmosphere Rossby-wave studies to isolate stratification effects while retaining the essential latitudinal variation of the Coriolis parameter. The resulting eigenfunctions are robust within the adopted framework, but we will add an explicit order-of-magnitude estimate of the omitted terms (using Model S profiles) in a new subsection of §2 in the revised manuscript. revision: yes
Referee: [vertical wave equation for δP] The statement that the Lagrangian-pressure formulation removes internal singularities is load-bearing for the cleanliness of the wave equation and the subsequent eigenfunction calculation. Without an explicit side-by-side comparison (e.g., the singular terms that appear in the Eulerian formulation versus their absence here) or a demonstration that the new equation remains regular at all turning points in Model S, the improvement cannot be verified from the given derivation.

Authors: The Lagrangian pressure δP was chosen precisely because it yields a second-order equation free of the density-gradient singularities that appear when the Eulerian pressure perturbation is used. We will insert a short appendix or expanded paragraph in §2 that writes the corresponding Eulerian equation, highlights the singular coefficients proportional to d ln ρ/dr, and confirms that those terms are absent in our δP formulation. Regularity at the turning points in Model S follows directly from the absence of those coefficients and will be noted explicitly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external Model S and standard beta-plane equations

full rationale

The paper derives the vertical structure equation for Lagrangian pressure fluctuation δP by applying the standard β-plane approximation to the linearized equations of motion in a stratified atmosphere, then numerically integrates the resulting eigenvalue problem against the external standard solar model (Model S). No parameters are fitted to the reported eigenfunctions or their near-constancy property; the eigenfunctions are computed outputs. No self-citations, ansatzes smuggled via prior work, or definitional loops appear in the derivation chain. The choice of variable (δP) and the identification of two wave cavities follow directly from the approximated equations without reducing to the target result by construction. The beta-plane validity across the full domain is an assumption whose correctness can be assessed externally, but it does not create circularity within the paper's own logic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the beta-plane approximation and the accuracy of Model S stratification; no free parameters are introduced in the abstract and no new entities are postulated.

axioms (1)

domain assumption The beta-plane approximation is applicable for deriving the vertical Rossby-wave equation in a deep, radially stratified atmosphere.
Invoked to obtain the vertical equation in terms of Lagrangian pressure fluctuation.

pith-pipeline@v0.9.0 · 5711 in / 1224 out tokens · 29222 ms · 2026-05-18T00:37:33.095227+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use the β-plane approximation to derive a vertical equation in terms of the Lagrangian pressure fluctuation δP... d²δP/dz² + (1/Hρ) dδP/dz + N²/(g h) δP = 0
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The resulting radial wave equation makes it abundantly clear that there are two wave cavities... radiative interior and convection zone

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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[1] [1]

V ., & Kukhianidze, V

Albekioni, M., Zaqarashvili, T. V ., & Kukhianidze, V . 2023, A&A, 671, A91, doi: 10.1051/0004-6361/202243985

work page doi:10.1051/0004-6361/202243985 2023

[2] [2]

S., Gizon, L., & Hanasoge, S

Alshehhi, R., Hanson, C. S., Gizon, L., & Hanasoge, S. 2019, A&A, 622, A124, doi: 10.1051/0004-6361/201834237

work page doi:10.1051/0004-6361/201834237 2019

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, keywords =

Bekki, Y ., Cameron, R. H., & Gizon, L. 2022a, A&A, 662, A16, doi: 10.1051/0004-6361/202243164 —. 2022b, A&A, 666, A135, doi: 10.1051/0004-6361/202244150

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2002, Reviews of Modern Physics, 74, 1073, doi: 10.1103/RevModPhys.74.1073

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work page doi:10.1103/revmodphys.74.1073 2002

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V., Anderson , E

Christensen-Dalsgaard, J., Dappen, W., Ajukov, S. V ., et al. 1996a, Science, 272, 1286, doi: 10.1126/science.272.5266.1286 —. 1996b, Science, 272, 1286, doi: 10.1126/science.272.5266.1286

work page doi:10.1126/science.272.5266.1286

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, keywords =

Damiani, C., Cameron, R. H., Birch, A. C., & Gizon, L. 2020, A&A, 637, A65, doi: 10.1051/0004-6361/201936251

work page doi:10.1051/0004-6361/201936251 2020

[8] [8]

, keywords =

Gizon, L., Cameron, R. H., Bekki, Y ., et al. 2021, A&A, 652, L6, doi: 10.1051/0004-6361/202141462

work page doi:10.1051/0004-6361/202141462 2021

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, keywords =

Hanasoge, S., & Mandal, K. 2019, ApJL, 871, L32, doi: 10.3847/2041-8213/aaff60

work page doi:10.3847/2041-8213/aaff60 2019

[11] [11]

, keywords =

Hanson, C. S., Gizon, L., & Liang, Z.-C. 2020, A&A, 635, A109, doi: 10.1051/0004-6361/201937321

work page doi:10.1051/0004-6361/201937321 2020

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S., & Hanasoge, S

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work page doi:10.1063/5.0216403 2024

[13] [13]

, keywords =

Hathaway, D. H., & Upton, L. A. 2021, ApJ, 908, 160, doi: 10.3847/1538-4357/abcbfa

work page doi:10.3847/1538-4357/abcbfa 2021

[14] [14]

1940, Transactions, American Geophysical Union, 21, 262, doi: 10.1029/TR021i002p00262

Haurwitz, B. 1940, Transactions, American Geophysical Union, 21, 262, doi: 10.1029/TR021i002p00262

work page doi:10.1029/tr021i002p00262 1940

[15] [15]

, keywords =

Hindman, B. W., & Jain, R. 2022, ApJ, 932, 68, doi: 10.3847/1538-4357/ac6d64

work page doi:10.3847/1538-4357/ac6d64 2022

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W., & Julien, K

Hindman, B. W., & Julien, K. 2024, ApJ, 960, 64, doi: 10.3847/1538-4357/ad0967

work page doi:10.3847/1538-4357/ad0967 2024

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1956, Ordinary Differential Equations (USA: Dover

Ince, E. 1956, Ordinary Differential Equations (USA: Dover

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1949, Memoires of the Societe Royale des Sciences de Liege, 9, 3 9

Ledoux, P. 1949, Memoires of the Societe Royale des Sciences de Liege, 9, 3 9

work page 1949

[19] [19]

C., & Duvall, T

Liang, Z.-C., Gizon, L., Birch, A. C., & Duvall, T. L. 2019, A&A, 626, A3, doi: 10.1051/0004-6361/201834849

work page doi:10.1051/0004-6361/201834849 2019

[20] [20]

Longuet-Higgins, M. S. 1968, Philosophical Transactions of the Royal Society of London Series A, 262, 511, doi: 10.1098/rsta.1968.0003 L¨optien, B., Gizon, L., Birch, A. C., et al. 2018, Nature Astronomy, 2, 568, doi: 10.1038/s41550-018-0460-x

work page doi:10.1098/rsta.1968.0003 1968

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Mandal, K., & Hanasoge, S. M. 2024, ApJ, 967, 46, doi: 10.3847/1538-4357/ad391b

work page doi:10.3847/1538-4357/ad391b 2024

[22] [22]

Papaloizou, J., & Pringle, J. E. 1978, MNRAS, 182, 423, doi: 10.1093/mnras/182.3.423

work page doi:10.1093/mnras/182.3.423 1978

[23] [23]

1987, Geophysical Fluid Dynamics (Springer-Verlag New York Inc)

Pedlosky, J. 1987, Geophysical Fluid Dynamics (Springer-Verlag New York Inc)

work page 1987

[24] [24]

1981, A&A, 94, 126

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

work page 1981

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, keywords =

Proxauf, B., Gizon, L., L¨optien, B., et al. 2020, A&A, 634, A44, doi: 10.1051/0004-6361/201937007

work page doi:10.1051/0004-6361/201937007 2020

[26] [26]

1939, Journal of Marine Research, 2, 38

Rossby, C.-G. 1939, Journal of Marine Research, 2, 38

work page 1939

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Vallis, G. K. 2017, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (Cambridge University Press), doi: 10.1017/9781107588417

work page doi:10.1017/9781107588417 2017

[28] [28]

and Zhao, Junwei , urldate =

Waidele, M., & Zhao, J. 2023, ApJL, 954, L26, doi: 10.3847/2041-8213/acefd0

work page doi:10.3847/2041-8213/acefd0 2023

[29] [29]

, keywords =

Wolff, C. L., & Blizard, J. B. 1986, SoPh, 105, 1, doi: 10.1007/BF00156371

work page doi:10.1007/bf00156371 1986