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Numerical simulations of oscillations for axisymmetric solar backgrounds with differential rotation and gravity

T0 review · 1 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper claims its HDG solver for axisymmetric solar oscillations—with gravity, attenuation, and differential rotation—produces power spectra whose rotation-induced splittings match observed a-coefficients to within 0.03 nHz.

desk verdict A careful, well-validated forward-modeling paper for axisymmetric solar oscillations; the f_ghost approximation is real but negligible for the modes tested, and the paper deserves serious refereeing. read the letter →

arxiv 2508.19386 v7 pith:24IMX6OR submitted 2025-08-26 astro-ph.SR astro-ph.IMmath.AP

Ha Pham , Florian Faucher , Damien Fournier , Hélène Barucq , Laurent Gizon This is my paper · ORCID
classification astro-ph.SRastro-ph.IMmath.AP MSC 65N3085-08 PACS 96.60.Ly02.70.Dh
keywords helioseismologystellaroscillationsdifferentialrotationHybridizableDiscontinuousGalerkinmodesplittingbuoyancyfrequencyGreen'sfunctionspowerspectra
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

This paper builds a numerical solver for the equations that govern linear adiabatic oscillations of the Sun in a realistic background that includes gravity, acoustic attenuation, and differential rotation, and shows that the simulated wavefields reproduce observed helioseismic data. The authors solve the oscillation equations in first-order form (displacement and pressure perturbation as unknowns), decompose the axisymmetric problem into azimuthal modes, and discretize each modal problem on the meridional plane with a Hybridizable Discontinuous Galerkin (HDG) method. The key quantitative result: for the observed solar rotation profile, the synthetic power spectrum gives frequency-splitting a-coefficients a1/2π = 442.86 nHz, a3/2π = 22.17 nHz, a5/2π = −3.22 nHz, matching measured values 442.85±0.05, 22.15±0.08, −3.21±0.10 nHz. If correct, the solver supplies accurate Green's kernels for local helioseismology—forward modeling of power spectra and cross-covariances that spherical eigensolvers cannot provide once rotation breaks spherical symmetry. The paper also shows that gravity changes the mathematical nature of the wave operator (elliptic versus hyperbolic regions set by the buoyancy frequency), which makes the choice of HDG stabilization load-bearing for accuracy.

What carries the argument

The load-bearing object is the first-order system A u + β1 w + ∇w = g, ∇·u + β2·u + ϱw = h, with u the rescaled Lagrangian displacement and w the pressure perturbation, obtained from the Gough–Thompson equations via a Liouville change of variables. Axisymmetry reduces it to azimuthal modal problems on the meridional half-disk, whose operator relates to a scalar PDE with leading term ∇m·A_m^{-1}∇m. The buoyancy frequency N decides the local type of that operator: elliptic where ω²>N², hyperbolic where ω²<N² (at zero attenuation). The HDG method couples cells through a face trace λ of w and a stabilization parameter τ = |A^{-1}β1·n| + iω α_tune |n^T A^{-1} n|, whose imaginary part receives its

What would settle it

Run the same pipeline for the lowest harmonic degrees (ℓ ≲ 20), whose acoustic paths reach the deep interior where the neglected ghost force peaks near 3% of the hydrostatic balance; a systematic departure of the fitted a-coefficients from observed values, beyond the few-hundredths-of-nHz accuracy obtained at ℓ = 85, would show the background inconsistency matters. A complementary check: rebuild the background to satisfy rotational hydrostatics (as done for polytropic rotating stars) and test whether the ℓ = 85 coefficients shift by more than 0.03 nHz.

Watch

Extended reading notes

Core claim

This paper claims a Hybridizable Discontinuous Galerkin (HDG) scheme for the first-order stellar oscillation equations computes Green's kernels for axisymmetric solar backgrounds with gravity, acoustic attenuation, and differential rotation. Working in the Gough–Thompson formulation (displacement and pressure perturbation as unknowns), the axisymmetric problem splits into independent 2D modal problems on the meridional half-disk. The central validation: for the observed solar rotation profile, the synthetic power spectrum yields a1/2π = 442.86 nHz, a3/2π = 22.17 nHz, a5/2π = −3.22 nHz, matching observed values 442.85±0.05, 22.15±0.08, −3.21±0.10 nHz.

Load-bearing premise

The observed rotation is overlaid on the non-rotating, spherically symmetric Model-S background, so the reference state does not satisfy rotational hydrostatics and the solver neglects the implied ghost force f_ghost = ρ0 Ω×Ω×x (up to 3% of the balance, largest near the centre, per Remark 7); the validated match to observed mode splitting depends on that inconsistency being negligible where the compared modes live.

Editorial extensions

If this is right

  • Rotation-induced splitting becomes a quantitative validation: the synthetic power spectrum for the observed solar rotation profile yields a1/2π = 442.86 nHz, a3/2π = 22.17 nHz, a5/2π = −3.22 nHz, within 0.02–0.03 nHz of measured values—below the observational noise level.
  • Green's kernels computed this way can serve as forward models for local helioseismic inversions (time-distance and holography) on axisymmetric backgrounds with realistic flows, which spherically symmetric eigensolver-based approaches cannot supply once rotation breaks spherical symmetry.
  • Without rotation, the power-spectrum ridges coincide with GYRE eigenfrequencies, cross-checking the Green's-function approach against an independent solver and reproducing the discrete mode structure.
  • Because the HDG solve provides the full 3D displacement field, synthetic Doppler signals can be improved by including projection and radiative-transfer effects rather than using the pressure perturbation as a proxy.
  • The same axisymmetric pipeline applies to other rotating bodies—rapidly rotating stars and gas giants—and, at low frequencies, to inertial modes whose sensitivity reaches the deep interior.

Reading between the lines

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

  • The strongest untested consequence of the background inconsistency sits in the deep interior: modes with low degree ℓ reach depths where the neglected ghost force peaks near 3% of the hydrostatic balance. If such modes were simulated, a systematic departure from observed a-coefficients beyond the 0.03 nHz accuracy seen at ℓ = 85 would finger the reference-state error rather than the numerical meth
  • The space-like/time-like stabilization rule for the hyperbolic region is a general recipe for HDG on first-order systems whose type changes across the domain; it could be transferred to any stratified-wave problem with an effective buoyancy frequency (atmospheric or oceanic internal waves) and tested on idealized profiles before committing to stellar backgrounds.
  • Because the HDG solve yields the full displacement vector, synthetic observables could move from the pressure proxy to line-of-sight velocity with projection and radiative transfer; the fidelity gain is not quantified in this paper, but the solver architecture makes it a direct next step.
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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

1 major / 7 minor

Summary. The paper develops a Hybridizable Discontinuous Galerkin (HDG) solver for the time-harmonic equations of linear stellar oscillations in axisymmetric backgrounds with differential rotation and gravity. The working equations are a first-order system in Lagrangian displacement and Eulerian pressure perturbation, equivalent to Galbrun-type formulations but with lower regularity requirements. The authors derive the azimuthal-mode decomposition, give a full HDG discretization with a stabilization strategy, and study the role of the buoyancy frequency in creating elliptic/hyperbolic regions. They then synthesize helioseismic observables (power spectra and cross-covariances) from the computed Green's kernels. Without rotation, the synthetic power spectrum matches GYRE eigenfrequencies; with a simplified rotation profile, fitted a-coefficients agree with analytic values; with the observed solar rotation profile of Larson & Schou, the simulated a1, a3, a5 agree with observed values to within roughly 0.01-0.03 nHz. The central claim is that the solver can accurately compute Green's kernels for realistic solar backgrounds including gravity and differential rotation.

Significance. If the results hold, the paper provides a validated, open-source 2.5D forward solver for local helioseismology with gravity and rotation, a substantial step beyond earlier no-gravity or spherically symmetric tools. The strongest points are the independent no-rotation validation against GYRE, the analytic simplified-rotation check, and the excellent agreement with observed rotation-induced mode splitting. The detailed empirical study of HDG stabilization in the presence of a non-zero buoyancy frequency is also valuable. Main caveats are the non-equilibrium background approximation (Remark 7), the partly circular nature of the solar-profile comparison, and the fact that the rotation validation is performed for a single (ℓ,n) mode; none of these appears to invalidate the quantitative agreement, but they should be addressed explicitly.

major comments (1)
  1. [§3.1 (Remark 7), §7.3.2, Table 1] The rotating validation uses the non-rotating Model-S background together with Ω≠0, so f_ghost=ρ0Ω×Ω×x is nonzero and not implemented. This is disclosed, and it is not fatal: the matched mode is ℓ=85, n=7 at ~3 mHz, a high-degree p-mode concentrated near the surface, while the hydrostatic deviation is largest in the deep interior; the fitted a1,a3,a5 are first order in Ω, whereas f_ghost is O(Ω^2) and affects even coefficients. Still, the paper should state this scaling and give a quantitative bound on the effect. In addition, because profile [39] is inferred from those same observed a-coefficients, the solar-profile comparison is partly a consistency check; the independent analytic simplified-profile validation in Table 1 should be presented as the primary quantitative evidence.
minor comments (7)
  1. [§1] The sentence 'ellipticity is lost when N^2>0 and for frequencies above N' appears to reverse the definitions in (3.49) and §3.5: the hyperbolic region is ω^2<N^2. Please correct the wording.
  2. [§7.1.2, Eqs. (7.17)-(7.22)] The cross-covariance definition omits a complex conjugate. The later derivation using the addition theorem implicitly assumes conjugation of spherical harmonics; without a stated convention, (7.22) is ambiguous. Please clarify the Fourier convention and the role of conjugation.
  3. [§5.5, §6.3] The stabilization scaling α_tune is a free parameter with empirically chosen values (α_tune=±10^6 in (6.5)). A brief statement on the sensitivity of the §7.3 a-coefficient results to this parameter, or a note that the source geometry used for the observables is stable across τa/τb, would improve reproducibility.
  4. [Remark 7] Typo: 'hystrostatic' should be 'hydrostatic'. Also, the 'at most 3%' deviation should be defined more precisely (which norm? at which radius?), since it is an important caveat for the rotating background.
  5. [§7.3.1, Eq. (7.44)] State explicitly that the observed a-coefficients quoted from [39] are for ℓ=85, n=7, and note any mode-selection details; currently the reader must infer this from Table 1 and the surrounding text.
  6. [§6.1] Please specify whether γ_att=10 μHz and 2 μHz are angular frequencies or ordinary frequencies, since the equations use angular frequency ω. This affects how readers interpret the attenuation relative to 3 mHz and 0.2 mHz.
  7. [Figure 16] The caption and legend should define 'Approx', 'Fit', 'Max' and the green/gray curves more explicitly; the current caption relies heavily on the body text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solver is validated against independent external benchmarks (GYRE and analytic a-coefficients), and no fitted parameter is renamed as a prediction.

full rationale

The paper's derivation chain is self-contained from first principles (Sections 2–3) to the HDG discretization (Section 5), and the helioseismic products are computed from the Green's kernel through explicit integral identities (Sections 4 and 7.1). The central validations are external: the no-rotation power spectrum is compared with eigenfrequencies computed by the independent GYRE code (Section 7.2, Figure 13), and the rotation validation includes a simplified profile with an analytic a-coefficient prediction (Remark 22, Equation 7.48) that does not depend on the present solver. For the solar rotation profile, the input is the published inversion product of [39], and the comparison in Table 1 (442.86 vs 442.85 nHz for a1/2π) is therefore a consistency check rather than a blind prediction; however, it is not equivalent by construction to the input, and no parameter of the HDG method is fitted to the observed a-coefficients. The self-citations to prior work [4, 50, 51] concern methodological choices (Galbrun/Liouville variants and HDG stabilization) and are not used as evidence for the numerical agreement. The flagged approximation in Remark 7 (f_ghost = ρ0 Ω×Ω×x not implemented when combining the non-rotating Model-S background with rotation) is a disclosed modeling limitation with an estimated small effect, not a circular step. No equation is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The solver relies on standard solar background inputs (Model-S) and known physical assumptions; the only hand-fitted numerical parameters are the attenuation, the stabilization scaling, and standard fitting choices for a-coefficients. No new physical entities are postulated.

free parameters (4)
  • attenuation gamma_att = 10 µHz (3 and 6 mHz), 2 µHz (0.2 mHz)
    Added to the wave operator for well-posedness and to avoid A = 0 at N^2 = omega^2; hand-chosen per frequency, Section 6.1.
  • stabilization scaling alpha_tune = 10^6
    Scales the diffusion-type HDG stabilization in Eq. (5.59); selected empirically after a comparative study over the complex plane, Section 6.3.
  • a-coefficient fit order j_max = 6
    Used to represent the rotation-induced frequency shift in (7.43); chosen following standard helioseismic practice [39].
  • outer boundary radius r_max = 1.001 or 1.01 (scaled solar radii)
    Computational domain truncation; the paper shows the solutions at the observation height are insensitive to this choice, Section 6.3.2.
assumptions (7)
  • domain assumption Cowling approximation: Eulerian perturbation of the gravitational potential is neglected.
    Standard in local helioseismology; stated in the title and used throughout Section 2.3.
  • domain assumption Adiabatic, unmagnetized ideal fluid with the equation of state (2.16).
    Introduced in Section 2.1 as the hydrodynamic description of the Sun.
  • domain assumption No-resonance assumption: relations (2.50b) are imposed without integrating the linearized continuity and energy equations.
    Invoked in Section 2.3, Remark 3, following [33]. Needed to close the perturbation system.
  • domain assumption Vacuum boundary condition: vanishing Lagrangian pressure perturbation on the outer boundary, Eq. (2.63).
    Standard boundary condition for stellar oscillations, stated in Section 2.3.
  • ad hoc to paper Spherically symmetric Model-S background is used even when rotation is added; f_ghost neq 0 is not implemented.
    Remark 7 acknowledges that adding rotation to Model-S violates rotational hydrostatics and introduces a ghost force up to 3% near the center, which is not implemented in (2.65).
  • standard math Well-posedness of the damped time-harmonic Galbrun-type equation is assumed from Halla & Hohage [34].
    Cited in Section 1 to justify that small attenuation guarantees well-posedness.
  • domain assumption The meriodional boundary satisfies grad_p0 parallel to the outer normal (4.2).
    Used to state the boundary condition in the azimuthal modal problem, Section 4.

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Pith. "Pith review of Numerical simulations of oscillations for axisymmetric solar backgrounds with differential rotation and gravity." pith.science (2026). https://pith.science/paper/24IMX6OR

@misc{pith2026250819386,
  author       = {Pith},
  title        = {Pith review of: Numerical simulations of oscillations for axisymmetric solar backgrounds with differential rotation and gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24IMX6OR}},
  note         = {Machine review of arXiv:2508.19386}
}
abstract

Local helioseismology comprises of imaging and inversion techniques employed to reconstruct the dynamic and interior of the Sun from correlations of oscillations observed on the surface, all of which require modeling solar oscillations and computing Green's kernels. In this context, we implement and investigate the robustness of the Hybridizable Discontinuous Galerkin (HDG) method in solving the equation modeling stellar oscillations for realistic solar backgrounds containing acoustic attenuation, gravity, and differential rotation. While a common choice for modeling stellar oscillations is the Galbrun's equation, our working equations are derived from an equivalent variant, involving less regularity in its coefficients, working with Lagrangian displacement and pressure perturbation as unknowns. Under differential rotation and axisymmetric assumption, the system is solved in azimuthal decomposition with the HDG method. Compared to no-gravity approximations, the mathematical nature of the wave operator is now linked to the profile of the solar buoyancy frequency $N$ which encodes gravity, and leads to distinction into regions of elliptic or hyperbolic behavior of the wave operator at zero attenuation. While small attenuation is systematically included to guarantee theoretical well-posedness, the above phenomenon affects the numerical solutions in terms of amplitude and oscillation pattern, and requires a judicious choice of stabilization. We investigate the stabilization of the HDG discretization scheme, and demonstrate its importance to ensure the accuracy of numerical results, which is shown to depend on frequencies relative to $N$, and on the position of the Dirac source. As validations, the numerical power spectra reproduce accurately the observed effects of the solar rotation on acoustic waves.

Figures

Figures reproduced from arXiv: 2508.19386 by the authors.

Figure 1
Figure 1. Representation of the reference flow χ0 and perturbed flow χ and displacement ∆L χ defined in (2.17). The velocities of the two trajectories are respectively denoted by, v0(a, t) := ∂tχ0 (a, t), v(a, t) := ∂tχ(a, t). (2.13) • Following [40], the notation ˜· is employed to rewrite a quantity in Lagrangian coordinates (a, t) to Eulerian coordinates (x, t) which are related by corresponding flows. For example, for a qu… view at source ↗
Figure 2
Figure 2. Illustration of the numerical domain for axisymmetry which corresponds to the meridional [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Radial solar background models for the density [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The buoyancy frequency (or Brunt–V¨ais¨al¨a frequency) is represented with a solid line, Lamb [PITH_FULL_IMAGE:figures/full_fig_p038_4.png]
Figure 5
Figure 5. Figure 5: Comparison of the solutions Re(w•) for the different formulations for a source positioned in (1, 0), at mode m = 0 and frequency 6mHz. The zoom near the source corresponds to the zone (0.99, 1.001) × (−0.005, 0.02), and uses a different scaling to improve visualization…
Figure 6
Figure 6. Figure 6: Simulations and relative difference e (6.2) for a Dirac source in (1, 0) at 6mHz using Mesh001 175k for different choices of HDG stabilization (6.5). The solutions on the entire domain are complemented with a zoom near source position on interval (0.9975, 1.001) × (−0.…
Figure 7
Figure 7. Figure 7: Simulations Re(wc) at 3mHz using Mesh001 175kfor a source positioned in (0.999, 0) (left), (0.999 876, 0) (middle) and (1, 0) (right), for different choices of stabilization (6.5). At 3mHz, the transition between Bhyp and Bell is in r = 0.999 999 5. The solutions on th…
Figure 8
Figure 8. Figure 8: Simulations are carried out at 3mHz for a source positioned in (1 [PITH_FULL_IMAGE:figures/full_fig_p044_8.png]
Figure 9
Figure 9. Figure 9: Comparisons of the solutions at 3mHz for fixed height [PITH_FULL_IMAGE:figures/full_fig_p045_9.png]
Figure 10
Figure 10. Figure 10: Simulations are carried out at 3mHz for a source positioned in (0 [PITH_FULL_IMAGE:figures/full_fig_p046_10.png]
Figure 11
Figure 11. Figure 11: Simulations at 0.2mHz, the reference solution is on the left computed with Mesh01 1200k, then we show the solutions using Mesh001 30k for different stabilization τa, τb and τc from left to right. In each subfigure, the first line shows the wavefield and the second one…
Figure 12
Figure 12. Figure 12: Comparisons of the solutions with and without gravity effects for a source positioned in [PITH_FULL_IMAGE:figures/full_fig_p048_12.png]
Figure 13
Figure 13. Figure 13: Power spectrum P 0 l (ω) (7.30) without rotation (background color) and comparison with eigenvalues (cyan crosses) computed with the Gyre code. The bottom right panels show slices with frequencies at fixed ℓ, with the Gyre eigenvalues indicated by the cyan vertical li…
Figure 14
Figure 14. Figure 14: Time-distance diagram showing the cross-covariance [PITH_FULL_IMAGE:figures/full_fig_p054_14.png]
Figure 15
Figure 15. Figure 15: Solar rotation Ω/(2π) in nHz. Remark 20. The derivatives of Ω with respect to the axisymmetric coordinates η and z are also needed in the numerical implementation, see Section A. As Ω is usually known as a function of r and θ, we use, dΩ dη = dΩ dθ dθ dη + dΩ dr dr dη…
Figure 16
Figure 16. Figure 16: Comparison of power spectra P m ℓ (ω) for different rotation profiles. With rotation, the power spectrum changes with azimuthal order m, leading to a shift in frequency in the position of maximal power, shift that can be related to the rotation model. On the spectra, …

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    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1 'skip if FUNCTION new.block.check...

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.