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Zones of Tayler Instability in Stars

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Pith's one-line read This paper establishes revised instability criteria for the Tayler instability in stellar interiors and shows that in low-mass red giants the instability is suppressed in the compositionally stratified shell around the helium core for…

desk verdict A clean extension of the authors' own linear-stability framework that yields a credible suppression layer in evolved low-mass stars, but the active-zone maps are more conditional than the abstract suggests. read the letter →

arxiv 2411.08492 v2 pith:RG4MRUBV submitted 2024-11-13 astro-ph.SR physics.plasm-ph

Valentin A. Skoutnev , Andrei M. Beloborodov This is my paper · ORCID
classification astro-ph.SRphysics.plasm-ph
keywords TaylerinstabilitytoroidalmagneticfieldsangularmomentumtransportcompositionalstratificationredgiantbranchPrandtlnumberstellarevolutionmagnetohydrodynamics
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 asks when and where the Tayler instability (TI) of toroidal magnetic fields can actually grow at the canonical rate assumed by angular-momentum transport models in stars. It extends the linear stability analysis of a rotating, stably stratified, magnetized fluid to include both thermal and compositional stratification, then distills the conditions into analytic criteria for four canonical wavenumbers. Applied to stellar models, the criteria show a sharp dichotomy: in a representative $1.5\,M_\odot$ red giant the TI is switched off in the compositionally stratified layer around the helium core during most of the RGB phase, leaving only a much slower instability there, while in stars above about $4\,M_\odot$ the TI can develop throughout the deep radiative interior, usually at the thermal- or composition-diffusion wavenumber. The result matters because existing Tayler-Spruit dynamo prescriptions assume the canonical growth rate in all radiative zones, and the paper identifies where that assumption fails.

What carries the argument

The central object is the sixth-order dispersion relation for wave perturbations of a toroidal field $B_\phi$ in a rotating star with two buoyancy variables, one thermal and one compositional. The paper identifies four canonical wavenumbers $k_\eta=(2\Omega/\eta)^{1/2}$, $k_\nu=(2\Omega/\nu)^{1/2}$, $k_{\kappa_{\rm th}}=(k_\theta^2 N_{\rm th}^2/(2\Omega\kappa_{\rm th}))^{1/4}$, and $k_{\kappa_\mu}=(k_\theta^2 N_\mu^2/(2\Omega\kappa_\mu))^{1/4}$, where the relevant diffusive timescale matches the rotation timescale and the growth rate reaches $\gamma_{\max}$. The instability criteria reduce to three dimensionless parameters—$P_m=\nu/\eta$, $C_m=\kappa_\mu/\eta$, and $N_\mu/N_{\rm th}$—which determine which mode, if any, is unstable and are implemented as a toggle switch in a stellar evolution code.

What would settle it

Measure the radial magnetic field in the hydrogen-burning shell of a low-mass red giant: if it exceeds the roughly 3-gauss threshold of Equation (18), the predicted TI-active zones in that layer are wrong; a second check is to solve the full sixth-order dispersion relation with finite radial field and differential rotation and see whether the $(N_\mu/N_{\rm th})^{-4}$ suppression and the peak shift toward $k_{N_\mu}$ appear.

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Extended reading notes

Core claim

The central claim is that, once both thermal and compositional stratification are included, the TI has four canonical modes—one enabled by magnetic diffusion, one by viscosity, one by thermal diffusion, and one by compositional diffusion—each with a well-defined interval of toroidal field strength where it grows at the maximum rate $\gamma_{\max}=\omega_A^2/4\Omega$. In the fiducial $1.5\,M_\odot$ model, the compositionally stratified layer around the helium core has magnetic Prandtl number $P_m\approx1$, diffusivity ratio $C_m=\kappa_\mu/\eta\approx1$, and $N_\mu/N_{\rm th}>1$, so all four canonical modes are stable for most of the RGB phase; the fastest surviving instability grows at wavenumber $k\sim k_{N_\mu}$ with rate reduced by roughly $(N_\mu/N_{\rm th})^{-4}$. In stars with $M\gtrsim4\,M_\odot$, $C_m>1$ holds throughout the deep interior, so the TI can develop everywhere, and the most unstable mode is typically $k_{\kappa_{\rm th}}$ or $k_{\kappa_\mu}$ rather than the magnetic-diffusion mode.

Load-bearing premise

The maps assume the star's radial magnetic field is weak (below about 3 gauss in the key layer of the 1.5-solar-mass model) and that differential rotation is weak enough not to shear the unstable waves; if real red-giant fields exceed this, the active-zone maps overestimate where the Tayler instability operates.

Editorial extensions

If this is right

  • Tayler-Spruit dynamo prescriptions that assume the canonical growth rate everywhere will overestimate angular-momentum transport in the compositionally stratified layer around the helium core of low-mass red giants for most of the red-giant branch.
  • In that layer the surviving instability grows only at the reduced rate $\gamma\sim\gamma_{\max}(N_\mu/N_{\rm th})^{-4}$, so transport is weaker but not entirely absent.
  • In stars above roughly four solar masses, the TI can be active throughout the deep radiative interior, with the most unstable mode typically the thermal- or composition-diffusion mode $k_{\kappa_{\rm th}}$ or $k_{\kappa_\mu}$.
  • The analytic criteria give evolution codes a toggle switch that tells them when the canonical growth rate $\gamma_{\max}$ applies and when it does not.
  • For about $10^7$ years near the end of the red-giant phase, rising $P_m$ and $C_m$ in the burning shell briefly restore canonical TI modes at $k_\nu$ and $k_{\kappa_\mu}$ in the core-envelope transition.

Reading between the lines

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

  • If the suppression layer persists as mapped, it can act as a barrier that decouples core and envelope angular momentum for most of the red-giant phase, which would show up as red-giant cores rotating faster or slower than canonical Tayler-Spruit models predict; this transport consequence is not quantified in the paper.
  • The clean divide near four solar masses suggests a testable prediction: rotation profiles of stars just above and below that mass should differ systematically if the $C_m>1$ criterion controls where the TI operates.
  • The same criteria imply that codes replacing the old effective Brunt-Vaisala shortcut will shift the location and wavenumber of active TI; comparing such codes against asteroseismic rotation data could indirectly test the revised criteria.
  • The strong radial fields inferred by asteroseismology may suppress the TI even where the maps show it active, pointing toward fossil-field or magnetic-web transport rather than Tayler-Spruit turbulence; the authors mention this possibility but do not model it.
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Editorial analysis

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

Referee Report

2 major / 5 minor

Summary. The manuscript extends the linear stability analysis of the Tayler instability (TI) in rotating, stratified stars from Skoutnev & Beloborodov (2024) to the case of simultaneous thermal and compositional stratification. It derives analytic instability criteria for the four canonical wavenumbers k_eta, k_nu, k_kappa_th, and k_kappa_mu, summarizes them in Tables 1 and 2, implements the criteria as a "toggle switch" in the MESA stellar evolution code, and maps the most unstable mode for stellar models from 1.5 to 32 solar masses. The two main astrophysical claims are: (i) in a 1.5 solar mass RGB star, a thin layer around the helium core has all four canonical TI modes stable, with the residual TI growth rate suppressed by a factor of order (N_mu/N_th)^{-4}; and (ii) in stars of about 4 solar masses and above, C_m = kappa_mu/eta exceeds unity throughout the deep interior, so the TI can develop there, typically at k_kappa_th or k_kappa_mu. The application is explicitly restricted to weak radial magnetic fields and weak differential rotation through Eqs. (11), (13), and (15).

Significance. If the criteria are correct, this paper provides a practical, numerically checked way to decide where the Tayler-Spruit dynamo operates with its canonical growth rate, directly usable in stellar evolution codes. The identification of a compositionally stratified suppression layer in low-mass RGB stars is a new and potentially important prediction, and the high-mass regime with P_m > 1 and C_m > 1 corrects earlier treatments that implicitly assumed the low-mass magnetic Prandtl number ordering. The paper is not circular: the criteria are derived from a dispersion relation and checked against numerical roots of the same relation, while the MESA maps are applications of those criteria rather than independent tests. The main limitation is that the maps are conditional on assumptions that are stated in Section 2.3 but not implemented as cuts in the maps or reflected in the abstract.

major comments (2)
  1. [§3.3, Eq. (18); Figs. 2, 3, 7] The radial-field condition Eq. (11) is load-bearing for the active-zone maps, and it is not implemented in the MESA toggle switch. The authors' own Eq. (18) gives B_R ≲ 3 G for the 1.5 solar mass compositionally stratified layer, while the asteroseismic values they quote are B_R ≳ 3 × 10^4 G there. A stronger radial field adds stabilizing magnetic tension, so the regions shown as active in Figs. 2, 3, and 7 are optimistic upper limits, and the abstract's statement that in M ≳ 4 solar mass stars "the TI can be active throughout their radiative zones" is not supported by the analysis as presented. The low-mass suppression result is not affected, since a strong radial field would only suppress the TI further, but the maps and the high-mass conclusion should either implement Eq. (11) or be explicitly relabeled as valid only for weak fossil radial fields.
  2. [§2.3, Eq. (15); Fig. 5; Figs. 2–3] The same conditional-status issue applies to differential rotation. For magnetostrophic-wave modes the q = 0 analysis is valid only for omega_A > omega_{A,q} (Eq. 15), yet the instability strips in Fig. 5 extend below the plotted omega_{A,q} curve at some radii, and Section 3.3 admits a mild violation in the deep core. The maps in Figs. 2 and 3 do not mark the regions where Eq. (15) is violated, so a reader cannot tell how much of the plotted "active" volume is actually within the regime of validity. Please add the omega_{A,q} boundary to the toggle switch or state explicitly how the main zone maps change when it is enforced.
minor comments (5)
  1. [§2.3, around Eq. (12)] The factor 2π in the definition of t_q is dropped in the order-of-magnitude condition Eq. (13); please state explicitly that Eq. (13) is an order-of-magnitude estimate.
  2. [Table 1] The formatting of the inequalities across the third and fourth columns is difficult to parse; a numbered list of conditions would improve readability.
  3. [Figure 4 caption] Please state explicitly that the prefactor 0.84 and exponent -8/3 in the plotted suppression expression are empirical fits, with the derived asymptotic limit (N_mu/N_th)^{-4} given in Appendix G.
  4. [§3.3] The word "astroseismology" should be "asteroseismology".
  5. [Section 3.1] The MESA implementation is described only verbally; a code/data availability statement, or a link to the inlists and run scripts, would help reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the instability criteria are derived analytically from the dispersion relation, the MESA maps evaluate those criteria without fitting, and the SB24 self-citations are independent evidence.

full rationale

The paper's central product is a collection of analytical instability criteria (Tables 1 and 2) obtained by solving the generalized dispersion relation in Appendix A (Eqs. A5-A6), with numerical solution of the six roots used as confirmation rather than as an input. The MESA implementation in Section 3 simply evaluates these criteria using standard microphysical diffusivities (Appendix E) and MESA structure; the dimensionless controls P m and Cm are not tuned to produce the suppression layer. The suppression of the canonical modes follows from the derived conditions (e.g. Table 2 gives no k_kappa_th instability for N_mu > N_th, and k_kappa_mu requires Cm > 1), so it is a consequence of the analysis rather than a fitted target. Citations to SB24 are foundational but not circular: SB24 is a separate, parameter-free derivation of the single-stratification dispersion relation, and the present paper extends it to the two-stratification case rather than importing the target result. The weak-radial-field condition Eq. (11) is an explicitly stated validity assumption, and the asteroseismic B_R limits in Section 3.3 are a caveat on where the maps apply, not an equation recycled as a conclusion. The one fitted expression in the Figure 4 caption is a phenomenological envelope for numerically computed growth rates that is explicitly matched to the analytic Appendix G scaling; it is not used to generate the active-zone maps. No circular step can be exhibited.

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

No new physical entities are introduced. The analysis rests on the prior SB24 dispersion relation, a specific toroidal field geometry, standard plasma diffusivities, and neglect of radial field and differential rotation. The only adjustable parameters are k_theta, the threshold alpha, and the interpolation coefficients for the suppression factor.

free parameters (3)
  • k_theta (latitudinal wavenumber) = set to 1/R
    Free parameter in the instability intervals; authors choose the strict lower bound k_theta=1/R, which gives the lowest unstable omega_A intervals. Larger k_theta shifts intervals to higher omega_A.
  • alpha (instability threshold ratio) = 2
    A mode is counted unstable if omega_TI,max_A / omega_TI,min_A > alpha, with alpha=2. Authors state results are insensitive for 1 <~ alpha <~ 3.
  • Suppression fit prefactor and exponent = 0.84 and -8/3
    The interpolation gamma/gamma_max = 0.84/(1+(N_mu/N_th)^3/2)^(-8/3) is fitted to numerical solutions of the dispersion relation; the analytic estimate gives (N_mu/N_th)^-4.
assumptions (5)
  • standard math The linearized MHD dispersion relation from SB24, with both stratification terms added additively, correctly describes TI perturbations in the Boussinesq and WKB limits.
    Equation (A5) is the starting point for all criteria and is taken from the authors' prior SB24 analysis; no independent formal verification is provided.
  • domain assumption The background magnetic field near the polar axis has B_phi proportional to r (p=1) and only m=1 modes are unstable.
    Used to simplify the dispersion relation to Equation (A6); follows from Stokes theorem for finite current density on the axis, but is a modeling choice for stellar fields.
  • domain assumption The star is uniformly rotating on spherical shells with q=0 during the stability analysis; differential rotation and radial field are neglected.
    Authors assume B_R weak enough for Eq. (11) and q small enough for Eqs. (13)-(15); these conditions are not guaranteed and can be violated by observed fields.
  • domain assumption Standard plasma diffusivity expressions (Spitzer, Jermyn et al. 2022, Garaud et al. 2015) apply in radiative zones.
    The values of P_m and C_m, which control the modes, are computed from these formulas in Appendix E; deviations would shift the mapped zones.
  • domain assumption kappa_th > kappa_mu, eta, nu in typical stellar interiors, and the fast-diffusion ordering used in Appendix C holds.
    The instability criteria in Table 2 assume kappa_th > kappa_mu and dominant thermal diffusivity; authors state this is typical for stars.

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Pith. "Pith review of Zones of Tayler Instability in Stars." pith.science (2026). https://pith.science/paper/RG4MRUBV

@misc{pith2026241108492,
  author       = {Pith},
  title        = {Pith review of: Zones of Tayler Instability in Stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RG4MRUBV}},
  note         = {Machine review of arXiv:2411.08492}
}
read the original abstract

The Tayler instability (TI) of toroidal magnetic fields is a candidate mechanism for driving turbulence, angular momentum (AM) transport, and dynamo action in stellar radiative zones. Recently \cite{Skoutnev_2024} revisited the linear stability analysis of a toroidal magnetic field in a rotating and stably stratified fluid. In this paper, we extend the analysis to include both thermal and compositional stratification, allowing for general application to stars. We formulate an analytical instability criterion for use as a ``toggle switch" in stellar evolution codes. It determines when and where in a star the TI develops with a canonical growth rate as assumed in existing prescriptions for AM transport based on Tayler-Spruit dynamo. We implement such a ``toggle switch" in the MESA stellar evolution code and map out the stability of each mode of the TI on a grid of stellar evolution models. In evolved lower mass stars, the TI becomes suppressed in the compositionally stratified layer around the hydrogen burning shell. In higher mass stars, the TI can be active throughout their radiative zones, but at different wavenumbers than previously expected.

Figures

Figures reproduced from arXiv: 2411.08492 by the authors.

Figure 1
Figure 1. Intervals of ωA/2Ω that give the MW instability peak at kTI = kκth (light blue) or kTI = kκµ (dark blue) vs. Nµ/Nth. The two limits of the key parameter Cm = κµ/η, Cm > 1 and Cm < 1, are shown on the left and right, respectively. The figure assumes the typical regime for stars: κth ≫ κµ, η (dominant thermal diffusivity) and kν ≫ kκth , kκµ (weak effects of viscosity). The axes are on logarithmic scales. ωA are deter… view at source ↗
Figure 2
Figure 2. Map of TI modes and three key dimensionless parameters during the evolution of a 1.5M⊙ star. Top left: the most unstable mode of the TI (kν, kη, kκth , or kκµ ) is identified and indicated by color for each mass shell of the star excluding the convection zone (gray). The TI is suppressed in the white region. Top right: Nµ/Nth, a proxy for the relative strength of compositional stratification. Bottom left and right: … view at source ↗
Figure 4
Figure 4. Radial profile of the maximum possible growth rate of the TI (normalized to the canonical γ max = ω 2 A/4Ω) in the 1.5M⊙ star with mHe = 0.25M⊙ (age t = 2.8 Gyr). The growth rate is obtained from numerical solutions of the dispersion relation at each radius R. It depends on ωA as a parameter, and the black curve shows the maximum possible γ found by scanning the interval of 0 < ωA < Ω. The deep pit observed outside … view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: Zoom-in of the region around the helium core during the post-main sequence in the fiducial 1.5M⊙ model. Top two panels display the most unstable TI mode, shown on the t-R and t-m planes. In the white layer, the TI is disabled at all four canonical wavenumbers kν, kη, k…
Figure 6
Figure 6. Figure 6: Numerical solution for the growth rate γ(k) vs wavenumber k at R/R⊙ = 0.05 in a 1.5M⊙ star at t = 2.852 Gyr (when mHe ≈ 0.33M⊙). This radius is marked by the vertical dashed black line in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 5
Figure 5. Figure 5: Radial profiles of a 1.5M⊙ star on the RGB when the helium core mass is mHe ≈ 0.33M⊙. This stellar model is marked by the vertical dashed line at t ≈ 2.852 Gyr in Fig￾ures 2 and 3. Gray region at R ∼ R⊙ indicates the convective envelope. Bottom: basic structure of the …
Figure 7
Figure 7. Figure 7: Stellar structure in the temperature-density plane against a color map of Cm(T, ρ) = κµ/η. Curves show the stellar structure of stars with M/M⊙ = 1.5, 4, 8, 16 and 32. The M = 1.5M⊙ stellar model is shown at two evolution phases: during the main sequence (t = 2.1 Gyr, …
Figure 8
Figure 8. Figure 8: Analysis of TI modes in the 1.5M⊙ star with mHe = 0.25M⊙ (age t = 2.8 Gyr), demonstrating the suppression of TI in the compositionally stratified layer above the helium core. Left: Radial dependence of the instability intervals of ωA/Ω for the canonical TI modes kν, kη…

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Works this paper leans on

65 extracted references · 45 canonical work pages · cited by 1 Pith paper

  1. [1]

    1979, Solar Physics, 62, 23

    Acheson, D. 1979, Solar Physics, 62, 23

    work page 1979

  2. [2]

    Aerts, C., Mathis, S., & Rogers, T. M. 2019, ARA&A, 57, 35

    work page 2019

  3. [3]

    2011, MNRAS, 412, 107 Barr` ere, P., Guilet, J., Raynaud, R., & Reboul-Salze, A

    Arlt, R., & R¨ udiger, G. 2011, MNRAS, 412, 107 Barr` ere, P., Guilet, J., Raynaud, R., & Reboul-Salze, A. 2023, Monthly Notices of the Royal Astronomical Society: Letters, 526, L88

    work page 2011

  4. [4]

    A., & Gusakov, M

    Becerra, L., Reisenegger, A., Valdivia, J. A., & Gusakov, M. E. 2022, MNRAS, 511, 732

    2022

  5. [5]

    G., Montalban, J., Kallinger, T., et al

    Beck, P. G., Montalban, J., Kallinger, T., et al. 2012, Nature, 481, 55

    2012

  6. [6]

    2023, MNRAS, 522, 1706

    Blouin, S., Mao, H., Herwig, F., et al. 2023, MNRAS, 522, 1706

    work page 2023

  7. [7]

    1957, Zhur

    Braginskii, S. 1957, Zhur. Eksptl’. i Teoret. Fiz., 33

    work page 1957

  8. [8]

    2006, A&A, 449, 451 —

    Braithwaite, J. 2006, A&A, 449, 451 —. 2009, MNRAS, 397, 763

    work page 2006

Show all 65 references
  1. [9]

    Braithwaite, J., & Spruit, H. C. 2017, Royal Society Open Science, 4, 160271

    2017

  2. [10]

    2021, A&A, 650, A53 22

    Bugnet, L., Prat, V., Mathis, S., et al. 2021, A&A, 650, A53 22

    2021

  3. [11]

    2016, ApJ, 824, 14

    Cantiello, M., Fuller, J., & Bildsten, L. 2016, ApJ, 824, 14

    2016

  4. [12]

    2014, ApJ, 788, 93

    Christensen-Dalsgaard, J., & Paxton, B. 2014, ApJ, 788, 93

    2014

  5. [13]

    2023, A&A, 670, L16

    Deheuvels, S., Li, G., Ballot, J., & Ligni` eres, F. 2023, A&A, 670, L16

    2023

  6. [14]

    A., Chaplin, W

    Deheuvels, S., Garcia, R. A., Chaplin, W. J., et al. 2012, ApJ, 756, 19

    2012

  7. [15]

    Denissenkov, P. A. 2010, ApJ, 723, 563 Di Mauro, M., Ventura, R., Cardini, D., et al. 2016, ApJ, 817, 65

    2010

  8. [16]

    2022, A&A, 664, L16

    Eggenberger, P., Moyano, F., & den Hartogh, J. 2022, A&A, 664, L16

    2022

  9. [17]

    2015, Sci, 350, 423

    Bildsten, L. 2015, Sci, 350, 423

    2015

  10. [18]

    2014, ApJ, 796, 17

    Fuller, J., Lecoanet, D., Cantiello, M., & Brown, B. 2014, ApJ, 796, 17

    2014

  11. [19]

    2018, AnRFM, 50, 275

    Garaud, P. 2018, AnRFM, 50, 275

    2018

  12. [20]

    2018, A&A, 616, A24

    Gehan, C., Mosser, B., Michel, E., Samadi, R., & Kallinger, T. 2018, A&A, 616, A24

    2018

  13. [21]

    2019, MNRAS, 490, 4281

    Smolarkiewicz, P. 2019, MNRAS, 490, 4281

    2019

  14. [22]

    1959, ApJ, 130, 276

    Hazlehurst, J., & Sargent, W. 1959, ApJ, 130, 276

    1959

  15. [23]

    2000, ApJ, 528, 368

    Heger, A., Langer, N., & Woosley, S. 2000, ApJ, 528, 368

    2000

  16. [24]

    2005, ApJ, 626, 350

    Heger, A., Woosley, S., & Spruit, H. 2005, ApJ, 626, 350

    2005

  17. [25]

    T., Kawaler, S

    Hermes, J., G¨ ansicke, B. T., Kawaler, S. D., et al. 2017, ApJS, 232, 23

    2017

  18. [26]

    1966, ApJ, 146, 858

    Hubbard, W. 1966, ApJ, 146, 858

    1966

  19. [27]

    1985, GApFD, 32, 273

    Hughes, D. 1985, GApFD, 32, 273

    1985

  20. [28]

    S., Anders, E

    Jermyn, A. S., Anders, E. H., Lecoanet, D., & Cantiello, M. 2022, ApJS, 262, 19

    2022

  21. [29]

    S., Bauer, E

    Jermyn, A. S., Bauer, E. B., Schwab, J., et al. 2023, ApJS, 265, 15

    2023

  22. [30]

    2023, MNRAS, 521, 5372

    Ji, S., Fuller, J., & Lecoanet, D. 2023, MNRAS, 521, 5372

    2023

  23. [31]

    2020, A&A, 641, A13

    Jouve, L., Ligni` eres, F., & Gaurat, M. 2020, A&A, 641, A13

    2020

  24. [32]

    Kawaler, S. D. 2014, arXiv preprint arXiv:1410.6934

    2014 arXiv

  25. [33]

    2015, ApJ, 808, 35

    Kissin, Y., & Thompson, C. 2015, ApJ, 808, 35

    2015

  26. [34]

    S., Hon, M., & Huber, D

    Kuszlewicz, J. S., Hon, M., & Huber, D. 2023, ApJ, 954, 152

    2023

  27. [35]

    2024, A&A, 688, A184

    Li, G., Deheuvels, S., & Ballot, J. 2024, A&A, 688, A184

    2024

  28. [36]

    2022, Nature, 610, 43

    Li, G., Deheuvels, S., Ballot, J., & Lignieres, F. 2022, Nature, 610, 43

    2022

  29. [37]

    2023, A&A, 680, A26

    Li, G., Deheuvels, S., Li, T., Ballot, J., & Ligni` eres, F. 2023, A&A, 680, A26

    2023

  30. [38]

    2019, MNRAS, 488, 4338

    Ma, L., & Fuller, J. 2019, MNRAS, 488, 4338

    2019

  31. [39]

    2008, Physics, formation and evolution of rotating stars (Springer Science & Business Media)

    Maeder, A. 2008, Physics, formation and evolution of rotating stars (Springer Science & Business Media)

    2008

  32. [40]

    2000, ARA&A, 38, 143

    Maeder, A., & Meynet, G. 2000, ARA&A, 38, 143

    2000

  33. [41]

    1993, in International Astronomical Union Colloquium, Vol

    Michaud, G., & Proffitt, C. 1993, in International Astronomical Union Colloquium, Vol. 137, Cambridge University Press, 246–259

    1993

  34. [42]

    2023, MNRAS, 521, 1415

    Smolarkiewicz, P. 2023, MNRAS, 521, 1415

    2023

  35. [43]

    2024, A&A, 681, L20

    Mosser, B., Dr´ eau, G., Pin¸ con, C., et al. 2024, A&A, 681, L20

    2024

  36. [44]

    2012, A&A, 548, A10

    Mosser, B., Goupil, M., Belkacem, K., et al. 2012, A&A, 548, A10

    2012

  37. [45]

    2010, ApJS, 192, 3

    Paxton, B., Bildsten, L., Dotter, A., et al. 2010, ApJS, 192, 3

    2010

  38. [46]

    2013, ApJS, 208, 4

    Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208, 4

    2013

  39. [47]

    2015, ApJS, 220, 15

    Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS, 220, 15

    2015

  40. [48]

    B., et al

    Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS, 234, 34

    2018

  41. [49]

    2019, ApJS, 243, 10

    Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243, 10

    2019

  42. [50]

    2023, Sci, 379, 300

    Petitdemange, L., Marcotte, F., & Gissinger, C. 2023, Sci, 379, 300

    2023

  43. [51]

    1985, MNRAS, 216, 139

    Pitts, E., & Tayler, R. 1985, MNRAS, 216, 139

    1985

  44. [52]

    2024, A&A Sch¨ urmann, C., Langer, N., Xu, X., & Wang, C

    Rosales, J., Petrovic, J., Mennickent, R., et al. 2024, A&A Sch¨ urmann, C., Langer, N., Xu, X., & Wang, C. 2022, A&A, 667, A122

    2024

  45. [53]

    A., & Beloborodov, A

    Skoutnev, V. A., & Beloborodov, A. M. 2024, ApJ, 974, 290 —. 2025, arXiv preprint arXiv:2504.07223

    2024

  46. [54]

    A., & Veronis, G

    Spiegel, E. A., & Veronis, G. 1960, ApJ, 131, 442

    1960

  47. [55]

    1962, Physics of Fully Ionized Gases

    Spitzer, L. 1962, Physics of Fully Ionized Gases

    1962

  48. [56]

    1999, A&A, 349, 189 —

    Spruit, H. 1999, A&A, 349, 189 —. 2002, A&A, 381, 923

    1999

  49. [57]

    2008, A&A, 481, L87

    Suijs, M., Langer, N., Poelarends, A.-J., et al. 2008, A&A, 481, L87

    2008

  50. [58]

    2021, A&A, 646, A19

    Takahashi, K., & Langer, N. 2021, A&A, 646, A19

    2021

  51. [59]

    G., Pinsonneault, M

    Tayar, J., Beck, P. G., Pinsonneault, M. H., Garcia, R. A., & Mathur, S. 2019, ApJ, 887, 203

    2019

  52. [60]

    1973, MNRAS, 161, 365

    Tayler, R. 1973, MNRAS, 161, 365

    1973

  53. [61]

    C., Bertolami, M

    Wachlin, F. C., Bertolami, M. M., & Althaus, L. G. 2011, A&A, 533, A139

    2011

  54. [62]

    1987, ApJ, 313, 284

    Wendell, C., Van Horn, H., & Sargent, D. 1987, ApJ, 313, 284

    1987

  55. [63]

    C., Kagan, D., & Chatzopoulos, E

    Wheeler, J. C., Kagan, D., & Chatzopoulos, E. 2015, ApJ, 799, 85

    2015

  56. [64]

    1992, A&A, 265, 115

    Zahn, J.-P. 1992, A&A, 265, 115

    1992

  57. [65]

    2007, A&A, 474, 145

    Zahn, J.-P., Brun, A., & Mathis, S. 2007, A&A, 474, 145

    2007

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