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Nonlinear internal waves breaking in stellar radiation zones. Parametrisation for the transport of angular momentum: bridging geophysical to stellar fluid dynamics

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper derives the first prescription for angular momentum transport by breaking internal gravity waves in stellar radiation zones, expressed as a minimum of three fluxes.

desk verdict A genuinely useful, mostly sound prescription for angular-momentum transport by breaking internal gravity waves, but the central cap equation uses min on signed fluxes and so fails for prograde modes. read the letter →

arxiv 2411.13925 v1 pith:5MWSA3LP submitted 2024-11-21 astro-ph.SR astro-ph.EPphysics.ao-ph

Stéphane Mathis This is my paper · ORCID
classification astro-ph.SRastro-ph.EPphysics.ao-ph
keywords internalgravitywavesstars:rotationevolutionmethods:analyticalangularmomentumtransportwavebreakingsaturationschemeasteroseismology
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 supplies the piece of physics that stellar evolution codes have been missing: what happens when internal gravity waves in stably stratified stellar radiation zones break because they overturn the stratification or because their own vertical shear goes unstable. The central claim is that each wave mode's angular momentum flux is the minimum of three values: the usual radiatively damped flux, the flux at convective-breaking saturation, and the flux at shear-breaking saturation. The prescription adapts a saturation model developed for the Earth's atmosphere, and the paper argues it brings stellar wave–mean-flow interaction to the same level of realism as atmospheric global circulation models. If correct, it gives stellar evolution codes a ready-to-implement term that could help close the gap between predicted and asteroseismically observed angular momentum extraction, which is about two orders of magnitude when waves and magnetic fields are neglected. The paper also extends the formula to rotating, magnetised stars, where the Coriolis and Lorentz forces reduce the convective-breaking flux.

What carries the argument

The carrying machinery is a saturation scheme adapted from atmospheric physics. For convective breaking, the wave amplitude is capped where the radial gradient of the temperature perturbation, $|\partial_r T'|\simeq k_r|T'|$, equals the background gradient scale $\Gamma=T N_T^2/(g\delta)$; this yields the saturated vertical velocity $\hat{\omega}/k_r$ and, through the wave polarisation relation, the saturated energy flux (Eq. 31) and angular momentum flux (Eq. 32). For shear breaking, the cap is set by the Richardson criterion $N^2/|d\langle \mathbf{u}_h\cdot\mathbf{u}_h^*\rangle/dr|^2\le Ri_c\,\beta_{l,m}$ with $\beta_{l,m}=m^2/[l(l+1)]$, giving saturated fluxes that, when heat diffusion is included, grow with the heat diffusivity $K$. The final formula puts the three transport channels into one minimum, so the limiting mechanism sets the deposit. In the rotating, magnetised generalisation, spherical harmonics are replaced by Hough functions and the dispersion relation becomes $k_r=(N/\omega_M)k_h$ with $\omega_M^2=\omega_s^2-m^2\omega_A^2$, which shifts the same machinery to magneto-gravito-inertial waves.

What would settle it

Run a direct numerical simulation of internal gravity waves in a spherical shell with stellar-like stratification and Prandtl number $10^{-6}$, measure the rms vertical velocity and angular momentum flux at the onset of breaking, and compare with $|\hat{u}_r|_{\mathrm{sat}}=\hat{\omega}/k_r$ and with Eq. (44): if breaking starts at a noticeably different amplitude or deposits momentum at a different radius, the saturation cap is wrong.

Watch

Extended reading notes

Core claim

The paper's central result is Eq. (44): the total angular momentum flux transported by internal gravity waves at radius $r$ is the sum over modes of the minimum of three fluxes, namely the standard quasi-adiabatic flux $-(m/\hat{\omega})\,F_{E;l,m}(r_0)\,e^{-\tau_{l,m}(r)}$, the convective-breaking flux $F^{\mathrm{CWB}}_{J;l,m}$, and the shear-breaking flux $F^{\mathrm{SWB}}_{J;l,m}$. The convective-breaking flux follows from the saturation rule that the wave's temperature gradient reaches the background gradient, $|\partial_r T'|\simeq k_r|T'|=\Gamma$, which fixes the saturated radial velocity to $|\hat{u}_{r;l,m}|_{\mathrm{sat}}=\hat{\omega}/k_r=\hat{\omega}^2/(N k_h)$. The shear-breaking flux follows from the Richardson criterion for the wave's own vertical shear, with the spherical-geometry factor $\beta_{l,m}=m^2/[l(l+1)]$, and with heat diffusion included through a Péclet number. The result is that breaking deposits angular momentum preferentially where the Doppler-shifted frequency $\hat{\omega}$ tends to zero, at high latitudinal degree $l$, at large $N/\hat{\omega}$, and where density is low. The paper states this is the first prescription for angular momentum transport triggered by nonlinear breaking of internal gravity waves in stellar radiation zones.

Load-bearing premise

The load-bearing premise is the saturation condition imported from atmospheric physics: a wave breaks when the radial gradient of its temperature perturbation, $k_r|T'|$, reaches the background temperature gradient $\Gamma$, fixing the saturated velocity to $\hat{\omega}/k_r$; this assumes the temperature perturbation survives to that point even though stellar radiation zones have extremely strong heat diffusion (Prandtl number $\sim10^{-6}$) that could soften it before breaking.

Editorial extensions

If this is right

  • The prescription implies that internal gravity waves in stars can break before reaching critical layers, so the saturation cap, not radiative damping alone, sets where angular momentum is deposited.
  • The breaking flux grows with latitudinal degree $l$, with $N/\hat{\omega}$, and as density drops, and it is deposited most efficiently where the Doppler-shifted frequency $\hat{\omega}$ vanishes; retrograde waves extract angular momentum from the radiative cores of solar-type stars while prograde waves deposit it, with the opposite signs in early-type stars.
  • The formula is analytically ready to be implemented in one-dimensional stellar structure and evolution codes, raising wave-driven angular momentum transport in stars to the same level of description as atmospheric global circulation models.
  • In rotating, magnetised stars the Coriolis and Lorentz forces reduce the convective-breaking flux, and the transport vanishes at the magnetic critical layer where $\omega_M=0$.
  • Near the low-density surfaces of early-type stars, the enhanced breaking-driven momentum deposit may participate in driving matter ejection, as in active Be stars.

Reading between the lines

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

  • The same min-of-three structure could be turned into a diagnostic: in a stellar evolution model, the radius where the saturated flux first caps the radiatively damped flux should coincide with the shallowest gradient in an asteroseismically inferred rotation profile, so disagreement would localise where the saturation model needs revision.
  • Because the convective-breaking flux scales with $\rho$ while the heat-diffusive shear-breaking flux has a strong dependence on $K$, the parameterisation could plausibly be extended to chemical transport by replacing the angular momentum flux with a tracer flux along the same saturation curves.
  • The dependence of the rotating, magnetised version on $\omega_M$ predicts an angular momentum 'dead zone' at the magnetic critical layer; comparing rotation profiles of stars with known magnetic fields against that prediction would provide a test independent of the hydrodynamic case.
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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

2 major / 5 minor

Summary. The manuscript adapts a saturation scheme originally developed for internal gravity waves in the Earth's atmosphere to deep spherical stellar radiation zones. Starting from the JWKB description of low-frequency IGWs, it derives saturated vertical and horizontal velocities for convective breaking (Eqs. 27-28) and for shear-driven breaking without and with thermal diffusion (Eqs. 36 and 40), from which it computes energy and angular-momentum fluxes (Eqs. 31-32, 37-38, 41-42). The central prescription, Eq. (44), is a sum over modes of the minimum of the quasi-adiabatic radiatively damped flux and the two nonlinearly saturated fluxes. A rotating, magnetised extension is given in Section 4 with Hough functions replacing spherical harmonics and culminating in Eq. (52). The paper claims to provide the first prescription for angular-momentum transport by nonlinearly breaking IGWs in stellar radiation zones.

Significance. If Eq. (44) can be corrected and the saturation criterion properly justified, the paper would supply a ready-to-implement expression for a missing ingredient in one-dimensional stellar evolution codes. The derivation is internally consistent and traceable: the saturated amplitudes and fluxes follow from stated instability criteria, no parameters are fitted to stellar data (the only free parameter is the critical Richardson number Ri_c), and the geophysical saturation scheme has been independently validated against stratospheric balloon measurements. The algebraic steps are explicit, and the physical discussion of the dependence on l, m, N, rho, K, and the Doppler-shifted frequency is useful. The main weaknesses are that the central formula as written does not implement the saturation statement it claims, and the convective saturation criterion is imported without a quantitative treatment of thermal diffusion in stellar conditions.

major comments (2)
  1. [Section 3.3, Eq. (44); also Section 4, Eq. (52)] The pointwise minimum in Eq. (44) is taken over signed fluxes, but for a fixed mode all three bracketed terms have the same sign: each is proportional to -m*epsilon times a positive factor, where the energy-flux sign epsilon is carried by the first term as well as by the two saturation terms, assuming the usual branch with hatomega > 0. Consequently, for the sign branch in which the flux is negative (e.g. prograde waves, m < 0, in the solar-type case epsilon = -1), min selects the most negative term, i.e. the largest magnitude. In precisely the regime where nonlinear breaking should cap the transport, the formula returns the quasi-adiabatic damped flux rather than the smaller saturated flux. The statement preceding Eq. (44) that the flux cannot exceed its value in the saturated regime is therefore not enforced for prograde modes. This is not a typographical issue: it changes the sign-selected angular-momentum transport and would affect any stellar-evolution implementation. The correct operation is a sign-preserving magnitude cap, for example F = sign(F_damped) * min(|F_damped|, |F_CWB|, |F_SWB|), or equivalently a branchwise max for the negative-sign branch; the same repair is needed in Eq. (52).
  2. [Section 3.1.1, Eqs. (24)-(27)] The saturation criterion for convective breaking uses the adiabatic heat-transport equation Dt T' + Gamma ur = 0 and identifies |dr T'| ~ kr |T'| = Gamma. In stellar radiation zones the Prandtl number is very small but the thermal diffusivity K is large, and heat diffusion is retained, not neglected, in the quasi-adiabatic damping treatment of Section 2.2. It is therefore not self-evident that the temperature perturbation reaches the gradient threshold before being diffusively damped or phase-shifted. The derivation of the saturated velocity |ur|_sat = hatomega / kr, and hence of Eqs. (27)-(32) and the corresponding terms in Eq. (44), needs an explicit timescale comparison (e.g. hatomega versus K kr^2) or a derivation starting from the linearised heat equation including K. If the diffusion-modified saturation differs, the saturated amplitudes and fluxes must be revised. The conclusion's later admission that the local criterion is an approximation does not by itself settle this quantitative point.
minor comments (5)
  1. [Throughout] There are several typographical errors: 'convectivelly-excited' in Section 1, 'adress' in Section 3.1, 'ballons' in Section 3.1.2, 'loose' for 'lose' in Section 3.1.2, and 'beyound' in Section 5.
  2. [Eqs. (14) and (21)] The typesetting of the damping integrand appears corrupted: 'K N N 2 T' should presumably be 'K N N_T^2' (or the intended combination of N and N_T). Please correct the displayed equations.
  3. [Section 2.1 and Eq. (44)] The sign convention for the Doppler-shifted frequency hatomega is not stated. The manuscript says prograde waves have m < 0 and retrograde waves have m > 0, but it does not state whether hatomega is positive for the propagating waves considered. Since every term in Eq. (44) changes sign with hatomega, the convention should be made explicit.
  4. [Section 3.1.1 and Section 5] The conclusion that the local saturation criterion is 'an approximation of the reality' once turbulence develops is useful, but it should be stated at the point where the criterion is introduced, near Eq. (25), with a brief indication of the expected validity domain in stellar parameters.
  5. [Section 4, Eq. (51)] The angular average in Eq. (51) is written with the notation <Theta^2>_theta, which averages only over colatitude, while the flux is described as a horizontal average over the sphere. Please clarify how the azimuthal dependence of the Hough functions is removed or normalised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the breaking fluxes are derived from explicit stability criteria and an externally validated saturation model, with no parameter fitted to stellar data.

full rationale

The paper's central prescription, Eq. (44), is a piecewise minimum of three fluxes, each defined independently: the quasi-adiabatic damped flux (first term) and the convective and shear saturation fluxes (second and third terms). The saturated velocities in Eqs. (27), (36), and (40) follow algebraically from the stated instability criteria (Eqs. (25), (33), and (39)) together with the linearized heat equation and dispersion relation; no quantity is defined in terms of the target flux, and no parameter is fitted to stellar observations. The geophysical saturation scheme is taken from Lott et al. (2012) and Lott & Guez (2013), whose predictions were independently compared with stratospheric balloon measurements (Lott et al. 2023); this is external support rather than self-citation. The self-citations that appear (Alvan et al. 2013, 2015; Mathis & de Brye 2011, 2012) supply standard dispersion relations, the Richardson criterion, and TARM eigenfunctions; they are inputs to the derivation, not conclusions that presuppose the breaking prescription, and none is a uniqueness claim used to forbid alternatives. The paper explicitly flags its own limitations ('once turbulence is developing, such a local criterion is an approximation of the reality', and the absence of a general criterion for rotating/magnetized shear instability in Sec. 4), which are correctness caveats, not circular steps. The sign-of-min issue in Eq. (44) raised in the skeptic note is an internal consistency or correctness concern, not a case of the prediction reducing to its inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the imported atmospheric saturation model and standard fluid-dynamics approximations. The only free parameter is the critical Richardson number, which is an input from prior literature, not fitted here. No new particles, forces, or conserved quantities are introduced.

free parameters (1)
  • Ri_c (critical Richardson number)
    Threshold for vertical shear instability used in the shear-breaking saturated amplitudes and fluxes (Eqs. 36-42). The paper does not set its value; it is an input from prior literature. The prescription's magnitude scales as 1/Ri_c for the non-diffusive shear case and 1/Ri_c^2 for the heat-diffusion case.
assumptions (6)
  • domain assumption The local saturation model: an IGW breaks when its temperature gradient overturns the background entropy gradient, and the saturated amplitude is set by Eq. (25).
    Imported from atmospheric physics (Lindzen 1981; Lott et al. 2012; Lott & Guez 2013) and applied to stellar interiors. This is the core of the adaptation and is not derived within the paper.
  • standard math JWKB dispersion relation and polarization relations for low-frequency IGWs, k_r = (N/hat_omega) k_h and u_h approximately (k_r/k_h) u_r.
    Used throughout Sections 2 and 3 to relate vertical and horizontal velocities and to evaluate breaking criteria. Standard for omega much less than N.
  • domain assumption Anelastic approximation and Cowling approximation (neglect of gravity fluctuations).
    Introduced in Section 2.1 to filter acoustic waves and simplify the momentum equation for low-frequency IGWs.
  • domain assumption Richardson criterion for vertical shear instability with geometric factor beta_l,m = m^2/(l(l+1)) (Eq. 33).
    Taken from Alvan et al. (2013) for spherical geometry. Used to derive the shear-breaking saturated velocities.
  • domain assumption Shellular rotation approximation.
    Assumes rotation depends only on radius, required for the 1D SSE-code target (Section 2.1).
  • domain assumption Traditional Approximation for Rotation and Magnetism (TARM) and the Malkus magnetic field with constant Alfven frequency (Eq. 46).
    Used in Section 4 to include Coriolis and Lorentz forces via the Mathis & de Brye (2011, 2012) model. This is a simplified magnetic geometry.

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Pith. "Pith review of Nonlinear internal waves breaking in stellar radiation zones. Parametrisation for the transport of angular momentum: bridging geophysical to stellar fluid dynamics." pith.science (2026). https://pith.science/paper/5MWSA3LP

@misc{pith2026241113925,
  author       = {Pith},
  title        = {Pith review of: Nonlinear internal waves breaking in stellar radiation zones. Parametrisation for the transport of angular momentum: bridging geophysical to stellar fluid dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MWSA3LP}},
  note         = {Machine review of arXiv:2411.13925}
}
read the original abstract

Internal gravity waves (hereafter IGWs) are one of the mechanisms that can play a key role to redistribute efficiently angular momentum in stars along their evolution. The study of IGWs is thus of major importance since space-based asteroseismology reveals a transport of angular momentum in stars, which is stronger by two orders of magnitude than the one predicted by stellar models ignoring their action or those of magnetic fields. IGWs trigger angular momentum transport when they are damped by heat or viscous diffusion, when they meet a critical layer or when they break. Theoretical prescriptions have been derived for the transport of angular momentum induced by IGWs because of their radiative and viscous dampings and of the critical layers they encounter along their propagation. However, none has been proposed for the transport triggered by their nonlinear breaking. In this work, we aim to derive such a physical and robust prescription, which can be implemented in stellar structure and evolution codes. We adapt an analytical saturation model, which has been developed for IGWs nonlinear convective breaking in the Earth atmosphere and has been successfully compared to in-situ measurements in the stratosphere, to the case of deep spherical stellar interiors. In a first step, we neglect the modification of IGWs by the Coriolis acceleration and the Lorentz force, which are discussed and taken into account in a second step. We derive a complete semi-analytical prescription for the transport of angular momentum by IGWs, which takes into account both their radiative damping and their potential nonlinear breaking because of their convective and vertical shear instabilities. This allows us to bring the physical prescription for the interactions between IGWs and the differential rotation to the same level of realism that the one used in global circulation models for the atmosphere.

Figures

Figures reproduced from arXiv: 2411.13925 by the authors.

Figure 1
Figure 1. Principle of convective wave breaking: an IGW is excited at r = r0. When propagating towards a critical layer located at r = rc , the radial gradient of its temperature fluctuation grows with kr until it overturns the stable equilibrium entropy stratification at the breaking radius r = rb. This triggers a convective instability. (2012) and Lott & Guez (2013). To derive its amplitude, we use the heat transport equati… view at source ↗

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Works this paper leans on

100 extracted references · 44 canonical work pages

  1. [1]

    J., Becker, E., et al

    Achatz, U., Alexander, M. J., Becker, E., et al. 2024, Journal of the Atmospheric Sciences, 81, 237

    2024

  2. [2]

    & Tkachenko, A

    Aerts, C. & Tkachenko, A. 2023, arXiv e-prints, arXiv:2311.08453

    arXiv 2023

  3. [3]

    2017, ApJ, 847, L7

    Aerts, C., Van Reeth, T., & Tkachenko, A. 2017, ApJ, 847, L7

    2017

  4. [4]

    2013, A&A, 553, A86

    Alvan, L., Mathis, S., & Decressin, T. 2013, A&A, 553, A86

    2013

  5. [5]

    S., Mathis, S., & Garcia, R

    Alvan, L., Strugarek, A., Brun, A. S., Mathis, S., & Garcia, R. A. 2015, A&A, 581, A112

    2015

  6. [6]

    2014, in Putting A Stars into Context: Evolution, Environment, and Re- lated Stars, ed

    Arlt, R. 2014, in Putting A Stars into Context: Evolution, Environment, and Re- lated Stars, ed. G. Mathys, E. R. Griffin, O. Kochukhov, R. Monier, & G. M. Wahlgren, 93–101

    2014

  7. [7]

    Augustson, K. C. & Mathis, S. 2019, ApJ, 874, 83

    2019

  8. [8]

    Banner, M. L. & Peregrine, D. H. 1993, Annual Review of Fluid Mechanics, 25, 373

    1993

Show all 100 references
  1. [9]

    Barker, A. J. 2011, MNRAS, 414, 1365

    2011

  2. [10]

    Barker, A. J. & Ogilvie, G. I. 2010, MNRAS, 404, 1849

    2010

  3. [11]

    Booker, J. R. & Bretherton, F. P. 1967, Journal of Fluid Mechanics, 27, 513

    1967

  4. [12]

    & Spruit, H

    Braithwaite, J. & Spruit, H. C. 2004, Nature, 431, 819

    2004

  5. [13]

    S., Browning, M

    Brun, A. S., Browning, M. K., & Toomre, J. 2005, ApJ, 629, 461

    2005

  6. [14]

    S., Miesch, M

    Brun, A. S., Miesch, M. S., & Toomre, J. 2011, ApJ, 742, 79

    2011

  7. [15]

    S., Strugarek, A., Noraz, Q., et al

    Brun, A. S., Strugarek, A., Noraz, Q., et al. 2022, ApJ, 926, 21

    2022

  8. [16]

    S., Strugarek, A., Varela, J., et al

    Brun, A. S., Strugarek, A., Varela, J., et al. 2017, ApJ, 836, 192

    2017

  9. [17]

    Brun, A. S. & Zahn, J. P. 2006, A&A, 457, 665

    2006

  10. [18]

    2021, A&A, 650, A53

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

    2021

  11. [19]

    2014, ApJ, 788, 93

    Cantiello, M., Mankovich, C., Bildsten, L., Christensen-Dalsgaard, J., & Paxton, B. 2014, ApJ, 788, 93

    2014

  12. [20]

    A., & Mathis, S

    Ceillier, T., Eggenberger, P., García, R. A., & Mathis, S. 2013, A&A, 555, A54

    2013

  13. [21]

    2013, A&A, 554, A40

    Charbonnel, C., Decressin, T., Amard, L., Palacios, A., & Talon, S. 2013, A&A, 554, A40

    2013

  14. [22]

    & Talon, S

    Charbonnel, C. & Talon, S. 2005, Science, 309, 2189

    2005

  15. [23]

    Cowling, T. G. 1941, MNRAS, 101, 367

    1941

  16. [24]

    G., et al

    Deheuvels, S., Ballot, J., Beck, P. G., et al. 2015, A&A, 580, A96

    2015

  17. [25]

    2020, A&A, 641, A117

    Deheuvels, S., Ballot, J., Eggenberger, P., et al. 2020, A&A, 641, A117

    2020

  18. [26]

    J., et al

    Deheuvels, S., Do˘gan, G., Goupil, M. J., et al. 2014, A&A, 564, A27

    2014

  19. [27]

    A., Chaplin, W

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

    2012

  20. [28]

    2022, A&A, 661, A133

    Dhouib, H., Mathis, S., Bugnet, L., Van Reeth, T., & Aerts, C. 2022, A&A, 661, A133

    2022

  21. [29]

    2021, A&A, 656, A122

    Dhouib, H., Prat, V ., Van Reeth, T., & Mathis, S. 2021, A&A, 656, A122

    2021

  22. [30]

    & Rieutord, M

    Dintrans, B. & Rieutord, M. 2000, A&A, 354, 86

    2000

  23. [31]

    & Mathis, S

    Duez, V . & Mathis, S. 2010, A&A, 517, A58

    2010

  24. [32]

    & Brun, A

    Emeriau-Viard, C. & Brun, A. S. 2017, ApJ, 846, 8 Espinosa Lara, F. & Rieutord, M. 2013, A&A, 552, A35 Fröman, N. & Fröman, P. O. 2005, Physical Problems Solved by the Phase- Integral Method

    2017

  25. [33]

    2014, ApJ, 796, 17

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

    2014

  26. [34]

    L., & Jermyn, A

    Fuller, J., Piro, A. L., & Jermyn, A. S. 2019, MNRAS, 485, 3661

    2019

  27. [35]

    & Bouvier, J

    Gallet, F. & Bouvier, J. 2013, A&A, 556, A36 García, R. A., Turck-Chièze, S., Jiménez-Reyes, S. J., et al. 2007, Science, 316, 1591 Garcia Lopez, R. J. & Spruit, H. C. 1991, ApJ, 377, 268

    2013

  28. [36]

    & Shrira, V

    Gerkema, T. & Shrira, V . I. 2005, Journal of Fluid Mechanics, 529, 195

    2005

  29. [37]

    Holton, J. R. & Zhu, X. 1984, Journal of the Atmospheric Sciences, 41, 2653

    1984

  30. [38]

    1999, ApJ, 520, 859

    Kumar, P., Talon, S., & Zahn, J.-P. 1999, ApJ, 520, 859

    1999

  31. [39]

    W., Saio, H., Takata, M., et al

    Kurtz, D. W., Saio, H., Takata, M., et al. 2014, MNRAS, 444, 102

    2014

  32. [40]

    Lazovik, Y . A. 2021, MNRAS, 508, 3408

    2021

  33. [41]

    G., Townsend, R

    Lecoanet, D., Zweibel, E. G., Townsend, R. H. D., & Huang, Y .-M. 2010, ApJ, 712, 1116

    2010

  34. [42]

    & Saio, H

    Lee, U. & Saio, H. 1997, ApJ, 491, 839

    1997

  35. [43]

    2024, arXiv e-prints, arXiv:2405.12116

    Li, G., Deheuvels, S., & Ballot, J. 2024, arXiv e-prints, arXiv:2405.12116

    2024 arXiv

  36. [44]

    R., et al

    Li, G., Van Reeth, T., Bedding, T. R., et al. 2020, MNRAS, 491, 3586 Lignières, F. 1999, A&A, 348, 933

    2020

  37. [45]

    Lindzen, R. S. 1981, J. Geophys. Res., 86, 9707

    1981

  38. [46]

    Longuet-Higgins, M. S. 1968, Philosophical Transactions of the Royal Society of London Series A, 262, 511

    1968

  39. [47]

    & Guez, L

    Lott, F. & Guez, L. 2013, Journal of Geophysical Research (Atmospheres), 118, 8897

    2013

  40. [48]

    2012, Geophys

    Lott, F., Guez, L., & Maury, P. 2012, Geophys. Res. Lett., 39, L06807

    2012

  41. [49]

    2023, Journal of Geophysical Research (Atmospheres), 128, e2022JD037585

    Lott, F., Rani, R., Podglajen, A., et al. 2023, Journal of Geophysical Research (Atmospheres), 128, e2022JD037585

    2023

  42. [50]

    MacGregor, K. B. & Rogers, T. M. 2011, Sol. Phys., 270, 417

    2011

  43. [51]

    2003, A&A, 399, 263

    Maeder, A. 2003, A&A, 399, 263

    2003

  44. [52]

    & Zahn, J.-P

    Maeder, A. & Zahn, J.-P. 1998, A&A, 334, 1000

    1998

  45. [53]

    Malkus, W. V . R. 1967, Journal of Fluid Mechanics, 28, 793

    1967

  46. [54]

    P., Goupil, M

    Marques, J. P., Goupil, M. J., Lebreton, Y ., et al. 2013, A&A, 549, A74

    2013

  47. [55]

    2009, A&A, 506, 811

    Mathis, S. 2009, A&A, 506, 811

    2009

  48. [56]

    & de Brye, N

    Mathis, S. & de Brye, N. 2011, A&A, 526, A65

    2011

  49. [57]

    & de Brye, N

    Mathis, S. & de Brye, N. 2012, A&A, 540, A37

    2012

  50. [58]

    2013, A&A, 558, A11

    Mathis, S., Decressin, T., Eggenberger, P., & Charbonnel, C. 2013, A&A, 558, A11

    2013

  51. [59]

    2014, A&A, 565, A47

    Mathis, S., Neiner, C., & Tran Minh, N. 2014, A&A, 565, A47

    2014

  52. [60]

    Mathis, S., Palacios, A., & Zahn, J. P. 2004, A&A, 425, 243

    2004

  53. [61]

    2018, A&A, 620, A22

    Mathis, S., Prat, V ., Amard, L., et al. 2018, A&A, 620, A22

    2018

  54. [62]

    & Zahn, J

    Mathis, S. & Zahn, J. P. 2004, A&A, 425, 229

    2004

  55. [63]

    & Maeder, A

    Meynet, G. & Maeder, A. 2000, A&A, 361, 101

    2000

  56. [64]

    M., Baruteau, C., Rieutord, M., & Ballot, J

    Mirouh, G. M., Baruteau, C., Rieutord, M., & Ballot, J. 2016, Journal of Fluid Mechanics, 800, 213

    2016

  57. [65]

    Mombarg, J. S. G., Rieutord, M., & Espinosa Lara, F. 2023, A&A, 677, L5 Montalbán, J. & Schatzman, E. 2000, A&A, 354, 943

    2023

  58. [66]

    2024, A&A, 681, L20

    Mosser, B., Dréau, G., Pinçon, C., et al. 2024, A&A, 681, L20

    2024

  59. [67]

    J., Belkacem, K., et al

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

    2012

  60. [68]

    J., Fossati, L., Bedding, T

    Murphy, S. J., Fossati, L., Bedding, T. R., et al. 2016, MNRAS, 459, 1201

    2016

  61. [69]

    2020, A&A, 644, A9

    Neiner, C., Lee, U., Mathis, S., et al. 2020, A&A, 644, A9

    2020

  62. [70]

    M., Marques, J

    Ouazzani, R. M., Marques, J. P., Goupil, M. J., et al. 2019, A&A, 626, A121

    2019

  63. [71]

    2003, A&A, 399, 603

    Palacios, A., Talon, S., Charbonnel, C., & Forestini, M. 2003, A&A, 399, 603

    2003

  64. [72]

    2021, A&A, 646, A64

    Park, J., Prat, V ., Mathis, S., & Bugnet, L. 2021, A&A, 646, A64

    2021

  65. [73]

    2023, Science, 379, 300 Pinçon, C., Belkacem, K., Goupil, M

    Petitdemange, L., Marcotte, F., & Gissinger, C. 2023, Science, 379, 300 Pinçon, C., Belkacem, K., Goupil, M. J., & Marques, J. P. 2017, A&A, 605, A31

    2023

  66. [74]

    & Lignières, F

    Prat, V . & Lignières, F. 2013, A&A, 551, L3

    2013

  67. [75]

    & Lignières, F

    Prat, V . & Lignières, F. 2014, A&A, 566, A110

    2014

  68. [76]

    Press, W. H. 1981, ApJ, 245, 286

    1981

  69. [77]

    2022, Journal of Ad- vances in Modeling Earth Systems, 14, e2021MS002563, e2021MS002563 2021MS002563

    Ribstein, B., Millet, C., Lott, F., & de la Cámara, A. 2022, Journal of Ad- vances in Modeling Earth Systems, 14, e2021MS002563, e2021MS002563 2021MS002563

    2022

  70. [78]

    1987, Geophysical and Astrophysical Fluid Dynamics, 39, 163

    Rieutord, M. 1987, Geophysical and Astrophysical Fluid Dynamics, 39, 163

    1987

  71. [79]

    2016, Journal of Computational Physics, 318, 277

    Rieutord, M., Espinosa Lara, F., & Putigny, B. 2016, Journal of Computational Physics, 318, 277

    2016

  72. [80]

    Rogers, T. M. 2015, ApJ, 815, L30

    2015

  73. [81]

    M., Lin, D

    Rogers, T. M., Lin, D. N. C., McElwaine, J. N., & Lau, H. H. B. 2013, ApJ, 772, 21

    2013

  74. [82]

    Rogers, T. M. & McElwaine, J. N. 2017, ApJ, 848, L1

    2017

  75. [83]

    Z., Ong, J

    Rui, N. Z., Ong, J. M. J., & Mathis, S. 2024, MNRAS, 527, 6346

    2024

  76. [84]

    W., Takata, M., et al

    Saio, H., Kurtz, D. W., Takata, M., et al. 2015, MNRAS, 447, 3264

    2015

  77. [85]

    & Cambon, C

    Salhi, A. & Cambon, C. 2010, Phys. Rev. E, 81, 026302

    2010

  78. [86]

    Spiegel, E. A. & Zahn, J. P. 1992, A&A, 265, 106

    1992

  79. [87]

    Spruit, H. C. 2002, A&A, 381, 923

    2002

  80. [88]

    & Macfarlane, C

    Stansell, P. & Macfarlane, C. 2002, Journal of Physical Oceanography, 32, 1269

    2002

  81. [89]

    Stevenson, D. J. 1979, Geophysical and Astrophysical Fluid Dynamics, 12, 139

    1979

  82. [90]

    Sutherland, B. R. 2001, Journal of Fluid Mechanics, 429, 343

    2001

  83. [91]

    & Charbonnel, C

    Talon, S. & Charbonnel, C. 2005, A&A, 440, 981

    2005

  84. [92]

    P., Maeder, A., & Meynet, G

    Talon, S., Zahn, J. P., Maeder, A., & Meynet, G. 1997, A&A, 322, 209

    1997

  85. [93]

    J., Christensen-Dalsgaard, J., Miesch, M

    Thompson, M. J., Christensen-Dalsgaard, J., Miesch, M. S., & Toomre, J. 2003, ARA&A, 41, 599

    2003

  86. [94]

    Townsend, A. A. 1958, Journal of Fluid Mechanics, 4, 361

    1958

  87. [95]

    Townsend, R. H. D. 2003, MNRAS, 340, 1020

    2003

  88. [96]

    Townsend, R. H. D. 2020, MNRAS, 497, 2670

    2020

  89. [97]

    Vadas, S. L. & Fritts, D. C. 2005, Journal of Geophysical Research (Atmo- spheres), 110, D15103 Van Reeth, T., Tkachenko, A., & Aerts, C. 2016, A&A, 593, A120

    2005

  90. [98]

    P., Vanon, R., Edelmann, P

    Varghese, A., Ratnasingam, R. P., Vanon, R., Edelmann, P. V . F., & Rogers, T. M. 2023, ApJ, 942, 53

    2023

  91. [99]

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

    1992

  92. [100]

    P., Talon, S., & Matias, J

    Zahn, J. P., Talon, S., & Matias, J. 1997, A&A, 322, 320 Article number, page 9 of 9

    1997

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